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Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 11 (2004) 149-180 Copyright c 2004 Watam Press

Dynamics of Continuous, Discrete and Impulsive SystemsSeries A: Mathematical Analysis 11 (2004) 149-180

Copyright c©2004 Watam Press

AN OVERVIEW OF THE COLLECTED WORKS OFPROFESSOR DRAGOSLAV SILJAK

Z. Gajic1 and M. Ikeda2

1Department of Electrical EngineeringRutgers University

Piscataway, NJ [email protected]

2Department of Computer-Controlled Mechanical SystemsOsaka University

Suita, Osaka 565-0871, [email protected]

Preface1. Parameter Space Design of Control Systems

2. Positivity of Uncertain Polynomials

3. Large-Scale Systems: Connective Stability

4. Competitive-Cooperative Systems

5. Parametric Stability

6. Decentralized Control of Complex Systems

7. Graph Theoretic Algorithms

8. Decentralized Control via Convex Optimization

9. The Inclusion Principle

10. Reliable Control

Books

Journal papers

Preface

Dragoslav D. Siljak was born in Belgrade, Serbia, on September 10, 1933.He started elementary school in the Fall of 1940 and enjoyed a few months ofschool, when, suddenly, one morning in the Spring of 1941 a barrage of bombsengulfed the apartment house where his family resided; the Second WorldWar arrived in Serbia, and especially Belgrade, with fury and destruction.For the next four years, his life and the life of his family would hang inthe balance. Hunger, extreme cold in Winter without heat, and persistent

150 Z. Gajic and M. Ikeda

bombing of Belgrade, made fear for life always present. To survive, the familywould escape from the city into the countryside to stay and work with theirrelatives, only to find themselves in the middle of the guerilla war that ragedin the Serbian heartland. The schools were closed in the city, and the onlyeducation he could get was in a rural area school, which was open seldomduring short stretches of peace.

After the end of the war, life started to improve steadily, schools wereopen, and Siljak went back to school. At fourteen, he started competitiveswimming and water polo at the local club, balancing study and practicealmost daily. In his senior year, at eighteen, he made the Yugoslav nationalwater polo team that won the silver medal at the 1952 Olympic Games inHelsinki, Finland. A month later, he entered the University of Belgrade tostudy electrical engineering and continue the balancing act of study and prac-tice in an increasingly demanding professional sport. He made the nationalteam again, which won the world cup in 1953, in Nijmegen, The Netherlands.At the University, Siljak liked mathematics, earning the highest grades inmath courses. In his senior year he took an exciting course on the theoryof automatic control that was taught by Professor Dusan Mitrovic. Fromthen on, his professional interest was set for life. He once again made theYugoslav Olympic team for the 1960 Olympic Games in Rome, Italy, but theteam took a disappointing fourth place.

Siljak started doing research in 1957, while working on his BS thesis un-der Professor Mitrovic in the Department of Electrical Engineering at theUniversity of Belgrade. The topic of the thesis was a controller design in theparameter space for sampled-data systems. He applied Mitrovic’s methodto obtain stability regions in the space of controller parameters, and pub-lished his first IEEE Transactions paper in 1961, based on his MS thesison control of sampled-data systems with time-delay. He continued this lineof research by fusing Mitrovic’s method with Neimark’s D-decomposition,and published his results in a sequence of three IEEE Transactions paperson the control of continuous, sampled-data, and nonlinear systems in theparameter space. An interesting novelty in this work was the introductionof Chebyshev’s polynomials into the computation of stability regions in theparameter space. Equally interesting was Siljak’s proof of the shading rulein D-decompositions, which was crucial in identifying the root distributionof characteristic equations in the parameter space. Siljak’s contributions tothe parameter space design of control systems, which served as a major partof his PhD dissertation in 1963, were not witnessed by his adviser ProfessorMitrovic, due to his tragic death in 1961. In 1963, Siljak was promoted tothe rank of Docent. Soon after promotion, he accepted an invitation by G.J. Thaler to visit Santa Clara University, and left for USA in 1964.

Immediately after arriving at Santa Clara, Siljak started building a re-search base centered around parameter space methods for the design of con-trol systems. He interacted with Thaler in their common research interest,and in 1965 they obtained a research grant from NASA Ames Research Center

An Overview of the Collected Works of Dragoslav Siljak 151

to develop the parameter space approach. This research activity attracted theattention of S. M. Seltzer from NASA Marshal Space Center, where Siljak’sparameter space methods were used in the control design of spacecraft, inparticular, the large booster Saturn V that carried man to the moon. NASAcontinued to support Siljak’s joint works with Thaler and Seltzer well intothe l970’s, resulting in a number of improvements that made the parame-ter space design competitive with well-established classical designs based onroot-locus, Bode, Nyquist and Nichols diagrams.

Below we present concise description of important problems (in our view)that were formulated and solved by Professor Siljak. The problems aregrouped in ten main research areas.

1 Parameter Space Design of Control Systems

To highlight a few interesting aspects of the parameter space approach,let us consider a real polynomial

f(s) =

n∑

k=0

ak(p)sk, an 6= 0

which is a characteristic polynomial of a linear control system, whose coef-ficients ak(p) are dependent on a parameter vector p. When we express thecomplex variable s as

s = −ωnζ + jωn

√

1 − ζ2

the real and imaginary parts of the polynomial can be expressed in the com-pact form

R(ζ, ωn; p) ≡n∑

k=0

(−1)kak(p)ωknTk(ζ)

I(ζ, ωn; p) ≡n∑

k=0

(−1)k+1ak(p)ωkn

√

1 − ζ2Uk(ζ)

where Tk(ζ) and Uk(ζ) are Chebyshev polynomials of the first and secondkind.

If the complex variable is replaced by

s = σ + jω

R and I can be reformulated as

R(σ, ω; p) ≡n∑

k=0

ak(p)Xk(σ, ω)

I(σ, ω; p) ≡n∑

k=0

ak(p)Yk(σ, ω)


where the polynomials Xk and Yk, which were introduced by Siljak, arecomputed by the recurrence formulas

Xk+1 − 2σXk + (σ2 + ω2)Xk−1 = 0

Yk+1 − 2σYk + (σ2 + ω2)Yk−1 = 0

with X0 = 1, X1 = σ, Y0 = 0 and Y1 = ω.

Both Chebyshev’s and Siljak’s forms of R and I play a useful role indelineating stability regions in the parameter space, which correspond todesired root regions in the complex s−plane. When Siljak’s polynomials areused, boundaries of the regions are defined by the equations

complex root boundaries: R(σ, ω; p) = 0

I(σ, ω; p) = 0

real root boundaries: a0(p) = 0

an(p) = 0

By plotting the boundaries in the parameter space, the designer can es-tablish a direct correlation between root regions in the complex plane andadjustable parameters appearing in the coefficients of the corresponding char-acteristic equation. Obviously, stability regions can be easily visualized whentwo or, at most, three parameters are allowed to be free. In the earlystages of the parameter space development, this was a considerable advantageover classical single parameter methods, especially because parameter spaceboundaries could be plotted accurately using modest computing machines,or even a hand calculator.

When parameter vector p has dimension larger than three, stability re-gions are difficult to visualize, and Siljak concluded that the problem ofrobust stability analysis is one of interpretation. In a joint work with J. S.Karmarkar, he imbedded a sphere within design constraints in the parameterspace and formulated a Chebyshev-like minimax problem,

minimize −ρ

subject to hi(p) − ρ ≥ 0, i = 1, 2, . . . , r

where hi(p) are continuous functions representing the design constraints, andρ is the radius of the sphere. This allowed for the use of mathematicalprogramming algorithms to maximize the radius, and get the largest estimateof the design region, which can be readily interpreted in the parameter space.Since there is a considerable freedom in choosing design constraints, usingthe computed sphere the designer can choose feedback gains which guaranteerobust stability among other relevant performance specifications. The paperwith Karmarkar was presented at the 1971 Allerton conference, where in thetitle they coined the term “computer-aided control design.” This work has


been recognized in the literature as a precursor to the modern use of convexoptimization and mathematical programming in control system design.

Among the interesting aspects of Siljak’s polynomials (e.g., that they obeythe Pascal triangle) is the fact that they played a significant role in a gradientroot solving scheme which Siljak initiated with P. Kokotovic. An importantfeature of the root solving algorithm is that, in minimizing a Lyapunov-likefunction,

V (σ, ω) = R2(σ, ω) + I2(σ, ω),

the algorithm converges globally and quadratically to a root in the complexplane independently of the disposition of the polynomial equation, the dis-tribution of its roots, or their distance from the starting point. J. Stolanimproved Siljak’s root-solver and compared it with the modern versions ofthe algorithms of Laguerre, Jenkins-Traub, and MATLAB to demonstratethe superiority of the improved Siljak’s scheme in solving difficult benchmarkproblems.

2 Positivity of Uncertain Polynomials

Let us consider a real polynomial g(s) and define positivity as

g(s) ≡n∑

k=0

gksk > 0, ∀s ∈ R+.

To provide an algebraic criterion for the positivity of g(s) in terms of thecoefficients gk, Siljak formulated the Modified Routh Array,

r0 = (−1)ngn (−1)n−1gn−1 . . . −g1 g0

r1 = (−1)nngn (−1)n−1(n − 1)gn−1 . . . −g1

...

r2n = g0

and proved that the number κ of positive zeros of g(s) is

κ = n − V (r0, r1, . . . , r2n),

where V is the number of sign variations in the coefficients of the first columnof the array. In particular, the positivity of g(s) follows from setting g0 > 0and requiring that κ = 0.

Siljak reformulated Popov’s frequency inequality in terms of polynomialpositivity and provided a test for absolute stability of the Lur’e-Postnikovnonlinear systems, which had the same numerical simplicity as the Routh-Hurwitz test for stability of linear time-invariant systems. He extended the


polynomial positivity test to determine positive realness of real rational func-tions and matrices. At the same time, he generalized the test for positivity ofreal polynomials on the unit circle, which led to efficient numerical algorithmsfor testing circle positive realness of rational functions and matrices.

In 1979, Siljak extended his positivity algorithm to testing stability oftwo-variable polynomials, which is required in the design of two-dimensionaldigital filters. A real two-variable polynomial

h(s, z) =

n∑

j=0

m∑

k=0

hjksjzk

where s, z ∈ C are two complex variables, is said to be stable if

h(s, z) 6= 0,

s ∈ CC−

∩

z ∈ CC−

.

CC− is the complement of C− = s ∈ C : Re s < 0 - the open left half of

complex plane C. As shown by H. G. Ansel in 1964, h(s, z) is stable if andonly if

h(s, 1) 6= 0, ∀s ∈ CC−

h(iω, z) 6= 0, ∀z ∈ CC−

Siljak showed that these conditions are equivalent to easily testable conditions

f(s) 6= 0, ∀s ∈ CC−

g(ω) > 0, ∀ω ∈ R+

H(0) > 0

where f(s) = h(s, 1), g(ω) = detH(ω), and H(ω) is the corresponding Her-mite matrix. In this way, to test a two-variable polynomial for stability, allone needs is to compute two Routh arrays, one standard array for the stabil-ity of f(s) and one modified array for the positivity of g(ω), and determineif a numerical matrix H(0) is positive definite. The extensions of the testto stability with respect to two unit circles, as well as the imaginary axisand unit circle, were shown to be equally elegant and efficient. Over theyears, the efficiency of Siljak’s stability test has been used as a measuringstick by people who have proposed alternative numerical tests for stability oftwo-variable polynomials.

For either modeling or operational reasons, control system parametersare not known precisely. Stability analysis must account for parametric un-certainty in order to provide useful information about system performance.This fact motivated Siljak to initiate a study of robust absolute stability ofthe Lur’e-Postnikov’s nonlinear control system

S : x = Ax + bu

y = cT x

u = −φ(t, y)


where the vectors b and c are parts of an uncertain parameter vector p. Herewrote Popov’s inequality as polynomial positivity

g(ω, p) ≡ det(ω2I + A2)2k − cT A(ω2I + A2)b−−[

cT A(ω2I + A2)b]

> 0, ∀ω ∈ R+

and showed that stability region P in the parameter space is convex withrespect to either b or c, and that the boundary of the region is an envelopedefined by the two equations

g(ω, p) = 0,∂

∂ωg(ω, p) = 0

accompanied by two additional equations

g0(p) = 0, gn(p) = 0.

Again, Siljak and Karmarkar interpreted absolute stability regions in theparameter space using a Chebyshev-like method of inequalities.

When in the 1980’s the work of Kharitonov revived interest in robust sta-bility, Siljak returned to stability under parametric perturbations and wrotehis 1989 survey paper reviewing the field of robust stability in the parameterspace. With his son Matija Siljak (then MSEE student at Stanford Uni-versity) he reconsidered positivity of uncertain polynomials and polynomialmatrices, and with his former student Dusan Stipanovic used Bernstein poly-nomials to solve the problems of robust D-stability, positive realness of realrational functions and matrices, as well as stability of two-variable uncertainpolynomials having polynomic uncertainty structures.

A comprehensive presentation of parameter space control design was pub-lished by Siljak in his first monograph Nonlinear Systems (Wiley, 1969),where a number of interesting topics were introduced and developed. Siljak’sresults on stability and positivity of polynomials have been referred to andused by many people, most notably by V. A. Yakubovich, E. I. Jury, Ya. Z.Tsypkin, G. J. Thaler, A. Tesi, I. B. Yunger, A. Vicino, B. T. Polyak, S.P. Bhattacharyya, Y. Bistritz, V. I. Skorodinskii, S. M. Seltzer, M. Dahleh,R. K. Yedavalli, X. Hu, J. Gregor, L. H. Keel, and D. Henrion.

3 Large-Scale Systems: Connective Stability

In the early 1970’s, Siljak started his research on large-scale systemsthat lasts to the present day. At the outset, he argued that structural un-certainty is the central problem of stability of large interconnected systems.At the 1971 Allerton conference, Siljak introduced the concept of connectivestability, which requires that the system remains stable in the sense of Lya-punov under structural perturbation whereby subsystems are disconnected


and again connected in unpredictable ways during operation. In control-ling large interconnected systems, connective stability has been particularlyuseful because, by accident or design, the controlled systems change theirinterconnection structure.

To sketch the concept of connective stability, let us consider a dynamicsystem

S : x = f(t, x)

where x(t) ∈ Rn is the state of S at time t ∈ R, and the function f : Rn+1 →Rn is piecewise continuous in both arguments. We assume that f(t, 0) = 0for all t ∈ R and that x = 0 is the unique equilibrium of S.

To study the stability of the equilibrium under structural perturbations,we assume that the system is S decomposed as

S : xi = gi(t, xi) + hi(t, x), i ∈ N

which is an interconnection of N subsystems

Si: xi = gi(t, xi), i ∈ N

where xi(t) ∈ Rni is the state of the i-th subsystem Si, function hi : Rn+1 →Rni is the interconnection of subsystem Si with the rest of the system S,and N = 1, 2, . . . , N. To describe the structure of S, we represent theinterconnection functions as

hi(t, x) ≡ hi(t, ei1x1, ei2x2, . . . , eiNxN ), i ∈ N

where the binary numbers eij are elements of the N ×N fundamental inter-connection matrix E = (eij) defined by

eij = 1, xj occurs in hi(t, x)

eij = 0, xj does not occur in hi(t, x)

which is the standard occurrence matrix, meaning eij = 1 if Sj acts on Si, oreij = 0 if it does not. Structural perturbations of S are described by the N×Ninterconnection matrix E = (eij) where the elements eij : Rn+1 → [0, 1] arepiecewise continuous in t and continuous in x. Matrix E is generated by E(denoted E ∈ E) when eij = 0 if and only if eij(t, x) ≡ 0, and describesquantitative and qualitative changes in the interconnection structure of S

defined by E.

A system S is said to be connectively stable if its equilibrium is globallyasymptotically stable for all E ∈ E.

Conditions for connective stability were obtained by Siljak using theMatrosov-Bellman concept of vector Lyapunov functions and the theory ofdifferential inequalities developed by V. Lakshmikantham and S. Leela. Witheach subsystems one associates a scalar function νi : Rni+1 → R+, which is


a continuous function that satisfies a Lipschitz condition in xi with a con-stant κi > 0. We also require that subsystem functions νi(t, xi) satisfy theinequalities

φi1(‖xi‖) ≤ νi(t, xi) ≤ φi2(‖xi‖)D+νi(t, xi)Si

≤ −φi3(‖xi‖), ∀(t, xi) ∈ Rni+1

where φi1, φi2 ∈ K∞, φi3 ∈ K are Hahn’s functions.As for the interconnections, we assume that there are numbers ξij such

that ξij ≥ 0 (i 6= j), and

‖hi(t, x)‖ ≤N∑

j=1

eijξijφi3(‖xj‖), ∀(t, x) ∈ Rn+1.

Finally, we form a scalar Lyapunov function

V (t, x) = dT ν(t, x),

where ν : Rn+1 → RN is the vector Lyapunov function ν = (ν1, ν2, . . . , νN )T

and d ∈ RN+ is a constant vector with positive components, and compute a

differential inequality

D+ν(t, x)S ≤ −dWφ3(x), ∀(t, x) ∈ Rn+1

where φ3 = (φ13, φ23, . . . , φN3)T , and the so-called N × N aggregate matrix

W = (wij) is defined by

wij = 1 − eiiκiξii, i = j

wij = −eijκiξij , i 6= j

The system S is connectively stable if the matrix W is an M-matrix.

A comprehensive early account of connective stability and large-scale sys-tems has been provided by Siljak in his book Large-Scale Dynamic System:Stability and Structure (North Holland, 1978). The initial results have beengreatly improved by works of many people, notably by M. Ikeda, G. S. Ladde,Y. Ohta, Lj. T. Grujic, V. Lakshmikantham, S. Leela, V. A. Matrosov, A. A.Martynyuk, X. Liu, and D. M. Stipanovic. Many of the improvements andgeneralizations have used the well-known stability results obtained by F. N.Bailey, A. Michel, and M. Araki.

4 Competitive-Cooperative Systems

Soon after proposing the connective stability concept, Siljak observed thefact that monotone functions (known also as Kamke’s functions) in differen-tial inequalities appear as gross-substitute excess demand functions in models


of competitive equilibrium in multiple-market systems, as well as models ofmutualism in Lotka-Volterra population models, and Richardson’s model ofarms race. Using this fact, he published a paper in Nature in 1974 proposingM -matrix conditions for stability of multi-species communities, arguing atthe same time that these conditions provide the proper context for resolvingthe stability vs. complexity problem in model ecosystems; the problem wasstudied extensively by many people including W. R. Ashby, R. C. Lewontin,R. M. May, G. S. Ladde, V. Grimm, B. S. Goh, and S. L. Pimm. In a seriesof papers in the 1970’s and in the 1978 monograph on large-scale systems,Siljak applied M -matrices to stability of the models of competition and co-operation in economics, multispecies communities, and arms race. In thiscontext he introduced nonlinear matrix systems,

S : x = A(t, x)x,

where the coefficients of the n × n system matrix A = (aij) were defined as

aij(t, x) = −δijϕi(t, x) + eij(t, x)ϕij(t, x)

δij is the Kronecker symbol, and ϕi(t, x) and ϕij(t, x) are continuous func-tions in both arguments. By properly bounding these functions, Siljak usedagain the M -matrix to establish the connective stability of nonlinear matrixsystems.

The obtained results have been extended by Ladde and Siljak to sta-bility analysis of population models in random environments described bystochastic equations of the Ito type,

dx = A(t, x)xdt + B(t, x)xdz,

where z ∈ R is a random Wiener process with E

[z(t1) − z(t2)]2

= |t1 − t2|.The elements aij(t, x) and bij(t, x) of nonlinear time-varying matrices A =(aij) and B = (bij) have been expressed in terms of the piecewise contin-uous elements of interconnection matrices E = (eij) and L = (lij), respec-tively. The interconnection matrices were used to model uncertain structuralchanges in both the deterministic and stochastic interactions among speciesin the community, and serve as a basis for connective stability of ecosystemsin a stochastic environment.

This line of research has been successfully generalized by G. S. Ladde forstability analysis of compartmental and chemical systems, systems with diffu-sion, and hybrid systems. Expositions of numerous results obtained by manypeople in the field of monotone systems (also known as positive systems) havebeen provided in monographs by G. S. Goh, and by E. Kaszkurewicz and A.Bhaya. In a seminal survey paper on ecosystem stability, V. Grimm and C.Wissel included connective stability in their survey and extensive discussionof concepts of ecological stability.


5 Parametric Stability

During his study of Lotka-Volterra models, Siljak observed that struc-tural perturbations result in changes of the nonzero equilibrium of the model.He called this phenomenon moving equilibrium and pointed out that the stan-dard preconditioning of stability analysis, whereby the equilibrium is shiftedto the origin, may obscure the fact that the original equilibrium is not fixedand its location is uncertain, causing liability in the end results. Siljak, incollaboration with M. Ikeda and Y. Ohta, formulated a new concept of para-metric stability which addresses simultaneously the twin problem of existenceand stability of a moving equilibrium.

When motions of a time-invariant dynamic system

S : x = f(x, p)

with state x(t) ∈ Rn at time t ∈ R, depend on a constant parameter vectorp ∈ Rl, we assume that there is a stable equilibrium state xe(p∗) ∈ Rn for anominal parameter value p∗ ∈ Rl.

We say that system S is parametrically stable at p∗ if there is a neigh-borhood Ω(p∗) ⊂ Rl such that:

(i) an equilibrium xe(p∗) ∈ Rn exists for any p ∈ Ω(p∗), and

(ii) equilibrium xe(p) is stable for any p ∈ Ω(p∗).

The concept has been applied to Lotka-Volterra models relying on M -matrix theory. A challenging parametric absolute stability problem has beensolved for Lur’e-Postnikov systems, which required special care due to un-certain nonlinearities. Two distinct solutions were obtained in collaborationwith T. Wada, one via Popov’s inequality and the other using linear matrixinequalities. Ohta and Siljak presented a scheme to quadratically stabilizeuncertain systems with a moving equilibrium using linear feedback. Recently,the concept has been generalized to include stability of moving invariant setsand dynamical systems on time scales by V. Lakshmikantham and S. Leela.A. I. Zecevic and Siljak showed how stabilization of the moving equilibriumcan be accomplished applying linear feedback control within the frameworkof convex optimization involving linear matrix inequalities.

6 Decentralized Control of Complex Systems

Complexity is a subjective notion, and so is the notion of complex sys-tems. Siljak argued that in controlling large complex systems the accumu-lated experience suggests that the following features are essential:

Dimensionality

Uncertainty


Information Structure Constraints

Most of the methods and results along these lines, which were obtainedin the field of decentralized control before 1991, have been surveyed and pre-sented by Siljak in his monograph Decentralized Control of Complex Systems(Academic Press, 1991).

To describe some of the most interesting ideas in this context, let usconsider a linear time-invariant dynamic system

S : x = Ax + Bu

y = Cx

where x(t) ∈ Rn, u(t) ∈ R

m, and y(t) ∈ Rl are state, input and output of

system S at time t ∈ R. For either conceptual or computational reasonssystem S is decomposed into an interconnection

S : xi = Aixi + Biui +N∑

j=1

(Aijxj + Bijuj)

yi = Cixi +N∑

j=1

Cijxj , i ∈ N

of N subsystemsSi: xi = Aixi + Biui

yi = Cixi

where xi(t) ∈ Rni , ui(t) ∈ Rmi , and yi(t) ∈ Rli are the state, input, andoutput of the subsystem Si at time t ∈ R, all matrices have appropriatedimensions. The decomposition of S is disjoint, that is, no components ofstate, input, and output of S are shared by the subsystems Si.

The crucial restriction in controlling system S is that the state feedbackcontrol law

u = Kx

must be decentralized. Each subsystem Si is controlled by using only thelocally available states xi, that is,

ui = Kixi, i ∈ N

implying the block-diagonal structure K = diagK1, K2, . . . , KN for thegain matrix.

This type of information structure constraint is common in applications.In controlling large power systems, for example, states are not shared betweenthe power areas because they are geographically distant from each other. Aplatoon of vehicles (or robots) is controlled by a decentralized control law toreduce the communication overhead required by a centralized control scheme.


Stabilizing control laws may not exist under decentralized informationstructure even though the system is otherwise controllable and observable.The constraints are non-classical because they cannot be generally incor-porated in standard optimization schemes. For these reasons, advances indecentralized control theory required non-orthodox concepts and methods tomake the resulting control laws work in practice.

7 Graph-Theoretic Algorithms

To improve the manageability of large dynamic systems, Siljak asso-ciated a directed graph (digraph) D =(U × X × Y, E) with system S. Heproposed to replace controllability and observability of S by new notions ofinput-reachability and output-reachability of digraph D, which are minimalrequirements for controllability and observability, but numerically attractivein the control of large systems. In collaboration with M. E. Sezer, Siljak de-veloped a graph-theoretic decomposition algorithm that transforms D intoan input- and output-reachable acyclic structure, which results in a lower-block-triangular form of S

S : xi = Aixi + Biui +i∑

j=1j 6=i

Aijxj +i∑

j=1j 6=i

Bijuj

yi = Cixi +i∑

j=1j 6=i

Cijxj , i ∈ N.

This form can serve as a preconditioner for subsystem-by-subsystem stabi-lization of S using decentralized controllers and observers.

Another favorable structure for decentralized control is epsilon decompo-sition. In this case, by permuting state, input, and output vertices of D thesystem S can be made to appear in the form

S : xi = Aixi + Biui + εN∑

j=1

(Aijxj + Bijuj)

yi = Cixi + εN∑

j=1

Cijxj , i ∈ N

where ε > 0 is a small number and the absolute values of elements of thematrices Aij , Bij , and Cij are smaller than one; the subsystems Si are weaklycoupled. The graph-theoretic algorithm decomposition algorithm, which wasinitiated by Sezer and Siljak, was extremely simple (linear in complexity)and allowed for the efficient use of decentralized control laws. Simply put, bystabilizing the subsystems independently using local feedback, one can stabi-lize the overall system because the interconnections between the subsystems


are weak. The algorithm opened up the possibility of using a large numberof results for weakly coupled systems, which have been obtained by P. V.Kokotovic, Z. Gajic, X. Shen, and many others.

Decentralized fixed modes of S were defined by Wang and Davison asmodes that cannot be shifted by constant decentralized output feedback.Existence of such modes in the right half of the complex plane means thatsystem S cannot be stabilized by decentralized control. Sezer and Siljakargued that a decentralized fixed mode can originate either from a perfectmatching of system parameters (in which case, a slight change of the param-eters can eliminate the mode), or it is due to special properties of digraphD representing the structure of the system (in which case, the mode remainsfixed no matter how much the parameters are perturbed, as long as the orig-inal structure is preserved). From a physical point of view, only the lattertype of fixed modes are important, not only because it is very unlikely tohave an exact matching of the parameters, but also because it is not possibleto know whether such a matching takes place in a given system.

In their 1981 paper, Sezer and Siljak introduced the notion of structurallyfixed modes by considering system S in the following form:

S : x = Ax +N∑

j=1

Biui

yi = Cix, i ∈ N

By defining composite matrices

BN = [B1 B2 . . . BN ] , CN =[

CT1 CT

2 . . . CTN

]T

and applying decentralized output feedback

ui = Kiyi, i ∈ N

one obtains the closed-loop system

S : x =(

A + BNKNCN)

x

where KN is a block-diagonal matrix

KN = diag(K1, K2, . . . , KN).

If Λ(A) denotes the set of eigenvalues of A, and K is a set of all block diagonalmatrices KN, then

ΛK =⋂

KN∈K

Λ(

A + BNKNCN)

is the set of fixed modes with respect to the decentralized control definedabove. A system S with a digraph D has structurally fixed modes with re-spect to a given decentralized control if every system, which has a digraph


isomorphic to D, has fixed modes with respect to the same decentralized con-trol. Sezer and Siljak formulated graph-theoretic (necessary and sufficient)conditions for system S to have structurally fixed modes, and an efficientalgorithm to test their existence.

Once we recognize the fact that decentralized control laws may not ex-ist, it is natural to ask, Can we delineate classes of decentrally stabilizablesystems? Siljak studied this problem first with M. Vukcevic, and later withM. Ikeda, K. Yasuda, and M. E. Sezer. They identified several structures ofinterconnected systems that can always be stabilized by decentralized stateand output feedback. These structures can be interpreted as generalizationsof the matching conditions formulated by G. Leitmann. A structure of thistype played a crucial role in the formulation of adaptive decentralized con-trol design that Siljak proposed with D. T. Gavel in their often referenced1989 paper. In contrast to then existing adaptive decentralized schemes,the Gavel-Siljak design allowed the decentralized feedback gains to rise towhatever level is necessary to ensure that stability of subsystems overridesperturbations caused by interconnection fluctuations, ultimately implyingthe stability of the interconnected system as a whole.

Over more than three decades, Siljak carried out research on decentral-ized stabilization of complex interconnected systems in collaboration withmany people, most of them visitors from the US and abroad, as well as hisgraduate students. The resulting papers and participating co-authors aredocumented in the books and papers listed at the end of this overview. Fromthe outset, Siljak advocated robust decentralized control to cope with bothmodeling and operational uncertainties always arguing that “a normal stateof a complex system is a failed state.” The concepts and techniques for ro-bust control design have been based on vector Lyapunov functions, linearquadratic control theory for both deterministic and stochastic systems, aswell as pole-positioning schemes.

8 Decentralized Control via Convex Optimiza-

tion

In this section, we will outline an interesting design scheme for decen-tralized control which was devised recently by Siljak in collaboration withD. M. Stipanovic and A. I. Zecevic. After more than three decades of re-search, Siljak succeeded in producing a scheme which provides algorithmicgeneration of decentralized control laws for robust stabilization of uncertaininterconnected systems. The scheme is formulated within the framework ofLinear Matrix Inequalities, which has been developed by many people, mostnotably by V. A. Yakubovich, Yu. Nesterov, A. Nemirovskii, and for controlapplications by S. Boyd, L. El Gaoui, E. Feron, and V. Balakrishnan. Most


relevant to the scheme are the papers on decentralized control published byJ. Bernussou, J. C. Geromel, P. Peres, and M. C. de Oliveira.

Let us consider the system

S : xi = Aixi + Biui + hi(t, x), i ∈ N

where hi : Rn+1 → Rn are uncertain piecewise continuous interconnections

hi(t, x) =N∑

j=1

eij(t, x)hij(t, x), i ∈ N

about which we only know that they satisfy a quadratic bound,

hTi (t, x)hi(t, x) ≤ α2

i xT HT

i Hix,

with αi > 0 as interconnection bounds and matrices Hi defined as

Hi =√

N diag ei1βi1In1, ei2βi2In2

, . . . , einβinInN ,

where βij are nonnegative numbers, and Iniare identity matrices of order

ni × ni.Applying decentralized control

u = KDx

where the gain matrix KD = diag(K1, K2, . . . , KN) has the familiar block-diagonal form, closed-loop system S can be written in a compact form

S : x = (AD + BDKD)x + h(t, x)

with matrices AD = diag(A1, A2, . . . , AN ), BD = diag(B1, B2, . . . , BN ), and

interconnection function h =(

hT1 , hT

2 , . . . , hTN

)Tconstrained as

hT (t, x)h(t, x) ≤ xT

(

N∑

i=1

α2i H

Ti Hi

)

x.

Gain matrix KD is computed in the factored form

KD = LDY −1D

where the matrices LD and YD are obtained by solving the following convexoptimization problem:

MinimizeN∑

i =1

γi +N∑

i =1

κLi +

N∑

i =1

κYi subject to:

YD > 0


ADYD + YDATD + BDLD + LT

DBTD I YDHT

1 · · · YDHTN

I −I 0 · · · 0H1YD 0 −γ1I · · · 0

......

.... . .

...HNYD 0 0 · · · −γN I

< 0

γi − 1/α2i < 0, i ∈ N

and[

−κLi I LT

i

Li −I

]

< 0;

[

Yi II κY

i I

]

> 0 i ∈ N

In this problem, αi is a desired (fixed) bound on the interconnectionhi(t, x), which we want to approach from below by minimizing γi. Thetwo positive numbers κL

i and κYi are constraints on the magnitude of the

decentralized gain Ki expressed via the magnitudes of the factors Li and Yi,satisfying

LTi Li < κL

i I, Y −1i < κY

i I.

If the problem is feasible, then the (positive definite) quadratic form

V (x) = xT PDx

is a Liapunov function for the closed-loop system S, where PD = Y −1D .

The optimization approach has been successfully applied to robust de-centralized stabilization of nonlinear multimachine power systems by Siljak,A. I. Zecevic, and D. M. Stipanovic. In this application, it is especially in-teresting that the approach can readily cope with the moving equilibriumof the power system, and secure parametric stability by linear decentralizedfeedback.

9 The Inclusion Principle

In a wide variety of applications, there is a need to deal with the mul-tiplicity of mathematical models having different dimensions but describingthe same dynamic system. An obvious example is the area of model reduc-tion, where we want to reduce the dimension of a system in order to gain inmanageability at the expense of accuracy. A less obvious benefit has beenthe expansion of a given system to gain conceptual and numerical advantagesthat a larger model can offer. Such is the case, for example, with the designof decentralized control for power systems. It is natural to consider dynamicsof a tie-line between two power areas as an inseparable part of both areas.A way to design local control laws is to let the two subsystems (power areas)share a common part (tie-line); this is an overlapping decomposition. By ex-panding the system in a larger space, the subsystems appear as disjoint and


decentralized control laws can be generated by standard methods for disjointsubsystems. After the control laws are obtained in the expanded space, theycan be contracted for implementation in the system residing in the originalsmaller space. The expansion-contraction design process was utilized in 1978by Siljak, when he proposed multi-controller schemes for building reliablesystems faced with controller uncertainty. Several years later, the InclusionPrinciple was formulated by Ikeda and Siljak in collaboration with Siljak’sgraduate student D. E. White, which offered a rich mathematical frameworkin an algebraic setting for comparing dynamic systems of different dimen-sions. The following brief account of the principle will suffice to convey themain concepts.

Let us consider a pair of linear dynamic systems

S : x = Ax + Bu

y = Cx

S : ˙x = Ax + Bu

y = Cx

where x ∈ X , u ∈ U and y ∈ Y are state, input, and output vectors of systemS having dimensions n, m and l, and x ∈ X , u ∈ U and y ∈ Y are state,input, and output vectors of system S having dimensions n, m and l. Thecrucial assumption is that the dimension of S is larger than or equal to thedimension of S, that is,

n ≤ n, m ≤ m, l ≤ l.

For t ≥ t0, the input functions u[t0,t] : [t0, t] → U and u[t0,t] : [t0, t] → Ubelong to the set of piecewise continuous functions Uf and Uf . By x(t; t0, x0,

u[t0,t]) and x(t; t0, x0, u[t0,t]) we denote the solutions of systems S and S forinitial time t0 ∈ R+, initial states x0 ∈ Rn and x0 ∈ Rn, and fixed controlfunctions u[t0,t] ∈ Uf and u[t0,t] ∈ Uf . It is further assumed that state, input,

and output spaces of S and S are related by linear transformations definedby monomorphisms

V : X → X , R : U → U , T : Y → Yand epimorphisms

U : X → X , Q : U → U , S : Y → Yhaving the full rank, so that

UV = In, QR = Im, ST = Il.

The inclusion principle is stated as follows:

S includes S, or S is included in S, that is, S is an expansion of S andS is a contraction of S, if there exists a quadruplet of matrices (U, V, R, S)such that, for any t0, x0, and u[t0,t] ∈ Uf , the choice

x0 = V x0

u(t) = Ru(t), for all t ≥ t0


impliesx(t; t0, x0, u[t0,t]) = Ux(t; t0, x0, u[t0,t])

y[x(t)] = Sy[x(t)], for all t ≥ t0.

Stated simply, this means that the solution space of system S includesthe solution space of the smaller system S. An immediate (and significant)consequence of this fact is that the stability of the expanded system S impliesthe same property for the contracted system S. This further implies that wecan, for conceptual or computational reasons, expand system S in a largerspace, determine a feedback control law

u = Kx

that stabilizes the expansion S, and contract the law from the expandedspace to the law

u = Kx,

which stabilizes the original system S.The expansion-contraction process has been used in a wide variety of ap-

plications including electric multi-machine power systems, large segmentedtelescope, complex rotor-bearing system, expert systems, platoons of vehicleson freeways, and formation flight of aerial vehicles. We should also mentionthe work involving solutions of large algebraic equations by overlapping de-compositions and parallel processing. Broadening of the theoretical base ofthe principle has included stochastic systems (optimization and stability),systems with hereditary elements, discrete-time, nonlinear and hybrid sys-tems, and systems with discontinuous control. Results have been obtainedby many people, including M. Ikeda, M. E. Sezer, S. S. Stankovic, A. Iftar, L.Bakule, A. I. Zecevic, U. Ozguner, Y. Ohta, R. Krtolica, A. A. Martynyuk,M. Hodzic, K. D. Young, Z. Gajic, M. F. Hassan, X.-B. Chen, and H. Ito.

10 Reliable Control

Due to the ever increasing complexity of engineering systems, reliabilityof control design has become a pressing issue. For this reason, in 1980 Siljakintroduced the concept of reliable control. A novel feature of the concept wasthe multi-controller structure whereby more than one controller was assignedto a plant. This essential redundancy of control, which was designed to copewith controller failures, imitated von Neumann’s dictum “in building complexcomputing machines, unreliability of components should be overcome not bymaking the components more reliable, but organizing them in such a waythat the reliability of the whole computer is greater than the reliability of itsparts.” The dictum was subsequently utilized by Moore and Shannon as aprinciple in synthesizing reliable circuits by employing functional redundancy.


In a generic multi-controller configuration, the designer attaches two con-trollers to a plant,

C1 : x1 = A11x1 + A12x2 + B1u1

u1 = K11x1 + K12x2

P : x2 = A21x1 + A22x2 + A23x3 + h(t, x2)

C2 : x3 = A32x2 + A33x3 + B2u2

u2 = K22x2 + K23x3

and designs the control laws in such a way that if any one of the controllers(C1 or C2) fails, the remaining controller would stabilize the (unstable) plantP. Out of the four possibilities

C1C2, 6C1C2, C1 6C2, 6C1 6C2

the first three are permissible, because there is at least one controller at-tached to the plant. Siljak introduced the notion of simultaneous stabilitythat requires that three distinct systems, which correspond to the first threefunctioning control structures, be stable at the same time.

By attaching the probability of controller failures, reliability of controlhas been formulated in probabilistic terms. To include maintenance of faultycontrollers, when a failure state is reached (the fourth possibility where bothcontrollers fail), reliable control was cast by Ladde and Siljak in the stochasticstability framework following the seminal work on stochastic dynamic systemsby I. Ya. Kats and N. N. Krasovskii, as well as the works of E. A. Lidskii,D. Sworder, V. Lakshmikantham, and D. T. Liu. The overall system wasdescribed as an interconnected system

S : x = (AD + BDKD)x + A(η(t))x

p = Πp

where η(t) is a Markov process with four states corresponding to the con-troller configurations above. The stochastic stability of system S, whichguaranteed reliable performance, was determined by applying the stochasticversion of the vector Lyapunov function. The concept of reliable and stochas-tic stability control has been further developed by many people, includingM. Vidyasagar, N. Viswanadham, A. N. Gundes, Z. Bien, A. B. Ozguler, A.Loparo, J. Medanic, W. R. Perkins, M. Mariton, P. Bertrand, P. V. Pakshin,E. K. Boukas, R. J. Veillette, G.-H. Yang, and X.-L. Tan.


Books

Decentralized Control of Complex SystemsAcademic Press, Boston, MA, 1991

Large-Scale Dynamic Systems: Stability and StructureNorth-Holland, New York, 1978

Stability of Control Systems(in Serbo-Croatian), Forum, Novi Sad, Serbia, 1974

Nonlinear Systems: The Parameter Analysis and DesignWiley, New York, 1969.

Journal Papers

Complex Dynamic Systems

Inclusion Principle for Linear Time-Varying Systems, SIAM Journal on Con-trol and Optimization, 42 (2003), 321-341 (with S. S. Stankovic).

Stabilization of Nonlinear Systems With Moving Equilibria, IEEE Transac-tions on Automatic Control, 48 (2003), 1036-1040 (with A. I. Zecevic).

Connective Stability of Discontinuous Dynamic Systems, Journal of Opti-mization Theory and Applications, 115 (2002), 711-726 (with D. M. Sti-panovic).

Model Abstraction and Inclusion Principle: A Comparison, IEEE Transac-tions on Automatic Control, 47 (2002), 529-532 (with S. S. Stankovic).

Contractibility of Overlapping Decentralized Control, Systems & Control Let-ters, 44 (2001), 189-200 (with S. S. Stankovic).

Robust Stability and Stabilization of Discrete-Time Non-Linear Systems: TheLMI Approach, International Journal of Control, 74 (2001), 873-879 (with D.M. Stipanovic).

Robust Stabilization of Nonlinear Systems: The LMI Approach, Mathemati-cal Problems in Engineering, 6 (2000), 461-493 (with D. M. Stipanovic).

Decentralized Overlapping Control of a Platoon of Vehicles, IEEE Transac-tions on Control Systems Technology, 8 (2000), 816-832 (with S. S. Stankovicand M. J. Stanojevic).

Large-Scale and Decentralized Systems, Wiley Encyclopedia of Electrical andElectronics Engineering, J. G. Webster (ed.), 11 (1999), 209-224 (with A. I.Zecevic).

Headway Control of a Platoon of Vehicles Based on the Inclusion Principle,In Complex Dynamical Systems With Incomplete Information, E. Reithmeier


and G. Leitmann, (eds.), Shaker Verlag, (1999), 153-163 (with S. S. Stankovicand S. M. Mladenovic).

Decentralized Control and Computations: Status and Prospects, AutomaticaReviews in Control, 20 (1996), 131-141.

Decentralized Control, The Control Handbook, W. S. Levine (Ed.), CRCPress, Boca Raton, FL, (1996), 779-793 (with M. E. Sezer).

Optimal Decentralized Control for Stochastic Dynamic Systems, Recent Trendsin Optimization Theory and Applications, R. P. Agarwal (ed.), World Scien-tific, Singapore, (1995), 337-352 (with S. V. Savastyuk).

Robust Stabilization of Nonlinear Systems via Linear State Feedback, RobustControl System Techniques and Applications, C. T. Leondes (ed.), AcademicPress, Boston, 51 (1992), 1-30 (with M. Ikeda).

Reliable Stabilization via Factorization Methods, IEEE Transactions on Au-tomatic Control, 37 (1992), 1786-1791 (with M. Ikeda and X. L. Tan).

Parametric Stability, New Trends in Systems Theory, G. Conte, A.M. Perdonand B. Wyman (eds.), Birkhauser, Boston, MA, (1991), 1-20 (with M. Ikedaand Y. Ohta).

A Stochastic Inclusion Principle, Differential Equations: Stability and Con-trol, S. Elaydi (ed.), Marcel Dekker, New York, (1991), 295-320 (with R.Krtolica and M. Hodzic).

Decentralized Multirate Control, IEEE Transactions on Automatic Control,35 (1990), 60-65 (with M. E. Sezer).

Optimality and Robustness of Linear-Quadratic Control for Nonlinear Sys-tems, Automatica, 26 (1990), 499-511 (with M. Ikeda).

Sequential LQG Optimization of Hierarchically Structured Systems, Auto-matica, 25 (1989), 545-559 (with S. S. Stankovic).

Decentralized Adaptive Control: Structural Conditions for Stability, IEEETransactions on Automatic Control, AC-34 (1989), 413-426 (with D. T.Gavel).

Robust Stability of Discrete Systems, International Journal of Control, 48(1988), 2055-2063 (with M. E. Sezer).

On Almost Invariant Subspaces of Structured Systems and Decentralized Con-trol, IEEE Transactions on Automatic Control, AC-33 (1988), 931-939 (Y.Hayakawa).

Reliability of Control, Systems and Control Encyclopedia, M. G. Singh (ed.),Pergamon Press, London, (1987), 4008-4011.

Overlapping Decentralized Control, Systems and Control Encyclopedia, M.G. Singh (ed.), Pergamon Press, London, (1987), 3568-3572.


Interconnected Systems: Decentralized Control, Systems and Control Ency-clopedia, M. G. Singh (ed.), Pergamon Press, London, (1987), 2557-2560.

Nested Epsilon-Decompositions and Clustering of Complex Systems, Auto-matica, 22 (1986), 321-331 (with M. E. Sezer).

Decentralized Control Using Quasi-Block Diagonal Dominance of TransferFunction Matrices, IEEE Transactions on Automatic Control, AC-31 (1986),420-430 (with Y. Ohta and T. Matsumoto).

Overlapping Decentralized Control with Input, State, and Output Inclusion,Control Theory and Advanced Technology, 3 (1986), 155-172 (with M. Ikeda).

Decentralized Estimation and Control with Overlapping Information Sets,IEEE Transactions on Automatic Control, AC-31 (1986), 83-86 (with M.Hodzic).

Overlapping Block Diagonal Dominance and Existence of Liapunov Func-tions, Journal of Mathematical Analysis and Applications, 112 (1985), 396-410 (with Y. Ohta).

Hierarchical Liapunov Functions, Journal of Mathematical Analysis and Ap-plications, 111 (1985), 110-128 (with M. Ikeda).

Estimation and Control of Large Sparse Systems, Automatica, 21 (1985),227-292 (with M. Hodzic).

Failure Detection in Large Sparse Systems, Large-Scale Systems, 9 (1985),83-100 (with M. Hodzic).

Overlapping Decentralized Control of Linear Time-Varying Systems, Advancesin Large Scale Systems, J.B. Cruz, Jr. (ed.), JAI Press, 1 (1984), 93-116 (withM. Ikeda).

An Inclusion Principle for Dynamic Systems, IEEE Transactions on Auto-matic Control, AC-29 (1984), 244-249 (with M. Ikeda and D. E. White).

A Graph-theoretic Characterization of Structurally Fixed Modes, Automat-ica, 20 (1984), 247-250 (with M. E. Sezer).

An Inclusion Principle for Hereditary Systems, Journal of MathematicalAnalysis and Applications, 98 (1984), 581-598 (with Y. Ohta).

A Graphical Test for Structurally Fixed Modes, Mathematical Modeling, 4(1983), 339-348 (with M. E. Sezer).

Multiparameter Singular Perturbations of Linear Systems with Multiple TimeScales, Automatica, 19 (1983), 385-394 (with G. S. Ladde).

Complex Dynamic Systems: Dimensionality, Structure, and Uncertainty,Large Scale Systems, 4 (1983), 279-294.

Optimality of Decentralized Control for Large-Scale Systems, Automatica, 19(1983), 309-316 (with M. Ikeda and K. Yasuda).


Multiplex Control Systems: Stochastic Stability and Dynamic Reliability, In-ternational Journal of Control, 38 (1983), 515-524 (with G. S. Ladde).

A Graph-Theoretic Algorithm for Hierarchical Decomposition of DynamicSystems with Applications to Estimation and Control, IEEE Transactionson Systems, Man, and Cybernetics, SMC-13 (1983), 197-207 (with V. Pichaiand M. E. Sezer).

Robust Decentralized Control Using Output Feedback, IEE Proceedings, 129(1982), 310-314 (with O. Huseyin and M. E. Sezer).

When is a Linear Decentralized Control Optimal? Analysis and Optimiza-tion of Systems, A. Bensoussan and J. L. Lions (eds.), Springer-Verlag, 24(1982), 419-431 (with M. Ikeda).

Validation of Reduced-Order Models for Control System Design, Journal ofGuidance, Control, and Dynamics, 5 (1982), 430-437 (with M. E. Sezer).

Generalized Decompositions of Dynamic Systems and Vector Lyapunov Func-tions, IEEE Transactions on Automatic Control, AC-26 (1981), 1118-1125(with M. Ikeda).

Sensitivity of Large-Scale Systems, Journal of the Franklin Institute, 312(1981), 179-197 (with M. E. Sezer).

Robustness of Suboptimal Control: Gain and Phase Margin, IEEE Transac-tions on Automatic Control, AC-26 (1981), 907-911 (with M. E. Sezer).

On Decentralized Stabilization and Structure of Linear Large Scale Systems,Automatica, 17 (1981), 641-644 (with M. E. Sezer).

Structurally Fixed Modes, Systems & Control Letters, 1 (1981), 60-64 (withM. E. Sezer).

Decentralized Control with Overlapping Information Sets, Journal of Opti-mization Theory and Applications, 34 (1981), 279-310 (with M. Ikeda).

Vulnerability of Dynamic Systems, International Journal of Control, 34 (1981),1049-1060 (with M. E. Sezer).

On Structural Decomposition and Stabilization of Large-Scale Control Sys-tems, IEEE Transactions on Automatic Control, AC-26 (1981), 439-444 (withM. E. Sezer).

Decentralized Stabilization of Large Scale Systems with Time Delay, LargeScale Systems, 1 (1980), 273-279 (with M. Ikeda).

On Decentrally Stabilizable Large-Scale Systems, Automatica, 16 (1980), 331-334 (with M. Ikeda).

Decentralized Stabilization of Linear Large-Scale Systems, IEEE Transactionson Automatic Control, AC-25 (1980), 106-107 (with M. Ikeda).

Overlapping Decompositions, Expansions, and Contractions of Dynamic Sys-


tems, Large Scale Systems, 1 (1980), 29-38 (with M. Ikeda).

Reliable Control Using Multiple Control Systems, International Journal ofControl, 31 (1980), 303-329.

Suboptimality of Decentralized Stochastic Control and Estimation, IEEE Trans-actions on Automatic Control, AC-25 (1980), 76-83 (with R. Krtolica).

Counterexamples to Fessas’ Conjecture, IEEE Transactions on AutomaticControl, AC-24 (1979), 670 (with M. Ikeda).

Overlapping Decentralized Control, Handbook of Large Scale Systems Engi-neering Applications, M.G. Singh and A. Titli (eds.), North-Holland, NewYork, (1979), 145-166.

On Decentralized Control of Large-Scale Systems, Dynamics of MultibodySystems, K. Magnus (ed.), Springer, New York, (1978), 318-330.

On Decentralized Estimation, International Journal of Control, 27 (1978),113-131 (with M. B. Vukcevic).

Decentrally Stabilizable Linear and Bilinear Large-Scale Systems, Interna-tional Journal of Control, 26 (1977), 289-305 (with M. B. Vukcevic).

On Pure Structure of Dynamic Systems, Nonlinear Analysis, Theory, Meth-ods, and Applications, 1 (1977), 397-413.

On Reachability of Dynamic Systems, International Journal of Systems Sci-ence, 8 (1977), 321-338.

Decentralization, Stabilization, and Estimation of Large-Scale Linear Sys-tems, IEEE Transactions on Automatic Control, AC-21 (1976), 363-366 (withM. B. Vukcevic).

Large-Scale Systems: Optimality vs. Reliability, Decentralized Control, Mul-tiPerson Optimization, and Large-Scale Systems, Y.C. Ho and S.K. Mitter(eds.), Plenum Press, New York, (1976), 411-425 (with M. K. Sundareshan).

A Multilevel Optimization of Large-Scale Dynamic Systems, IEEE Transac-tions on Automatic Control, AC-21 (1976), 79-84 M. K. Sundareshan).

Multilevel Control of Large-Scale Systems: Decentralization, Stabilization,Estimation, and Reliability, Large-Scale Dynamical Systems, R. Saeks (ed.),Point Lobos Press, Los Angeles, Calif., (1976), 33-57 (with M. B.Vukcevic).

Large-Scale Systems: Stability, Complexity, Reliability, Journal of the FranklinInstitute, 301 (1976), 49-69.

Connective Stability of Large-Scale Discrete Systems, International Journalof Control, 6 (1975), 713-721 (with Lj. T. Grujic).

Exponential Stability of Large-Scale Discrete Systems, International Journalof Control, 19 (1974), pp. 481-491 (with Lj. T. Grujic).

Asymptotic Stability and Instability of Large-Scale Systems, IEEE Transac-


tions on Automatic Control, AC-18 (1973), 636-645 (with Lj. T. Grujic).

On Stability of Large-Scale Systems Under Structural Perturbations, IEEETransactions on Systems, Man and Cybernetics, SMC-3 (1973), 415-417.

On Stability of Discrete Composite Systems, IEEE Transactions on Auto-matic Control, AC-18 (1973), 552-524 (with Lj. T. Grujic)..

Stability of Large-Scale Systems Under Structural Perturbations, IEEE Trans-actions on Systems, Man and Cybernetics, SMC-2 (1972), 657-663.

Stability, Positivity, and Control

SPR Criteria for Uncertain Rational Matrices via Polynomial Positivity andBernstein’s Expansions, IEEE Transactions of Circuits and Systems I, 48(2001), 1366-1369 (with D. M. Stipanovic).

Book Review of “Matrix Diagonal Stability in Systems and Computations,”by E. Kaszkurevicz and A. Bhaya, SIAM Review, 43 (2001), 225-228

Stability of Polytopic Systems via Convex M-Matrices and Parameter-DependentLiapunov Functions, Nonlinear Analysis, 40 (2000) 589-609 (with D. M. Sti-panovic).

Parametric Absolute Stability of Multivariable Lur’e Systems, Automatica,36 (2000), 1365-1372 (with T. Wada, M. Ikeda and Y. Ohta).

Robust D-Stability via Positivity, Automatica, 35 (1999), 1477-1484(with D.M. Stipanovic).

Parametric Absolute Stability of Lur’e Systems, IEEE Transactions on Auto-matic Control, 43 (1998), 1649-1653 (with T. Wada, M. Ikeda and Y. Ohta).

Nonnegativity of Uncertain Polynomials, Mathematical Problems in Engi-neering, 4 (1998), 135-163 (with M. D. Siljak).

Extensions of Ladas’ Oscillation Theorem to Polytopic Difference Equations,Journal of Difference Equations and Applications, 3 (1998), 473-483.

Stability Regions of Equilibria in High-Gain Neural Networks, Nonlinear World,4 (1997), 403-416 (with Y. Ohta).

Parametric Absolute Stability of Multivariable Lur’e Systems: A Popov-TypeCondition and Application of Polygon Interval Arithmetic, Nonlinear Anal-ysis, Theory, Methods & Applications, 30 (1997), 3713-3723(with T. Wada,M. Ikeda and Y. Ohta).

A Learning Scheme for Dynamic Neural Networks: Equilibrium Manifoldand Connective Stability, Neural Networks, 8 (1995), 853-864 (H. C. Tseng).

Parametric Quadratic Stabilizability of Uncertain Nonlinear Systems, Sys-tems and Control Letters, 22 (1994), 437-444 (with Y. Ohta).

On Stability of Interval Matrices, IEEE Transactions on Automatic Control,


39 (1994), 368-371 (with M. E. Sezer).

A Robust Control Design in the Parameter Space, Robustness of DynamicSystems with Parameter Uncertainties, M. Mansour, S. Balemi, and W. Truol(eds.), Birkhauser, Basel, Switzerland, (1992), 229-240.

Polytopes of Nonnegative Polynomials, Recent Advances in Robust Control,P. Dorato and R. K. Yedavalli (eds.) IEEE Press, New York, (1990), 71-77.

Parameter Space Methods for Robust Control Design: A Guided Tour, IEEETransactions on Automatic Control, 34 (1989), 674-688.

A Note on Robust Stability Bounds, IEEE Transactions on Automatic Con-trol, 35 (1989), pp. 1212-1215 (with M. E. Sezer).

Routh’s Algorithm: A Centennial Survey, SIAM Review, 19 (1977), 472-489(S. Barnett).

Stability Criteria for Multivariable Polynomials, Electronic Letters, 11 (1975),217-218.

Maximization of Absolute Stability Regions by Mathematical ProgrammingMethods, Regelungstechnik, 23 (1975), 59-61 (J. S. Karmarkar).

Stability Criteria for Two-Variable Polynomials, IEEE Transactions on Cir-cuits and Systems, CAS-22 (1975), 185-189.

Comments on Stability Test for Two-Dimensional Recursive Filters, IEEETransactions on Acoustics, Speech, and Signal Processing, ASSP-32 (1974),473.

On Exponential Absolute Stability, International Journal of Control,19 (1974),481-491.

Regions of Absolute Ultimate Boundedness for Discrete-Time Systems, Regelung-stechnik, 21 (1973), 329-332 (S. Weissenberger).

Algebraic Criteria for Positive Realness Relative to the Unit Circle, Journalof the Franklin Institute, 295 (1973), 469-482.

Singular Perturbation of Absolute Stability, IEEE Transactions on AutomaticControl, AC-17 (1972), 720.

Absolute Stability of Attitude Control Systems for Large Boosters, Journal ofSpacecraft and Rockets, 9 (1972), 506-510 (with S. M. Seltzer).

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STABILITY THEORY FOR SET DIFFERENTIALEQUATIONS

V. Lakshmikantham, S. Leela and J. Vasundhara Devi1

Department of Mathematical SciencesFlorida Institute of Technology, Melbourne, FL 32901, USA.

To our dear friend Professor D. D. Siljak on his 70th birthday.

Abstract. The formulation of set differential equations has an intrinsic disadvantage that

the diameter of the solution is nondecreasing as time increases and therefore the behavior of

solutions, in some cases, do not match with the solutions of ordinary differential equations

from which set differential equations can be generated. In this paper an approach is

provided to remove the disadvantage.

Keywords. Set differential equations, stability.

AMS (MOS) subject classification: 34D20, 34G20.

1 Preliminaries

Let K(Rn)(Kc(Rn)) denote the collection of all nonempty, compact (com-

pact, convex) subsets of Rn. Define the Hausdorff metric

D[A, B] = max

[

supx∈B

d(x, A), supy∈A

d(y, B)

]

(1.1)

where d[x, A] = inf[d(x, y) : y ∈ A], A, B are bounded sets in Rn. We notethat K(Rn), (Kc(R

n)), with the metric is a complete metric space.It is known that if the space Kc(R

n) is equipped with the natural algebraicoperations of addition and nonnegative scalar multiplication, then Kc(R

n)becomes a semilinear metric space which can be embedded as a completecone into a corresponding Banach space [2, 14].

The Hausdorff metric (1.1) satisfies the following properties.

D[A + C, B + C] = D[A, B] and D[A, B] = D[B, A], (1.2)

D[λA, λB] = λD[A, B], (1.3)

D[A, B] ≤ D[A, C] + D[C, B], (1.4)

for all A, B, C ∈ Kc(Rn) and λ ∈ R+.

1on leave from GVP College for PG courses, Visakhapatnam, India.

182 V. Lakshmikantham, S. Leela and J. Vasundhara Devi

Let A, B ∈ Kc(Rn). The set C ∈ Kc(R

n) satisfying A = B + C is knownas the Hukuhara difference of the sets A and B and is denoted by the symbolA−B. We say that the mapping F : I → Kc(R

n) has a Hukuhara derivativeDHF (t0) at a point t0 ∈ I, if there exists an element DHF (t0) ∈ Kc(R

n)such that the limits

limh→0+

F (t0 + h) − F (t0)

h, and lim

h→0+

F (t0) − F (t0 − h)

h(1.5)

exist in the topology of Kc(Rn) and are equal to DHF (t0). Here I is any

interval in R.

Proposition 1.1 If F : I → Kc(Rn) is Hukuhara differentiable, then the

real valued function t → diam F (t), t ∈ I is nondecreasing on I.

We note that the existence of the limits (1.5) was not used in the proofof the Proposition 1.1. See [1]. In fact, instead of the hypothesis that F isHukuhara differentiable on I, one could substitute the assumption that foreach t ∈ I, the Hukuhara differences

F (t + h) − F (t) and F (t) − F (t − h)

both exist for sufficiently small h > 0. We should also note that the fact F (t)is Hukuhara differentiable on I and diam F (t) > 0, t ∈ I need not implythat F (t) is monotone relative to the set inclusion. See [1].

By embedding Kc(Rn) as a complete cone in a corresponding Banach

space and taking into account the result on differentiation of Bochner integral,we find that if

F (t) = X0 +

∫ t

0

Φ(s) ds, X0 ∈ Kc(Rn), (1.6)

where Φ : I → Kc(Rn) is integrable in the sense of Bochner, then DHF (t)

exists and the equality

DHF (t) = Φ(t), a.e on I, (1.7)

holds. Also, the Hukuhara integral

∫

I

F (s) ds =

[∫

I

f(s) ds : f is a continuous selector of F

]

,

for any compact set I ⊂ R+. With these preliminaries, we consider the setdifferential equation

DHU = F (t, U), U(t0) = U0 ∈ Kc(Rn), t0 ≥ 0, (1.8)

where F ∈ C [R+ × Kc(Rn), Kc(R

n)].

Stability Theory for Set Differential Equations 183

The mapping U ∈ C1 [J, Kc(Rn)], J = [t0, t0 + a] is said to be a solution

of (1.8) on J if it satisfies (1.8) on J . Since U(t) is continuously differentiable,we have

U(t) = U0 +

∫ t

t0

DHU(s) ds, t ∈ J. (1.9)

Thus we associate with the initial value problem(IVP)(1.8) the following

U(t) = U0 +

∫ t

t0

F (s, U(s))ds, t ∈ J, (1.10)

where the integral is the Hukuhara integral [8, 9]. Observe also that U(t) isa solution of (1.8) iff it satisfies (1.10) on J .

The investigation of set differential equation (1.8) as an independent sub-ject has some advantages. For example, when U(t) is a single valued mapping,it is easy to see that Hukuhara derivative and the integral reduce to the or-dinary vector derivative and the integral, and therefore, the results obtainedin this new framework of (1.8) become the corresponding results of ordinarydifferential systems. Also, we have only semilinear complete metric spaceto work with, in the present setup, compared to the complete normed linearspace one employs in the study of ordinary differential systems. Furthermore,set differential equations, that are generated by multivalued differential inclu-sions, when the multivalued functions involved do not possess convex values,can be used as a tool for studying multivalued differential inclusions. SeeTolstonogov [14].

In this paper, we shall investigate the stability criteria for set differentialequation (1.8). For this purpose, we shall utilize needed known results forsuch equations. Recently, the study of set differential equations is initiatedas an independent subject and several basic results on existence, uniqueness,comparison result, global existence and continuous dependence are discussedin [2-7, 11-13].

2 Results in Set Differential Equations

In order to discuss the stability theory of set differential equations (1.8), weneed some known results [3, 11] relative to such equations. We start with acomparison result.

Theorem 2.1 Assume that F ∈ C[R+ × Kc(Rn), Kc(R

n)] and for t ∈ R+,

A, B ∈ Kc(Rn)

D[F (t, A), F (t, B)] ≤ g(t, D[A, B]),

where g ∈ C[R2+, R+]. Suppose further that the maximal solution r(t, t0, w0)

of the scalar differential equation

w′ = g(t, w), w(t0) = w0 ≥ 0,


exists for t ≥ t0. Then, if U(t) = U(t, t0, U0), V (t) = V (t, t0, V0) are any twosolutions of equation (1.8)such that U(t0) = U0, V (t0) = V0, U0, V0 ∈ Kc(R

n)existing for t ≥ t0, we have:

D[U(t), V (t)] ≤ r(t, t0, w0), t ≥ t0,

provided D[U0, V0] ≤ w0.

The next result is an existence and uniqueness Theorem more general thanLipschitz type condition, the proof of which exhibits the idea of comparisonprinciple.

Theorem 2.2 Assume that

(a) F ∈ C[R0, Kc(Rn)], where R0 = J × B(U0, b), J = [t0, t0 + a], a > 0,

B(U0, b) = [U ∈ Kc(Rn) : D[U, U0] ≤ b] and D[F (t, U), θ] ≤ M0 on

R0, where θ is the zero element of Rn regarded as a one point set;

(b) g ∈ C [J × [0, 2b], R+] , g(t, w) ≤ M1 on J × [0, 2b], g(t, 0) ≡ 0, g(t, w)is nondecreasing in w for each t ∈ J and w(t) ≡ 0 is the only solutionof

w′ = g(t, w), w(t0) = 0;

(c) D[F (t, U), F (t, V )] ≤ g(t, D[U, V ]) on R0.

Then the successive approximations defined by

Un+1(t) = U0 +

∫ t

t0

F (s, Un(s))ds, n = 0, 1, 2, . . . ,

exist on J0 = [t0, t+α], α = min(a, bM

], M = max(M0, M1), as continuousfunctions and converge uniformly to the unique solution U(t) = U(t, t0, U0)of IVP (1.8) on J0.

The following global existence result is a special case of Theorem 5.2 in[11] which serves our purpose.


(1) F ∈ C[R+ × Kc(Rn), Kc(R

n)] and for (t, A) ∈ R+ × Kc(Rn),

D[F (t, A), θ] ≤ q(t, D[A, θ]),

where q ∈ C[R2+, R+], q(t, w) is nondecreasing in w for each t ∈ R+

and the maximal solution r(t, t0, w0) of

w′ = q(t, w), w(t0) = w0,

exists for t ≥ t0 and for every w0 > 0;


(2) there exists a local solution U(t) = U(t, t0, U0) of (1.8) for every(t0, U0) ∈ R+ × Kc(R

n).

Then for every U0 ∈ Kc(Rn) such that D[U0, θ] ≤ w0, the IVP (1.8) possesses

a solution U(t) = U(t, t0, U0) defined for t ≥ t0, satisfying

D[U(t), θ] ≤ r(t, t0, w0), t ≥ t0.

3 Lyapunov-Like Functions And Stability Criteria

To investigate the stability criteria of (1.8) the following comparison theoremin terms of Lyapunov-like functions is very important, which can be provedvia the theory of differential inequalities.


(i) V ∈ C[R+ × Kc(Rn), R+] and |V (t, A) − V (t, B)| ≤ LD[A, B], where

L is the local Lipschitz constant, for A, B ∈ Kc(Rn), t ∈ R+;

(ii) g ∈ C[R2+, R] and for t ∈ R+, A ∈ Kc(R

n),

D+V (t, A) ≡ lim suph→0+

1

h[V (t + h, A + hF (t, A))−V (t, A)] ≤ g (t, V (t, A)) .

Then, if U(t) = U(t, t0, U0) is any solution of (1.8) existing on [t0,∞) suchthat V (t0, U0) ≤ w0, we have

V (t, U(t) ≤ r(t, t0, w0), t ∈ [t0,∞),

where r(t, t0, w0) is the maximal solution of

w′ = g(t, w), w(t0) = w0 ≥ 0, (3.1)

existing on [t0,∞).

Proof: Let U(t) = U(t, t0, U0) be any solution of (1.8) existing on [t0,∞).Define m(t) = V (t, U(t)) so that m(t0) = V (t0, U0) ≤ w0. Now for smallh > 0,

m(t + h) − m(t) = V (t + h, U(t + h)) − V (t, U(t))

= V (t + h, U(t + h)) − V (t + h, U(t) + hF (t, U(t)))

+V (t + h, U(t) + hF (t, U(t))) − V (t, U(t))

≤ LD [U(t + h), U(t) + hF (t, U(t))]

+V ((t + h, U(t) + hF (t, U(t))) − V (t, U(t)) ,


using the Lipschitz condition given in (i). Thus

D+m(t) = lim suph→0+

1

h[m(t + h) − m(t)]

≤ D+V (t, U(t)) + L lim suph→0+

1

h[D[U(t + h), U(t) + hF (t, U(t))]] .

Let U(t + h) = U(t) + Z(t), where Z(t) is the Hukuhara difference ofU(t + h), U(t) for small h > 0, where Z(t) is assumed to exist. Hence em-ploying the properties of D[·, ·], we see that

D[U(t + h), U(t) + hF (t, U(t))] = D[U(t) + Z(t), U(t) + hF (t, U(t))]

= D[Z(t), hF (t, U(t))] = D[U(t + h) − U(t), hF (t, U(t))].

As a result, we find that

1

h[D[U(t + h), U(t) + hF (t, U(t))] = D

[

U(t + h) − U(t)

h, F (t, U(t))

]

,

and therefore,

lim suph→0+

1

h[D[U(t + h), U(t) + hF (t, U(t))]]

= lim suph→0+

D

[

U(t + h) − U(t)

h, F (t, U(t))

]

= D [U ′

H(t), F (t, U(t))] ≡ 0,

since U(t) is a solution of (1.8). We therefore have the scalar differentialinequality

D+m(t) ≤ g (t, m(t)) , m(t0) ≤ w0,

which yields the desired estimate in view of Theorem 1.4.1 in [10],

m(t) ≤ r(t, t0, w0), t ∈ [t0,∞).

This proves the claimed estimate of the Theorem.Recall that the set differential equation (SDE)(1.8) reduces to ordinary

differential equation (ODE), when U(t) is single valued and (SDE)(1.8) canbe generated from ODE as well. The latter is done as follows. Set F (t, A) =Cof(t, A), A ∈ Kc(R

n), where f : R+ × Rn → Rn arising from the ODE,

u′ = f(t, u), u(t0) = u0 ∈ Rn. (3.2)

Consequently, the solution of ODE(3.2) are imbedded in the solutions of theSDE(1.8). However, to consider the stability criteria of the trivial solutionof (1.8), one needs to employ, in a natural way, the measure D[U(t), θ] =‖U(t)‖ = diam U(t), t ≥ t0; which implies by Proposition 1.1 that thediam U(t) is nondecreasing in t. This is due to the fact, in the generation


of SDE from ODE, undesirable elements may enter the solution and themeasure employed, namely, ‖U(t)‖ is therefore unsuitable to develop stabil-ity theory. Since the cause of the problem in SDE is due to the requirementof Hukuhara differences (1.5) for formulating the SDE(1.8), we need to utilizethe existence of Hukuhara difference in the initial conditions also, in order tomatch the behavior of solutions of SDE with the corresponding ODE. Thisis precisely what we plan to do. Suppose that Hukuhara difference exists forany given initial values U0, V0 ∈ Kc(R

n) so that U0 − V0 = W0 is defined.Then we consider the stability of the solution U(t, t0, U0 −V0) = U(t, t0, W0)of (1.8).

We are now in a position to prove the following theorem concerning thestability of the trivial solution of (1.8). For this purpose we shall assume thatthe system (1.8) has the trivial solution.

Theorem 3.2 Assume that the conditions (i) and (ii) of Theorem 3.1 holdon R+ × S(ρ) instead of Kc(R

n), where S(ρ) = [U ∈ Kc(Rn) : ‖U‖ < ρ].

Suppose that

b(‖U‖) ≤ V (t, U) ≤ a(‖U‖) on R+ × S(ρ),

where a, b ∈ C[[0, ρ), R+] are the usual K class functions. Then the stabilityproperties of the trivial solution of (3.1) imply the corresponding stabilityproperties of the trivial solution of (1.8) respectively, when we utilize thesolutions U(t, t0, U0 − V0) = U(t, t0, W0).

We omit the proof of Theorem 3.2, since, once we have the comparisonTheorem 3.1, it is not difficult to construct the proofs of the correspondingstability criteria following the standard proofs of known results with appro-priate modifications. See [10].

We shall next present examples to illustrate our approach.Consider the ODE

u′ = −u, u(0) = u0 ∈ R (3.3)

and the corresponding SDE

DHU = −U, U(0) = U0 ∈ Kc(R). (3.4)

Since the values of the solution of (3.4) are interval functions, the equation(3.4) can be written as

[u′

1, u′

2] = (−1)U = [−u2, − u1], (3.5)

where U = [u1, u2]. The relation (3.5) is equivalent to the system of equations

u′

1 = −u2, u1(0) = u10,

u′

2 = −u1, u2(0) = u20,


whose solution is

u1(t) = 1

2[u10 + u20]e

−t + 1

2[u10 − u20]e

t,

u2(t) = 1

2[u20 + u10]e

−t + 1

2[u20 − u10]e

t, t ≥ 0.

(3.6)

Given U0 ∈ Kc(R), if there exist V0, W0 ∈ Kc(R) such that U0 = V0 + W0,

then the Hukuhara difference U0 − V0 = W0. Let us choose

U0 = [u10, u20], V0 =

[

(u10 − u20)

2,(u20 − u10)

2

]

,

so that

W0 =

[

(u10 + u20)

2,(u20 + u10)

2

]

.

Then it follows assuming u10 6= −u20, that

U(t, U0) =

[

−1

2(u20 − u10),

1

2(u20 − u10)

]

et

+

[

1

2(u10 + u20),

1

2(u10 + u20)

]

e−t, t ≥ 0.

U(t, V0) =

[

1

2(u10 − u20),

1

2(u20 − u10)

]

et, and

U(t, W0) =

[

1

2(u10 + u20),

1

2(u20 + u10)

]

e−t, t ≥ 0.

If on the other hand, u10 = −u20, then it is enough to change the roles ofU0 and V0. We note that for any general initial value U0, the solution ofSDE(3.4) contains both the desired and the undesired parts compared to thesolution of the ODE(3.3). In order to isolate the desired part of the solutionU(t, U0) of (3.4) which matches the solution of the ODE(3.3), we need to usethe initial values satisfying the desired Hukuhara difference of the given twoinitial values. This is necessary only when the generated SDE from ODE hasthe property that no translate of DHU, U is included in the set θ, whichis the case in (3.4), when we write (3.4) as

DHU + U = θ.

If, on the other hand, we have the SDE as

DHU = λ(t)U, U(0) = U0, (3.7)

which is generated byu′ = λ(t)u, u(0) = u0, (3.8)

where λ(t) > 0 is a real valued function from R+ → R+ such thatλ ∈ L1(R+), then we see that, with similar computation,

U(t, U0) = U0 exp

[∫ t

0

λ(s)ds

]

, t ≥ 0,


for any U0 ∈ Kc(R). Hence we get the stability of the trivial solution of (3.7).In this case, we note that the solutions of both equations (3.7) and (3.8)match providing the same stability results. There is no necessity, therefore,to choose the initial values as before, since the undesirable part of the solutiondoes not exist among solutions of (3.7). Consequently, it does not matter,whether we use the Hukuhara difference or not, we get the same conclusion.In order to be consistent and take care of all the situations, the stabilityresults are formulated in terms of Hukuhara differences of the initial values.

Remark 3.1 We wish to point out that in earlier paper [12], the expressionfor the solution Uβ(t) of DHUβ = −Uβ, which is similar to (3.4) for each β

is not correct, unless we follow the present approach of computation.

References

[1] H. T. Banks and M. G. Jacobs, A differential Calculus for multifunctions,J. M. A. A, 29(1970), 246 - 272.

[2] A. J. Brandao Lopes Pinto, F. S. De Blasi and F. Iervolino, Uniqueness and exis-tence theorems for differential equations with compact convex-valued solutions, Boll.Unione. Mat. Italy, 3 (1970) 47-54.

[3] T. Gnana Bhaskar and V. Lakshmikantham, Set Differential Equations and FlowInvariance, Applicable Analysis 82 (2003), 357-368.

[4] T. Gnana Bhaskar and V. Lakshmikantham, Generalized Flow Invariance for Differ-ential Equations, to appear.

[5] T. Gnana Bhaskar and V. Lakshmikantham, Lyapunov Stability for Set DifferentialEquations, to appear.

[6] T. Gnana Bhaskar and V. Lakshmikantham, Generalized Proximal Normals and FlowInvariance for Differential Inclusions, Nonlinear Studies, to appear.

[7] T. Gnana Bhaskar and V. Lakshmikantham, Generalized Proximal Normals and Lya-punov Stability, to appear.

[8] M. Hukuhara, Sur l’aplication Semicontinue Dont la Valeur est un Compact Convexe,Funkcial Ekvac, 10 (1967), 43-68.

[9] M. Hukuhara, Integration des Applications Mesurables Dont la Valeur est un CompactConvex, Funkcial. Ekvac. 10 (1967), 205-229.

[10] V. Laksmikantham and S. Leela, Differential and Integral Inequalities, Vol. I, Aca-demic Press, New York, 1969.

[11] V. Laksmikantham, S. Leela and A. S. Vatsala, Setvalued Hybrid Differential Equa-tions and Stability in Terms of Two Measures, Journal of Hybrid Systems, 2 (2002),169-187.

[12] V. Laksmikantham, S. Leela and A. S. Vatsala, Interconnection between Set andFuzzy Differential Equations, Nonlinear Analysis, 54 (2003), 351-360.

[13] V. Laksmikantham and A. Tolstonogov, Existance and Interrelation between Set andFuzzy Differential Equations, Nonlinear Analysis, to appear.

[14] A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publish-ers, Dordrecht, 2000.


STABILITY THEORY FOR SET DIFFERENTIALEQUATIONS

V. Lakshmikantham, S. Leela and J. Vasundhara Devi1

Department of Mathematical SciencesFlorida Institute of Technology, Melbourne, FL 32901, USA.

To our dear friend Professor D. D. Siljak on his 70th birthday.

Abstract. The formulation of set differential equations has an intrinsic disadvantage that

the diameter of the solution is nondecreasing as time increases and therefore the behavior of

solutions, in some cases, do not match with the solutions of ordinary differential equations

from which set differential equations can be generated. In this paper an approach is

provided to remove the disadvantage.

Keywords. Set differential equations, stability.

AMS (MOS) subject classification: 34D20, 34G20.

1 Preliminaries

Let K(Rn)(Kc(Rn)) denote the collection of all nonempty, compact (com-

pact, convex) subsets of Rn. Define the Hausdorff metric

D[A, B] = max

[

supx∈B

d(x, A), supy∈A

d(y, B)

]

(1.1)

where d[x, A] = inf[d(x, y) : y ∈ A], A, B are bounded sets in Rn. We notethat K(Rn), (Kc(R

n)), with the metric is a complete metric space.It is known that if the space Kc(R

n) is equipped with the natural algebraicoperations of addition and nonnegative scalar multiplication, then Kc(R

n)becomes a semilinear metric space which can be embedded as a completecone into a corresponding Banach space [2, 14].

The Hausdorff metric (1.1) satisfies the following properties.

D[A + C, B + C] = D[A, B] and D[A, B] = D[B, A], (1.2)

D[λA, λB] = λD[A, B], (1.3)

D[A, B] ≤ D[A, C] + D[C, B], (1.4)

for all A, B, C ∈ Kc(Rn) and λ ∈ R+.

1on leave from GVP College for PG courses, Visakhapatnam, India.


Let A, B ∈ Kc(Rn). The set C ∈ Kc(R

n) satisfying A = B + C is knownas the Hukuhara difference of the sets A and B and is denoted by the symbolA−B. We say that the mapping F : I → Kc(R

n) has a Hukuhara derivativeDHF (t0) at a point t0 ∈ I, if there exists an element DHF (t0) ∈ Kc(R

n)such that the limits

limh→0+

F (t0 + h) − F (t0)

h, and lim

h→0+

F (t0) − F (t0 − h)

h(1.5)

exist in the topology of Kc(Rn) and are equal to DHF (t0). Here I is any

interval in R.

Proposition 1.1 If F : I → Kc(Rn) is Hukuhara differentiable, then the

real valued function t → diam F (t), t ∈ I is nondecreasing on I.

We note that the existence of the limits (1.5) was not used in the proofof the Proposition 1.1. See [1]. In fact, instead of the hypothesis that F isHukuhara differentiable on I, one could substitute the assumption that foreach t ∈ I, the Hukuhara differences

F (t + h) − F (t) and F (t) − F (t − h)

both exist for sufficiently small h > 0. We should also note that the fact F (t)is Hukuhara differentiable on I and diam F (t) > 0, t ∈ I need not implythat F (t) is monotone relative to the set inclusion. See [1].

By embedding Kc(Rn) as a complete cone in a corresponding Banach

space and taking into account the result on differentiation of Bochner integral,we find that if

F (t) = X0 +

∫ t

0

Φ(s) ds, X0 ∈ Kc(Rn), (1.6)

where Φ : I → Kc(Rn) is integrable in the sense of Bochner, then DHF (t)

exists and the equality

DHF (t) = Φ(t), a.e on I, (1.7)

holds. Also, the Hukuhara integral

∫

I

F (s) ds =

[∫

I

f(s) ds : f is a continuous selector of F

]

,

for any compact set I ⊂ R+. With these preliminaries, we consider the setdifferential equation

DHU = F (t, U), U(t0) = U0 ∈ Kc(Rn), t0 ≥ 0, (1.8)

where F ∈ C [R+ × Kc(Rn), Kc(R

n)].


The mapping U ∈ C1 [J, Kc(Rn)], J = [t0, t0 + a] is said to be a solution

of (1.8) on J if it satisfies (1.8) on J . Since U(t) is continuously differentiable,we have

U(t) = U0 +

∫ t

t0

DHU(s) ds, t ∈ J. (1.9)

Thus we associate with the initial value problem(IVP)(1.8) the following

U(t) = U0 +

∫ t

t0

F (s, U(s))ds, t ∈ J, (1.10)

where the integral is the Hukuhara integral [8, 9]. Observe also that U(t) isa solution of (1.8) iff it satisfies (1.10) on J .

The investigation of set differential equation (1.8) as an independent sub-ject has some advantages. For example, when U(t) is a single valued mapping,it is easy to see that Hukuhara derivative and the integral reduce to the or-dinary vector derivative and the integral, and therefore, the results obtainedin this new framework of (1.8) become the corresponding results of ordinarydifferential systems. Also, we have only semilinear complete metric spaceto work with, in the present setup, compared to the complete normed linearspace one employs in the study of ordinary differential systems. Furthermore,set differential equations, that are generated by multivalued differential inclu-sions, when the multivalued functions involved do not possess convex values,can be used as a tool for studying multivalued differential inclusions. SeeTolstonogov [14].

In this paper, we shall investigate the stability criteria for set differentialequation (1.8). For this purpose, we shall utilize needed known results forsuch equations. Recently, the study of set differential equations is initiatedas an independent subject and several basic results on existence, uniqueness,comparison result, global existence and continuous dependence are discussedin [2-7, 11-13].

2 Results in Set Differential Equations

In order to discuss the stability theory of set differential equations (1.8), weneed some known results [3, 11] relative to such equations. We start with acomparison result.

Theorem 2.1 Assume that F ∈ C[R+ × Kc(Rn), Kc(R

n)] and for t ∈ R+,

A, B ∈ Kc(Rn)

D[F (t, A), F (t, B)] ≤ g(t, D[A, B]),

where g ∈ C[R2+, R+]. Suppose further that the maximal solution r(t, t0, w0)

of the scalar differential equation

w′ = g(t, w), w(t0) = w0 ≥ 0,


exists for t ≥ t0. Then, if U(t) = U(t, t0, U0), V (t) = V (t, t0, V0) are any twosolutions of equation (1.8)such that U(t0) = U0, V (t0) = V0, U0, V0 ∈ Kc(R

n)existing for t ≥ t0, we have:

D[U(t), V (t)] ≤ r(t, t0, w0), t ≥ t0,

provided D[U0, V0] ≤ w0.

The next result is an existence and uniqueness Theorem more general thanLipschitz type condition, the proof of which exhibits the idea of comparisonprinciple.


(a) F ∈ C[R0, Kc(Rn)], where R0 = J × B(U0, b), J = [t0, t0 + a], a > 0,

B(U0, b) = [U ∈ Kc(Rn) : D[U, U0] ≤ b] and D[F (t, U), θ] ≤ M0 on

R0, where θ is the zero element of Rn regarded as a one point set;

(b) g ∈ C [J × [0, 2b], R+] , g(t, w) ≤ M1 on J × [0, 2b], g(t, 0) ≡ 0, g(t, w)is nondecreasing in w for each t ∈ J and w(t) ≡ 0 is the only solutionof

w′ = g(t, w), w(t0) = 0;

(c) D[F (t, U), F (t, V )] ≤ g(t, D[U, V ]) on R0.

Then the successive approximations defined by

Un+1(t) = U0 +

∫ t

t0

F (s, Un(s))ds, n = 0, 1, 2, . . . ,

exist on J0 = [t0, t+α], α = min(a, bM

], M = max(M0, M1), as continuousfunctions and converge uniformly to the unique solution U(t) = U(t, t0, U0)of IVP (1.8) on J0.

The following global existence result is a special case of Theorem 5.2 in[11] which serves our purpose.


(1) F ∈ C[R+ × Kc(Rn), Kc(R

n)] and for (t, A) ∈ R+ × Kc(Rn),

D[F (t, A), θ] ≤ q(t, D[A, θ]),

where q ∈ C[R2+, R+], q(t, w) is nondecreasing in w for each t ∈ R+

and the maximal solution r(t, t0, w0) of

w′ = q(t, w), w(t0) = w0,

exists for t ≥ t0 and for every w0 > 0;


(2) there exists a local solution U(t) = U(t, t0, U0) of (1.8) for every(t0, U0) ∈ R+ × Kc(R

n).

Then for every U0 ∈ Kc(Rn) such that D[U0, θ] ≤ w0, the IVP (1.8) possesses

a solution U(t) = U(t, t0, U0) defined for t ≥ t0, satisfying

D[U(t), θ] ≤ r(t, t0, w0), t ≥ t0.

3 Lyapunov-Like Functions And Stability Criteria

To investigate the stability criteria of (1.8) the following comparison theoremin terms of Lyapunov-like functions is very important, which can be provedvia the theory of differential inequalities.


(i) V ∈ C[R+ × Kc(Rn), R+] and |V (t, A) − V (t, B)| ≤ LD[A, B], where

L is the local Lipschitz constant, for A, B ∈ Kc(Rn), t ∈ R+;

(ii) g ∈ C[R2+, R] and for t ∈ R+, A ∈ Kc(R

n),

D+V (t, A) ≡ lim suph→0+

1

h[V (t + h, A + hF (t, A))−V (t, A)] ≤ g (t, V (t, A)) .

Then, if U(t) = U(t, t0, U0) is any solution of (1.8) existing on [t0,∞) suchthat V (t0, U0) ≤ w0, we have

V (t, U(t) ≤ r(t, t0, w0), t ∈ [t0,∞),

where r(t, t0, w0) is the maximal solution of

w′ = g(t, w), w(t0) = w0 ≥ 0, (3.1)

existing on [t0,∞).

Proof: Let U(t) = U(t, t0, U0) be any solution of (1.8) existing on [t0,∞).Define m(t) = V (t, U(t)) so that m(t0) = V (t0, U0) ≤ w0. Now for smallh > 0,

m(t + h) − m(t) = V (t + h, U(t + h)) − V (t, U(t))

= V (t + h, U(t + h)) − V (t + h, U(t) + hF (t, U(t)))

+V (t + h, U(t) + hF (t, U(t))) − V (t, U(t))

≤ LD [U(t + h), U(t) + hF (t, U(t))]

+V ((t + h, U(t) + hF (t, U(t))) − V (t, U(t)) ,


using the Lipschitz condition given in (i). Thus

D+m(t) = lim suph→0+

1

h[m(t + h) − m(t)]

≤ D+V (t, U(t)) + L lim suph→0+

1

h[D[U(t + h), U(t) + hF (t, U(t))]] .

Let U(t + h) = U(t) + Z(t), where Z(t) is the Hukuhara difference ofU(t + h), U(t) for small h > 0, where Z(t) is assumed to exist. Hence em-ploying the properties of D[·, ·], we see that

D[U(t + h), U(t) + hF (t, U(t))] = D[U(t) + Z(t), U(t) + hF (t, U(t))]

= D[Z(t), hF (t, U(t))] = D[U(t + h) − U(t), hF (t, U(t))].

As a result, we find that

1

h[D[U(t + h), U(t) + hF (t, U(t))] = D

[

U(t + h) − U(t)

h, F (t, U(t))

]

,

and therefore,

lim suph→0+

1

h[D[U(t + h), U(t) + hF (t, U(t))]]

= lim suph→0+

D

[

U(t + h) − U(t)

h, F (t, U(t))

]

= D [U ′

H(t), F (t, U(t))] ≡ 0,

since U(t) is a solution of (1.8). We therefore have the scalar differentialinequality

D+m(t) ≤ g (t, m(t)) , m(t0) ≤ w0,

which yields the desired estimate in view of Theorem 1.4.1 in [10],

m(t) ≤ r(t, t0, w0), t ∈ [t0,∞).

This proves the claimed estimate of the Theorem.Recall that the set differential equation (SDE)(1.8) reduces to ordinary

differential equation (ODE), when U(t) is single valued and (SDE)(1.8) canbe generated from ODE as well. The latter is done as follows. Set F (t, A) =Cof(t, A), A ∈ Kc(R

n), where f : R+ × Rn → Rn arising from the ODE,

u′ = f(t, u), u(t0) = u0 ∈ Rn. (3.2)

Consequently, the solution of ODE(3.2) are imbedded in the solutions of theSDE(1.8). However, to consider the stability criteria of the trivial solutionof (1.8), one needs to employ, in a natural way, the measure D[U(t), θ] =‖U(t)‖ = diam U(t), t ≥ t0; which implies by Proposition 1.1 that thediam U(t) is nondecreasing in t. This is due to the fact, in the generation


of SDE from ODE, undesirable elements may enter the solution and themeasure employed, namely, ‖U(t)‖ is therefore unsuitable to develop stabil-ity theory. Since the cause of the problem in SDE is due to the requirementof Hukuhara differences (1.5) for formulating the SDE(1.8), we need to utilizethe existence of Hukuhara difference in the initial conditions also, in order tomatch the behavior of solutions of SDE with the corresponding ODE. Thisis precisely what we plan to do. Suppose that Hukuhara difference exists forany given initial values U0, V0 ∈ Kc(R

n) so that U0 − V0 = W0 is defined.Then we consider the stability of the solution U(t, t0, U0 −V0) = U(t, t0, W0)of (1.8).

We are now in a position to prove the following theorem concerning thestability of the trivial solution of (1.8). For this purpose we shall assume thatthe system (1.8) has the trivial solution.

Theorem 3.2 Assume that the conditions (i) and (ii) of Theorem 3.1 holdon R+ × S(ρ) instead of Kc(R

n), where S(ρ) = [U ∈ Kc(Rn) : ‖U‖ < ρ].

Suppose that

b(‖U‖) ≤ V (t, U) ≤ a(‖U‖) on R+ × S(ρ),

where a, b ∈ C[[0, ρ), R+] are the usual K class functions. Then the stabilityproperties of the trivial solution of (3.1) imply the corresponding stabilityproperties of the trivial solution of (1.8) respectively, when we utilize thesolutions U(t, t0, U0 − V0) = U(t, t0, W0).

We omit the proof of Theorem 3.2, since, once we have the comparisonTheorem 3.1, it is not difficult to construct the proofs of the correspondingstability criteria following the standard proofs of known results with appro-priate modifications. See [10].

We shall next present examples to illustrate our approach.Consider the ODE

u′ = −u, u(0) = u0 ∈ R (3.3)

and the corresponding SDE

DHU = −U, U(0) = U0 ∈ Kc(R). (3.4)

Since the values of the solution of (3.4) are interval functions, the equation(3.4) can be written as

[u′

1, u′

2] = (−1)U = [−u2, − u1], (3.5)

where U = [u1, u2]. The relation (3.5) is equivalent to the system of equations

u′

1 = −u2, u1(0) = u10,

u′

2 = −u1, u2(0) = u20,


whose solution is

u1(t) = 1

2[u10 + u20]e

−t + 1

2[u10 − u20]e

t,

u2(t) = 1

2[u20 + u10]e

−t + 1

2[u20 − u10]e

t, t ≥ 0.

(3.6)

Given U0 ∈ Kc(R), if there exist V0, W0 ∈ Kc(R) such that U0 = V0 + W0,

then the Hukuhara difference U0 − V0 = W0. Let us choose

U0 = [u10, u20], V0 =

[

(u10 − u20)

2,(u20 − u10)

2

]

,

so that

W0 =

[

(u10 + u20)

2,(u20 + u10)

2

]

.

Then it follows assuming u10 6= −u20, that

U(t, U0) =

[

−1

2(u20 − u10),

1

2(u20 − u10)

]

et

+

[

1

2(u10 + u20),

1

2(u10 + u20)

]

e−t, t ≥ 0.

U(t, V0) =

[

1

2(u10 − u20),

1

2(u20 − u10)

]

et, and

U(t, W0) =

[

1

2(u10 + u20),

1

2(u20 + u10)

]

e−t, t ≥ 0.

If on the other hand, u10 = −u20, then it is enough to change the roles ofU0 and V0. We note that for any general initial value U0, the solution ofSDE(3.4) contains both the desired and the undesired parts compared to thesolution of the ODE(3.3). In order to isolate the desired part of the solutionU(t, U0) of (3.4) which matches the solution of the ODE(3.3), we need to usethe initial values satisfying the desired Hukuhara difference of the given twoinitial values. This is necessary only when the generated SDE from ODE hasthe property that no translate of DHU, U is included in the set θ, whichis the case in (3.4), when we write (3.4) as

DHU + U = θ.

If, on the other hand, we have the SDE as

DHU = λ(t)U, U(0) = U0, (3.7)

which is generated byu′ = λ(t)u, u(0) = u0, (3.8)

where λ(t) > 0 is a real valued function from R+ → R+ such thatλ ∈ L1(R+), then we see that, with similar computation,

U(t, U0) = U0 exp

[∫ t

0

λ(s)ds

]

, t ≥ 0,


for any U0 ∈ Kc(R). Hence we get the stability of the trivial solution of (3.7).In this case, we note that the solutions of both equations (3.7) and (3.8)match providing the same stability results. There is no necessity, therefore,to choose the initial values as before, since the undesirable part of the solutiondoes not exist among solutions of (3.7). Consequently, it does not matter,whether we use the Hukuhara difference or not, we get the same conclusion.In order to be consistent and take care of all the situations, the stabilityresults are formulated in terms of Hukuhara differences of the initial values.

Remark 3.1 We wish to point out that in earlier paper [12], the expressionfor the solution Uβ(t) of DHUβ = −Uβ, which is similar to (3.4) for each β

is not correct, unless we follow the present approach of computation.

References

[1] H. T. Banks and M. G. Jacobs, A differential Calculus for multifunctions,J. M. A. A, 29(1970), 246 - 272.

[2] A. J. Brandao Lopes Pinto, F. S. De Blasi and F. Iervolino, Uniqueness and exis-tence theorems for differential equations with compact convex-valued solutions, Boll.Unione. Mat. Italy, 3 (1970) 47-54.

[3] T. Gnana Bhaskar and V. Lakshmikantham, Set Differential Equations and FlowInvariance, Applicable Analysis 82 (2003), 357-368.

[4] T. Gnana Bhaskar and V. Lakshmikantham, Generalized Flow Invariance for Differ-ential Equations, to appear.

[5] T. Gnana Bhaskar and V. Lakshmikantham, Lyapunov Stability for Set DifferentialEquations, to appear.

[6] T. Gnana Bhaskar and V. Lakshmikantham, Generalized Proximal Normals and FlowInvariance for Differential Inclusions, Nonlinear Studies, to appear.

[7] T. Gnana Bhaskar and V. Lakshmikantham, Generalized Proximal Normals and Lya-punov Stability, to appear.

[8] M. Hukuhara, Sur l’aplication Semicontinue Dont la Valeur est un Compact Convexe,Funkcial Ekvac, 10 (1967), 43-68.

[9] M. Hukuhara, Integration des Applications Mesurables Dont la Valeur est un CompactConvex, Funkcial. Ekvac. 10 (1967), 205-229.

[10] V. Laksmikantham and S. Leela, Differential and Integral Inequalities, Vol. I, Aca-demic Press, New York, 1969.

[11] V. Laksmikantham, S. Leela and A. S. Vatsala, Setvalued Hybrid Differential Equa-tions and Stability in Terms of Two Measures, Journal of Hybrid Systems, 2 (2002),169-187.

[12] V. Laksmikantham, S. Leela and A. S. Vatsala, Interconnection between Set andFuzzy Differential Equations, Nonlinear Analysis, 54 (2003), 351-360.

[13] V. Laksmikantham and A. Tolstonogov, Existance and Interrelation between Set andFuzzy Differential Equations, Nonlinear Analysis, to appear.

[14] A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publish-ers, Dordrecht, 2000.


A DIRECT METHOD OF OPTIMIZATION AND

ITS APPLICATION TO A CLASS OFDIFFERENTIAL GAMES

George Leitmann

University of California

Berkeley, California 94720, USA

Abstract: A coordinate transformation approach is employed to deduce a direct, non-

variational method to obtain extremizing solutions for some classes of integrals as well as

optimal control and differential game problems.

AMS (MOS) subject classification: 49k.

1 Introduction

Many problems in economics and engineering can be posed as dynamic op-timization problems involving the extremization of an integral over a givenclass of functions subject to prescribed end conditions. These problems areusually addressed via the Calculus of Variations or the Maximum Principleof optimal control theory, applying necessary conditions to obtain candidateoptimal solutions, and then assuring optimality via sufficient conditions ifavailable. These methods are variational in that they employ the compari-son of solutions in a neighborhood of the optimal one.

A different approach was first proposed in the 1960’s and more recentlyexpanded in [6]-[9]. This approach permits the direct derivation of globalextrema for some classes of dynamic optimization problems without the useof comparison techniques. Instead, it employs coordinate transformationsand the imposition of a functional identity. The direct method is readilyapplicable to a class of open-loop differential games, as shown in [5].

Finally, in order to enlarge the class of problems which can be treatedby the proposed direct optimization method, a modification of the methodis introduced.

2 CALCULUS OF VARIATIONS

The Basic Problem

192 G. Leitmann

Given

I(y(·)) :=

∫ b

a

F [x, y(x), y′(x)]dx (2.1)

with F (·) : [a, b] × R2n → R continuous, and ()′ := d()dx

, extremize I(y(·))over all piecewise smooth (pws) y(·) : [a, b] → Rn satisfying prescribed endconditions

y(a) = ya, y(b) = yb. (2.2)

A Basic Lemma

The following lemma was deduced in [7]-[9], together with various corol-laries:Lemma 2.1 Let y = z(x, y) be a transformation having a unique inverse

y = z(x, y), with z(·) of class C1 on [a, b]×Rn, such that there is a one-to-onecorrespondence

y(x) ⇐⇒ y(x) (2.3)

for all pws y(·) satisfying (2.2) and all pws y(·) satisfying

y(a) = z(a, ya), y(b) = z(b, yb) (2.4)

Let H(·) : [a, b]×Rn → R be a C1 function such that the functional identity

F [x, y(x), y′(x)] − F [x, y(x), y′(x)] =dH [x, y(x)]

dx(2.5)

is met for all pws y(·) satisfying (2.4), where y(x) = z[x, y(x)].If y∗(·) extremizes

I(y(·)) =

∫ b

a

F [x, y(x), y′(x)]dx (2.6)

over all pws y(·) satisfying (2.4), then y∗(·) with y∗(x) = z[x, y∗(x)] extrem-izes I(y(·)) over all pws y(·) satisfying (2.2).

The Relation to Caratheodory’s Basic Theorem

The following remark is quoted from [2]:The theory for direct sufficient conditions for solving problems in the

calculus of variations and optimal control is not nearly as developed as thetheory for necessary conditions or that of the direct methods for the exis-tence of minimizers (or maximizers). Indeed, most results of this type maybe traced back to the earliest roots with very little change. In this paper wereport on some recent developments on two such methods appearing in Carl-son (2002a).1 These two methods are compared and contrasted, and finallycombined to arrive at what is apparently a new method. The first of these

1[3]

A Direct Method of Optimization and its Application 193

two methods appears in 1935 in the book by Caratheodory (1982)2 while thesecond is due to Leitmann (1967).3 These two methods have striking similar-ities, yet are clearly different. The first perturbs the objective functional toobtain a new functional which has the same set of minimizers (or maximiz-ers) as the original problem. The feasible set is the same for both problems.In the second method, a transformation is defined which establishes a one-to-one correspondence between the original set of feasible trajectories anda new set of trajectories. With this transformation, the original problem isreplaced by a new one in which the objective functional remains the samebut the feasible set is replaced by the new trajectories.

Corollaries

The Lemma 2.1 has a corollary from which all subsequent corollariesfollow, namely,

Corollary 2.1. The functional identity (2.5) implies the identity

F [x, z(x, y),∂z(x, y)

∂x+

n∑

i=1

∂z(x, y)

∂yi

pi]−F (x, y, p) ≡∂H(x, y)

∂x+

n∑

i=1

∂H(x, y)

∂yi

pi

(2.7)on (a, b) ×R2n.

An important consequence of Corollary 1 isCorollary 2.2. The left-hand-side of identity (2.7) is linear in the pi,

that is, it is of the form

θ(x, y) +

n∑

i=1

ψi(x, y)pi (2.8)

and∂H(x, y)

∂x= θ(x, y),

∂H(x, y)

∂yi

= ψi(x, y) (2.9)

on [a, b] ×Rn.Remark. Provided z(·) and H(·) are such that Corollary 2.2 is met,

identity (2.7) applies for x on the closed interval [a,b]. Furthermore, (2.7) isthen necessary and sufficient for the satisfaction of functional identity (2.5).The latter is readily seen by letting y = y(x) and p = y′(x) for any given pwsy(·).

Other corollaries and remarks, as well as numerous examples, can befound in [6]-[9]. In addition, the extension of the direct method to a class ofdifferential games is treated in [5].

Differential Side Conditions

Given

I(y(·)) =

∫ b

a

F [x, y(x), y′(x)]dx

2[1]3[10]

194 G. Leitmann

extremize I(y(·)) over all pws y(·) : [a, b] → Rn satisfying prescribed condi-tions

si[x, y(x), y′(x)] = 0, i = 1, 2, · · · ,m < n

with si(·) continuous on [a, b] ×R2n, and

y(a) = ya, y(b) = yb.

The Infinite Horizon Case

Given

I(y(·)) :=

∫

∞

a

F [x, y(x), y′(x)]dx

with F (·) : [a,∞) × R2n → R continuous, extremize I(y(·)) over all pws

y(·) : [a,∞) → Rn satisfying given initial condition

y(a) = ya.

These extensions to the basic problem are treated in [9]. Inequality con-

straints are treated in [4].

3 OPEN-LOOP DIFFERENTIAL GAMES WITH

SEPARATED STATE EQUATIONS

Statement of the Problem

In order to simplify the notation, we restrict the discussion to two-player

games; however, the discussion is equally applicable to many-player games.

Furthermore, we make an obvious change of notation in order to adhere to

the notation usually employed in the theory of differential games.

Consider two players who compete over time. Each player i = 1, 2 chooses

a sequence of actions so as to extremize a given integral

extru(·)

J1 =

∫ t2

t1

I1[t, x(t), y(t), u(t), v(t)]dt

(3.1)

given v(·), and

extrv(·)

J2 =

∫ t2

t1

I2[t, x(t), y(t), u(t), v(t)]dt

(3.2)


given u(·), and subject to the dynamic constraints of the form

dx(t)

dt=: x′(t) = u(t)l1[t, x(t)] +m1[t, x(t)], (3.3)

dy(t)

dt=: y′(t) = v(t)l2[t, y(t)] +m2[t, y(t)] (3.4)

with continuous and non-zero li(·) and continuous mi(·), and the boundary

conditions

x(t1) = x1, x(t2) = x2, (3.5)

y(t1) = y1, y(t2) = y2. (3.6)

In the differential game (3.1)-(3.6) [x(t), y(t)] is the vector of state vari-

ables of players 1 and 2, respectively and [u(t), v(t)] is the vector of control

variables. Player 1 chooses his actions in terms of the control u(t) and player

2 chooses v(t). Ii[t, x(t), y(t), u(t), v(t)] is the integrand of the objective func-

tion of player i. The important characteristic of the differential game (3.1)-

(3.6) is that neither the opponent’s state nor his control variable enter the

state equation of either player. Differential games with this property are

referred to as games with separated state equations.

In a game with separated state equations a given control function of the

rival results in a given state variable of the rival. For instance let us look

at player 1 who treats v(·) as given. For a given control v(·) there exists

a unique solution of the state equation (3.4) so that the statements “for a

given v(·)” and “a given y(·)” are equivalent. An analogous situation obtains

for player 2. Thus, the dynamic game can be restated as a problem of the

Calculus of Variations by substituting (3.3) and (3.4) into (3.1) and (3.2) to

obtain

extrx(·)

J1 =

∫ t2

t1

I1

t, x(t), y(t),x′(t) −m1[t, x(t)]

l1[t, x(t)],y′(t) −m2[t, y(t)]

l2[t, x(t)]

dt

given y(·), and

extry(·)

J2 =

∫ t2

t1

I2

t, x(t), y(t),x′(t) −m1[t, x(t)]

l1[t, x(t)],y′(t) −m2[t, y(t)]

l2[t, x(t)]

dt

196 G. Leitmann

given x(·), and subject to the boundary conditions (3.5) and (3.6).

In what follows we will be dealing with problems that can be reduced

to the following two player games with scalar state variables and finite time

horizon. For the sake of convenience we restrict out attention to minimization

problems

minx(·)

J1 =

∫ t2

t1

F1[t, x(t), x′(t), y(t), y′(t)]dt

(3.7)

for a given y(·), and

miny(·)

J2 =

∫ t2

t1

F2[t, x(t), x′(t), y(t), y′(t)]dt

(3.8)

for a given x(·), and subject to the prescribed boundary conditions

x(t1) = x1, x(t2) = x2, (3.9)

y(t1) = y1, y(t2) = y2. (3.10)

Definition 3.1. A time path of state variables [x∗(·), y∗(·)] is called

an open-loop Nash equilibrium of the game (3.7)-(3.10) if and only if x∗(·)

minimizes

J1 =

∫ t2

t1

F1[t, x(t), x′(t), y∗(t), y∗′(t)]dt

subject to the boundary condition (3.9), and y∗(·) minimizes

J2 =

∫ t2

t1

F2[t, x∗(t), x∗′(t), y(t), y′(t)]dt

subject to the boundary condition (3.10).

A Basic Lemma

Lemma 3.1. Let x = z1(t, x) be a transformation having a unique inverse

x = z1(t, x) for all t ∈ [t1, t2] such that there is a one-to-one correspondence

x(t) ⇐⇒ x(t)


for all x(·) satisfying boundary condition (3.9) and x(·) satisfying

x(t1) = z1(t1, x1), x(t2) = z2(t2, x2).

If for a given y∗(·) the transformation x = z1(t, x) is such that there exists a

functional identity of the form

F1[t, x(t), x′(t), y∗(t), y∗′(t)] − F1[t, x(t), x

′(t), y∗(t), y∗′(t)] =d

dtH1[t, x(t)]

(3.11)

then, if x∗(·) yields an extremum of J1 with x∗(·) satisfying the trans-

formed boundary conditions, x∗(·) with x∗(t) = z1[t, x∗(t)] yields an ex-

tremum for J1 with the boundary conditions (3.9).

Let y = z2(t, y) be a transformation having a unique inverse y = z2(t, y)

for all t ∈ [t1, t2] such that there is a one-to-one correspondence

y(t) ⇐⇒ y(t)

for all y(·) satisfying boundary condition (3.10) and y(·) satisfying

y(t1) = z2(t1, y1), y(t2) = z2(t2, y2).

If for a given x∗(·) the transformation y = z2(t, y) is such that there exists a

functional identity of the form

F2[t, x∗(t), x∗′(t), y(t), y′(t)] − F2[t, x

∗(t), x∗′(t), y(t), y′(t)] =d

dtH2[t, y(t)]

(3.12)

then, if y∗(·) yields an extremum of J2 with y∗(·) satisfying the trans-

formed boundary conditions, y∗(·) with y∗(t) = z2[t, y∗(t)] yields an extremum

for J2 with the boundary conditions (3.10). Moreover then [x∗(·), y∗(·)] with

x∗(t) = z1[t, x∗(t)], and y∗(t) = z2[t, y

∗(t)] is an open-loop Nash equilibrium.

Corollaries

Corollary 3.1. The integrands Fi(·) and the transformation in Lemma

3.1 must be such that the left-hand sides of (3.11) and (3.12) are linear in

x′(t) and y′(t), respectively.

198 G. Leitmann

Corollary 3.2. For integrands Fi(·) of the form

F1[t, x(t), x′(t), y∗(t), y∗′(t)] = a1[t, x(t), y

∗(t)][x′(t)]2 + b1[t, x(t), y∗(t)]x′(t)

+c1[t, x(t), y∗(t)], a1[t, x, y

∗(t)] 6= 0∀(t, x) ∈ [t1, t2] ×R

(3.13)

F2[t, x∗(t), x∗′(t), y(t), y′(t)] = a2[t, y(t), x

∗(t)][y′(t)]2 + b2[t, y(t), x∗(t)]y′(t)

+c2[t, y(t), x∗(t)], a2[t, y, x

∗(t)] 6= 0∀(t, y) ∈ [t1, t2] ×R

(3.14)

the class of admissible transformations must satisfy the following partial dif-

ferential equations

[

∂z1(t, x)

∂x

]2

a1[t, z1(t, x), y∗(t)] = a1[t, x, y

∗(t)], (3.15)

[

∂z2(t, y)

∂y

]2

a2[t, z2(t, y), x∗(t)] = a2[t, y, x

∗(t)]. (3.16)

If the coefficients of [x′(t)]2 and [y′(t)]2 are nonzero functions of t only, i.e.,

a1[t, y∗(t)] and a2[t, x

∗(t)], then

∂z1(t, x)

∂x= ±1,

∂z2(t, y)

∂y= ±1,

so that admissible transformations must be of the form

x = z1(t, x) = ±x+ f(t),

y = z1(t, y) = ±y + g(t).

Examples

Example 3.1. Consider the following simple linear quadratic differential

game

minx(·)

J1 =

∫ t2

t1

[x2(t) + y2(t) + [x′(t)]2dt

(3.17)


given y(·), and

miny(·)

J2 =

∫ t2

t1

[x2(t) + y2(t) + [y′(t)]2dt

(3.18)

given x(·), and subject to the boundary conditions (3.9) and (3.10). Now we

apply Lemma 3.1 in order to deduce directly the open-loop Nash equilibrium

[x∗(·), y∗(·)] of the game (3.17)-(3.18).

Applying Corollary 3.2, and choosing the + sign, we have transformations

of the form

x(t) = x(t) + f(t), (3.19)

y(t) = y(t) + g(t). (3.20)

Using these transformations in the functional identities (3.11) and (3.12), the

identities resulting from (3.11) and (3.12) (recall Corollary 2.2) imply

H1t (t, x) = 2xf(t) + f2(t) + f ′2(t),

H1x(t, x) = 2f ′(t),

and

H2t (t, y) = 2yg(t) + g2(t) + g′2(t),

H2y (t, x) = 2g′(t).

Now, if we apply the identities

H1tx(t, x) ≡ H1

xt(t, y), H2ty(t, y) ≡ H2

yt(t, y),

we obtain a differential equation for f(·) and g(·), respectively,

f ′′(t) − f(t) = 0,

g′′(t) − g(t) = 0.

The general solutions of these equations are given by

f(t) = c1et + c2e

−t,

200 G. Leitmann

g(t) = d1et + d2e

−t,

where ci and di are constants of integration.

With (3.17) and (3.18) for (x, y) it is easily seen that x∗(t) ≡ 0 is the best

response of player 1 to y∗(·), and y∗(t) ≡ 0 is the best response of player 2

to x∗(·). Hence, we get

x∗(t) = f(t) = [c1et + c2e

−t],

y∗(t) = g(t) = [d1et + d2e

−t],

as the open-loop Nash equilibrium for the original problem, where the con-

stants of integration are determined through the boundary conditions (3.9)

and (3.10). This solution corresponds exactly to the one derived via the

Maximum Principle.

Example 3.2. As a second example we consider the game

minx(·)

J1 =

∫ t2

t1

G1[x(t), y(t)] + [x′(t)]2dt

(3.21)

given y(·), and

miny(·)

J2 =

∫ t2

t1

G2[x(t), y(t)] + [y′(t)]2dt

(3.22)

given x(·), and with boundary conditions (3.9) and (3.10). The differentiable

functions Gi(·, ·) are assumed to satisfy

G1(x, y) ≥ 0 ∀(x, y), G1(0, y) = 0,∂G1(x, y)

∂x

∣

∣

∣

∣

x=0

= 0 ∀y;

G2(x, y) ≥ 0 ∀(x, y), G2(x, 0) = 0,∂G2(x, y)

∂y

∣

∣

∣

∣

y=0

= 0 ∀x.

The differential game (3.21), (3.22) can be given the following economic in-

terpretation. x(t) and y(t), respectively, are the current stocks of pollution

in two countries. Each country incurs costs depending on the two stocks

of pollution. These costs are given by Gi[x(t), y(t)]. u(t) = −x′(t) is the

current effort by which country 1 tries to reduce the stock of pollution x(t),


and v(t) = −y′(t) is the current effort by which country 2 tries to reduce the

stock of pollution y(t). The abatement policies u(·) and v(·) are costly, with

the cost functions assumed to be quadratic. Given initial levels of pollution

x1 and y1, each country tries to achieve given levels x2 and y2, respectively,

at time t2 by minimizing the sum of the costs over the planning horizon;

weighting the costs is easily accommodated in the functions Gi(·, ·). That

the objective of each player is to reach a terminal state and simultaneously

to minimize costs over the planning horizon corresponds exactly to the agree-

ment reached by individual countries in the Kyoto treaty.

The Gi(·, ·) are cost functions for non-negative pollution levels x(t), y(t),

respectively. However, for negative values of x(t), y(t), which are not ad-

missible since negative pollution levels are devoid of physical meaning, the

Gi(·, ·) serve as penalty functions. In particular, using an argument by con-

tradiction, it can be shown readily that the equilibrium values x∗(t), y∗(t)

must stay non-negative on the interval [t1, t2].

To solve this problem, we again make use of Corollary 3.2 and introduce

the coordinate transformations

x(t) = x(t) + f(t),

y(t) = y(t) + g(t).

Then

H1xt(t, x) ≡ H1

tx(t, x) ⇒ 2f ′′(t) ≡∂G1[x+ f(t), y∗(t)] −G1[x, y

∗(t)]

∂x,

H2yt(t, y) ≡ H1

tyt(t, x) ⇒ 2g′′(t) ≡∂G2[x

∗(t), y + g(t)] −G2[x∗(t), y]

∂y.

The assumptions on Gi(·, ·) guarantee that x∗(t) ≡ 0 is the best response

of player 1 to y∗(·) and y∗(t) ≡ 0 is the best response of player 2 to x∗(·).

Hence, f(·) and g(·) must satisfy the differential equations

2f ′′(t) =∂G1[f(t), y∗(t)]

∂x,

2g′′(t) =∂G2[x

∗(t), g(t)]

∂y.

202 G. Leitmann

Since for an open-loop Nash equilibrium of the original problem we have

x∗(t) = f(t),

y∗(t) = g(t),

x∗(·) and y∗(·) need to satisfy the following system of differential equations:

2x∗′′(t) =∂G1[x

∗(t), y∗(t)]

∂x,

2y∗′′(t) =∂G2[x

∗(t), y∗(t)]

∂y,

subject to the boundary conditions (3.9) and (3.10).

Conclusions

We have used coordinate transformations as introduced in [7] and ex-

tended in Ref. [9] to derive open-loop Nash equilibria for finite time horizon

differential games. We presented the general theory for the case of differential

games with separated state equations (see [5]), and discussed specific exam-

ples. One of the strengths of coordinate transformations was demonstrated

by means of a simple transboundary pollution game. Even without global

curvature assumptions on the functions of the model we were able to derive a

Nash equilibrium. While this property of coordinate transformations is very

attractive there are many open issues for future research. In particular we

plan to generalize the theory of coordinate transformations beyond the class

of differential games with separated state equations and to explore its use in

deriving closed-loop Nash equilibria.

4 A MODIFIED BASIC LEMMA

The basic lemma of sec. 2 and its corollaries provide an attractive optimiza-

tion method if an extremizing solution y∗(·) in the class of the transformed

variables y(·) can be deduced directly (usually by inspection ), that is, with-

out recourse to variation-based theories. All of the examples treated in the

references allow one to do this. Here we introduce a modification of the basic


lemma in order to enlarge the class of problems which can be treated by

means of the lemma and its corollaries.

Consider a continuous F (·) : [a, b] ×R2n → R and the integral

I(y(·)) :=

∫ b

a

F [x, y(x), y′(x)]dx (4.1)

and replace the functional identity (2.5) by

F [x, y(x), y′(x)] − F [x, y(x), y′(x)] =dH [x, y(x)]

dx. (4.2)

Then by employing a proof analogous to that of the original basic lemma,

it can be shown that, if y∗(·) extremizes I(y(·)) over all pws y(·) satisfying

(2.4), then y∗(·) with y∗(x) = z, y∗(·) extremizes I(y(·)) over all pws y(·)

satisfying (2.2).

Except for the modified functional identity, the new basic lemma is iden-

tical to the original. Furthermore, with this modification taken into account,

all consequences of the basic lemma, namely, its corollaries, remain in force.

The potential advantage of the proposed modification consists in allow-

ing one to choose F (·) such that a minimizer y∗(·) of I(y(·)) can be deduced

directly. To illustrate this we consider a simple example. Of course, with

F (·) = F (·) one recovers the original basic lemma.

Example

Consider y ∈ R and

F (x, y, p) = p2 + xp+ y

F (x, y, p) = p2 (4.3)

so that, as a consequence of Corollary 2.2, an admissible transformation z(·)

must be of the form

y = y ± f(x) (4.4)

where f(·) : [a, b] → R is a smooth integration function. Then, in view of

(2.9)

204 G. Leitmann

∂H(x,y)∂x

= f ′2(x) + xf ′(x) + f(x) + y

∂H(x,y)∂y

= 2f ′(x) + x

(4.5)

so that H(·) is of class C2 and hence

∂2H(x, y)

∂y∂x≡∂2H(x, y)

∂x∂y. (4.6)

Consequently

1 = 2f ′′(x) + 1 (4.7)

whence

f ′′(x) = 0. (4.8)

Finally, in view of F (·), it follows by inspection that

y∗(x) = constant (4.9)

so that, from (4.4), (4.8) and (4.9),

y∗(x) = c1 + c2x (4.10)

where constants c1 and c2 are such that the given end conditions on y(·) are

met.5 REFERENCES

[1] C. Caratheodory, Calculus of Variations and Partial Differential Equations, Chelsea,New York, New York, 1982.

[2] D.A. Carlson, Two Direct Methods for Obtaining Solutions to Variational Problems,International Game Theory Review, to appear.

[3] D.A. Carlson, An Observation on Two Methods of Obtaining Solutions to Varia-tional Problems, J. of Optimization Theory and Applic., 115 (1), 2002, 345-361.

[4] P. Cartigny, and Ch. Deissenberg, Leitmann’s Direct Method with Inequality Con-straints, International Game Theory Review, to appear.

[5] E.J. Dockner and G. Leitmann, Coordinate Transformations and the Derivation ofOpen-Loop Nash Equilibria, J. of Optimization Theory and Applic., 110 (1), 2001,1-15.

[6] E.A. Galperin, Translational Symmetry of Certain Integrals with Application to theCalculus of Variations and Optimal Control, Mathematical and Computer Mod-elling, 36, 2002, 717-728.

[7] G. Leitmann, On a Class of Direct Optimization Problems, J. of OptimizationTheory and Applic., 108, (3), 2001, 467-481.

[8] G. Leitmann, On a Method of Direct Optimization, J. of Computational Technolo-gies, 7, 2002. Special Issue.


[9] G. Leitmann, Some Extensions of a Direct Optimization Method, J. of OptimizationTheory and Applic., 111 (1), 2001, 1-6.

[10] G. Leitmann, A Note on Absolute Extrema of Certain Integrals, International J. ofNon-Linear Mechanics, 2, 1967, 55-59.


DIRECT LIAPUNOV’S MATRIX FUNCTIONSMETHOD AND OVERLAPPING DECOMPOSITION

OF LARGE SCALE SYSTEMS

A.A. Martynyuk

Institute of MechanicsNational Academy of Siences of Ukraine, Kiev, Ukraine

To the 70th birthday of Professor D.D.Siljak

Abstract. In the framework of the method of matrix-valued Liapunov functions stability

problems are considered for the dynamical system extended (reduction) in terms of a linear

transformation. The examples are presented which show how generalized decomposition

together with matrix-valued function extend the possibilities of application of the second

Liapunov method in stability investigation of dynamical system.

Keywords. Large scale systems; overlapping decomposition; Liapunov’s matrix function;

stability; asymptotic stability.

Mathematics Subject Classification (2000): 93A15, 93D05; 93D30.

1 Preliminary results

Let the system of perturbed motion equations

Sx :dx

dt= f(t, x), x(t0) = x0, (1.1)

be given, where x ∈ Rn and f ∈ C(R+×Rn, Rn), which possesses under ini-tial conditions (t0, x0) ∈ int (R+ ×Rn) a unique solution x(t) = x(t; t0, x0)determined for all t ≥ t0, t0 ≥ 0.

Together with system (1.1) we consider a nonlinear continuously differe-ntiable transformation [1, 2]

y = Φ(t, x), Φ ∈ C(1,1)(R+ ×Rn, Rm) (1.2)

for which a reverse one exists and is determined by the formula

x = Π(t, y), Π ∈ C(1,1)(R+ ×Rm, Rn). (1.3)

Using (1.2), and (1.3) we reduce system (1.1) to the form

Sy :dy

dt= f(t, y), y(t0) = Φ(t0, x0), (1.4)

208 A.A. Martynyuk

where y ∈ Rm, f : R+ ×Rm → Rm and

f(t, y) =∂Φ

∂t(t,Π(t, y)) +

∂Φ

∂x(t,Π(t, y)) f(t,Π(t, y)). (1.5)

In view of assumptions on system (1.1) and transformations (1.2) and(1.3) it is clear that f ∈ C(R+ ×Rm, Rm).

By the appropriate choice of transformation (1.2) one can obtain either anextention of system (1.1) or its contraction. Taking into account the resultsof papers [1, 2] we set out the following definition.

Definition 1.1 The system (1.4) is an extention of system (1.1), iffor m > n there exists the transformation (1.2) together with the reversetransformation (1.3) and for any initial conditions (t0, x0) ∈ int (R+ × Rn)for the system (1.1) the condition y0 = Φ(t0, x0) implies

x(t; t0, x0) = Π(t, y(t; t0, y0)) (1.6)

for all t ≥ t0.

Definition 1.2 The system (1.4) is a contraction of the system (1.1), iffor m < n there exist the transformations (1.2) and (1.3) and relation (1.6)holds true under the conditions mentioned in Definition 1.1.

Note that for m = 2n, transformation (1.2) yields maximal nondege-nerate extension of system (1.1) and for m = 1, transformation (1.2) makesmaximal contraction of system (1.1). Recall that the Liapunov function V =V (t, x), V ∈ C(1,1)(R+ × Rn, R+), V (t, 0) = 0, makes maximal contractionof system (1.1) up to scalar equation.

Partial case of transformation (1.2) is the linear transformation

y = Tx, (1.7)

where T is m× n constant matrix with full rank columns. The extention ofsystem (1.1) by means transformation (1.7) was proposed in [3] and developedin [4] in context with vector Liapunov function. In this case vector function(1.5) is of the form

f(t, y) = Tf(t, T Iy) + m(t, y), (1.8)

where m : R+ ×Rm → Rm is a correcting function.

Theorem 1.1 System (1.4) with vector function (1.8) is an extentionof system (1.1), if there exists transformation (1.7) with generalized reversetransformation T I and one of the conditions below is satisfied

(a) m(t, Tx) = 0 for all (t, x) ∈ R+ ×Rn;

(b) T Im(t, y) = 0 for all (t, y) ∈ R+ ×Rm.

Direct Liapunov’s Matrix Functions Method 209

For the proof of this Theorem see [3].

Theorem 1.2 (cf. [3]) Assume that

(1) conditions of Theorem 1.1 are satisfied;

(2) systems (1.1) and (1.4) with vector function (1.8) have a unique equi-librium state x = 0 and y = 0 respectively, i.e. t ∈ R+;

(3) (t, y) ∈ R+ ×Rm Π(t, y) = 0, (y = 0) ∈ Rm.

Then certain type of stability (asymptotic stsbility) of the equilibrium statey = 0 of system(1.4) implies the corresponding type of stability (asymptoticstsbility) of the equilibrium state x = 0 of system (1.1) for both m > n, andm < n.

The proof of Theorem 1.2 is based on the use of correlation (1.7) andthe fact that for the extended (contraction) system (1.4) the assumption onstability (asymptotic stability) of its equilibrium state implies the existenceof corresponding Liapunov function (see [5], etc.).

2 Application of Liapunov’s matrix function

to extended systems

The methods of stability analysis of large-scale dynamical systems via one-level decomposition of the system and a vector Liapunov functions were sum-marized in a series of monographs [6–9], etc. In this section a method ofconstruction of matrix-valued Liapunov function for extended system is pro-posed.

We consider an extended system (1.4) with finite number of degrees offreedom whose motion is described by the equations

dyi

dt= fi(yi) +

m∑

j=1, j 6=i

gij(t, yi, yj) +Gi(t, y), i = 1, 2, . . . , s. (2.1)

where yi ∈ Rmi, yj ∈ Rmj, fi ∈ C(Rmi, Rmi), gij ∈ C(R+ × Rmi ×Rmj, Rmj ), Gi ∈ C(R+ × Rm, Rmi), and besides fi(0) = yij(t, 0, 0) =Gi(t, 0) for all i, j = 1, 2, . . . , s, t ∈ R+. System (2.1) is a partial caseof system (1.4) whose independent subsystems are autonomous.

In order to extend the method of matrix Liapunov functions to systems(2.1) it is necessary to estimate variation of matrix-valued function elementsand their total derivatives along solutions of the corresponding systems. Suchestimates are provided by the assumptions below.

Assumption 2.1 There exist open connected neighborhoods Ni ⊆ Rni

of the equilibriums state yi = 0, functions vii ∈ C1(Rni , R+), the comparisonfunctions ϕi1, ϕi2 and ψi of class K(KR) and real numbers c

¯ii > 0, cii > 0and γii such that

210 A.A. Martynyuk

(1) vii(yi) = 0 for all (yi = 0) ∈ Ni;

(2) c¯iiϕ

2i1(‖yi‖) ≤ vii(yi) ≤ ciiϕ

2i2(‖yi‖);

(3) (Dyivii(yi))

Tfi(yi) ≤ γiiψ2i (‖yi‖) for all yi ∈ Ni,

i = 1, 2, . . . ,m.

It is clear that under conditions of Assumption 2.1 the equilibrium statesyi = 0 of nonlinear isolated subsystems

dyi

dt= fi(yi), i = 1, 2, . . . ,m (2.2)

are

(a) uniformly asymptotically stable in the whole, if γii < 0 and(ϕi1, ϕi2, ψi) ∈ KR-class;

(b) stable, if γii = 0 and (ϕi1, ϕi2) ∈ K-class;

(c) unstable, if γii > 0 and (ϕi1, ϕi2, ψi) ∈ K-class.

The approach proposed in this section takes large scale systems (2.1) intoconsideration, subsystems (2.2) having various dynamical properties specifiedby conditions of Assumption 2.1.

Assumption 2.2 There exist open connected neighborhoods Ni ⊆ Rni

of the equilibrium states yi = 0, functions vij ∈ C1,1,1(Tτ × Rni × Rnj , R),comparison functions ϕi1, ϕi2 ∈ K(KR), positive constants (η1, . . . , ηm)T∈Rm, ηi > 0 and arbitrary constants c

¯ij , cij , i, j = 1, 2, . . . ,m, i 6= j suchthat

(1) vij(t, yi, yj) = 0 for all (yi, yj) = 0 ∈ Ni × Nj , t ∈ Tτ , i, j =1, 2, . . . ,m, i 6= j;

(2) c¯ijϕi1(‖yi‖)ϕj1(‖yj‖) ≤ vij(t, yi, yj) ≤ cijϕi2(‖yi‖)ϕj2(‖yj‖) for all(t, yi, yj) ∈ Tτ ×Ni ×Nj , i 6= j;

(3) Dtvij(t, yi, yj) + (Dyivij(t, yi, yj))

Tfi(yi)

+ (Dyjvij(t, yi, yj))

Tfj(yj) +ηi

2ηj(Dyi

vii(yi))Tgij(t, yi, yj)

+ηj

2ηi(Dyj

vjj(yj))Tgji(t, yi, yj) = 0; (2.3)

Remark 2.1 It is easy to notice that first order partial equations (2.3)are a somewhat variation of the classical Liapunov equation proposed in [10]for determination of auxiliary function in the theory of his direct methodof motion stability investigation. In a particular case these equations aretransformed into the systems of algebraic equations whose solutions can beconstructed analytically.


Assumption 2.3 There exist open connected neighbourhoods Ni ⊆Rni of the equilibrium states yi = 0, comparison functions ψi ∈ K(KR),i = 1, 2, . . . ,m, real numbers α1

ij , α2ij , α

3ij , ν

1ki, ν

1kij , µ

1kij and µ2

kij , i, j, k =1, 2, . . . ,m, such that

(1) (Dyivii(yi))

TGi(t, y) ≤ ψi(‖yi‖)m∑

k=1

ν1kiψ(‖yk‖) +R1(ψ)

for all (t, yi, yj) ∈ Tτ ×Ni ×Nj ;

(2) (Dyivij(t, ·))Tgij(t, yi, yj) ≤ α1

ijψ2i (‖yi‖) + α2

ijψi(‖yi‖)ψj(‖yj‖) +

α3ijψ

2j (‖yj‖) +R2(ψ) for all (t, yi, yj) ∈ Tτ ×Ni ×Nj ;

(3) (Dyivij(t, ·))TGi(t, y) ≤ ψj(‖yj‖)

m∑k=1

ν2ijkψk(‖yk‖) +R3(ψ)

for all (t, yi, yj) ∈ Tτ ×Ni ×Nj ;

(4) (Dyivij(t, ·))Tgik(t, yi, yk) ≤ ψj(‖yj‖)(µ1

ijkψk(‖yk‖) + µ2ijkψi(‖yi‖)) +

R4(ψ) for all (t, yi, yj) ∈ Tτ ×Ni ×Nj .

Here Rs(ψ) are polynomials in ψ = (ψ1(‖y1‖, . . . , ψm(‖ym‖)) in a powerhigher than three, Rs(0) = 0, s = 1, . . . , 4.

Under conditions (2) of Assumptions 2.1 and 2.2 it is easy to establishfor function

v(t, y, η) = ηTU(t, y)η =

m∑

i,j=1

vij(t, ·)ηiηj (2.4)

the bilateral estimate (cf. [11])

uT1H

TC¯Hu1 ≤ v(t, y, η) ≤ uT

2HTCHu2, (2.5)

where

u1 = (ϕ11(‖y1‖, . . . , ϕm1(‖ym‖))T,

u2 = (ϕ12(‖y1‖, . . . , ϕm2(‖ym‖))T

which holds true for all (t, y) ∈ Tτ ×N , N = N1 × · · · × Nm.

Based on conditions (3) of Assumptions 2.1, 2.2 and conditions (1) – (4) ofAssumption 2.3 it is easy to establish the inequality estimating the auxiliaryfunction variation along solutions of system (2.1). This estimate reads

Dv(t, y, η)∣∣(2.1)

≤ uT3Mu3, (2.6)

where u3 = (ψ1(‖y1‖), . . . , ψm(‖ym‖) and holds for all (t, y) ∈ Tτ ×N .

212 A.A. Martynyuk

Elements σij of matrix M in the inequality (2.6) have the following struc-ture

σii = η2i γii + η2

i νii +

m∑

k=1, k 6=i

(ηkηiν2kii + η2

i ν2kii) + 2

m∑

j=1, j 6=i

ηiηj(α1ij + α3

ji);

σij =1

2(η2

i ν1ji + η2

j ν1ij) +

m∑

k=1, k 6=j

ηkηjν2kij +

m∑

k=1, k 6=i

ηiηjν2kij

+ ηiηj(α2ij + α2

ji) +

m∑

k=1, k 6=i,j

(ηkηjµ1kji + ηiηjµ

2ijk + ηiηkµ

1kij + ηiηjµ

2jik),

i = 1, 2, . . . ,m, i 6= j.

Sufficient criteria of various types of stability of the equilibrium statey = 0 of system (2.1) are formulated in terms of the sign definiteness ofmatrices C, C and M from estimates (3.4), (3.5). We shall show that thefollowing assertion is valid.

Theorem 2.1 Assume that the perturbed motion equations are such thatall conditions of Assumptions 2.1 – 2.3 are fulfilled and moreover

(1) matrices C and C in estimate (2.4) are positive definite;

(2) matrix M in inequality (2.5) is negative semi-definite (negative defi-nite).

Then the equilibrium state y = 0 of system (2.1) is uniformly stable(uniformly asymptotically stable).

If, additionally, in conditions of Assumptions 2.1 – 2.3 all estimates aresatisfied for Ni = Rni ,Rk(ψ) = 0, k = 1, ...4 and comparison functions(ϕi1, ϕi2) ∈ KR-class, then the equilibrium state of system (2.1) is uniformlystable in the whole (uniformly asymptotically stable in the whole).

Proof is analogous to the proof of Theorem 2.1 from the paper [12]. Theconclusion about stability of the system (1.1) is implies by Theorem 1.2.

3 Stability of Linear Large Scale Systems

We consider the dynamical system

dx

dt= Ax, x(t0) = x0, (3.1)

where x(t) ∈ Rn, t ∈ R+, A is n × n-constant matrix. Vector x is dividedinto three subvectors xi, i = 1, 2, 3, so that x = (xT

1 , xT2 , x

T3 )T ∈ Rn and

xi ∈ Rni , n = n1 + n2 + n3.


Matrix A of system (3.1) is represented in the block form

A =

A11 A12 A13

A21 A22 A23

A31 A32 A33

, (3.2)

where submatrices Aij ∈ Rni×nj for all i, j = 1, 2, 3. Further three compo-nents of vector x are transformed into two components of vector y accordingto the rule: y1 = (xT

1 , xT2)

T, and y2 = (xT2 , x

T3 )T. Besides y = (yT

1 , yT2 )T∈ Rn,

where n = n1 + 2n2 + n3.By means of linear nondegenerate transform

y = Tx, (3.3)

where T is n× n-matrix of the form

T =

I1 0 0

01

2I2 0

01

2I2 0

0 0 I3

,

I1, I2, and I3 are identity matrices whose dimensions correspond to the di-mensions of subvectors x1, x2, and x3 of vector x, we reduce system (3.1) tothe form (see [6])

dy1dt

= A11y1 + A12y2,

dy2dt

= A21y1 + A22y2.(3.4)

Here

A22 =

(A11 A12

A21 A22

), A12 =

(0 A13

0 A23

),

A22 =

(A22 A23

A32 A33

), A21 =

(A21 0A31 0

).

Designate A = (Aij), i, j = 1, 2 and note that

A = TATT+M, (3.5)

where

T =

I1 0 0

01

2I2 0

01

2I2 0

0 0 I3

, T I =

I1 0 0 00 I2 I2 0

0 0 0 I3

,

214 A.A. Martynyuk

and

M =

01

2A12 −

1

2A12 0

01

2A22 −

1

2A22 0

01

2A22 −

1

2A22 0

01

2A32 −

1

2A32 0

. (3.6)

Definition 3.1 System (3.4) is an extension of system (3.1) if thereexists a linear transformation of maximal rank (3.3) such that for y0 = Tx0

x(t, x0) = T Iy(t, y0), t ≥ t0, (3.7)

where y(t, y0) is a solution of system (3.4).

It is proved (see [3]), that system (3.4) is an extension of (3.1) in the senseof Definition 3.1 iff one of conditions MT = 0 or T IM = 0 is satisfied.

Remark 3.1 It is noted by Ikeda and Siljak [3], and Siljak [4] that theextension procedure of system (3.1) or the general nonlinear system can re-sult in the applicability of the vector Liapunov function while its immediateapplication to the initial system is difficult or impossible. Below we cite anexample showing that this is not a common case, i.e. there exist systems of(3.1) type to which it is impossible to apply the vector Liapunov functioneven after their extension in the sense of Definition 3.1 to the form (3.4).

We select out of the extended system (3.4) two independent subsystems

dz1dt

= A11z1, z1(t0) = z10,

dz2dt

= A22z2, z2(t0) = z20

(3.8)

and assume that for the given sign-definite matricesG11 andG22 the algebraicLiapunov equations

AT11P11 + P11A11 = G11, (3.9)

AT22P22 + P22A22 = G22 (3.10)

have the solutions in the form of sign-definite symmetric matrices P11 andP22 respectively.

Let matrices P11 and P22 be determined by equations (3.9) and (3.10).Assume that there exist constants η1, η2 > 0 such that the algebraic equa-tion [13]

AT11P12 + P12A22 = −

η1η2P11A12 −

η2η1AT

21P22 (3.11)


has bounded matrix P12 as its solution.

Proposition 3.1 If and only if matrices A11 and −A22 do not havecommon eigenvalues, then the algebraic equation (11) has a unique solutionin the form of bounded matrix P12.

Proposition 3.1 follows from Theorem 85.1 of Lankaster [14].Equations (3.9) – (3.11) make the basis of the proposed new method of

Liapunov matrix-valued function construction.Now we construct a two-index system of functions

U(y) = [vij(y1, y2)], i, j = 1, 2, (3.11)

with elements

v11(y1) = yT1P11y1, v22(y2) = yT

2P22y2, (3.12)

v12(y1, y2) = v21(y1, y2) = yT1P12y2. (3.13)

Here matrices P11, P22, and P12 are determined by the equations (3.9) –(3.11).

For quadratic forms (3.13) and bilinear form (3.14) the estimates

v11(y1) ≥ λm(P11)‖y1‖2,

v22(y2) ≥ λm(P22)‖y2‖2,

v12(y1, y2) ≥ −λ1/2m (P12P

T12)‖y1‖ ‖y2‖,

(3.15)

are true, where λm(P11), and λm(P22) are minimal eigenvalues of matrices

P11 and P22, and λ1/2M (·) is the norm of matrix P12P

T12, coordinated with the

vector norm.It is easy to show that the function

v(y, η) = ηTU(y)η, η ∈ R2+, η > 0 (3.16)

satisfies for all y ∈ Rn the estimate

v(y, η) ≥ uTHTCHu, (3.17)

where u = (‖y1‖, ‖y2‖)T, H = diag (η1, η2), and

C =

(λm(P11) −λ

1/2M (P12P

T12)

−λ1/2M (P12P

T12) λm(P22)

).

The variation of the total derivative of function (3.17) along solutions ofsystem (3.4)

Dv(y, η) = yT1

(AT

11P11 + P11A11

)y1η

21

+ 2yT1

[(AT

11P12 + P12A22

)η1η2 + η2

1P11A11 + η22A22P22

]y2

+ + yT2

(AT

22P22 + P22A22

)y2 + η1η2y

T1

(P12A21 + +AT

21PT12

)y1

+ η1η2yT2

(AT

12P12 + PT12A12

)y2

(3.18)

216 A.A. Martynyuk

is estimated in view of equation (3.11). Denote by

λM = λM

(AT

11P11 + P11A11

); βM = βM

(AT

22P22 + P22A22

);

κM = κM

(P12A21 + AT

21PT12

); χM = χM

(AT

12P12 + PT12A12

) (3.19)

the maximal eigenvalues of the corresponding matrices. In view of designa-tions (3.19) for all (y1, y2) ∈ N1 ×N2

Dv(y, η)∣∣(4)

≤ λMη21‖y1‖

2 + κMη1η2‖y1‖2

+ βMη22‖y2‖

2 + χMη1η2‖y2‖.

Hence it follows that for all (y1, y2) ∈ N1 ×N2 the inequality

Dv(y, η) ≤ uTSu, (3.20)

holds true, where matrix S = [σij ], i, j = 1, 2, has the elements

σ11 = λMη21 + κMη1η2, σ22 = βMη2

2 + χMη1η2, σ12 = σ21 = 0.

Estimates (3.17) of function (3.16) and its total derivative (3.20) enableus to establish new stability test for system (3.1) as follows.

Theorem 3.1 Assume that the equations (3.1) are such that the follow-ing conditions are satisfied:

(1) system (3.4) is the extension of system (3.1) in the sense of Defini-tion 3.1;

(2) there exist solutions to algebraic equations (3.9) – (3.11);

(3) in estimate (3.17) matrix C is positive definite;

(4) in estimate (3.18) matrix S is negative semi-definite (negative definite).

Then the equilibrium state x = 0 of system (3.1) is uniformly stable inthe whole (uniformly asymptotically stable in the whole).

For the proof see [13].

Theorem 3.2 Assume that conditions (1) – (3) of Theorem 3.1 are sat-isfied and there exists a vector η = (η1, η2) > 0 such that instead of condition(4) of Theorem 3.1 the following inequalities are fulfilled

η21

(AT

11P11 + P11A11

)+ η1η2

(P12A21 + AT

21PT12

)< 0, (3.21)

η22

(AT

22P22 + P22A22

)+ η1η2

(AT

12P12 + PT12A12

)< 0. (3.22)

Then the equilibrium state x = 0 of system (3.1) is uniformly asymptoticallystable in the whole.

Proof Together with conditions (1) – (3) of Theorem 3.1 inequalities(3.21), and (3.22) ensure satisfaction of all conditions of Theorem 2.11 byIkeda and Siljak [3] and Theorem 25.2 by Hahn [15]. Hence the assertion ofTheorem 3.2.


4 Numerical example

The proposed technique of Liapunov matrix-valued function construction forthe extended system is illustrated by the example of the third order system

dx

dt=

−3 4 64 −5 4

−10 −4 −3

x, (4.1)

where x = (x1, x2, x3)T. Diagonal blocks of the matrix are taken as the

matrices of coefficients of independent subsystems of the extended system,i.e. system (4.1) is extended to

dy

dt=

−3 4 0 64 −5 0 4

4 0 −5 4

−10 0 −4 −3

y, (4.2)

where y = (y1, y2)T, y1 = (x1, x2)

T, y2 = (x2, x3)T are the state vectors of

two second order subsystems

dy1dt

=

(−3 44 −5

)y1 +

(0 60 4

)y2, (4.3)

dy2dt

=

(−5 4−4 −3

)y2 +

(4 0

−10 0

)y1. (4.4)

According to the adopted notation we have

A11 =

(−3 44 −5

), A22 =

(−5 4−4 −3

),

A12 =

(0 60 4

), A21 =

(4 0

−10 0

).

Assume that P11 = P22 = E, η = (1, 1)T and take

v11(y1) = yT1y1, v22(y2) = yT

2y2,

v12(y1, y2) = v21(y1, y2) = yT1P12y2.

as the elements of matrix-valued function (3.12). Here matrix P12 is deter-mined by the equation

(−3 44 −5

)P12 + P12

(−5 4−4 −3

)=

(−4 40 −4

)

218 A.A. Martynyuk

corresponding to equation (2.11). It is easy to verify that P12 = 12E, where E

is 2×2 identity matrix. Since v11 = ‖y1‖2, v22 = ‖y2‖2, v12 ≥ − 12 ‖y1‖ ‖y2‖,

the matrix C in estimate (3.17) reads

C =

(1 −1/2

−1/2 1

)

and is positive definite. It is easy to see that conditions (1) – (3) of Theo-rem 3.2 are satisfied and conditions (3.21), and (3.22) of this theorem havethe form

(−6 88 −10

)+

1

2

(8 −10

−10 0

)=

(−2 33 −10

)= S1,

(−10 00 −6

)+

1

2

(0 66 8

)=

(−10 33 −2

)= S2.

One can easily check that the matrices S1 and S2 are negative definite. Con-sequently, the equilibrium state y = 0 of system (4.2) is uniformly asympto-tically stable in the whole. Since all conditions of Theorem 2.2, are fulfilled,the equilibrium state x = 0 of system (4.1) possesses the same type of sta-bility.

5 Conclusion remarks

Example (4.1) with the extension (4.2) is the one to which vector Liapunovfunction is not applicable. This can be verified easily by the way proposed byIkeda and Siljak [3] for the proof of vector Liapunov function nonapplicabilityto the non-extended system in one case. The method proposed by us forLiapunov matrix-valued function construction (see [12, 13, 17]) in conjunctionwith the extension method of the state space of dynamical systems (see [18])enlarges the area of the Liapunov’s second method application in linear andnonlinear dynamics of systems.

References

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[2] A.A. Martynyuk, A Contraction principle. In: Advances in Nonlinear Dynamics,Gordon and Breach Science Publishers, Amsterdam, (1997) 99-105.

[3] M. Ikeda, D.D. Siljak, Generalized decomposition of dynamic systems and vectorLyapunov functions. IEEE Trans. Autom. Control, AC-26, No 5, (1981) 1118–1125.

[4] D.D. Siljak, Decentralized Control of Complex Systems. Boston, Academic Press,1991.


[5] T. Yoshizawa, Stability Theory by Liapunov’s Second Method The MathematicalSosiety of Japan, Tokyo, 1966.

[6] A.A. Martynyuk, Stability of Motion of Complex Systems, Kiev, Naukova Dumka,1975. (Russian)

[7] A.N. Michel, R.K. Miller, Qualitative Analysis of Large Scale Dynamical Systems,New York, Academic Press, 1977.

[8] D.D. Siljak, Large-Scale Dynamic Systems: Stability and Structure, New York,North Holland, 1978.

[9] A.A. Voronov, V.M. Matrosov (Eds.), Method of Vector Lyapunov Functions in

Stability Theory, Moscow, Nauka, 1987. (Russian)

[10] A.M. Liapunov, General Problem of Motion Stability. Moscow, Gostekhizdat, 1950.(Russian).

[11] M.Z. Djordjevic, Stability analysis of interconnected systems with possibly unstablesubsystems. Systems and Control Letters, 3, (1983) 165–169.

[12] A.A. Martynyuk, V.I. Slyn’ko, Solution of the problem of constructing Liapunovmatrix functions for a class of large-scale systems. Nonlinear Dynamics and Systems

Theory 1 (2) (2001) 193-203.

[13] A.A. Martynyuk, V.I. Slyn’ko, Matrix-valued Lyapunov function and overlappingdecomposition of dynamical system. Int. Appl. Mech. .

[14] P. Lankaster, Matrices Theory. Moscow, Nauka, 1978. (Russian).

[15] W. Hahn, Stability of Motion. Berlin, Springer-Verlag, 1967.

[16] A.A. Martynyuk, Stability by Liapunov’s Matrix Functions Method with Applica-

tions. New York, Marcel Dekker, Inc., 1998.

[17] A.A. Martynyuk, Qualitative Methods in Nonlinear Dynamics. Novel Approaches

to Liapunov’s Matrix Functions, New York, Marcel Dekker, 2002.

[18] A.A. Martynyuk, Extension of the state space of dynamical systems and the problemof stability. In: Colloquia Mathematica Societaties Janos Bolyai 47. Differential

Equations: Qualitative Theory, Szeged (Hungary), (1984) 711-749.

email:[email protected]://www.sciencearea.com.ua


L2 GAIN ANALYSIS OF SWITCHED SYMMETRICSYSTEMS WITH TIME DELAYS

Anthony N. Michel1, Guisheng Zhai2, Xinkai Chen3 and Ye Sun1

1 Department of Electrical Engineering, University of Notre Dame, USA2 Department of Opto-Mechatronics, Wakayama University, JAPAN

3 Department of Intelligent Systems, Kinki University, JAPAN

Abstract: In this paper, we study L2 gain property for a class of switched systems which

are composed of a finite number of linear time-invariant (LTI) symmetric subsystems with

time delays in system states. We show that when all subsystems have L2 gain γ in the

sense of satisfying an LMI, the switched system has the same L2 gain γ under arbitrary

switching. The key idea is to establish a common Lyapunov function for all subsystems in

the sense of L2 gain.

Keywords: Switched symmetric system, time delay, arbitrary switching, L2 gain, com-

mon Lyapunov function, linear matrix inequality (LMI).

AMS (MOS) subject classification: 93C15, 93D25, 93D30.

1 Introduction

In the last two decades, there has been increasing interest in the stabilityanalysis and controller design for switched systems; for recent progress andperspectives in the field of switched systems, see the survey papers [1, 2] andthe references cited therein. It is agreed that there are three basic problems instability and design of switched systems: (i) find conditions for stabilizabilityunder arbitrary switching; (ii) identify the limited but useful class of stabiliz-ing switching signals; and (iii) construct a stabilizing switching signal. Thereare many existing works on Problem (ii) and (iii). For example, Refs. [3]-[5]considered Problem (ii) using piecewise Lyapunov functions, and Refs. [6]-[8]considered Problem (ii) for switched systems with pairwise commutation orLie-algebraic conditions. Ref. [9, 10] considered Problem (iii) by dividing thestate space associated with appropriate switching depending on state, andRefs. [11]-[13] considered quadratic stabilization, which belongs to Problem(iii), for switched systems composed of a pair of unstable linear subsystemsby using a linear stable combination of unstable subsystems. However, wesee very few dealing with the first problem, though it is desirable to permitarbitrary switching in many real applications. Ref. [14] showed that whenall subsystems are stable and commutative pairwise, the switched system is

222 A.N. Michel, G. Zhai, X. Chen and Y. Sun

stable under arbitrary switching. There are some other results (e.g., [15])concerning common Lyapunov functions for the subsystems in a switchedsystem, but we do not find any explicit answer to Problem (i) except [14].

For switched systems, there are a few results concerning L2 gain analysis.Hespanha considered such a problem in his Ph.D. dissertation [16], by usinga piecewise Lyapunov function approach. In [17], a modified approach hasbeen proposed for more general switched systems and more exact results havebeen obtained. In that context, it has been shown that when all subsystemsare Hurwitz stable and have L2 gains smaller than a positive scalar γ0, theswitched system under an average dwell time scheme [3] has a weighted L2

gain γ0, and the weighted L2 gain approaches normal L2 gain if the averagedwell time is chosen sufficiently large. However, the results obtained in [16]and [17] are conservative, and it is supposed that the main reason is in theuse of piecewise Lyapunov functions. Recently, Ref. [18] considered thecomputation of L2 gain of a switched linear system with large dwell time,and gave an algorithm by considering the separation between the stabilizingand antistabilizing solutions to a set of algebraic Riccati equations. Noticingthat these papers deal with the class of switching signals with (average)dwell time, we are motivated to ask the following question: Are there switched

systems that preserve their subsystems’ stability and L2 gain properties under

arbitrary switching?

For the above question concerning stability and L2 gain, we have given aclear (though not complete) answer in a recent paper [19]. In that context,we have shown that a class of switched systems, which are composed of afinite number of linear time-invariant (LTI) symmetric subsystems (withouttime delay) and called shortly switched symmetric systems, will preserve theirsubsystems’ stability and L2 gain properties under arbitrary switching. Wetake such symmetric systems into consideration since they appear quite oftenin many engineering disciplines (for example, RC or RL electrical networks,viscoelastic materials and chemical reactions) [20], and thus belong to animportant class in control engineering.

In the present paper, we extend the discussion in [19], especially the partof L2 gain analysis, to an important class of switched symmetric time delaysystems described by equations of the form

x(t) = Aσ(t)x(t) + Aσ(t)x(t − τ) + Bσ(t)w(t)

x(t) = φ(t) , ∀t ∈ [−τ, 0]

z(t) = Cσ(t)x(t) + Dσ(t)w(t) ,

(1)

where x(t) ∈ ℜn is the state, w(t) ∈ ℜm is the input, z(t) ∈ ℜp is theoutput, τ > 0 is the time delay in the state, and φ(·) is the initial con-dition. σ(·) : ℜ+ → IN = 1, 2, · · · , N is a piecewise constant function,called a switching signal, which is assumed to be arbitrary throughout thispaper, except that we assume no state jump occurs when switching from onesubsystem to another system. Here, ℜ+ denotes the set of all nonnegative

L2 Gain Analysis 223

numbers, Ai, Ai, Bi, Ci, Di (i ∈ IN ) are constant matrices of appropriatedimension denoting the subsystems, and N > 1 is the number of subsystems.In the present case, we assume that all subsystems in (1) are symmetric inthe sense of satisfying

Ai = ATi , Ai = AT

i , Bi = CTi , Di = DT

i , ∀i ∈ IN . (2)

It is noted that the condition (2) does not cover all symmetric subsystems,which are usually defined in the form of transfer functions, and a more gen-eral definition is that TiAi = AT

i Ti , TiBi = CTi , Di = DT

i holds for somenonsingular symmetric matrix Ti [20, 21, 22]. However, since (2) representsan interesting class of symmetric systems [23, 19], we will first consider suchkind of symmetric systems, and then extend the discussion to the case con-cerning a more general condition.

The motivation of including time delays in (1) is from that fact that timedelays always exist in most real control systems. For example, in networkedcontrol systems [24, 25] where all information (reference input, plant output,control input, etc.) is exchanged through a network among control systemcomponents (sensors, controller, actuators, etc.), there exist various timedelays inevitably due to busy and/or unreliable transmission path. Therehave been extensive publications concerning time delay systems. For goodreferences on time delay systems and discussions of the importance of suchsystems, the readers may want to refer to, e.g., [26], [27].

This paper is organized as follows. In Section 2, we state and prove somepreliminary results concerning L2 gain analysis for linear time delay systems.In Section 3, we analyze L2 gain property for the switched system (1) with(2). Assuming that all subsystems have L2 gain γ in the sense of satisfyingan LMI, we prove that there exists a common Lyapunov function for allsubsystems in the sense of L2 gain, and that the switched system has the sameL2 gain γ under arbitrary switching. In Section 4, we consider the switchedsystem (1) in a more general case rather than (2), and derive correspondingresult for L2 gain analysis of the switched system under arbitrary switching.Finally, Section 5 concludes the paper.

2 Preliminary Results

In this section, we state and prove some preliminary results which will beused later. Throughout this paper, we say a system has L2 gain γ when∫ t

0zT (s)z(s)ds ≤ γ2

∫ t

0wT (s)w(s)ds holds for any w(t) 6≡ 0 and any t > 0.

Lemma 1. Consider the linear time delay system

x(t) = Ax(t) + Ax(t − τ) + Bw(t)

x(t) = 0 , ∀t ∈ [−τ, 0]

z(t) = Cx(t) + Dw(t) ,

(3)


where x(t) ∈ ℜn is the state, w(t) ∈ ℜm is the input, z(t) ∈ ℜp is the output,and τ > 0 is the time delay. If there exist P > 0 and P > 0 satisfying theLMI [28]

AT P + PA + P P A PB CT

AT P −P 0 0

BT P 0 −γI DT

C 0 D −γI

≤ 0 , (4)

then the system (3) has L2 gain γ.Proof. For the Lyapunov function candidate

V (x, t) = xT (t)Px(t) +

∫ τ

0

xT (t − s)P x(t − s)ds , (5)

we compute its derivative along the trajectories of (3) as

d

dtV (x, t) = xT (t)P

[

Ax(t) + Ax(t − τ) + Bw(t)]

+[

Ax(t) + Ax(t − τ) + Bw(t)]T

Px(t)

+xT (t)P x(t) − xT (t − τ)P x(t − τ)

= xT

AT P + PA + P P A PB

AT P −P 0BT P 0 0

x , (6)

where xT = [xT (t) xT (t − τ) wT (t)]. Since (4) is equivalent to

AT P + PA + P +1

γCT C PA PB +

1

γCT D

AT P −P 0

BT P +1

γDT C 0 −γI +

1

γDT D

≤ 0 , (7)

we obtain from (6) and (7) that

d

dtV (x, t) ≤ −xT

1

γCT C 0

1

γCT D

0 0 01

γDT C 0 −γI +

1

γDT D

x

= −1

γ

(

zT (t)z(t) − γ2wT (t)w(t))

. (8)

Integrating both sides of the inequality (8) and using V (x(t), t) ≥ 0, V (x(0), 0) =0, we obtain

∫ t

0

zT (s)z(s)ds ≤ γ2

∫ t

0

wT (s)w(s)ds , (9)


which implies the system (3) has L2 gain γ.

Next, we state and prove a lemma which plays an important role in thispaper. We note that the idea of this lemma and its proof are motivated byRef. [23].

Lemma 2. Assume that the linear time delay system (3) satisfies A = AT ,A = δIn, B = CT , D = DT . If there exists P > 0 satisfying the LMI

AP + PA + P δP PB B

δP −P 0 0

BT P 0 −γI D

BT 0 D −γI

≤ 0 , (10)

then the system (3) has L2 gain γ . Furthermore,

2A + I δI B B

δI −I 0 0

BT 0 −γI D

BT 0 D −γI

≤ 0 . (11)

Proof. Since (10) means that (4) is satisfied with P = P , we knowimmediately from Lemma 1 that the system (3) has L2 gain γ. Now, weproceed to prove (11).

Since P > 0, there always exists a nonsingular matrix U satisfying UT =U−1 such that

UT PU = Σ0 = diagσ1, σ2, · · · , σn , (12)

where σi > 0, i = 1, 2, · · · , n. Pre- and post-multiplying the first and secondrows and columns in (10) by UT and U , respectively, we obtain

AUΣ0 + Σ0AU + Σ0 δΣ0 Σ0BU BU

δΣ0 −Σ0 0 0

BTUΣ0 0 −γI D

BTU 0 D −γI

≤ 0 , (13)

where AU = UT AU , BU = UT B. Furthermore, pre- and post-multiplyingthe first and second rows and columns in (13) by Σ−1

0 , and then exchangingthe third and the fourth rows and columns, leads to

AUΣ−10 + Σ−1

0 AU + Σ−10 δΣ−1

0 Σ−10 BU BU

δΣ−10 −Σ−1

0 0 0

BTUΣ−1

0 0 −γI D

BTU 0 D −γI

≤ 0 . (14)


Since σ1 > 0, there always exists a scalar λ1 such that

0 < λ1 < 1 , λ1σ1 + (1 − λ1)σ−11 = 1 . (15)

Then, by computing λ1×(13) + (1 − λ1)×(14), we obtain


δΣ1 −Σ1 0 0

BTUΣ1 0 −γI D

BTU 0 D −γI

≤ 0 , (16)

where

Σ1 = diagλ1σ1 + (1 − λ1)σ−11 , λ1σ2 + (1 − λ1)σ

−12 ,

· · · , λ1σn + (1 − λ1)σ−1n

= diag1 , σ2 , · · · , σn > 0 . (17)

In the similar way to obtain (14), we can obtain from (16) that

AUΣ−11 + Σ−1

1 AU + Σ−11 δΣ−1

1 Σ−11 BU BU

δΣ−11 −Σ−1

1 0 0

BTUΣ−1

1 0 −γI D

BTU 0 D −γI

≤ 0 . (18)

Since σ2 > 0, there exists a scalar λ2 such that

0 < λ2 < 1 , λ2σ2 + (1 − λ2)σ−12 = 1 . (19)

Then, the linear convex combination of (16) and (18) results in


δΣ2 −Σ2 0 0

BTUΣ2 0 −γI D

BTU 0 D −γI

≤ 0 , (20)

where

Σ2 = diag1 , λ2σ2 + (1 − λ2)σ−12 , · · · , λ2σn + (1 − λ2)σ

−1n

= diag1 , 1 , σ3 , · · · , σn > 0 . (21)

By repeating this process, we see that Σn = I also satisfies (13), i.e.,

2AU + I δI BU BU

δI −I 0 0

BTU 0 −γI D

BTU 0 D −γI

≤ 0 . (22)


Pre- and post-multiplying this inequality respectively by diagU , U , I , Iand diagUT , UT , I , I, we obtain (11).

Remark 1. Lemma 2 implies that if for the linear symmetric time delaysystem (3) satisfying A = AT , A = δIn, B = CT and D = DT there is aLyapunov function (5) where P = P satisfies (10), then

VI(x, t) = xT (t)x(t) +

∫ τ

0

xT (t − s)x(t − s)ds (23)

serves also as a Lyapunov function.

3 L2 Gain Analysis

In this section, we assume φ(t) = 0, ∀t ∈ [−τ, 0], and study L2 gain propertyfor the switched symmetric system (1) with (2) under arbitrary switching.

We now state and prove the main result of the present section.

Theorem 1. Assume that (2) holds with Ai = δiI, ∀i ∈ IN . If all subsys-tems in (1) have L2 gain γ in the sense that there exists Pi > 0 satisfyingthe LMI

AiPi + PiAi + Pi δiPi PiBi Bi

δiPi −Pi 0 0

BTi Pi 0 −γI Di

BTi 0 Di −γI

≤ 0 , (24)

then (23) is a common Lyapunov function for all subsystems in the sense ofL2 gain, and the switched system (1) has the same L2 gain γ under arbitraryswitching.

Proof. If (24) is true, then according to Lemma 2,

2Ai + I δiI Bi Bi

δiI −I 0 0

BTi 0 −γI Di

BTi 0 Di −γI

≤ 0 (25)

holds for all i ∈ IN . From Lemma 1, Lemma 2 and their proofs, we obtaineasily that the derivative of (23) along the trajectories of any subsystemsatisfies

d

dtVI(x, t) ≤ −

1

γΓ(t) , (26)

where

Γ(t)= zT (t)z(t) − γ2wT (t)w(t) . (27)

This implies that (23) is a common Lyapunov function for all subsystems inthe sense of L2 gain.


Now, for any piecewise constant switching signal σ(t) and any given in-teger t > 0, we let t1, · · · , tr (r ≥ 1) denote the switching points of σ(t) overthe interval [0, t). From (26), we obtain

VI(t, x(t)) − VI(tr, x(tr)) ≤ −1

γ

∫ t

tr

Γ(s)ds

VI(tr, x(tr)) − VI(tr−1, x(tr−1)) ≤ −1

γ

∫ tr

tr−1

Γ(s)ds

· · · · · · · · · · · ·

VI(t1, x(t1)) − VI(0, x(0)) ≤ −1

γ

∫ t1

0

Γ(s)ds ,

(28)

and thus

VI(t, x(t)) − VI(0, x(0)) ≤ −1

γ

∫ t

0

Γ(s)ds . (29)

We use the assumption of φ(t) = 0 (−τ ≤ t ≤ 0) and the fact of VI(t, x(t)) ≥ 0to obtain

∫ t

0

zT (s)z(s)ds ≤ γ2

∫ t

0

wT (s)w(s)ds . (30)

We note that the above inequality holds for any t > 0 including the case oft → ∞, and we did not add any limitation on the switching signal. Therefore,the switched system (1) with (2) has L2 gain γ under arbitrary switching.

Remark 2. Theorem 1 can be extended easily to multi-delay systems. Sup-pose that the subsystems are described as

x(t) = Aix(t) +

ℓ∑

j=1

Aijx(t − τj) + Biw(t)

z(t) = Cix(t) + Diw(t) ,

(31)

where Ai = ATi , Aij = δijI , Bi = CT

i , Di = DTi , j = 1, · · · , ℓ, and τj ’s are

time delays. If there exists Pi > 0 satisfying the LMI

AiPi + PiAi + ℓPi δi1Pi · · · δiℓPi PiBi Bi

δi1Pi −Pi · · · 0 0 0

......

. . .... 0 0

δiℓPi 0 · · · −Pi 0 0

BTi Pi 0 · · · 0 −γI Di

BTi 0 · · · 0 Di −γI

≤ 0 , (32)


then all the subsystems of (31) have L2 gain γ, and

2Ai + ℓPi δi1I · · · δiℓI Bi Bi

δi1I −I · · · 0 0 0

......

. . .... 0 0

δiℓI 0 · · · −I 0 0

BTi 0 · · · 0 −γI Di

BTi 0 · · · 0 Di −γI

≤ 0 . (33)

Furthermore,

V ℓI (x, t) = xT (t)x(t) +

ℓ∑

j=1

∫ τj

0

xT (t − s)x(t − s)ds (34)

is a common Lyapunov function for all the subsystems in the sense of L2 gain,and the switched system composed of (31) has L2 gain γ under arbitraryswitching.

4 An Extension

In this section, we extend our discussion to the switched symmetric system(1) where instead of (2),

TAi = ATi T , T Ai = AT

i T , TBi = CTi , Di = DT

i , ∀i ∈ IN (35)

are satisfied for some T > 0. It is noted here that the condition (35) coversthe symmetric systems considered in [23] and [19].

To proceed, we first prove a preliminary result.

Lemma 3. Assume that the linear time delay system (3) satisfies AT T =TA, A = δIn, TB = CT , D = DT for some T > 0. If there exists P > 0satisfying the LMI

AT P + PA + P δP PB CT

δP −P 0 0

BT P 0 −γI D

C 0 D −γI

≤ 0 , (36)

then the system (3) has L2 gain γ. Furthermore,

2TA + T δT TB CT

δT −T 0 0

BT T 0 −γI D

C 0 D −γI

≤ 0 . (37)


Proof. Since (36) means that (4) is satisfied with P = P , we knowimmediately from Lemma 1 that the system (3) has L2 gain γ. We nowproceed to prove (37).

Using T > 0, we consider the similarity transformation

A⋆ = T1

2 AT−1

2 , B⋆ = T1

2 B , C⋆ = CT−1

2 , D⋆ = D . (38)

Then, we can confirm easily that

A⋆ = AT⋆ , B⋆ = CT

⋆ , DT⋆ = D⋆ . (39)

Next, pre- and post-multiplying the first and second rows and columns in(36) by T−

1

2 , we obtain

A⋆P⋆ + P⋆A⋆ + P⋆ δP⋆ P⋆B⋆ B⋆

δP⋆ −P⋆ 0 0

BT⋆ P⋆ 0 −γI D⋆

BT⋆ 0 D⋆ −γI

≤ 0 , (40)

where P⋆ = T−1

2 PT−1

2 . Hence, using the same technique as in the proof ofLemma 2, we can prove easily that P⋆ in (40) can be replaced by the identitymatrix, i.e.,

2A⋆ + I δI B⋆ B⋆

δI −I 0 0

BT⋆ 0 −γI D⋆

BT⋆ 0 D⋆ −γI

≤ 0 . (41)

Pre- and post-multiplying this inequality by diagT1

2 , T1

2 , I , I, we obtain(37).

Noticing that in the proof of Lemma 3 replacing P⋆ with I means re-placing P with T for the original system, we obtain the following importantobservation.

Remark 3. If for the linear symmetric time delay system (3) satisfyingwith AT T = TA, A = δIn, TB = CT , D = DT for some T > 0 there is aLyapunov function (5) in the sense of L2 gain where P = P satisfies (36),then

VT (x, t) = xT (t)Tx(t) +

∫ τ

0

xT (t − s)Tx(t − s)ds (42)

serves also as a Lyapunov function.

Now, we state and prove the main result in this section.

Theorem 2. Assume that (35) holds with some T > 0 and Ai = δiI,∀i ∈ IN . If all subsystems in (1) have L2 gain γ in the sense that there exists


Pi > 0 satisfying the LMI

ATi Pi + PiAi + Pi δiPi PiBi CT

i

δiPi −Pi 0 0

BTi Pi 0 −γI Di

Ci 0 Di −γI

≤ 0 , (43)

then (42) is a common Lyapunov function for all subsystems in the sense ofL2 gain, and the switched system (1) has the same L2 gain γ under arbitraryswitching.

Proof. If (43) is true, then according to Lemma 3,

2TAi + T δT TBi CTi

δT −T 0 0

BTi T 0 −γI Di

Ci 0 Di −γI

≤ 0 (44)

holds for all i ∈ IN . Then, from Lemma 1, Lemma 3 and their proofs,we obtain easily that the derivative of (42) along the trajectories of anysubsystem satisfies

d

dtVT (x, t) ≤ −

1

γΓ(t) , (45)

which implies that (42) is a common Lyapunov function for all subsystemsin the sense of L2 gain.

Now, for any piecewise constant switching signal σ(t) and any given timet > 0, we let t1, · · · , tr (r ≥ 1) denote the switching points of σ(t) over theinterval [0, t). From (45), we obtain

VT (t, x(t)) − VT (tr, x(tr)) ≤ −1

γ

∫ t

tr

Γ(s)ds

VT (tr, x(tr)) − VT (tr−1, x(tr−1)) ≤ −1

γ

∫ tr

tr−1

Γ(s)ds

· · · · · · · · · · · ·

VT (t1, x(t1)) − VT (0, x(0)) ≤ −1

γ

∫ t1

0

Γ(s)ds ,

(46)

and thus

VT (t, x(t)) − VT (0, x(0)) ≤ −1

γ

∫ t

0

Γ(s)ds . (47)

We use the assumption of φ(t) = 0 (−τ ≤ t ≤ 0) and the fact of VT (t, x(t)) ≥

0 to obtain∫ t

0 zT (s)z(s)ds ≤ γ2∫ t

0 wT (s)w(s)ds . We note that this inequalityholds for any t > 0 including the case of t → ∞, and that we did not add


any limitation on the switching signal. Therefore, the switched system (1)has L2 gain γ under arbitrary switching.

Remark 4. Theorem 2 can be extended easily to multi-delay systems. Sup-pose that the subsystems are described as (31) where TAi = AT

i T , Aij =δijI , TBi = CT

i , Di = DTi , j = 1, · · · , ℓ, and τj ’s are time delays. If there

exists Pi > 0 satisfying the LMI

ATi Pi + PiAi + ℓPi δi1Pi · · · δiℓPi PiBi CT

i

δi1Pi −Pi · · · 0 0 0

......

. . .... 0 0

δiℓPi 0 · · · −Pi 0 0

BTi Pi 0 · · · 0 −γI Di

Ci 0 · · · 0 Di −γI

≤ 0 , (48)

then all the subsystems of (31) have L2 gain γ, and

2TAi + ℓT δi1T · · · δiℓT TBi CTi

δi1T −T · · · 0 0 0

......

. . .... 0 0

δiℓT 0 · · · −T 0 0

BTi T 0 · · · 0 −γI Di

Ci 0 · · · 0 Di −γI

≤ 0 . (49)

Furthermore,

V ℓT (x, t) = xT (t)Tx(t) +

ℓ∑

j=1

∫ τj

0

xT (t − s)Tx(t − s)ds (50)

is a common Lyapunov function for all the subsystems in the sense of L2 gain,and the switched system composed of (31) has L2 gain γ under arbitraryswitching.

5 Conclusion

In this paper, we have studied L2 gain property for a class of switched sys-tems which are composed of a finite number of linear time-invariant (LTI)symmetric systems with time delays. Assuming that all subsystems have L2

gain γ in the sense of satisfying a nonstrict linear matrix inequality, we haveshown that there exists a common Lyapunov function for all subsystems inthe sense of L2 gain, and that the switched system has the same L2 gain γ

under arbitrary switching.


6 Acknowledgement

The authors would like to thank Professor Masao Ikeda, Osaka University,for his valuable comments. This research is supported in part by Grants-in-Aid for Scientific Research, No.15760320, by The Ministry of Education,Culture, Sports, Science and Technology, JAPAN.

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STABILITY OF STOCHASTIC DISTRIBUTEDPARAMETER LARGE-SCALE CONTROL SYSTEMS

UNDER RANDOM STRUCTURALPERTURBATIONS

G. S. Ladde

Department of Mathematics

The University of Texas at Arlington, Arlington, Texas 76019 USA

Abstract. By employing feedback control laws, the robust stochastic stability of dis-

tributed parameter large-scale control systems under random structural perturbations is

investigated. The results are obtained by using vector Lyapunov-like functionals in the

context of the block-comparison theorem, the theory of systems of differential inequalities,

and decomposition-aggregation method. The complexity versus stability and stochastic

versus stability issues are also studied. Examples are given to illustrate the usefulness of

the results. The byproduct of this research has generated an open mathematical problem,

namely, the boundary value problem for first order nonlinear partial matrix differential

inequalities/equations.

AMS (MOS) subject classification: 60H10, 34D20, 93A15, 93E15 and 93E03

1 Introduction

In a number of applied problems about a system being studied is limited tooutput information. The method of stochastic approximation has proved tobe a useful mathematical tool in the study of such systems. The main featureof this method is that it does not require the knowledge about the input ofthe system [10, 18, 20]. Only output information which is readily available forthese systems, is required. In this work, stochastic approximation technique[12, 13] is used to investigate the stability of stochastic distributed parameterlarge-scale control system under random structural perturbations.

In Section 2, a brief outline about “A Guided Tour” [19] by Profes-sor Siljak is presented. Section 3 deals with the problem description. InSection 4, we formulate a block comparison theorem in the context of dis-tributed parameter systems under both internal and external random per-turbations, vector Lyapunov functional, and systems of block differential in-equalities. The presented theorem generalizes and extends comparison the-orems in [3, 4, 5, 6, 7, 12, 13, 15, 18] in a natural way. By employing thecomparison theorem of Section 4, Section 5 deals with sufficient conditions

236 G. S. Ladde

that assure stability properties of stochastic systems of partial differentialequations of Ito-type under Markovian structural perturbations. The Resultsof Sections 4 and 5 were presented in an international conference [9] and areextensions of the results in [3, 4, 5, 6, 7, 12, 13, 14, 15, 18]. In Section 6,the mean-square stability property of isolated subsystems is investigated inthe framework of results of Sections 4 and 5. In Section 7, we apply blockvector Lyapunov functionals, block comparison theorem and block system ofdifferential equations [5, 9, 13] to conclude the mean-square connective sta-bility [15, 18, 20] of large-scale closed-loop stochastic distributed parametersystems under structural uncertainties both random and deterministic para-metric perturbations. In Section 8, a controlled chemical reaction in a tubularreactor [2, 16] with axial diffusion under random structural perturbation isillustrated. The sufficient conditions are given in terms of rate functions andnoise parameters. The mathematical conditions are algebraically simple andeasy to verify [18, 20].

2 “A Guided Tour” - by Professor Siljak

In this section, a brief outline about “A Guided Tour”-by Professor Siljakis presented. It consists of two methods, namely, the classical approach viacharacteristic and modern method based upon Lyapunov functions. We beginwith a parameter space method for robust control design via characteristicapproach.

Algebraic Method: It is well-known fact that this method is applicableto time-invariant control system design problem. The goal of parameter spacemethods in the framework of algebra is to establish the relationship betweenroots of characteristic equation and adjustable parameters appearing in thecoefficients of the equation. The main features of this is illustrated in [19]by the characteristic equations associated with a positional servo: withouta tachometer feedback, with a tachometer feedback and with a tachometercoupled with additional feedbacks and corresponding parameters associatedwith the plant and controllers. For further details, see [19].

A few remarks concerning this approach are as: (i) The method doesnot favor controller over plant uncertainties. (ii) The main assumption thatcoefficients of the characteristic equation are linear functions of adjustableparameters is overly restrictive. (iii) Beyond 3D, it is difficult to analyze,geometrically.

To overcome the above stated difficulties, the state-space and frequencydomain specifications, for examples: (i) stability, (ii) steady state error,(iii) quadratic performance index for linear systems, and (iv) etc., were usedto determine a set of inequalities delineating a feasible region in the space de-sign and plant parameters (parameter space). This kind of design approachis robust. The given specification is fulfilled in determined region of parame-ter space. The diameter of this region determines the measure of robustness.

Large Scale Control Systems 237

For further details see [19].Lyapunov Second Method: The Lyapunov Second Method plays a

very significant role in the study of qualitative analysis of solutions of non-linear nonstationary systems of differential equations [10, 18, 20]. It is a natu-ral alternative method for “parameter analysis” and “robust design of control

systems” under structural perturbations. This method does not require theknowledge of the characteristic polynomial/transfer function. Moreover, it isapplicable to nonlinear and nonstationary control systems. The concept offeedback provides a basis to select a control function of time and state so that:(i) it stabilizes the nominal part of the system and (ii) the given system is sta-ble under certain class of uncertainty parameters. The concept of connectivestability, vector Lyapunov functions and the decomposition-aggregation tech-nique [18] provide a suitable framework to undertake the study of structuraluncertainty in the system in a systematic and unified way. The decentralizedcontrol schemes [20] provide a powerful framework to handle uncertainties.The use of decentralized control law [20] to insure the connective stabilityillustrates the fundamental feature of decentralized control in robust designwith structural uncertainties. The robustness of stabilized systems cruciallydepend upon the following mutually dependent factors, namely, a structureof the plant, a choice of feedback, and a choice of Lyapunov function. Forfurther details, see [18, 19, 20].

3 Problem Formulation

For the sake of simplicity, we consider the following stochastic linear dis-tributed parameter control system under Markovian structural perturbations

∂yk = [Lk(x, η)yk + Bk(η)uk + Ak(x, η, E)y]dt + [σk(x, η)yk

+Σk(x, η, E)y]dzk

By(t, x, ω) = 0 on Γt0 = [t0,∞) × ∂H

y(t0, x) = y0 on H ⊆ Rm, k ∈ I(1, p)

(3.1)

which is composed of interconnected isolated subsystems

∂yk = [Lk(x, η)yk + Bk(η)uk]dt + σk(x, η)ykdzk

Bky(t, x, ω) = 0 on Γt0 = [t0,∞) × ∂H

yk(t0, x) = y0k on H ⊆ Rm, k ∈ I(1, p)

(3.2)

where

Lk(x, η) =

m∑

i,j=1

∂

∂xi

(

Dkij(x, η)

∂

∂xj

)

+

m∑

j=1

Ckj (x, η)

∂

∂xj

+ Ak(x, η), (3.3)

Bk =

m∑

i,j=1

Dkij(x, η) cos(ν, xi)

∂

∂xj

+ ∆k(x, η) (3.4)

238 G. S. Ladde

cos(ν, xi) is the i-th direction cosine of ν, ν being the normal at Γ directedoutward of H ; ∆k(x, η) is nk × nk symmetric matrix function; yk ∈ Rnk,y ∈ Rn, y = (yT

1 , yT2 , . . . , yT

p )T , n =∑p

k=1 nk; η = η(t, ω) is a right con-tinuous Markov chain with a set of states S = 1, 2, . . . , s and s × s statetransition matrix π = (πij)s×s; coefficient matrices Dk

ij , Ckj , Ak, Bk, and σk

are defined on S × H into the space of matrices of appropriate dimensions;Dk = (Dk

ij)m×m is symmetric block matrix whose elements, Dkij are nk × nk

symmetric positive definite matrices; matrices Σk and Ak are defined onS ×Mnk×n with appropriate dimensions, and E = (ekh)p×p is a p× p inter-connection matrix representing uncertainty (deterministic) about the system;zk is a normalized Wiener process with independent increments; y0, η and z

are mutually independent for each t ≥ t0; uk ∈ Rlk is an admissible controlvector applied to k-th subsystem; l =

∑pk=1 lk; k ∈ I(1, p) = 1, 2, . . . , p.

We apply a local decentralized control law

uk(x, η) = −Kk(x, η)yk, k ∈ I(1, p), (3.5)

where Kk(x, η) is a gain matrix with the state-space specification such asthe mean square exponential stability/robustness characteristics. Under thisconsideration, one obtains the following closed-loop systems correspondingto (3.1) and (3.2)

∂yk = [Lck(x, η)yk + Ak(x, η, E)y]dt

+[σk(x, η)yk + Σk(x, η, E)y]dzk

Bkyk(t, x, ω) = 0 on Γt0 = [t0,∞) × ∂H

y(t0, x) = y0 on H ⊆ Rm, k ∈ I(1, p)

(3.6)

and

∂yk = Lck(x, η)ykdt + σk(x, η)ykdzk

Bkyk(t, x, ω) = 0 on Γt0 = [t0,∞) × ∂H

y(t0, x) = y0 on H ⊆ Rm, k ∈ I(1, p)

(3.7)

respectively, where,

Lck(x, η) =

m∑

i,j=1

∂

∂xi

(

Dkij(x, η)

∂

∂xj

)

+

m∑

j=1

Ckj (x, η)

∂

∂xj

+ Ack(x, η), (3.8)

whereAck(x, η) = (Ak(x, η) − Bk(x, η)Kk(x, η)). (3.9)

The formulated problem generalizes various types of problems [6, 7, 13, 14,15, 18, 20] in systematic and unified way. This is illustrated by the followingexample in the context of distributed parameter systems.

Example 3.1 a) If elements Dkij of symmetric block matrix Dk = (Dk

ij)m×m

are of the form Dkij = dk

ijInk×nk, then the differential operator Lk(x, η) in


(3.1) reduces to

Lk(x, η) =

m∑

i,j=1

∂

∂xi

(

dkij(x, η)

∂

∂xj

)

+

m∑

j=1

Ckj (x, η)

∂

∂xj

+ Ak(x, η), (3.10)

where, Ink×nk is nk × nk identity matrix. This is a well-known special case[1, 11, 21].

b) If elements Dkij of symmetric block matrix Dk = (Dk

ij)m×m are of the

form Dkij = dk

ijInk×nk and dkij = 0 for i 6= j, then the differential operator

Lk(x, η) in (3.1) reduces to

Lk(x, η) =

m∑

j=1

∂

∂xj

(

dkjj(x, η)

∂

∂xj

)

+

m∑

j=1

Ckj (x, η)

∂

∂xj

+ Ak(x, η). (3.11)

4 Comparison Method

In this section, we formulate a block comparison theorem without proof inthe context of distributed parameter systems under both internal and ex-ternal random perturbations, vector Lyapunov functional, and systems ofdifferential inequalities [6, 7, 10, 12, 13, 15, 18]. The proofs of results canbe constructed by imitating the arguments in [3, 4, 12, 13, 15]. Therefore,details are left to the reader as exercises. These results are generalizationsand extensions of results in [3, 4, 6, 7, 10, 12, 13, 14, 15]. These results willbe utilized, subsequently. For this purpose, let us consider a following systemof stochastic partial differential equations under random environmental andstructural perturbations:

∂yk = fk(t, x, y, ykx, ykxx, η)dt + Fk(t, x, y, ykx, ykxx, η)dz

Bky(t, x, ω) = φ(t, x, ω) on Γt0 = [t0,∞) × ∂H

y(t0, x, ω) = y0(x, ω) on H ⊆ Rm, k ∈ I(1, p)

(4.1)

where Bk is boundary operator defined by Bkyk(t, x, ω) = αk(t, x, ω)yk +βk(t, x, ω)∂yk

∂ν; z is a normalized Wiener field; B, z, y0, φ and η are mutually

independent processes; further assume that fk, Fk, and B are smooth enoughto assure the existence of solution process defined on a complete probabilityspace (Ω,F , P ), and other variables are as defined before.

In the following, we state a lemma that connects the vector flow de-scribed by a system of stochastic partial differential equations of Ito-typeunder Markovian structural perturbations in the context of vector Lyapunovfunctional.

Lemma 4.1 Let V ∈ C[J × C2[H, Rn] × S, RN+ ] and partial Frechet deriva-

tives Vt, Vy and Vyy exist and continuous in (t, y, η). Then

D+V (t, y, S) ≤ LV (t, y, S)+MV (t, y, S), for (t, y) ∈ J×C2[H, Rn] (4.2)

240 G. S. Ladde

where linear operators D+, L and M are defined by

D+V (t, y, i) = limh→0+

sup1

hE[

V (t + h, y + f(t, x, y, yx, yxx, η(t))h

+ F (t, x, y, yx, yxx, η)∆z, η(t + h)) (4.3)

− V (t, y, η(t))∣

∣η(t) = i,Ft

]

where ∆z = z(t + h) − z(t) and

D+V (t, y, S) = (D+V (t, y, 1), D+V (t, y, 2), . . . , D+V (t, y, i), D+V (t, y, s))T ,

LV (t, y, i) = Vt(t, y, i) + Vy(t, y, i)f(t, x, y, yx, yxx, i) (4.4)

+1

2tr(Vyy(t, y, k)A(t, y, i))

where A(t, y, i) = F (t, x, y, yx, yxx, i)F (t, x, y, yx, yxx, i)T ,

MV (t, y, S) = πbV (t, y, S), (4.5)

where V (t, y, S) = (V (t, y, 1), V (t, y, 2), . . . , V (t, y, i), V (t, y, s))T and πb =(πb

ij) is the block matrix with πbii = πiiIN×N for j = i and πb

ij = πijIN×N for

j 6= i, πij are the elements of the intensity matrix associated with the Markov

chain η and Is×s is s × s identity matrix.

Now, we present a comparison theorem in the context of vector-Lyapunovlike functionals.

Theorem 4.1 Let the hypotheses of Lemma 4.1 be satisfied. Furthermore,

assume that

(i) LV (t, y, i) defined in (4.4) satisfies the following inequality

LV (t, y, i) ≤ gi(t, V (t, y, i)) for (t, y) ∈ J × C2[H, Rn]; (4.6)

(ii) gi ∈ C[J × RN+ , RN ], gi(t, u) is concave and quasi-monotone nonde-

creasing in y for fixed t ∈ J ;

(iii) r(t) = r(t, t0, r0) is the maximal solution of the block comparison system

of differential equations

u′ = W (t, u), u(t0) = r0 (4.7)

existing t ≥ t0, where

W (t, r) = g(t, r) + πbr; (4.8)


(iv) the solution process, y(t, x) = y(t, x, t0, i0) of (4.1) and E[V (t, y(t, x),η(t))] exist for t ≥ t0 and

V (t0, y(t0, x), S) ≤ r0. (4.9)

Then

E[V (t, y(t, x), S) | y(t0, x) = y0, η(t0) = i0] ≤ r(t), for t ≥ t0.

(4.10)

Remark 4.1 Due to limited space, results similar to Lemma 4.1 and Theo-rem 4.1 in the context of vector Lyapunov function will appear, elsewhere.

5 Stability Analysis

In this section, sufficient conditions are given to establish the stability proper-ties of the trivial solutions process of stochastic systems of partial differentialequations of Ito-type under Markovian structural perturbations (4.1). Thepresented result generalizes and extends the earlier results [3, 4, 6, 7, 10,12, 13, 14, 15] in a systematic and unified way. Again, we simply present aresult without proof. Proof can be constructed by imitating the argumentsin [3, 4, 12, 13, 15].

In the following, we are ready to present a (p, q)-th moment stability andasymptotic stability [3, 4] result in the context of Lyapunov-like functionals.

Theorem 5.1 Assume that all the hypotheses of Theorem 4.1 are satisfied.

Further assume that

(i) V (t, y, η(t)) ≡ 0, f(t, x, 0, 0, 0) ≡ 0, F (t, x, 0, 0, 0) ≡ 0, and W (t, 0) ≡ 0

(ii) for (t, y, i) ∈ J × C2[H, Rn] × S and V (t, y, i) satisfies

b(‖y‖p

q) ≤s∑

i=1

N∑

k=1

Vk(t, y, i) ≤ a(t, ‖y‖p

q), (5.1)

where b ∈ VK[R+, R+] and a ∈ CK[J × R+, R+] such that a(t, 0) = 0and a(t, r) is strictly increasing in r for each t ∈ R+.

Then

a) (SM∗) implies ((p, q)-S),

b) (ASM∗) implies ((p, q)-AS).

Remark 5.1 Due to limited space, results similar to Theorem 5.1 in thecontext of vector Lyapunov function will appear, elsewhere.

242 G. S. Ladde

6 Isolated Subsystems

In this section, by choosing control laws (3.5) for stochastic distributed pa-rameter isolated control subsystems under Markovian structural perturba-tions (3.2), the mean-square exponential stability of the equilibrium stateof corresponding closed-loop system (3.7) is investigated. This is achievedin the context of Lyapunov functional, differential inequalities, comparisontheorem and sufficient conditions that are developed in Sections 4 and 5.

To test the mean-square exponential stability, we present an auxiliaryresult that will be used subsequently. Here after, in order to ease the typing,we suppress the dependence of independent variables of underlying rate andauxiliary functions in our subsequent discussion.

Lemma 6.1 Let Dklz and Ck

z be matrices as defined in (3.1). Let Rk and Φk

be a symmetric matrices of suitable size defined on I(1, s) and I(1, s) × H,

respectively. Then

(a)

m∑

l,z=1

[

∂

∂xl

yTk RkDk

lz

∂

∂xz

yk +∂

∂xz

yTk Dk

lzRk ∂

∂xl

yk

]

=

m∑

l,z=1

∂

∂xl

yTk P k

lz

∂

∂xz

yk;

(6.1)(b)

m∑

l,z=1

∂

∂xl

yTk Plz

∂

∂xz

yk =

m∑

l,z=1

(

∂

∂xl

yk + Φkyk

)T

P klz

(

∂

∂xz

yk + Φkyk

)

−m∑

l,z=1

∂

∂xl

yTk Λk

lzΦkyk −

m∑

l,z=1

yTk ΦkΛkT

lz

∂

∂xl

yk

−m∑

l,z=1

∂

∂xz

yTk ΛkT

lz Φkyk −m∑

l,z=1

yTk ΦkΛk

lz

∂

∂xz

yk

− yTk Φk

m∑

l,z=1

P klz

Φkyk;

(c)

m∑

l,z=1

[

∂

∂xl

yTk Rk

(

Dklz

∂

∂xz

yk + Ckl yk

)

+

(

Dklz

∂

∂xz

yk + Ckl yk

)T

Rk ∂

∂xl

yk

]

=

m∑

l,z=1

(

∂

∂xl

yk + Φkyk

)T

P klz

(

∂

∂xz

yk + Φkyk

)

−m∑

l,z=1

1

2

∂

∂xl

(yTk Φk1

lz yk) +

m∑

l,z=1

∂

∂xz

(yTk Φk2

lz yk)


+ yTk

m∑

l,z=1

(

1

2

[

∂

∂xl

Φk1lz +

∂

∂xz

Φk2lz

]

− Φk(P klzΦk)

)

yk (6.2)

where

Λklz ≡ Λk

lz(x, i) = Rk(i)Dklz(x, i); P k

lz ≡ Plz(x, i) = (Λklz(x, i) + ΛkT

lz (x, i));

Θkl ≡ Θk

l (x, i) =1

2(Rk(i)Ck

l (x, i) + CTkl (x, i)Rk(i)); Φk ≡ Φk(x, i);

Φk1lz ≡ Φk1

lz (x, i) = Λklz(x, i)Φk(x, i) + Φk(x, i)ΛkT

lz (x, i) − Θkl (x, i);

Φk2lz ≡ Φk2

lz (x, i) = Φk(x, i)Λklz(x, i) + ΛkT

lz (x, i)Φk(x, i) − Θkl (x, i).

Proof. The proof of this lemma follows by employing the properties ofsymmetric matrices together with the fact, A = 1

2 (A + AT ) + 12 (A−AT ) for

any matrix with real entries, and elementary calculus of several variables aswell as algebra. The details are left to the reader.

Example 6.1 The following special cases exhibit the significance and theflexibility of the above elementary but very powerful result to study thelinear stochastic distributed parameter system.

1. a) Λklz = ΛkT

lz ⇒ P klz = 2RkDk

lz and Φk1lz = Φk2

lz = ΛklzΦ

k + ΦkΛklz − Θk

l .

b) ΦkΛklz = Λk

lzΦk ⇒ Φk1

lz = Φk2lz = Λk

lzΦk + ΦkΛkT

lz − Θkl .

c) Λklz = ΛkT

lz and ΦkΛklz = Λk

lzΦk ⇒ P k

lz = 2RkDklz and Φk1

lz = Φk2lz =

2ΛklzΦ

k − Θkl .

2. a) Λklz = ΛkT

lz and Dklz = 0 for l 6= z ⇒ Φk1

lz = Φk2lz = 0 for l 6= z and

Φk1ll = Φk2

ll ΦkΛkll + Λk

llΦk − Θl for l = z.

b) m = 1 and Λk = ΛkT ⇒ P k = 2RkDk and Φk1 = Φk2 = ΛkΦk +ΦkΛk − Θk. This case generalizes the work in [17].

c) m = 1 and nk = 1 ⇒ P k = 2RkDk and Φk1 = Φk2 = Φk = 2ΛkΦk −Θk = 2R(DkΦk − 1

2Ck).

To test the mean-square exponential stability, we choose a quadratic form-like Lyapunov functional

Vk(t, yk, η(t)) =

∫

H

yTk Rk(η(t))yk dx (6.3)

where Rk(η(t)) is a given symmetric positive-definite matrix. Further assumethat this Rk, matrices Dk

lz , Ckz , F k and Φk satisfying the hypotheses of

244 G. S. Ladde

Lemma 6.1 satisfy the following condition:

−1

2

m∑

l,z=1

[

∂

∂xl

Φk1lz +

∂

∂xz

Φk2lz

]

+ Φk

m∑

l,z=1

P klz

Φk = Qk, on H

m∑

l,z=1

1

2

[

cos(ν, xl)(Φk1lz + 2Θk

l )

+ cos(ν, xz)(Φk2lz + 2Θk

z)]

+ 2∆k = 0, on ∂H

(6.4)

where Υk = Rk(

Ak − BkKk −∑m

l=1∂

∂xl

Ck)

= Rk(

Ack −∑m

l=1∂

∂xl

Ck)

,

Qk = −[

Υk + ΥkT + σkT Rkσk +√

Rk Gk√

Rk

]

, (6.5)

and Gk(x, i) is nk ×nk positive definite matrix for k ∈ I(1, p) and i ∈ I(1, s).

Remark 6.1 Equation (6.4) is a boundary value problem for first ordernonlinear partial matrix differential equations. Very little is known aboutthis problem which is related to boundary value problems for hyperbolicequations [11]. It is a mathematical problem that is interesting to investigate.For m = 1, this problem is well-known and well-developed [8, 11, 17].

Example 6.2 In the context of Example 6.1 relative to 1.a) (or 1.b)/1.c)),2.a), 2.b), and 2.c), (6.4) can be recasted as:

−1

2

m∑

l,z=1

[

∂

∂xl

Φk1lz +

∂

∂xz

Φk1lz

]

+ Φk

m∑

l,z=1

P klz

Φk = Qk, on H

m∑

l,z=1

cos(ν, xl)[Φk1lz + Θk

l ] + 2∆k = 0, on ∂H ;

(6.6)

−m∑

l=1

∂

∂xl

[ΛkllΦ

k + ΦkΛkll − Θl] + Φk

m∑

l,z=1

P klz

Φk = Qk, on H

m∑

l=1

cos(ν, xl)[ΛkllΦ

k + ΦkΛkll + Θk

l ] + 2∆k = 0, on ∂H ;

(6.7)

− ∂

∂x(ΛkΦk + ΦkΛk − Θk) + Φk(2RkDkΦk) = Qk, on H

([ΛkΦk + ΦkΛk] + Θk + 2∆k)(µ) = 0, for µ = 0, 1,(6.8)

and

− ∂

∂x

(

2Rk

(

DkΦk − 1

2Ck

))

+ Φk(2RkDkΦk) = Qk, on H(

2RkDkΦk +1

2Ck + 2∆k

)

(µ) = 0, for µ = 0, 1,

(6.9)


respectively.The Ito differential [10] of Vk is

dVk(t, yk, i) =

∫

H

(

yTk Rk(i)∂yk + (∂yk)T Rk(i)yk + (∂yk)T Rk(i)∂yk

)

dx

+

s∑

j=1

πij [Vk(t, yk, j) − Vk(t, yk, i)]dt.

¿From this, we have

D+Vk(t, yk, i) =

∫

H

(

yTk Rk(i)Lc

k(x, i)yk + (Lck(x, i)yk)T Rk(i)yk

)

dx

+

∫

H

(

(σk(x, i)yk)T Rk(x, i)σk(x, i)yk

)

dx

+

s∑

j=1

πij [Vk(t, yk, j) − Vk(t, yk, i)] (6.10)

for k ∈ I(1, p) and i ∈ I(1, s). This implies that

D+Vk(t, yk, S) ≤ LkVk(t, yk, S) + MkV (t, yk, S), (6.11)

for (t, yk) ∈ J × C2[

H, Rnk]

, where Mk is similar to M in (4.2), and Lk isdefined by

LkVk(t, yk, i) =

∫

H

(

yTk Rk(i)Lc


)

dx

+

∫

H

(

(σk(x, i)yk)T Rk(x, i)σk(x, i)yk

)

dx. (6.12)

Now, we find an estimate on LkVk(t, yk, i). For this purpose, we use thedefinition of Lc

k(x, i) and conditions of Lemma 6.1. First we decompose thefirst term in (6.12) as follows

∫

H

(

yTk Rk(i)Lc


)

dx

=

∫

H

yTk Rk(i)Lc

k(x, i)ykdx +

∫

H

(Lck(x, i)yk)T Rk(i)yk dx. (6.13)

Because of integrating by parts, using the calculus of several variablesand Divergence Theorem [21], we note that

m∑

l,z=1

∫

H

[

yTk Rk ∂

∂xl

(

Dklz

∂

∂xz

yk

)

+∂

∂xl

(

Dklz

∂

∂xz

yk

)T

Rkyk

]

+

m∑

j=1

∫

H

[

yTk Rk ∂

∂xj

(Ckj yk) +

∂

∂xj

(Ckj yk)T Rkyk

]

dx

246 G. S. Ladde

= −m∑

l,z=1

∫

H

[

∂

∂xl

yTk Rk

(

Dklz

∂

∂xz

yk + Ckl yk

)

+

(

Dklz

∂

∂xz

yk + Ckl yk

)T

Rk ∂

∂xl

yk

]

+

∫

∂H

m∑

l,z=1

cos(ν, xl)

[

yTk RkDk

lz

∂

∂xz

yk +∂

∂xz

yTk Dk

lz Rk

]

yk

dx

+

∫

∂H

yTk

m∑

j=1

cos(ν, xj)[RkCk

j + CkTj Rk]yk

dx. (6.14)

From (3.8), (6.5), (6.13), (6.14), Lemma 6.1 and algebraic manipulations,(6.12) reduces to

LkVk(t, yk, i) = −m∑

l,z=1

∫

H

(

∂

∂xl

yk + Φkyk

)T

P klz

(

∂

∂xz

yk + Φkyk

)

dx

+1

2

m∑

l,z=1

∫

H

∂

∂xl

(yTk Φk1

lz yk) +

m∑

l,z=1

∂

∂xz

(yTk Φk2

lz yk)

dx

−∫

H

yk

m∑

l,z=1

1

2

[

∂

∂xl

Φk1lz +

∂

∂xz

Φk2lz

]

− Φk

m∑

l,z=1

P klzr

Φk

yk

−∫

H

yTk

[√RkGk

√Rk + Qk

]

yk dx

+

m∑

l,z=1

∫

∂H

m∑

l,z=1

cos(ν, xl)

[

yTk P k

lz

∂

∂xz

yk +∂

∂xz

yTk P k

lzyk

]

dx

+

∫

∂H

yTk 2

m∑

j=1

cos(ν, xj)Θkj yk

dx.

This together with Divergence Theorem [21], algebraic manipulations, bound-ary conditions, non-negative definite property of quadratic form∑m

l,z=1 ulPklzuz and (6.4), yields

LkVk(t, yk, i) ≤ −αk(i)Vk(t, yk, i), (6.15)

where αk(i) is the minimum of the smallest eigenvalue of Gk. This satisfies(4.6) with gi

k(t, uk) = −αk(i)uik, i ∈ I(1, s) for each k ∈ I(1, p). In this case,

for each k ∈ I(1, p), comparison function W in (4.7) is

Wk(t, uk) = gk(t, uk) + πuk = Gkuk, (6.16)


where Gk is s × s matrix defined by

gijk =

−

αk(i) +

s∑

j 6=i

πij

for j = i,

πij for j 6= i,

(6.17)

all for k ∈ I(1, p). It is clear that giik < 0. In the context of (6.16) and (6.17).

It is well-known [18] that the trivial solution of (4.7) is exponentially stableif Gk satisfies the following quasi-dominant diagonal condition:

αk = max

αk(i) +

s∑

j 6=i

πij − d−1j (k)

s∑

j=1

di(k)πij

> 0, all for k ∈ I(1, p).

(6.18)Relations (6.3), (6.11), (6.15), (6.16), and (6.18) fulfill all hypotheses of The-orems 5.1. Hence by the application of Theorem 5.1, the mean-square expo-nential stability of the trivial solution of the closed-loop isolated subsystems(3.7) follows, immediately.

Remark 6.2 From the above discussion, it is obvious that an existence ofsolution of boundary value problem (6.4) has played an important role. How-ever, this condition can be replaced by more feasible condition, namely, ma-trix differential inequality:

−1

2

m∑

l,z=1

[

∂

∂xl

Φk1lz +

∂

∂xz

Φk2lz

]

+Φk

m∑

l,z=1

P klz

Φk ≤ Qk, on H

m∑

l,z=1

1

2

[

cos(ν, xl)(Φk1lz + 2Θk

l )

+ cos(ν, xz)(Φk2lz + 2Θk

z)]

≤ −2∆k on ∂H

(6.19)

that is a solution of boundary value problem for first order nonlinear par-tial matrix differential inequalities. This generates an open mathematicalproblem.

Remark 6.3 In this work, Rk considered to be independent of x. Moregeneral case will appear elsewhere.

7 Large-Scale Systems

To conclude the connective stability of original distributed parameter stochas-tic control system (3.1) under Markovian structural perturbations, we choose

248 G. S. Ladde

the decentralized control laws (3.5) [20] that establish the mean-square ex-ponential stability of the corresponding isolated systems. In this section,we apply block vector Lyapunov functionals, block comparison theorem andblock system of differential equations to conclude the mean-square connec-tive stability of large-scale closed-loop distributed parameter systems (3.6)corresponding to (3.1) under structural uncertainties both random and deter-ministic parametric perturbations. By following the ideas of Siljak [19, 20],we choose a prototype form of (3.6) as

∂yk =

[

Lck(x, η)yk +

p∑

h=1

ekhAkh(x, η)yh

]

dt

+

[

σk(x, η)yk +

p∑

h=1

ekhΣkh(x, η)yh

]

dzk

By(t, x, ω) = 0 on Γt0 = [t0,∞) × ∂H

y(t0, x) = y0 on H ⊆ Rm, k ∈ I(1, p)

(7.1)

which can be rewritten as

∂y = [Lc(x, η)y + EA(x, η)y]dt + [σ(x, η)y + EΣ(x, η)y]dz

By(t, x, ω) = 0 on Γt0 = [t0,∞) × ∂H

y(t0, x) = y0 on H ⊆ Rm,

(7.2)

where (yT1 , yT

2 , . . . , yTp )T = y ∈ Rn; diag(Lc

k(x, η),Lc2(x, η), . . . ,Lc

p(x, η)) =Lc(x, η), diag(z1, z2, . . . , zp) = z and diag(σ1(x, η), σ2(x, η), . . . , σp(x, η)) =σ(x, η); A(x, η) = (Akh(x, η))p×p and Σ(x, η) = (Σkh(x, η))p×p are blockmatrices; E is interconnection matrix as defined before and its entries arebetween 0 and 1.

For System (7.2), we associate the Lyapunov functional V ∈ C[J ×C2[H, Rn] × S, RN

+ ] where V (t, y, η) = (V1(t, y1, η), V2(t, y2, η), . . . ,Vp(t, yp, η))T , where Vk(t, yk, η) is defined in (6.3).

To compute D+V (t, x, y, S), LV (t, x, y, S) and MV (t, x, y, S) with re-spect to (3.6), first we introduce a few notations to connect relationshipbetween operators computed in Section 6 with regard to isolated stochasticdistributed parameter system (3.7).

Let us denote linear operators D+, Lk and Mk defined in Section 6 byD+

(3.7), LDk and MD

k . The linear operators corresponding to k-th isolated sub-

systems (3.7) for k-th component of interconnected system (7.1) are denotedby D+

(7.2), LCk and MC

k , respectively. Without further details, by following

Ladde and Lawrence [12, 13], computations and representations of D+(7.2), LC

k


and MCk as follows

LCk Vk(t, y, i) = LD

k Vk(t, yk, i)

+

p∑

h=1

ekh

∫

H

(

yTk Rk(x, i)Akh(x, i)yh + yT

h ATkh(x, i)Rk(x, i)yk

)

dx

+

∫

H

[

Fk(x, y, i)T Rk(x, i)Fk(x, y, i) − (σk(x, i)yk)T Rk(x, i)σk(x, i)yk

]

dx

(7.3)

where

Λk(x, y, i) =

(

p∑

h=1

ekhΣkh(x, i)yh + σk(x, i)yk

)T

× Rk(x, i)

(

p∑

h=1

ekhΣkh(x, i)yh + σk(x, i)yk

)

(7.4)

and

MCk = MD

k = MkVk(t, yk, i) =

s∑

j 6=1

Vk(t, yk, i) −s∑

j 6=1

πijVk(t, yk, j). (7.5)

To find an estimate on LCk Vk(t, y, i), we assume that the deterministic

interconnection and stochastic interconnection rate functions in (7.3) satisfythe following conditions

∫

H

(

p∑

h=1

ekhyTk Rk(i)Akh(x, i)yh + yT

h ATkh(x, i)Rk(i)yk

)

dx

≤p∑

h=1

ekhγkh(i)Vh(t, yh, i) (7.6)

and∫

H

(

Λk(x, y, i)T Rk(x, i)Λk(x, y, i) − (σk(x, i)yk)T Rk(x, i)σk(x, i)yk

)

dx

≤p∑

h=1

ekhνkh(i)Vh(t, yh, i), (7.7)

where γkh(i) ≥ 0 and νkh(i) ≥ 0, k 6= h, k, h ∈ I(1, p), i ∈ I(1, s);E ≤ E. From (6.11), (6.18), (7.3), (7.5), (7.6) and (7.7), we conclude thatLC

k Vk(t, y, I) satisfies the (4.6) with

LV (t, y, i) ≤ gi(t, V (t, y, i)) for (t, y) ∈ J × C2[H, Rn]; (7.8)

250 G. S. Ladde

where gi(t, ui) = Giui − αk(i)Ip×pui for i ∈ I(1, s). In this case, comparison

system in (4.7) reduces to

u′ = W (t, u), u(t0) = r0, (7.9)

whereW (t, u) = Gbu + πbu, (7.10)

and Gb is s × s block diagonal matrix whose elements are p × p matrices,Gi = (gi

kh)p×p which is defined by

gikh =

−

αk(i) +

s∑

j 6=i

πij − ekk(νkk(i) + γkk(i))

, for h = k,

ekh (γkh(i) + νkh(i)) , for j 6= i,

all for k, h ∈ I(1, p) and i ∈ I(1, s), and πb is s × s block matrix whoseelements are p × p matrices, π

ijb that are defined by

πijb =

0 for i = j,

πijIp×p for j 6= i,

where πij ’s are as defined in Lemma 4.1.In order to conclude the exponential mean-square connective stability of

the trivial solution of (7.9), we assume that Gi satisfies the following quasi-dominant diagonal condition

µ(Gi) = max

gikk + d−1

k (i)

p∑

h 6=1

dh(i)|gikh|

≤ −α(i), (7.11)

all for k ∈ I(1, p) and i ∈ I(1, s), where

α(i) > 0, i ∈ I(1, s), (7.12)

and further assume that α(i) in (7.11) fulfills condition similar to (6.18)in the context of G which is defined similar to in Gk in (6.17) by simplyreplacing αk(i) by α(i). In the light of this, conditions (7.11), (7.12), and(6.18) imply that the trivial solution of block comparison system (7.9) isexponentially asymptotically stable. From this, (7.8) and the applicationof Theorems 4.1 and 5.1, we conclude that the trivial solution of (7.2) ismean-square exponentially connectively stable.

8 Chemical Reaction Systems

In this section we consider the chemical reaction

p−1∑

k=1

akYk = 0 (8.1)


where Yk denotes the chemical species and ak the corresponding stoichiomet-ric coefficient, positive products and negative for reactants. It is assumed thatthe controlled chemical reaction takes place in a tubular reactor with axialdiffusion. It is further assumed that a significant heat of reaction and therate can be expressed in Arrhenius form. The energy and material balancesare described by the following partial differential equations with boundaryand initial conditions:

∂

∂θCk = δk(η)

∂2

∂z2Ck − v

∂

∂zCk + bku + fk,

ρcP

∂

∂θT = κk(η)

∂2

∂z2T − ρvcP

∂T

∂x+ bpu

+(−∆Hk)fk +2h

a(∆Tad − T );

(8.2)

with boundary and initial conditions:

δk(η)∂

∂xCk = vk(Ck − Cfk), for z = 0 and θ > 0,

κk(η)∂

∂zT = ρvcP (T − ∆Tad), for z = 0 and θ > 0,

∂

∂zCk = 0, for z = l and θ > 0,

∂

∂zT = 0, for z = l and θ > 0,

Ck(0, z) = C0k(z), x ∈ [0, l],

T (0, z) = T0(z), z ∈ [0, l],

(8.3)

where k ∈ I(1, p), Ck stands for the concentration of k-th chemical speciesYk; T stands for temperature; let C0k be initial concentration of Ck and letT0 initial temperature of T ; Cfk denotes influent concentration (feed concen-tration in the mixture) of Yk; ∆Tad denotes influent temperature (adiabatictemperature rise/fall); variables θ and z are real time and spatial (axial)variables, respectively; l and a stand for the length and the radius of thereactor, respectively; δk(η) and κk(η) are effective mass diffusivity and ef-fective thermal conductivity, respectively; v is linear velocity of fluid in thereactor; ρ stands for the density of fluid; cP and h denote the specific heatand heat transfer coefficient, respectively; u is control variable; fk standsfor the reaction rate function of concentration, temperature, and the totalpressure, in general, and takes values in moles of Yk formed per unit volumeper unit of time; ∆Hk = ak∆Hk and ∆Hk denotes the heat of reaction permole Yk and ∆Hk is the heat reaction for the reaction, η is Markov chain asdefined before.

Let C∗k and T ∗

k denote the steady state of system (8.2) and (8.3). Byusing this, we define the dimensionless variables and constants. They are as

252 G. S. Ladde

follows:

yk = C−1fk (Ck − C∗

k), yp = ∆T−1ad (T − T ∗),

x =z

l, t = θl−2, f = a−1

k fk,

rk2 (η) = vl(δk(η))−1, rk

1 (η) = lρvcP (κk(η))−1,

m = 2hl(aρvcP )−1,

Bk = l2bk(Cfk)−1, γk(η) = κk(η)(ρcP )−1,

Bp = l2(ρcP ∆Tad)−1,

F k = −∆Hk(ρcP )−1l2f,

νk(η) = ρcP (−∆Hk)−1, k ∈ I(1, p − 1).

(8.4)

By using the above change of variables defined in (8.4), equations (8.2)and (8.3) can be rewritten as

∂

∂tyk = δk(η)

∂2

∂x2yk − δk(η)rk

2 (η)∂

∂xyk + Bku + νk(η)C−1

fk F k,

∂

∂typ = γk(η)

∂2

∂x2yp − κk(η)γk

1 (η)∂

∂xyp − myp + Bpu + F k∆T−1

ad ;

(8.5)

boundary and initial conditions:

δk(η)∂

∂xyk = δk(η)rk

2 (η)yk, for x = 0 and t > 0,

γk(η)∂

∂xyp = κk(η)rk

1 (η)yp, for x = 0 and θ > 0,

∂

∂xyk = 0, for x = 1 and t > 0,

∂

∂xyp = 0, for x = 1 and t > 0,

yk(0, x) = y0k(x), x ∈ [0, l],

yp(0, x) = y0p(x), x ∈ [0, l].

(8.6)

Here rk1 (η) and rk

2 (η) are Peclet number processes for heat and mass transfers,respectively; yk, yp t and x as defined in (8.4).

For the sake of simplicity, reaction terms in (8.5) assumed to be in itslinearized form:

∂

∂tyk = δk(η)

∂2

∂x2yk − δk(η)rk

2 (η)∂

∂xyk + Bku +

p∑

l=1

aklyl,

∂

∂typ = γp(η)

∂2

∂x2yp − κp(η)rk

1 (η)∂

∂xyp − myp + Bpu +

p∑

l=1

ypl,

(8.7)


where elements of akl of p × p matrix are defined by:

akl(η) =

νk(η)∂

∂Cl

F (C∗, T ∗) for k, l ∈ I(1, p − 1),

νk(η)C−1fk ∆Tad

∂

∂TF (C∗, T ∗), for l = p, k ∈ I(1, p − 1),

Cfk(∆Tad)−1 ∂

∂Cl

F (C∗, T ∗), for k = p, l ∈ I(1, p − 1),

∂

∂TF (C∗, T ∗), for l = k = p.

(8.8)

By using control law similar to (3.5), the closed loop interconnected sys-tem corresponding to (8.7) can be written as follows:

∂

∂tyk = Lc

k(x, η)yk +

p∑

l 6=k

akl(η)yl, k ∈ I(1, p) (8.9)

where

Lck(x, η) =

∂

∂x

(

Dk(η)∂

∂x

)

+ Ck(η)∂

∂x+ (Ak(η) − Bk(η)Kk(x, η)), (8.10)

for k ∈ I(1, p − 1),

Dk(η) =

δk(η), for k ∈ I(1, p − 1),

γp(η), for k = p;

Ak(η) =

akk, for k ∈ I(1, p− 1),

app − m, for k = p;

and

Ck(η) =

−δk(η)rk2 (η), for k ∈ I(1, p − 1),

−γp(η)rk1 (η), for k = p.

(8.11)

From (8.12) and (8.13), it is obvious that the hypotheses of Lemma 6.1 andconditions of Example 6.1:2.c) are satisfied.

We choose a norm-square-like Lyapunov functional similar to (6.3)

Vk(t, yk, η(t)) =

∫ 1

0

Rk(η(t))y2k dx (8.12)

where Rk(η(t)) is positive function. We note that Qk = −2(Υk +RkGk) andΥk = Rk(Ak − BkKk) = RkAck. Moreover, in addition to Rk, Dk, Ck andQk are constants. Independent variable x ∈ [0, 1]. In this case, boundaryvalue problem (6.9) or (6.19) is

− ∂

∂x

(

2Rk

(

DkΦk − 1

2Ck

))

+ Φk(2RkDkΦk) = Qk

(

2RkDkΦk +1

2Ck

)

(µ) = 0, for µ = 0, 1(8.13)

254 G. S. Ladde

which is equivalent to

d

dxΦk − (Φk)2 = −(Ack + Gk)(Dk)−1

Φk(0) ≥ Ck(2Dk)−1

Φk(1) ≤ −Ck(2Dk)−1.

(8.14)

Set Ψk = −(Ack +Gk)(Dk)−1 and define αk = |Ψk|. For example, a solutionΦk of (8.14) is given

((αk)−1Φk(x)) = tan(αkx + arctan((αk)−1Φk0)),

wheneverΨk > 0 and 0 < Φk

0 < −Ck(2Dk)−1,

and

(αk)−1Φk(x) =(βk coshx − αk sinhx)

(αk coshx − βk sinhx),

whenever

Ψk < 0 and 0 < αk < βk ≤ Φk0 < −Ck(2Dk)−1.

From this, the mean-square exponential stability of isolated tubular chem-ical reaction system corresponding to (8.9) follows by imitating the argumentand discussions with regard to (6.15) and (6.18).

To conclude the connective stability of original distributed parameterstochastic control chemical system (8.2) under Markovian structural pertur-bations, we follow the argument similar to the argument used in Section 7.We rewrite (8.9) as interconnected closed loop system.

∂

∂ty = Lc(x, η)y + EA(x, η)y, (8.15)

where

Lc(x, η) =∂

∂x

(

D(η)∂

∂x

)

+ C(η)∂

∂x+ (A(η) − B(η)K(x, η)), (8.16)

where (y1, y2, . . . , yp)T = y ∈ Rp; diag(Lc

1(x, η),Lc2(x, η), . . . ,Lc

p(x, η)) =Lc(x, η), A(x, η) = (akl(x, η))p×p with akk = 0 and E is interconnectionmatrix as defined before and its entries are between 0 and 1. By imitatingthe rest of the argument of Section 7, we can arrive at a comparison systemsimilar to (7.9) except some elements, namely, Gi of block matrix Gb in (7.10)are replaced by matrices Gi = (gi

k1)p×p are defined as:

gikl =

−

αk(i) +s∑

j 6=i

πij − ekkγkk(i)

for l = k,

ekl (γkl(i)) for l 6= k,


all for k, l ∈ I(1, p) and i ∈ I(1, s).In the context of these estimates, under condition (7.11), (7.12) and

(6.18), and by following the argument described in Section 7, one can con-clude that the trivial solution process of (8.15) is mean-square exponentiallystable.

9 Conclusions

In this paper, connective stability of stochastic distributed parameter large-scale control system under random structural perturbations in the frameworkof Lyapunov second method, comparison theorems, differential inequalities,and decomposition-aggregation technique [10, 18, 20] is initiated. Presentedresults generalize and extend the existing results [6, 7, 14, 15, 18, 20] in sys-tematic and unified way. The stability conditions (6.18), (7.11) and (7.12) arealgebraically simple and easy to verify. These conditions depend on coefficientmatrices, random environmental parameters σ, random structural perturba-tion parameters (elements of intensity matrix (πij)s×s), elements of intercon-nected matrix E = (ekh)p×p, parameters associated with feedback controllersKk, deterministic diffusion coefficients Dk

lz , and so on. This information pro-vides partial answers to several issues in the qualitative analysis of controlleddynamic systems, such as: “complexity versus stability,” “stochastic versusdeterministic,” and so on. In fact, conditions provide information for charac-terization of random excitations and their robustness. These statements canbe illustrated by playing games with presented conditions (6.18), (7.11), and(7.12). Moreover these conditions are systems of algebraic inequalities. Theyare insensitive to changes in both deterministic as well as stochastic param-eters as long as conditions remain valid. For more detailed analysis aboutthese various particular issues, we refer to [3, 4, 6, 7, 12, 13, 15, 18, 19, 20].

10 Acknowledgement and Dedication

This research work is dedicated to honor Professor Dragoslav D. Siljak’s 70thbirthday and also to acknowledge his research contributions.

References

[1] N. U. Ahmed, Stability of a Class of Stochastic Distributed Parameter Systems withRandom Boundary Conditions, J. of Math Anal. Appl. Vol. 91 (1983) 274–298.

[2] Near R. Amundson, Some Further Observations on Tubular Reactor Stability, Can.

J. Chem. Eng., Vol. 43 (1965) 49–55.

[3] M. J. Anabtawi and G. S. Ladde, Convergence and Stability Analysis of PartialDifferential Equations under Markovial Structural Perturbations-I, Stoch. Anal. and

Appl. Vol. 18 (2000) 493–524.

256 G. S. Ladde

[4] M. J. Anabtawi and G. S. Ladde, Convergence and Stability Analysis of PartialDifferential Equations under Markovial Structural Perturbations-II, Stoch. Anal.

and Appl. Vol. 18 (2000) 671–696.

[5] M. J. Anabtawi and G. S. Ladde, Block Systems of Parabolic Differential Inequalitiesand Comparison Theorems, Proc. of Dyn. Sys. and Appl., Dynamic Publishers, Inc.,Atlanta, GA, Vol. 3 (2001) 23–34.

[6] M. J. Anabtawi, G. S. Ladde, and S. Sathananthan, Convergence and Stability ofLarge-scale Parabolic Systems under Markovial Structural Perturbations-I, Int. J.

of Appl. Math. Vol. 2 (2000) 57–85.

[7] M. J. Anabtawi, G. S. Ladde, and S. Sathananthan, Convergence and Stability ofLarge-scale Parabolic Systems under Markovial Structural Perturbations-II, Int. J.

of Appl. Math. Vol. 2 (2000) 87–111.

[8] Stephen R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Bound-ary Value Problems, Academic Press, Inc., New York, 1974.

[9] G. S. Ladde, Qualitative Analysis of Systems of Partial Differential Equations underRandom Excitations, ABSTRACT: Proc. of Int. Conf. on Dynamics of Continuous,

Discrete and Impulsive Systems, London, Ontario, Canada (2001) 78.

[10] G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, AcademicPress, New York, 1980.

[11] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniquefor Nonlinear Differential Equations, Pitman Publishing, Inc., Marshfield, MA, 1985.

[12] G. S. Ladde and Bonita A. Lawrence, Stability and Convergence of Stochastic Ap-proximation Procedures under Markovial Structural Perturbations, Dyn. Sys. and

Appl. Vol. 10 (2001) 145–175.

[13] G. S. Ladde and Bonita A. Lawrence, Stability and Convergence of Large-scaleStochastic Approximation Procedures under Markovial Structural Perturbations,Proc. of Current Trends in Differential Equations and Dynamical Systems, to ap-pear (2003).

[14] G. S. Ladde and I-Tsung Li, Stability of Large-Scale Distributed Parameter Systems,Dyn. Sys. and Appl. Vol. 11 (2002) 311–324.

[15] G. S. Ladde and D. D. Siljak, Connective Stability of Large-Scale Stochastic Systems,Int. J. Systems Sci. Vol.6 (1975) 713–721.

[16] C. McGreavy and M. A. Soliman, Stability of Tubular Reactor, AIChE J. Vol. 19(1972) 174–176.

[17] Y. Nishimura and M. Matsubara, Stability Conditions for Class of Distributed Pa-rameter Systems and Their Applications to Chemical Reaction Systems, Chem. En-

gng. Sci. Vol. 24 (1969) 1424–1440.

[18] D. D. Siljak, Large-scale Dynamic Systems: Stability and Structure, North-Holland,New York, 1978.

[19] D. D. Siljak, Parameter Space Methods for Robust Control Design: A Guided Tour,IEEE Trans. on Auto. Cont. Vol. 34 (1989) 674–688.

[20] D. D. Siljak, Decentralized Control of Complex Systems, Academic Press, New York,1991.

[21] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge,1987.


CONVEXITY OF THE REACHABLE SET OFNONLINEAR SYSTEMS UNDER L2 BOUNDED

CONTROLS

B.T.Polyak1

1Institute for Control Science, Moscow 117997, Russia,

[email protected]

Abstract. Recently [7, 8] the new convexity principle has been validated. It states that

a nonlinear image of a small ball in a Hilbert space is convex, provided that the map is

C1,1 and the center of the ball is a regular point of the map. We demonstrate how the

result can be exploited to guarantee the convexity of the reachable set for some nonlinear

control systems with L2-bounded control. This provides existence and uniqueness of the

solution in some optimal control problems as well as necessary and sufficient optimality

conditions and effective iterative methods for finding the solution.

Keywords. Convexity, image, nonlinear transformation, reachable set, nonlinear control,

optimality conditions, optimal control.

AMS (MOS) subject classification: 49K15, 47H, 52A05.

1 Introduction

Convexity plays a key role in functional analysis, optimization and controltheory. For instance, if a mathematical programming problem is convex, thennecessary optimality conditions coincide with sufficient ones, duality theo-rems hold and effective numerical methods can be constructed [12]. However,convex problems are just a small island in the ocean of non convex ones.

In the present paper we describe the technique, which is useful for es-tablishing convexity in nonlinear control problems. It is based on the recentresult [7, 8], asserting convexity of a nonlinear image of a small ball in aHilbert space. This result is addressed in Section 2, while all other Sectionsdeal with its applications to nonlinear control systems. Namely, in Section3 we prove the convexity of the reachable set for nonlinear systems providedthat the linearized system is controllable while the control is bounded andsmall enough in L2 norm. For linear systems the explicit description of thereachable set with L2-bounded control goes back to Kalman [4]. Lee andMarkus [6] (see also [5]) proved that controllability of the linearized systemimplies local controllability of the nonlinear system. However we don’t know

258 B.Polyak

any result on the convexity of the reachable set for nonlinear case; some re-lated ideas can be found in the earlier paper of the author [9]. In Section 4 weapply the statements of the previous Section to get necessary and sufficientoptimality conditions in related optimal control problems and provide a sim-ple iterative algorithm to solve the problems. An example (a mathematicalpendulum) is considered in Section 5.

2 The convexity principle

Let X,Y be two Hilbert spaces, let f : X → Y be a nonlinear map withLipschitz derivative on a ball B(a, r) = x ∈ X : ||x − a|| ≤ r, thus

||f ′(x) − f ′(z)|| ≤ L||x− z|| ∀x, z ∈ B(a, r). (1)

Suppose that a is a regular point of f , i.e. the linear operator f ′(a) maps Xonto Y ; then there exists ν > 0 such that

||f ′(a)∗y|| ≥ ν||y|| ∀y ∈ Y. (2)

For instance, for X,Y finite dimensional: X = Rn, Y = Rm, this conditionholds if rankf ′(a) = m; for this case ν = σ1(f

′(a)) — the least singular valueof f ′(a).

Theorem 1 If (1), (2) hold and ε < minr, ν/(2L), then the image of a ballB(a, ε) = x ∈ X : ||x−a|| ≤ ε under the map f is convex, i.e. F = f(x) :x ∈ B(a, ε) is a convex set in Y . Moreover, the set is strictly convex andits boundary is generated by boundary points of the ball: ∂F ⊂ f(∂B(a, ε)).

The rigorous proof of Theorem 1 is involved and can be found in [7, 8]while its idea is very simple. The ball B(a, ε) is strongly convex, thus itsimage under the linear map f ′(a) is strongly convex as well. But it can notloose convexity for a nonlinear map f , which is close enough to its lineariza-tion. The same reasoning explains that the result can not be extended toan arbitrary Banach space, where a ball is not strongly convex. Howeverthe extension of the principle to uniformly convex Banach spaces (such asLp, 1 < p < ∞) is an open problem. On the other hand, the result remainstrue, if we replace the ball by any other strongly convex set (e.g. by a nonde-generate ellipsoid). For a particular case f : Rn → Rn Theorem 1 has beenextended in [1] for strictly convex (not necessarily strongly convex) sets. Notethat smoothness assumptions of Theorem 1 can not be seriously relaxed. Forinstance, A.Ioffe constructed a counterexample with f continuously differ-entiable but not in C1,1. Then the result is false. Theorem 1 relates to socalled mapping theorems, the recent survey of investigations in this field canbe found in [3]. The simplest mapping teorem by Graves [2] claims that aball can be inscribed in F , and its radius linearly depends on ε; it is true

Convexity of the Reachable Set 259

in Banach (not necesserily Hilbert) spaces and under weeker assumptions onsmoothness of f .

Now we describe how the boundary of the image set ∂F can be effectivelygenerated. Consider the optimization problem for c ∈ Y

min||x−a||≤ε

(c, f(x)) = minf∈F

(c, f). (3)

If F is strictly convex, then the solution of this problem for arbitraryc 6= 0 is a boundary point of F and vice verse, any boundary point of F is asolution of (3) for some c 6= 0. The necessary and sufficient condition for apoint x∗ to be a solution of (3) is

x∗ = a− εf ′(x∗)∗c

||f ′(x∗)∗c|| . (4)

A simple iterative method to solve equation (4) reads:

xk+1 = a− εf ′(xk)∗c

||f ′(xk)∗c|| . (5)

We summarize the technique for constructing boundary points of F as follows.

Theorem 2 Under assumptions of Theorem 1 for any c 6= 0a)The solution x∗ of (3) exists and is unique, it belongs to ∂F , ||x∗−a|| =

ε and the necessary and sufficient optimality condition (4) holds. Moreoverthe set of solutions of (3) for all c ∈ Y, ||c|| = 1 coincides with ∂F .

b)Method (5) converges with linear rate of convergence for any x0 ∈B(a, ε):

||xk − x∗|| ≤ qk||x0 − x∗||, q = O(ε) =εL

ν − εL< 1. (6)

The proof follows immediately from the strict convexity of F provided byTheorem 1 and necessary and sufficient optimality condition for minimizationof a differentiable function with nonvanishing gradient on a ball; see detailsin [7, 8].

Another characterization of the boundary points of the set F (with ε, notprescribed in advance), can be obtained by the following result.

Theorem 3 If (1), (2) hold and d ∈ Y, ||d− f(a)|| is small enough thena) d belongs to the boundary of F with ε = minf(x)=d ||x− a||.b) The solution x∗ of the optimization problem

minf(x)=d

||x− a|| (7)

exists, is unique and satisfies necessary and sufficient optimality condition:there exists y ∈ Y such that

x∗ = a+ f ′(x∗)∗y (8)

260 B.Polyak

Say it another way, find

g(y) = minf(x)=y

||x− a|| (9)

then F is the level set of this function F = y : g(y) ≤ ε while ∂F = y :g(y) = ε.

Note that in contrast with Theorem 1 there is no explicit bound for ”closeenough” distance in Theorem 3.

In many cases the conditions of Theorem 1 can be effectively checked,and the radius ε of the ball can be estimated. One of such examples is aquadratic transformation.

Example. Let x ∈ Rn and f(x) = (f1(x), ...fm(x))T where fi(x) arequadratic functions:

fi(x) = (1/2)(Aix, x) + (ai, x), Ai = ATi ∈ Rn×n, ai ∈ Rn, i = 1, ...m.

(10)Take a = 0, that is B = x : ||x|| ≤ ε. Then f ′

i(x) = Aix + ai and (1)is satisfied on Rn with L = (

∑mi=1 ||Ai||2)1/2 where ||Ai|| stands for the

spectral norm of matrices Ai. Consider the matrix A with columns ai: A =(a1|a2|...|am). Then f ′(0)T y = Ay, and if rankA = m, then (2) holds withν = σ1(A) — the minimal singular value of A, that is ν = (min λ1(A

TA))1/2,where λ1 is the minimal eigenvalue of the corresponding matrix. Hence,Theorem 1 implies:

Proposition 1 If ε < ν/(2L), then the image of the ball B under the map(10) is convex:

F = f(x) : ||x|| ≤ εis a convex set in Rm.

This is in a sharp contrast with the results on images of arbitrary ballsunder quadratic transformations, where the convexity can be validated [10]just under very restrictive assumptions.

For instance, let n = m = 2 and

f1(x) = x1x2 − x1, f2(x) = x1x2 + x2. (11)

Then the estimates above guarantee that F is convex for ε < ε∗ = 1/(2√

2) ≈0.3536. It can be proved directly for this case that F is convex for ε ≤ ε∗

and looses convexity for ε > ε∗. Thus the estimate provided by Proposition1 is tight for this example.

Figure 1 shows the images of the ε-discs x ∈ R2 : ||x|| ≤ ε under themapping (11) for various values of ε.

Other examples, related to optimization problems and to linear algebracan be found in [7, 8, 11]. In what follows, we focus on control applicationsof the convexity principle.


−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ε=0.3

ε=0.2

ε=0.4

ε=0.5

ε=0.6

Figure 1: Image of a quadratic map

3 Convexity of the reachable set

A general nonlinear control system

x = φ(x, u, t), x ∈ Rn, u ∈ Rm, 0 ≤ t ≤ T, x(0) = b (12)

with L2-bounded control

u ∈ U = u : ||u||2 =

∫ T

0

|u(t)|2dt ≤ ε2 (13)

defines a reachable set

F = x(T ) : x(t) is a solution of (12), u ∈ U. (14)

¿From here on we denote finite dimensional Euclidean norms of vectors andmatrices as |.| (e.g.|u(t)|) while L2 norm is denoted ||u||. Our aim is to provethat under some assumptions the reachable set is convex for ε small enough.We do this relying on Theorem 1 with X = L2, Y = Rn, a = 0, f : u→ x(T ).Thus we should check conditions of the Theorem for this situation. Considerthe linearized system

z = φx(x0, 0, t)z + φu(x0, 0, t)u, y(0) = 0 (15)

262 B.Polyak

here x0 is the nominal trajectory, corresponding to nominal control u0 = 0:

x0 = φ(x0, 0, t), x0(0) = b.

Now we formulate some technical assumptions to verify conditions of Theo-rem 1.

A 1 The function φ(x, u, t) is continuous in x, u, measurable in t and satis-fies the growth condition for all x, u, 0 ≤ t ≤ T :

|φ(x, u, t)| ≤ k1 + k2|x| + k3|u|2.

In what follows all constants will be denoted ki. The above growth con-dition is restrictive (for instance, a term like xu is not allowed in φ). Underabove assumption we can guarantee the boundedness of solutions of (12).

Lemma 1 If A1 holds, then solutions of (12) are bounded:

|x(t)| ≤ k0, 0 ≤ t ≤ T

for all u ∈ U .

Proof We have from (12)

|x(t)| = |b+

∫ t

0

φ(x(τ), u(τ), τ)dτ | ≤ |b| + k1T + k3ε2 + k2

∫ t

0

|x(τ)|dτ |.

Applying Gronwall lemma, we get the desired inequality.Introduce the new growth condition:

A 2 For all |x| ≤ k0, |x+ x| ≤ k0, u, u+ u, 0 ≤ t ≤ T

|φ(x, u, t) − φ(x + x, u+ u, t)| ≤ k4|x| + k5|u| + k6|u2|.

Then we can estimate the distance between two solutions of the equation(12).

Lemma 2 If A1, A2 hold, then for two solutions x(t), x(t) + x(t) of (12)corresponding to two controls u(t), u(t) + u(t), u ∈ U, u + u ∈ U, u ∈ U wehave

|x(t)| ≤ k7||u||, 0 ≤ t ≤ T.

Proof Indeed

˙x = φ(x+ x, u+ u, t) − φ(x, u, t), x(0) = 0.

Integrating and applying A2 we obtain

|x(t)| ≤ k4

∫ t

0

|x(τ)|dτ+k5||u||t1/2+k6||u||2 ≤ k4

∫ t

0

|x(τ)|dτ+(k5T1/2+k6ε)||u||.

Now Gronwall lemma implies the assertion of the Lemma.Under more tight conditions we can estimate how close are the solutions

of the perturbed system and the linearized system.


A 3 Function φ(x, u, t) is differentiable in x, u and for all |x| ≤ k0, |x+ x| ≤k0, u, u+ u, 0 ≤ t ≤ T

|φx(x, u, t)| ≤ k8 (16)

|φ(x, u, t) − φ(x+ x, u+ u, t) − φx(x, u, t)x− φu(x, u, t)u| ≤ k9|x|2 + k10|u2|.

Consider the linearized system

z = φx(x, u, t)z + φu(x, u, t)u, z(0) = 0 (17)

where x(t) is a solution of (12) for a control u ∈ U while u(t) is a perturbationof control. Linearized system (15) is a particular case of (17) for nominalcontrol u = 0.

Lemma 3 If x(t), x(t) + x(t) are two solutions of (12) corresponding to twocontrols u(t), u(t) + u(t), u ∈ U, u+ u ∈ U, u ∈ U while z(t) is the solution ofthe linearized system (17), then under A1- A3 the following estimate holds

|x(t) − z(t)| ≤ k11||u||2, 0 ≤ t ≤ T.

This result ensures the differentiabilty of the map f : u → x(T ). Indeed,consider z(T ) - the solution of (17), then z(T ) is the linear map of u : z(T ) =Lu. On the other hand the result of Lemma 3 can be rewritten (for t = T )as

|f(u+ u) − f(u) − Lu| ≤ ||u||2.

This means that f(u) is differentiable and f ′(u)u = z(T ).The next asumption guarantees that f ∈ C1,1.

A 4 For all |x| ≤ k0, u, 0 ≤ t ≤ T partial derivatives φx(x, u, t), φu(x, u, t)|are Lipschitz in x, u.

Lemma 4 Under A1 - A4 the map f : u → x(T ) is differentiable withLipshitz derivative on U .

We skip the proof, it follows the same lines as before.The last assumption implies the regularity of of the map f ; it states that

f ′(0)u = z(T ) coincides with the whole space Rn.

A 5 The linearized system (15) is controllable.

Combining all the above lemmas, we see that the assumptions of Theorem1 hold true for our map, thus we validated the main result of the paper:

Theorem 4 If A1 -A5 are satisfied and ε > 0 is small enough, then thereachable set (14) is convex.

264 B.Polyak

Let us discuss the result. First, we do not provide the explicit boundfor ε > 0 which guarantees convexity. In principle it can be done: we canestimate all constants introduced above (for instance, in Lemma 1 we getk0 ≤ (|b| + k1T + k3ε

2) exp(k2T ), thus we obtain L in Theorem 1, and wecan estimate ν via controllability Grammian for equation (15) (see analysisof the linear case below). However such bound would be conservative andof little value. Second, most of the assumptions A1 – A4 are just naturalsmoothness and growth conditions and they are not very restrictive. Themain exception is the boundedness condition |φx(x, u, t)| ≤ k. It does notpermit to have terms like h(x)u in φ(x, u, t). However, terms like h(x)g(u)with g(u) bounded or a(t)u+ b(t)u2 are suitable.

The general description of the reachable set for linear systems has beenprovided by Kalman and coauthors [4] - it is an ellipsoid even without con-trollability assumption. Of course this result is obvious - the reachable setis a linear finite-dimensional image of L2 ball and hence is an ellipsoid. Theexplicit description of this ellipsoid for controllable case, given in [4], writesas follows. Let

x = A(t)x +B(t)u, x(0) = 0 (18)

and

F = x(T ) : x(t) is a solution of (18),

∫ T

0

|u(t)|2dt ≤ ε2. (19)

Then

F = x : xTW (T )−1x ≤ ε2 (20)

where W (T ) is the controllability Grammian:

W (T ) =

∫ T

0

Φ(t)B(t)BT (t)ΦT (t)dt (21)

which is assumed to be nondegenerate, while Φ(t) is the fundamental matrixof (18).

4 Optimal control

Relying on the convexity result, we are able to analyse some optimal con-trol problems. Consider the terminal optimization problem for system (12)subject to constraints (13):

min(c, x(T )) (22)

x = φ(x, u, t), x ∈ Rn, u ∈ Rm, 0 ≤ t ≤ T, x(0) = b∫ T

0

|u(t)|2dt ≤ ε2,


where c ∈ Rn 6= 0. Then this optimal control problem is equivalent tofinite-dimensional one:

minx∈F

(c, x)

which is convex under assumptions of Theorem 4. This leads to the followingresult.

Theorem 5 If A1- A5 are satisfied and ε > 0 is small enough, then:a) The solution x∗, u∗ of the problem (22) exists and is unique, ||u∗|| = ε;b) It satisfies the necessary and sufficient optimality condition

u∗(t) =−εφT

u (x∗, u∗, t)ψ(t)∫ T

0 |φTu (x∗, u∗, τ)ψ(τ)|2dτ

,

where ψ(t) is the solution of the adjoint system:

ψ = −φTx (x∗, u∗, t)ψ(t), ψ(T ) = −c;

c) The iterative method

xk = φ(xk, uk, t), xk(0) = b (23)

ψk

= −φTx (xk, uk, t)ψk(t), ψk(T ) = −c

uk+1(t) =−εφT

u (xk, uk, t)ψk(t)∫ T

0|φT

u (xk, uk, τ)ψk(τ)|2dτ,

converges to the solution with linear rate: ||uk − u∗|| ≤ Cqk, q < 1.

Proof It is a direct corollary of Theorem 2 and Theorem 4.This result can be easily extended to more general optimization problem

where the linear function (c, x(T )) is replaced with nonlinear Lipschitz differ-entiable function h(x(T )) with the only assumption h′(x0(T )) 6= 0. Theorem4 remains true with c replaced by h′(x∗(T )) in optimality conditions and byh′(xk(T )) in the iterative method.

On the other hand, instead of (22) we can consider another optimal controlproblem with fixed terminal point and quadratic performance:

min

∫ T

0

|u(t)|2dt (24)

x = φ(x, u, t), 0 ≤ t ≤ T, x(0) = b, x(T ) = d.

This can be covered by Theorem 3 which immediately implies the followingresult.

Theorem 6 If A1- A5 are satisfied and ||d− x0(T )|| is small enough, then:a) The solution x∗, u∗ of the problem (24) exists and is unique,

266 B.Polyak

b) It satisfies the necessary and sufficient optimality condition: there existy ∈ Rn such that

u∗(t) = φTu (x∗, u∗, t)ψ(t)

where ψ(t) is the solution of the adjoint system:

ψ = −φTx (x∗, u∗, t)ψ(t), ψ(T ) = y.

¿From the above optimality conditions we can conclude that if we assumeadditionally that

|φu(x, u, t)| ≤ k

for all all bounded x, all u and all 0 ≤ t ≤ T then optimal controls inboth problems (22), (24) are continuous and bounded. This means thatthe reachable set in (22) is the same as in similar problem with an extraconstraint |u(t)| ≤ M, 0 ≤ t ≤ T for M large enough. This allows to relaxthe most restrictive condition (16). Indeed it suffices to assume that

|φx(x, u, t)| ≤ k8 (25)

for all bounded x, u. Note that A1 must be satisfied for all x, u. Then, forinstance, affine in u equation

φ(x, u, t) = h0(x)+

m∑

i=0

uihi(x), |h0(x)| ≤ k12+k13|x|, |hi(x)| ≤ k14+k15

√

|x|

(26)is acceptable.

5 Example - a pendulum

Consider a pendulum forced by small energy control:

x1 = x2, x1(0) = 0 (27)

x2 = −ω2 sinx1 + u, x2(0) = 0∫ T

0

|u(t)|2dt ≤ ε2.

For simplicity of calculations we accept zero initial conditions, thus nominalsolution is u0(t) = 0, x0(t) = 0. Our aim is to describe the reachable set forthis problem. First of all we can easily check that growth and smoothnessassumptions A1-A4 are satisfied. The linearized system along the nominaltrajectory

x1 = x2, x1(0) = 0 (28)

x2 = −ω2x1 + u, x2(0) = 0


is obviously controllable. Thus we conclude that for ε small enough thereachable set of the system (27) is convex.

Before constructing F itself, we describe the reachable ellipsoid for thelinearized system (28) in accordance with (20). In our case

A =

(

0 1−ω2 0

)

, B =

(

01

)

, eAt =

(

cosωt (1/ω) sinωt−ω sinωt cosωt

)

,

and calculation of the Grammian provides

W (T ) =

(

T2ω2 − sin 2ωT

4ω3

1−cos 2ωT4ω2

1−cos 2ωT4ω2

T2 + sin 2ωT

4ω

)

.

If we choose T = 2π/ω, then W (T ) becomes diagonal and the reachable setis the ellipsoid

F = x : 4ω3x21 + 4ωx2

2 ≤ πε2.For ω = 1 this is a disc.

Now, to draw F for the nonlinear pendulum (27) we can apply algorithm(23) for various c ∈ R2, |c| = 1. Thus c is one parametric family, for instancewe can take c = (sin θ, cos θ)T , 0 ≤ θ ≤ 2π. However, on this way we canconstruct F only if it is convex. To handle the case of large ε, when Flooses the convexity, another technique was applied. Concider optimizationproblem of the type (24):

min

∫ T

0

|u(t)|2dt, (29)

x1 = x2, x1(0) = 0, x1(T ) = d1,

x2 = −ω2 sinx1 + u, x2(0) = 0, x2(T ) = d2.

Its solution satisfies optimality condition

u(t) = ψ2(t)

where ψ is the solution of the adjoint system:

ψ1 = ω2 cos(x1)ψ2, (30)

ψ2 = −ψ1.

Thus to construct the reachable set for (27) we solve initial point problemfor ODE

x1 = x2, x1(0) = 0, (31)

x2 = −ω2 sinx1 + ψ2, x2(0) = 0,

ψ1 = ω2 cos(x1)ψ2, ψ1(0) = a1,

ψ2 = −ψ1, ψ2(0) = a2

268 B.Polyak

with some initial conditions a and calculate x(T ) and v =∫ T

0|ψ2(t)|2. By

gridding a we calculate the function of two variables v(x(T )) on some grid.The level sets of this function are exactly the boundary of F , correspondingto v = ε2 (see Theorems 3, 6). The results of calculations are presented onFigure 2 for ε2 = 0.6 : 0.1 : 1.0. It can be seen as F looses the convexitywhen ε increases.

−3 −2 −1 0 1 2 3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

X1

X2

0.6

1.0

Figure 2: Reachable set for the pendulum

6 Conclusions

The new “image convexity” principle is a promising tool for analysis of variouscontrol problems. It validates the convexity of the reachable set for systemswith L2-bounded controls, guarantees uniqueness of the solution of energy-optimal controls, provide necessary and sufficient conditions and efficientnumerical methods for these controls. Similar technique can be applied tosome other optimal control problems with more general performance indexor with discrete time.


7 Acknowledgement

The paper has been prepared during stimulating visit of the author to Insti-tute Mittag-Leffler.

References

[1] S.V.Emelyanov, S.K.Korovin, N.A.Bobylev, On convexity of images of convex setsunder smooth transformations, Doklady RAN, 385, No 3, (2002), 302–304.

[2] L.M.Graves, Some mapping theorems, Duke Math, J., 17, (1950), 111–114.

[3] A.Ioffe, Metric regularity and subdifferential calculus, Russ. Math. Surveys, 55, No3, (2000), 501–556.

[4] R.E.Kalman, Y.C.Ho, K.S.Narendra, Controllability of linear dynamical systems,Contributions to Differential Equations, 1, No 2, (1963), 189–213.

[5] E.B.Lee, L.Markus, Foundations of Optimal Control Theory, John Wiley, New York,1970.

[6] L.Markus, E.B.Lee, On the existence of optimal controls, J. Basic Engr. (Trans.ASME), 84D, (1962), 1–20.

[7] B.Polyak, Convexity of nonlinear image of a small ball with applications to optimiza-tion, Set-Valued Analysis, 9, No 1/2, (2001), 159–168.

[8] B.Polyak, Local programming, Comp. Math. and Math. Phys., 41, No. 9, (2001),1259–1266.

[9] B.T.Polyak, On the theory of nonlinear control problems, Vestnik Mosc. Univ., Ser.Math., Mech., No 2, (1968), 30–40 (in Russian).

[10] B.T.Polyak, Convexity of quadratic transformations and its use in control and opti-mization, Journ. Optim. Th. and Appl., 99, No 3, (1998), 553–583.

[11] B.T.Polyak, The convexity principle and its applications, Bull. Brazilian Math. Soc.,34, No 1, (2003), 1–17.

[12] R.T.Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.


ON THE USE OF PIECEWISE LINEAR M- ANDM0-FUNCTIONS FOR STABILITY ANALYSIS OF

NONLINEAR COMPOSITE HEREDITARYSYSTEMS

Yuzo Ohta1, Kohkichi Tsuji2, and Tadashi Matsumoto3

1Department of Computer and Systems Engineering, Kobe University,Rokkodai, Nada, Kobe 657-8501, Japan

2Department of Information Systems, Aichi Prefectual UniversityNagakute-cho, Aichi-gun, Aichi, 480-1198, Japan

3Department of Electrical and Electronics Engineering, Fukui University,

Bunkyo, Fukui 910-8507, Japan

Abstract. In this paper, we study Lyapunov stability of nonlinear hereditary compos-

ite systems using vector Lyapunov functions method with nonlinear comparison systems,

and reduce conservatism of stability condition in previous papers. Moreover, we consider

piecewise linear test functions, and derive stability conditions which are easy to examine.

Keywords. Vector Lyapunov method, M- and M0-function, M-matrix, Piecewise linear

systems, Quasi-monotone systems, Composite systems, Hereditary systems

AMS (MOS) subject classification:

1 Introduction

Since the first paper by Bailey was published in 1966 [5], composite-systemmethod to analyze stability of large-scale systems was extensively investi-gated. Excellent surveys of work in this area was presented by Michel andMiller [11], Siljak [21], Araki [4], Vidyasagar [23], and Michel [10]. It wassaid that stability theory for composite systems reached a reasonably com-plete and mature level [10]. We agree this as long as we use linear comparisonsystems; however, we would like to point out that more efforts are neededwhen we use nonlinear comparison systems to reduce conservatism.

In this paper, we study Lyapunov stability of nonlinear hereditary compositesystems using vector Lyapunov functions method with nonlinear compari-son systems. Ladde [7], Amemiya [2], Ohta [14], and Ohta, Douseki andMatsumoto [15] tried to reduce conservatism by arrowing the nonlinearity inthe test function which determine the behavior of the comparison system.However, unfortunately, the reported results are not easy to check stabilityconditions because of the nonlinearity of the test function.

272 Y. Ohta, K. Tsuji and T. Matsumoto

In this paper, we derive less conservative stability results. Moreover, we con-sider the case when the test function is a piecewise linear function and deriveconditions which are easy to examine by introducing the notion of piecewiselinear M0-function.

2 Preliminaries

2.1 Notation and Definitions

Throughout the paper, Rn denotes the set of all real n-vectors v with compo-nents v1, v2, · · · , vn, Rn×m denotes the set of all real n×m matrices A withcomponents aij ’s. For v ∈ Rn and A ∈ Rn×m, vT ∈ R1×n and AT ∈ Rm×n

are the transposes of v and A, respectively. Fn denotes the set of all real n-vector valued piecewise continuous functions x with components x1, x2, · · · ,xn, which are defined on some interval I . For a given τ > 0, Cn denotes theset of all continuous functions ψ ∈ Fn defined on the interval I = [−τ, 0].For any continuous function x ∈ Fn defined on [−τ, T ), T > 0, and for anyfixed t ∈ [0, T ), xt denotes the restriction of x to the interval [t − τ, t],i.e., xt is an element of Cn defined by xt(s) = x(t + s) for s ∈ [−τ, 0]. Forany positive integer n, n stands for the index set 1, 2, · · · , n. ei ∈ Rn,i ∈ n, denotes the ith unit basis vector; while e ∈ Rn denotes the vectorwhose component ei = 1 for all i. 0 denotes the zero vector in Rn or the zeromatrix in Rn×m. We sometimes use a vector v ∈ Rn to denote an element xin Fn such that x(t) = v for all t ∈ I . For u, v ∈ Rn, u > v (u >= v) meansui > vi (ui >= vi) for all i ∈ n; similarly, for A = (aij), B = (bij) ∈ Rn×m,A > B (A >= B) means aij > bij (aij >= bij) for all i ∈ n and j ∈ m. Forx, y ∈ Fn, x > y (x >= y) means x(t) > y(t) (x(t) >= y(t)) for all t ∈ I . We useRn

+, Fn+ and Cn+ to denote the sets u ∈ Rn : u >= 0, x ∈ Fn : x >= 0, and

ψ ∈ Cn : ψ >= 0, respectively. For u, v ∈ Rn such that v <= u, [v, u] denotesthe set w ∈ Rn : v <= w <= u. A rectangle Q in Rn is the Cartesian prod-uct of n intervals on the real line, each of which may be either open, closed, orsemi-open; in particular, any of these intervals may be unbounded, and thusa rectangle may be all of Rn. For u ∈ Rn, |u| denotes a norm of u, while, forthe index set J = j1, j2, · · · , jm, m = |J | denotes the cardinality of J .For ψ ∈ Cn, ‖ψ‖ = sup |ψ(t)| : t ∈ [−τ, 0]. Let I = i1, i2, · · · , ik ⊆ n,J = j1, j2, · · · , j` ⊆ m. For u ∈ Rn, uI = [ui1 ui2 · · · uik ]T ∈ Rk; forA = (aij) ∈ Rn×m, AIJ denotes the k × ` matrix whose (p, q)-th elementis aip,jq . K (or K) denotes the set of all functions α : [0,∞) → R suchthat α(0) = 0 and increasing (or non-decreasing); K∞ denotes the set of allfunctions α ∈ K such that lim

s→∞α(s) = ∞.

Definition 1

(a) A matrix A = (aij) ∈ Rn×n belongs to the class Z if aij <= 0 for alli 6= j.(b) A matrix A ∈ Rn×n belongs to the class P (or P0) if its principal minors

On the Use of M and M0-Functions 273

are all positive (or non-negative).(c) A matrix A ∈ Rn×n is an M -matrix (or M0-matrix) if A ∈ (Z ∩ P ) (orA ∈ (Z ∩ P0)).(d) A matrix A is reducible if there exists a set of indices I ⊆ n such thatAIJ = 0 where J = n− I; A matrix A is irreducible if it is not reducible.

Definition 2

(a) A mapping f : D ⊆ Rn → Rn is isotone (or antitone) on D if v >= w,v, w ∈ D, implies that f(v) >= f(w) (or f(v) <= f(w)).(b) A mapping f : D ⊆ Rn → Rn is inverse isotone on D if f(v) >= f(w),v, w ∈ D, implies that v >= w.(c) A mapping f : D ⊆ Rn → Rn is off-diagonally isotone (or off-diagonallyantitone) on D if, for any i ∈ n, fi(v) >= fi(w) (or fi(v) <= fi(w)) holdswhenever v, w ∈ D, v >= w, vi = wi.(d) A mapping F : Cn → Rn is quasi-monotone if, for any i ∈ n, Fi(ψ) >=Fi(ψ) holds whenever ψ, ψ ∈ Cn, ψ >= ψ, ψi(0) = ψi(0).(e) A mapping f : D ⊆ Rn → Rn is an M -function on D if it is off-diagonally antitone and inverse isotone on D.

Definition 3 A mapping f : D ⊆ Rn → Rn is a P -function (or P0-function) on D if for any v, w ∈ D, v 6= w, there is an index k = k(v, w) ∈ nsuch that (vk −wk)[fk(v)− fk(w)] > 0 (or (vk −wk)[fk(v)− fk(w)] >= 0 andvk 6= wk) .

The following has been shown (Theorem 4.4 of [12]).

Lemma 1 The mapping f : Q ⊆ Rn → Rn on the rectangle Q is an M -function on Q if and only if f is an off-diagonally antitone P -function onQ.

In view of Lemma 1 and Definition 3, we define the following:

Definition 4 A mapping f : D ⊆ Rn → Rn is an M0-function if it is anoff-diagonally antitone P0-function on D.

2.2 System Description and Basic Results

Let us consider a nonlinear hereditary composite system S given by

S : ˙x(t) = G(xt, t) +H(xt, t), (1)

where x = [xT1 xT

2 · · · xTn ]T ∈ Fd, xi ∈ Fdi , d =

∑ni=1 di, ˙x(t) denotes the

right-hand derivative of x(·) at t,

G(xt, t) = [G1([x1]t, t) G2([x2]t, t) · · · Gn([xn]t, t)]T,

H(xt, t) = [H1(xt, t) H2(xt, t) · · · Hn(xt, t)]T,

Gi : Cdi ×R → Rdi and Hi : Cd ×R → Rdi are continuous.


The i-th isolated subsystem Si, i ∈ n, is given by

Si : ˙xi(t) = Gi([xi]t, t), i ∈ n (2)

For our discussion to have meaning, we assume the existence and uniquenessof the solution x = x(t0, φ) of (1) for arbitrary initial condition φ ∈ Cd att = t0. We denote the value of x at t >= t0 − τ by x(t) = x(t; t0, φ).

We assume that

Gi(0, t) = 0, Hi(0, t) = 0 ∀ i ∈ n, ∀ t ∈ R (3)

and consider Lyapunov stability of the zero solution of S.

To analyze stability, we will use vector Lyapunov functions method. As usual,we assume the following:(H.1). For each i ∈ n, there exist a locally Lipschitz continuous functionVi : Rdi → R+, functions αi, βi ∈ K, locally Lipschitz continuous functionsgi, gi ∈ K, and a locally Lipschitz continuous isotone function hi : Rn

+ → R+

satisfying the following properties:

αi(|wi|) <= Vi(wi) <= βi(|wi|) ∀ wi ∈ Rdi , (4)

D+(2)Vi[φi(0)] = lim

∆↓0

Vi[φi(0) + ∆Gi(φi, t)] − Vi[φi(0)]

∆

<= −gi(Vi[φi(0)]) + gi(‖νi‖) ∀ φi ∈ Cdi , (5)

|Vi[wi + ∆Hi(φ1, φ2, · · · , φn, t)] − Vi(wi)|

<= ∆hi(‖ν1‖, ‖ν2‖, · · · , ‖νn‖) + o(∆)

∀ wi ∈ Rdi , ∀ φj ∈ Cdj , j ∈ n, (6)

hi(0,0, · · · , 0) = 0, (7)

where νi ∈ C1 and νi(s) = Vi[φi(s)].Given any φ ∈ Cd, and let x = x(t0, φ) be the solution of (1) with initial

condition φ at t = t0. Let vi(t) = Vi(xi(t)) = Vi([xi]t(0)). Since Vi(·) islocally Lipschitz continuous, we have

lim∆↓0

vi(t+ ∆) − vi(t)

∆= lim

∆↓0

Vi[xi(t) + ∆Gi([xi]t, t) +Hi(xt, t)] − Vi[xi(t)]

∆.

Let v(t) = [v1(t) v2(t) · · · vn(t)]T. Then, by (5) and (6), we have thefollowing comparison inequality

lim∆↓0

v(t+ ∆) − v(t)

∆<= F (vt) (8)

where F = [F1 F2 · · · Fn]T and Fi : Cn+ → R is defined by

Fi(ψ) = −gi[ψi(0)] + gi(‖ψi‖) + hi(‖ψ1‖, ‖ψ2‖, · · · , ‖ψn‖). (9)


From (H.1), F (0) = 0 and F is quasi-monotone and locally Lipschitz contin-uous. We further introduce the test function f = [f1(w) f2(w) · · · fn(w)]T,where fi : Rn

+ → R is given by

fi(w) = −gi(wi) + gi(wi) + hi(w1, w2, · · · , wn), (10)

which is off-diagonally isotone and f(0) = 0. We note that f is the restrictionof the functional F given by (9) to the space of constant functions. We alsonote that F is defined if f is defined because of (10) and (9).

We show the basic stability results.

Theorem 1 In addition to (H.1), we assume the following:

(H.2). ∃ q ∈ Rn, q > 0 : ∀ w ∈ [0, q], w 6= 0, ∃ i ∈ n : fi(w) < fi(0) = 0,

and(H.3). ∀ µ ∈ (0, 1], ∃ q ∈ Rn : 0 < q <= µq, f(q) < f(0) = 0.

Then, the zero solution of S is uniform-asymptotically stable.

The condition (H.2) is the condition for boundness and convergence of solu-tions. The condition (H.3) is the additional condition to show the stability ofsolutions. We note here that it can be shown that (H.3) holds if (H.2) holds.

Lemma 2 Assume that f : [a, q] ⊆ Rn+ → Rn, q > a, is off-diagonally

isotone. If for any w ∈ [a, q], w 6= a, there exists an integer i ∈ n such thatfi(w) < (or <=) fi(a). Let µ = minqi−ai, i ∈ n. Then, for any µ ∈ (0, µ],there exists u >= 0 and σ > (or >=) 0 such that

f(u+ a) = −σe+ f(a), eTu = µ.

Therefore, (H.3) holds if (H.2) holds.

Similarly, the following result is easily obtained.

Theorem 2 Assume the following:(H.1)′. (H.1) holds except that the condition that αi, βi ∈ K is replaced bythe condition that αi, βi ∈ K∞;

(H.2)′. ∀ w ∈ Rn+, w 6= 0, ∃ i ∈ n : fi(w) < 0;

and(H.4). ∀ w ∈ Rn

+, ∃ q : q >= w, f(q) <= 0.

Then, the zero solution of S is uniform-asymptotically stable in the large.

In [3], [14], [15], similar results are shown under more conservative conditionsthat F is continuously differentiable and −f is an M -function. 1

1It is assumed that −f is an M-matrix on a rectangle Q, where Q = [0, u] or, Q = Rn+.

Since f(0) = 0, this condition is a sufficient condition for (H.2) or (H.2)′ (see Lemma 1and Definition 3).


The condition (H.4) might be restrictive in some case. Before we show animprovement of Theorem 2, we illustrate the idea.

Example 1 Suppose that n = 2, g1(w1) = h2(w1, w2) = σ1(w1), h1(w1, w2) =σ2(w2), and g2(w2) = 2σ2(w2) where

σ1(w1) =

w1, if w1 ∈ [0, 1],1, if w1 >= 1,

σ2(w2) =

w2, if w2 ∈ [0, 2],2, if w2 >= 2.

Then F : C2+ → R2 and f : R2

+ → R2 are given by

F (ψ) =

[

−σ1(ψ1(0)) + σ2(ψ2)−2σ2(ψ2(0)) + σ1(ψ1)

]

, f(w) =

[

−σ1(w1) + σ2(w2)−2σ2(w2) + σ1(w1)

]

.

PSfrag replacements

w1

w2

f1 = 0

f2 = 0

f1 > 0, f2 < 0

f1 < 0, f2 < 0

f1 < 0, f2 > 0

Fig. 1

In this case, there is no vector q satisfying (H.4) when w2 > 1. But, weobserve the following:1) If w2 < 1, then there is a vector q such that q >= w and f(q) <= 0.2) Let I(0) = 1, 2, I(1) = 2, J(1) = I(0) − I(1) = 1,

F 1I(1)(ψI(1)) = sup

ψJ(1)>=0FI(1)(ψ) = −2σ2(ψI(1)(0)) + 1, (11)

and

f1I(1)(wI(1)) = sup

wJ(1)>=0fI(1)(w) = −2σ2(wI(1)) + 1. (12)

Then, f1I(1)(a

1I(1)) = 0 for a1

I(1) = 0.5; for any wI(1) >= a1I(1), wI(1) 6= a1

I(1),

we have f12 (wI(1)) < 0; and for any w >= 0 there is a vector q1 such that

q1I(1) >= wI(1) and f1I(1)(q

1I(1)) <= 0. Therefore, if we consider

x1I(1) = F 1

I(1)([x1I(1)]t)

then x1I(1) = x(t0, [xI(1)]t0) will converges to a1

I(1).

3) Choose r1I(1) ∈ (a1I(1), 1] so that [x1

I(1)]t1 < r1I(1) for some t1 > t0. If


wI(1) < r1I(1), then there is a vector q such that q >= w and f(q) <= 0, and x

= x(t1, [x(t0, x0)]t1) = x(t0, x0) will converges to 0.In this way, we will show that the zero solution of S is asymptotically

stable in the large.

Theorem 3 In addition to (H.1)′ and (H.2)′ in Theorem 2, we assumethe following:(H.5). For each x0 ∈ Cn+, (18) has a unique solution x = x(t0, x0) such thatx >= 0;and(H.6). There exists a strictly decreasing sequence n = I(0) % I(1) % · · · %I(p) and quasi-monotone and locally Lipschitz continuous functions F ` :Cn`

+ → Rn` , ` ∈ p, where n` = |I(`)|, satisfying the following five condi-tions a)-e):(a) F `−1

I(`) (ψI(`−1)) <= F Ì(`)(ψI(`)) for all ` ∈ p and ψ ∈ Cn+, where F 0 = F ;

(b) for each ` ∈ p, there exists a vector rÌ(`) ∈ Rn` , rÌ(`) > 0I(`) so that

for any w ∈ Rn+ such that wI(`) < rÌ(`) there exists a vector q`−1 ∈ Rn,

q`−1 >= w satisfying f `−1I(`−1)(q

`−1I(`−1)) <= 0I(`−1) and q`−1

I(`)<= rÌ(`), where f `−1

I(`−1)

is the restriction of F `−1I(`−1) to the space of constant functions;

(c) for each ` ∈ p− 1, there exists a vector aÌ(`) ∈ Rn`

+ such that aÌ(`) <

rÌ(`), fÌ(`)(a

Ì(`)) = 0I(`), and for any wI(`) ∈ [aÌ(`), r

Ì(`)], wI(`) 6= aÌ(`), we

have f ì (wI(`)) < 0 for some i ∈ I(`);(d) there exists a vector apI(p) ∈ R

np

+ such that apI(p) < rpI(p), fpI(p)(a

pI(p)) =

0I(p) and for any wI(p) >= apI(p), wI(p) 6= apI(p), we have fpi (wI(p)) < 0 for

some i ∈ I(p); and(e) for any ψ ∈ Cn+, there exists a vector qp ∈ Rn such that qp >= ψ andfpI(p)(q

pI(p)) <= 0I(p).

Then, the zero solution of S is asymptotically stable in the large.

Theorem 3 is an improvement version of Theorem 3 in [15] in the sense thatTheorem 3 does not require that −f `’s are M -functions.

3 Main Results

Theorems 1, 2 and 3 in the previous section are highly useful to investigatestability of the composite system S. However, they are not of much practicalvalue in determining whether the conditions are satisfied or not. In thisrespect, it is important to have various sufficient conditions under whichstability of S is guaranteed.

In this section, we consider the case when the test function f is piecewiselinear. We assume the following:(H.7). The functions gi, gi, hi in (H.1) are continuous piecewise linear


functions. The function f defined by (10) is a piecewise linear function. Thedomain Rn

+ of f is divided by M number of hyperplanes into K number ofconvex regions R(k)Kk=1, each R(k) is given by

R(k) = w ∈ Rn+ : 〈nm, w〉 <= cm, m ∈ U(k),

〈nm, w〉 >= cm, m ∈ L(k), (13)

where nm ∈ Rn is the normal vector of the m-th hyperplane and nm = e`

(`-th coordinate vector) for some ` ∈ n, and cm >= 0, and U(k) ∪ L(k) ⊆M ,and f is given by f(w) = J(k)w + b(k), w ∈ R(k). 2 for all m, and let R(1)be the only one region including 0.

When (H.7) is satisfied, F defined by (9) is Lipschitz continuous and (H.5) inTheorem 3 is satisfied automatically since F is quasi-monotone and F (0) = 0[14]. Therefore, the remaining issues are to derive practical conditions for theconditions (H.2)′, (H.4) or (H.6). Concerning (H.2)′, we have the following:

Lemma 3 If −f is an M0-function on Rn+ and if (H.2) is satisfied, then

(H.2)′ is satisfied.

Note that Lemma 3 does not require that f is piecewise linear. When (H.7)is satisfied (and, hence, f is piecewise linear), (H.2) holds if and only if −J(1)is an M -matrix, and the condition f to be M0-function is the following.

Lemma 4 Let f : Rn+ → Rn be a continuous piecewise linear function.

Then, f is an M -function (or M0-function) if and only if Jacobian matrixof f is an M -matrix (or, M0-matrix) in each region.

To derive conditions for (H.4) or (H.6), we need the following.

Lemma 5 Let A ∈ Rn×n be an irreducible M0-matrix and let w >= 0 be avector such that Aw >= 0. 3 Then, we have(a) if w 6= 0, then w > 0;(b) if detA = 0, then Aw = 0.

Lemma 6 Let A ∈ Rn×n be an M0-matrix and w ∈ Rn+. When A is

reducible, we suppose that A and w are represented by

[A w] =

A11 A12 · · · A1m w1

A21 A22 · · · A2m w2

......

. . ....

...Am1 Am2 · · · Amm wm

, (14)

2 The condition that nm = e` for some `, is usually satisfied when we analyze stabilityof composite systems by using the vector Lyapunov functions method.

3 If A is an irreducible M0-matrix, then there exists a w > 0 such that Aw >= 0 (see(5.8) in [6]).


where Aij ∈ Rni×nj , wj ∈ Rnj

+ , Aij = 0 if i > j, Aij <= 0 if i < j, and Aii’sare irreducible M0-matrices.

Then there exists w >= 0, w 6= 0, satisfying the following three conditions.

Aw >= 0 (15)

∀ w >= 0 : Aw >= 0 ∃α > 0 : w <= αw (16)

∃ I ⊆ m : wi = 0 ∀ i ∈ I, wj > 0 ∀ j ∈ J = m− I

⇒ Aij = 0 ∀ i ∈ I, ∀ j ∈ J. (17)

Example 2 Let a reducible M0-matrix A be give by

A =

1 −1 0 00 0 0 00 0 0 −20 0 0 3

,

Then,

w1 =

2110

, w2 =

2100

, w3 =

3010

, w4 =

0450

are vectors such that Awk >= 0 and wk 6= 0. The vector w1 is a vector whichsatisfies (15)- (17).

Corollary 1 Let A be an M0-matrix. Then, there exists a w > 0 such thatAw >= 0 only if one of the following conditions holds.(a) detA 6= 0;(b) detA = 0 and A is irreducible; or(c) detA = 0, A is reducible and represented by (14), and Aij = 0 for allj 6= i whenever detAii = 0.

Moreover, we have:(a)’ If detA 6= 0, then A is an M -matrix, and there exists a w > 0 suchthat Aw > 0.(b)’ If detA = 0 and A is irreducible, then Aw = 0.

By using Lemma 6 and Corollary 1, we can derive conditions which guaranteethat (H.4) or (H.6) is satisfied.

Theorem 4 In addition to (H.1)′ and (H.7), we assume the following:(H.8). In R(1), −J(1) is an M -matrix,and(H.9). In each region R(k) not containing 0, −J(k) is an M0-matrix.

Then, the zero solution of S is asymptotically stable in the large. More-over, if, in addition, one of the conditions of Corollary 1 is satisfied in eachunbounded region, then the zero solution of S is uniform-asymptotically stablein the large.


Example 3 Let us consider again f given in Example 1. Domain R2+ of

f is divided into 4 regions R(1), R(2), R(3) and R(4). Let n1 = n2 = [1 0]T,n3 = n4 = [0 1]T, c1 = 0, c2 = 1, c3 = 0, c4 = 2, U(1) = 2, 4, L(1) =1, 3, U(2) = 4, L(2) = 2, 3, U(3) = 2, L(3) = 1, 4, U(4) = ∅,L(4) = 2, 4. In each R(k), k ∈ 4, f(w) = J(k)w + b(k), where

J(1) =

[

−1 11 −2

]

, b(1) =

[

00

]

, J(2) =

[

0 10 −2

]

, b(2) =

[

−11

]

,

J(3) =

[

−1 01 0

]

, b(3) =

[

2−4

]

, J(4) =

[

0 00 0

]

, b(4) =

[

1−3

]

.

Since f is continuous piecewise linear function, (J(k), b(k))’s of contiguousregions are related each other, that is, when R(k)∩R(k′) ⊆ Hm = w ∈ R2

+ :〈nm, w〉 − cm = 0, we have [J(k) b(k)] = [J(k′) b(k′)] + a[(nm)T − cm]for some vector a ∈ Rn. For example, we have J(4) = J(2) + a(n4)T,b(4) = b(2) + a(−c4), where a = [−1 2]T.

Since −J(1) is an M -matrix, there is a vector w1 > 0 such that J(1)w1 <0 (for example, w1 = [1 0.7]T). The half line 0 + αw1, α >= 0 intersectswith a boundary hyperplane H2 = w : 〈n2, w〉 − c2 = 0, 2 ∈ U(1), atp1 ∈ (R(1) ∩ R(2)). Choose w2 = [1 0]T, a vector satisfying (15)- (17) forA = −J(2). The half line p1 + αw2 is included in R(2), and (H.4) does nothold since w2 6> 0 (w2

2 = 0). Then, we set r1 = p1, I(1) = i ∈ I(0) :w2i = 0 = 2, J(1) = I(0) − I(1) = 1, where I(0) = 1, 2. Then,

J(2)I(1)J(1) = 0, and we define f1I(1) by

f1I(1)(wI(1)) = sup

wJ(1)>=0fI(1)(w) = −2σ2(wI(1)) + 1.

PSfrag replacements

w1

w2

w1w2

p1

0 1

2

R(1)

R(2)

R(3) R(4)

Fig. 2


4 An Example of a Composite System

Let us consider a composite systems S given by (1), in which n = 4, Gi andHi are given by

Gi(ψi, t) =

Aiψi(0), i = 1, 3κi[ψi(0)], i = 2, 4

and

Hi(ψ, i) =

biκi+1[ψi+1(0)] + diρi[αi1cT1 ψ1(−τ1) + αi3c

T3 ψ3(−τ3)], i = 1, 3

cTi−1ψi−1(0), i = 2, 4

where Ai’s are stable ni × ni matrices, bi, di, ci ∈ Rni , τi > 0, αij ∈ R,κi and ρi are continuous nonlinear functions satisfying κi(0) = ρi(0) = 0,0 <= µi(s) <= |κi(s)| <= πi(s), and |ρi(s)| <= σi|s| for all s ∈ R, in which µi andπi are nondecreasing piecewise linear functions given by

µi(s) =

µis if 0 <= s <= ξiµiξi if s >= ξi

πi(s) =

πis if 0 <= s <= ξiπiξi if s >= ξi

and πi >= µi >= 1.Let Pi and Qi are symmetric positive definite matrices satisfying PiAi +

ATi Pi = −Qi, for i = 1, 3. We choose vector Lyapunov function Vi(xi) =

[xTi Pixi]

1/2 for i = 1, 3, and Vi(xi) = |xi| for i = 2, 4. Then, gi, gi andhi in (H.1) are given by gi(wi) = −λiwi, gi(wi) = 0, hi(w) = βiπi(wi+1) +δiσi(|αi1|γ1w1 + |αi3|γ3w3) for i = 1, 3, gi(wi) = −µi(wi), gi(wi) = 0,hi(w) = γi−1wi−1 for i = 2, 4, where λi = minyTQiy/2y

TPiy : y 6= 0,βi = [bTi Pibi]

1/2, δi = [dTi Pidi]

1/2, and γi = [cTi P−1i ci]

1/2.

R4+ is divided into four regions, and boundary hyperplanes are w2 = ξ2

and w4 = ξ4. Jacobian matrices are given by

J(1) =

−λ1 + δ1σ1|α11|γ1 β1π2 δ1σ1|α13|γ3 0γ1 −µ2 0 0

δ3σ3|α31|γ1 0 −λ3 + δ3σ3|α33|γ3 β3π4

0 0 γ3 −µ4

,

J(2) =

−λ1 + δ1σ1|α11|γ1 0 δ1σ1|α13|γ3 0γ1 0 0 0

δ3σ3|α31|γ1 0 −λ3 + δ3σ3|α33|γ3 β3π4

0 0 γ3 −µ4

,

J(3) =

−λ1 + δ1σ1|α11|γ1 β1π2 δ1σ1|α13|γ3 0γ1 −µ2 0 0

δ3σ3|α31|γ1 0 −λ3 + δ3σ3|α33|γ3 00 0 γ3 0

,

and

J(4) =

−λ1 + δ1σ1|α11|γ1 0 δ1σ1|α13|γ3 0γ1 0 0 0

δ3σ3|α31|γ1 0 −λ3 + δ3σ3|α33|γ3 00 0 γ3 0

.


When −J(1) is an M -matrix, it is easy to see that −J(k), k = 2, 3, 4, areM0-matrices, and by Theorem 4, we can conclude that S is asymptoticallystable in the large.

Moreover, we note that −J(4) is not a M -matrix, and, hence, −f is notan M -function. Therefore, the previous results [3], [2], [8], [14], [15], [16]fails to establish stability in the large. Moreover, we note that the methodsusing Lyapunov functionals [7], [9] can not be applied to this example sinceisolated systems do not contains time-delayed parts.

5 Conclusion

In this paper, we analyzed stability of nonlinear hereditary composite systemsusing vector Lyapunov functions method with nonlinear comparison systems.We derived less conservative stability results. Moreover, we considered thecase when the test function is a piecewise linear function and derived condi-tions which are easy to examine using piecewise linear M0-function concept.

References

[1] T. Amemiya, Stability analysis of generalized Volterra equation with saturated invari-ant set with the aid of M-function, Trans. Soc. Instrum. & Contr. Eng., 15 (1979)429-435 (in Japanese).

[2] T. Amemiya, Macroscopic asymptotic stability of nonlinear interconnected systemswith delays, Preprint 23rd JAACE Meeting on Syst. and Contr., (1979) 19-20.

[3] T. Amemiya, On the stability of nonlinear interconnected systems, Int. J. Contr., 34(1981) 513-27.

[4] M. Araki, Stabilty of large-scale nonlinear systemsquadratic order theory of compositeSystem method using M-matrices, IEEE Trans. Automat. Contr., AC-7 (1978) 129-142.

[5] F. N. Bailey, The application of Lyapunov’s second method to interconnected systems,SIAM J. of Control, 1(1966) 443-462.

[6] M. Fiedler and V. Ptak, On matrices with nonnegative off-diagonal elements andpositive principal minors, Czech. Math. J., 12 (1962) 382-400.

[7] G. S. Ladde, Stability of large-scale hereditary systems under structural perturba-tions, Proc. IFAC Symposium on Large-Scale Systems Theory and Applications,Udine Italy (1976) 215-226.

[8] R. M. Levis nd B. D. O. Anderson, Insensitivity of a class of nonlinear CompartmentalSystems to the introduction of arbitrary time delays, IEEE Trans. Circuits Syst.,CAS-27 (1980) 604-612.

[9] A. N. Michel, Stability analysis or interconnected systems, SIAM J. Contr. 12 (1974)554-579.

[10] A. N. Michel, On the status of stability of interconnected systems, IEEE Trans.

Circuits Syst., CAS-30 (1983) 326-40.

[11] A. N. Michel and R. K. Miller, Quantative Analysis of Large Scale Dynamical Sys-

tems, Academic Press, New York 1977.

[12] J. More and W. C. Reinboldt, On P- and S-functions and related class of n-dimensional nonlinear mappings, Linear Algebra Appl., 6 (1973) 45-68.


[13] Y. Ohta, Stability criteria for off-diagonally monotone nonlinear dynamical systems,IEEE Trans. Circuits Syst., CAS-27 (1980) 956-962.

[14] Y. Ohta, Qualitative analysis of nonlinear quasimonotone dynamical systems de-scribed by functional differential equations, IEEE Trans. Circuits Syst., CAS-28(1981) 138-144.

[15] Y. Ohta, T. Douseki and T. Matsumoto, Stability analysis of nonlinear compositesystems including time delays, Proc. 8th IFAC World Congr., 2 (1981) 41-46.

[16] Y. Ohta and D. D. Siljak, An inclusion principle for hereditary systems, J. Math.

Anal. Appl., 98 (1984) 581-597.

[17] W. C. Reinboldt, On M-functions and their applications to nonlinear Gauss-Seideliterations and to network flows, J. Math. Anal. Appli., 32 (1970) 274-307.

[18] I. W. Sandberg, A note on market equilibrium with fixed supply, J. Economic Theory,11 (1975) 456-461.

[19] I. W. Sandberg, A criterion for the global stability of a price adjustment process,Modeling and Simulation, 8 (1977) 1055-1057.

[20] I. W. Sandberg, A theorem concerning properties of an arms-race model, IEEE Trans.

Syst. Man, Cybern., SMC-81 (1978) 29-31.

[21] D. D. Siljak, Large-Scale Dynamical Systems: Stability and Structure, North-Holland,New York, 1978.

[22] T. Tokumaru, N. Adachi and T. Amemlya, Macroscopic Stability of interconnectedsystems, Proc. IFAC 6th World Congr. (1975) Pt.ID, 44.4.

[23] M. Vidyasagar, Input-Output Analysis of Large-Scale Interconnected Systems, NewYork Springer-Verlag. 1981.

Appendix

Proof of Theorem 1. Let us consider the comparison system given by

x(t) = F (xt). (18)

We note that (H.2) means that 0 is the only one equilibrium point of (18)in [0, q]. By (H.3), there is a vector q such that f(q) <= 0. Since F is quasi-monotone and locally Lipschitz continuous, and since F (0) = 0, by Theorem2 of [14], for any x0 ∈ Cn such that 0 <= x0 <= q, (18) has a unique solutionx(t0, x0) on [t0,∞), which satisfies

0 <= x(t2) <= x(t1) <= q ∀ t2 >= t1 >= t0, limt→∞

x(t) = 0. (19)

Moreover, if vt0 <= x0, by Theorem 3 of [14], we have

v(t) <= x(t) <= q ∀ t >= t0. (20)

Since the vector q in (19) and (20) can be chosen arbitrary small by (H.3), itis easy to show that the zero solution of S is uniform-asymptotically stable.

Proof of Lemma 2. The proof is quite similar to that of Theorem 4.7 in[12]. We omit the proof.


Proof of Theorem 2. The proof is almost same with that of Theorem 1,and we omit the proof.

Proof of Theorem 3. Given arbitrary φ ∈ Cd, and let x = x(t0, φ)be the solution of (1) with initial condition φ at t = t0. Define ψ ∈ Cn+by ψi(s) = Vi[φi(s)], s ∈ [−τ, 0], and let vi(t) = Vi(xi(t)) = Vi([xi]t(0)),v(t) = [v1(t) v2(t) · · · vn(t)]T as in the proof of Theorem 1. Then, vsatisfies inequality (8).

Since F (0) = 0, by (a), we have F pI(p)(0I(p)) >= F 0I(p)(0) = 0I(p). By

(e), there is a vector qp >= ψ such that qpI(p) >= apI(p) and F pI(p)(qpI(p)) =

fpI(p)(qpI(p)) <= 0I(p). Consider a functional-differential equation

xpI(p)(t) = F pI(p)([xpI(p)]t), [xpI(p)]t0 = qpI(p). (21)

Since F pI(p)(0I(p)) >= 0I(p) = F pI(p)(apI(p)) >= F pI(p)(q

pI(p)) and since 0I(p) <=

apI(p) <= qpI(p), by Lemma 3 and Remark 4 in [14], (21) has a unique solution

xpI(p) = xpI(p)(t0, qpI(p)) such that 0 <= xpI(p) <= qpI(p) and xpI(p) converges apI(p).

Since apI(p) < rpI(p) by (d), we have [xpI(p)]t1 < rpI(p) for some t1 > t0.

On the other hand, by (8) and (a), we have

vI(p) <= F p([vI(p)]t), [vI(p)]t0 = ψI(p) <= qpI(p).

and, from Theorem 3 in [14], we have [vI(p)]t1 <= [xpI(p)]t1 < rpI(p).

Then, by (b), there is a vector qp−1 >= vt1 such that F p−1I(p−1)(q

p−1I(p−1)) <=

0I(p−1), and by (c) we similarly have [vI(p−1)]t2 < rp−1I(p−1) for some t2 > t1.

Repeating this process, we have [vI(1)]tp < r1I(1) for some tp > t0. Then, by

(b) and Lemma 3 in [14], we can conclude v converges 0. Then, it is easy toshow that the zero solution of S is asymptotically stable in the large.

Proof Lemma 3. By contradiction. Suppose (H.2)′ is not satisfied, thatis, that there is a w >= 0 such that w 6= 0 and that f(w) >= 0. Let I ⊆ nbe the set of index i such that wi > 0 and let J = n− I . Let us consider asub-function fI : R

|I|+ → R|I| given by fI(wI ) = fI(wI , 0). Then, −fI(wI ) is

again an M0-function, and fI(wI) = fI(wI , wJ ) = fI(wI , 0) >= 0. Moreover,since f(0) = f(0I , 0J) = 0 and since f is off-diagonally monotone, we havefj(wI , 0J) >= fj(0I , 0J) = 0 for all j ∈ J , and, hence, by (H.2), we have

∀ wI ∈ [0I , qI ], wI 6= 0, ∃ i ∈ I : fi(wI ) < 0. (22)

Choose µ > 0 such that (µq)I < wI . Then by (22) and by Lemma 2, we have

∃ qI ∈ R|I| : 0 < qI <= µqI , fI(qI ) < 0.

and we have wI > qI , fI(wI ) >= 0 > fI(qI ), that is,

(wi − qi)[fi(wI ) − fi(qI)] > 0 ∀ i ∈ I,


which contradicts to the condition that −fI is an M0-function.

Proof of Lemma 4. For the case of M -function, see [1]. For the case ofM0-function, see Corollary 5.3 in [12].

Proof of Lemma 5.

(a) Suppose that w 6= 0 and w 6> 0. Then there exists a set of indices I ⊆ nsuch that wI > 0 and wJ = 0, where J = n − I . Let us decompose A asfollows:

A =

[

AII AIJAJI AJJ

]

From Aw >= 0, we have

AJIwI +AJJwJ >= 0

On the other hand, wJ = 0, wI > 0 AJI <= 0, and AJI 6= 0 (since A isirreducible) means that

AJIwI +AJJwJ = AJIwI <= 0, AJIwI 6= 0,

which is a contradiction. Therefore, if w 6= 0, then w > 0.(b) It is shown that if A is an irreducibleM0-matrix such that Aw >= 0, Aw 6=0 for some w > 0 then A is an M -matrix (Lemma 4.2 in [17]). Therefore, ifdetA = 0, then we have Aw = 0.

Proof of Lemma 6. If A is an irreducible M0-matrix, then there exists aw > 0 such that Aw >= 0, and nothing remains to prove.

Suppose that A is reducible and given by (14). We define w by thefollowing algorithm:Step 1. for each i such that detAii = 0, do beginStep 2. for each j such that Aij 6= 0, set wj = 0;Step 3. end;Step 4. i := m;Step 5. if i <= 0 stop;Step 6. if wi was set to 0, then goto Step 9;Step 7. if detAii = 0, then choose wi > 0 so that Aiiwi = 0;Step 8. else 4 choose wi > 0 such that Aiiwi +

∑

j>i Aij wj > 0;Step 9. i := i− 1 go to Step 5;Step 10. end.Then. it is easy to see that (16) and (17) are satisfied.

Proof of Corollary 1. Corollary 1 is obvious from Lemmas 5, 6. We omitthe proof.

Proof Theorem 4. For each k, let wk >= 0 be a vector such that wk > 0 orwk 6> 0 but satisfies (15)-(17) for A = −J(k). By (H.8), J(1) is an M -matrix,and, hence, w1 > 0 and f(w1) = J(1)w1 + b(1) < 0.

4 Note that Aii is an M-matrix at Step 8 since det Aii > 0 and Aii is an M0-matrix.


Let p0 = 0, k1 = 1. Suppose that the half line p0 +αwk1 , α >= 0 intersectswith a boundary hyperplaneHm = w : 〈nm, w〉 = cm,m ∈ U(k1), of R(k1)at p1 = p0 + α1w

k1 when α = α1 > 0, and suppose p1 ∈ (R(k1) ∩ R(k2)).Similarly, suppose that the half line p1 + αwk2 , α > 0, intersects with aboundary hyperplane Hm, m ∈ U(k2) of R(k2) at p2 = p1 + α2w

k2 whenα = α2, and suppose p2 ∈ (R(k2) ∩ R(k3)). We repeat this process as longas the half line pj + αwkj+1 intersects with a boundary Hm, m ∈ U(kj+1) ofR(kj+1) at pj+1 = pj + αj+1w

kj+1 when α = αj+1.5 This stops after some

finite number, say j, of iteration, that is, pj−1 +αwkj does not intersect withany boundary of unbounded region R(kj).

If wkj > 0, we have (H.4).

When wkj 6> 0, we will show that we have (H.6). Let r1 = pj−1, I(0) = n,

I(1) = i ∈ I(0) : wk

j

i = 0, and J(1) = I(0) − I(1). Observe the following:1) By Lemma 6, J(kj)I(1)J(1) = 0, in R(kj),

f1I(1)(wI(1)) = J(kj)I(1)I(1)wI(1) + bI(1)(kj)

2) Let Hm = w : 〈nm, w〉 = cm, m ∈ U(kj). Then, for any m ∈ U(kj)

nmI(1) = 0 because if nmI(1) 6= 0 the half line pj−1 +αwkj , α >= 0 must intersectwith Hm.3) Let R(k′) be any unbounded region contiguous with R(kj) and let R(k′)∩R(kj) ⊆ Hm, for some m ∈ U(kj), then

[J(k′) b(k′)] = [J(kj) b(kj)] + a[(nm)T − cm]

hold for some a ∈ Rn since f is a continuous piecewise linear function. Then,we again have J(k′)I(1)J(1) = 0 because nmI(1) = 0.

4) From above observations, we can define f 1I(1) by

f1I(1)(wI(1)) = sup

wJ(1)>=0fI(1)(wI(1), wJ(1))

= J(k)I(1)I(1)wI(1) + bI(1)(k), w ∈ R(k), (23)

where R(k) is the region such that w + αwkj ∈ R(k) for sufficiently largeα > 0.5) Then, obviously, we have (H.6) (a) and (b) for ` = 1.6) Since f1

I(1)(0) >= fI(1)(0) = 0 and since f1I(1)(r

1I(1)) = fI(1)(r

1I(1)) < 0, by

Lemma 3 in [14], we can see that there exists a vector a1I(1) satisfying (H.6)

(c) for ` = 1.Let R(k1) be the region which includes a1

I(1). Note that f1I(1) is an M0-

function but J(k1)I(1)I(1) may not be M -matrix. However, by the fact (H.6)(c) holds for ` = 1, the condition corresponding to (H.2) is met for f 1

I(1), and,

5 Since wkj+1 >= 0, wkj+1 6= 0, and since nm >= 0, pj + αwkj+1 intersects a boundary〈nm, w〉 = cm, m ∈ U(kj+1) of R(kj+1) for α ∈ (0,∞) if R(kj+1) is a bounded region.


hence, by Lemma 2, there is a vector wk1I(1) = µ(r1I(1) − a1I(1)) > 0 such that

J(k1)I(1)I(1)wk1I(1) < 0I(1). Using this fact, we can generate again w

kj

I(1), pjI(1),

j = 1, 2, · · · , j.

We will repeat this process until we have the case when wk

j

I(p) > 0.

If, in addition, one of the conditions of Corollary 1 is satisfied in eachunbounded region, then we have wkj > 0, and, hence, we have (H.4).


A STUDY OF SET DIFFERENTIAL EQUATIONSWITH DELAY

J. Vasundhara Devi 1 and

A.S. Vatsala2

1Department of Mathematical SciencesFlorida Institute of Technology, Melbourne, Florida 32901, USA

2Department of Mathematics

University of Louisiana at Lafayette, Lafayette, LA 70504, USA

Abstract. Existence and global existence results are proved for set differential equations

with delay. Also nonuniform practical stability and nonuniform boundedness results have

been obtained using Lyapunov functions on product spaces.

Keywords. Set Differential Equations with delay, Boundedness, Practical Stability, Ex-

istence.

AMS subject classification: 34K20, 34K07.

This paper is written in honor of Professor Siljak’s 70th birth-

day.

1 Introduction

Recently, the study of set differential equations has attracted much atten-tion, [1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 16, 17] as an independent area because ofits inherent advantages. Also, the differential equations with delay are bet-ter models for physical phenomena as there is a time lag between the causeand the effect. The method of Lyapunov functions on product spaces wasinitiated in [11, 15] so as to appropriately study these differential equations.

In this paper, we study set differential equations with delay and obtainexistence results. Further, as the diameter of solutions of set differential equa-tions is nondecreasing as time increases, we study the nonuniform practicalstability and nonuniform boundedness results. See [8, 10] for other relatedresults.

1On leave from Dept. of Mathematics, G. V. P. College for P. G. Courses, Visalehap-atnam, India

290 J. Vasundhara Devi & A.S. Vatsala

2 Preliminaries

We start with the necessary preliminaries required to define a set differentialequation.

Let Kc(RN ) denote the collection of all nonempty compact and convex

subsets of RN . Define the Hausdorff metric

D[A, B] = max[supx∈B

d(x, A), supy∈A

d(y, B)] (2.1)

where d(x, A) = inf[d(x, y) : y ∈ A], A, B are bounded sets in RN . We

observe that Kc(RN ) is a complete metric space.

Suppose that the space Kc(RN ) is equipped with the natural algebraic

operations of addition and nonnegative scalar multiplication. Then Kc(RN )

becomes a semilinear metric space, which can be embedded as a completecone into a corresponding Banach space.

We note that the Hausdorff metric (2.1) satisfies the following properties:

D[A + C, B + C] = D[A, B] and D[A, B] = D[B, A] (2.2)

D[λA, λB] = λD[A, B] (2.3)

D[A, B] ≤ D[A, C] + D[C, B] (2.4)

for all A, B, C ∈ Kc(RN ) and λ ∈ R+.

Given any two sets A, B ∈ Kc(RN ), if there exists a set C ∈ Kc(R

N )satisfying A = B + C, then A − B is defined as the Hukuhara difference ofthe sets A and B.

The mapping F : I → Kc(RN ) has a Hukuhara derivative DHF (t0) at a

point t0 ∈ I, if

limh→0+

F (t0 + h) − F (t0)

hand lim

h→0+

F (t0) − F (t0 − h)

h

exists in the topology of Kc(RN ) and are equal to DHF (t0). Here I is any

interval in R.By embedding Kc(R

N ) as a complete cone in a corresponding Banachspace and talking into account the result on the differentiation of Bochnerintegral, we find that if

F (t) = X0 +

∫ t

t0

Φ(s)ds, X0 ∈ Kc(RN ) (2.5)

where Φ : I → Kc(RN ) is integrable in the sense of Bochner, then the DHF (t)

exists and equality

DHF (t) = Φ(T ), a.e. on I (2.6)

holds.

Set Differential Equations with Delay 291

Also, the Hukuhara integral of F is given by

∫

I

F (s)ds = [

∫

I

f(s)ds : f is a continuous selector of F ]

for any compact set I ⊂ R+.

Now we can consider the set differential equation

DHU = F (t, U), U(t0) = U0 ∈ Kc(RN ), t0 ≥ 0 (2.7)

where F ∈ C[R+ × Kc(RN ), Kc(R

N )].

Definition 2.1. The mapping U ∈ C1[J, Kc(RN )], J = [t0, t0 +a] satisfying

the equation (2.7) is said to be a solution of (2.7) on J.

Since U(t) is continuously differentiable, we have

U(t) = U0 +

∫ t

t0

DHU(s)ds, t ∈ J (2.8)

Hence , we can associate with the IVP (2.7) the Hukuhara Integral

U(t) = U0 +

∫ t

t0

F (s, U(s))ds, t ∈ J (2.9)

The following properties are useful tools for working our results.

If F : [t0, T ] → Kc(RN ) is integrable, we have

∫ t2

t0

F (t)dt =

∫ t1

t0

F (t)dt +

∫ t2

t1

F (t)dt, t0 ≤ t1 ≤ t2 ≤ t0 + T (2.10)

and ∫ T

t0

λF (t)dt = λ

∫ T

t0

F (t)dt, λ ∈ R+ (2.11)

Also, if F, G : [t0, T ] → Kc(RN ) are integrable, then D[F (·), G(·)] : [t0, T ] →

R is integrable and

D[

∫ t

t0

F (s)ds,

∫ t

t0

G(s)ds] ≤

∫ t

t0

D[F (s), G(s)]ds (2.12)

We observe that

D[A, θ] = ‖A‖ = supa∈A

‖a‖ (2.13)

for A ∈ Kc(RN ), where θ is the zero element of R

n which is regarded as aone point set.


3 Existence Results

We start by describing the set differential equation with delay.

Given any τ > 0, consider C0 = C[[−τ, 0] → Kc(RN )]]. For any Φ, Ψ ∈ C0,

define the metric D0[Φ, Ψ] = max−τ≤s≤0 D[Φ(s), Ψ(s)]. In other words, wecan write ‖Φ‖0 = D0[Φ, θ].

Suppose that J0 = [t0 − τ, t0 + a], a > 0. Let U ∈ C[J0, Kc(RN )] for

any t ≥ t0, t ∈ J0, let Ut denote a translation of the restriction of U to theinterval [t − τ, t], that is , Ut ∈ C0 defined by Ut(s) = U(t + s), −τ ≤ s ≤ 0.

Consider the set differential equation with finite delay given by

DHU = F (t, Ut),

Ut0 = Φ ∈ C0

(3.1)

where U ∈ Kc(RN ) and F ∈ C[J × C0, Kc(R

N )] and J = [t0, t0 + a].

The following existence result is obtained using the contraction principle.

Theorem 3.1. Assume that

D[F (t, Φ), F (t, Ψ)] ≤ KD0[Φ, Ψ], K > 0 (3.2)

for t ∈ J, Φ, Ψ ∈ C0. Then the IVP (3.1) possesses a unique solution U(t)on J0.

Proof. Consider the space of functions U ∈ C[J0, Kc(RN )] such that U(t) =

Φ0(t), t0 − τ ≤ t ≤ t0 and U ∈ C[J, Kc(RN )] with U(t0) = Φ0(0) with

Φ0(t) ∈ Kc(RN ), −τ ≤ t ≤ 0.

Define the metric on C[J0, Kc(RN )] by

D1(U, V ) = maxt0−τ≤t≤t0+a

D[U(t), V (t)]e−λt, t > 0 (3.3)

Next, define the operator T on C[J0, Kc(RN )] by

TU(t) = Φ0(t), t0 − τ ≤ t ≤ t0

TU(t) = Φ0(0) +

∫ t

t0

F (s, Us)ds, t ∈ J(3.4)

then

D[TU(t0 + s), TV (t0 + s)], −τ ≤ s ≤ 0,

= D[Φ0(s), Φ0(s)], t0 − τ ≤ s ≤ t0

= 0 ,


We get, using the properties of Hausdorff metric (2.2), (2.4) in

D[TU(t), TV (t)]

= D[Φ0(0) +

∫ t

t0

F (s, Us)ds, Φ0(0) +

∫ t

t0

F (s, Vs)ds]

= D[

∫ t

t0

F (s, Us)ds,

∫ t

t0

F (s, Vs)ds]

≤

∫ t

t0

D[F (ξ, Uξ), F (ξ, Vξ)]dξ

≤ K

∫ t

t0

D0[(Uξ, Vξ)]dξ.

Consider∫ t

t0

D[U(ξ + s), V (ξ + s)]dξ =

∫ t

t0−τ

D[U(σ), V (σ)]dσ

=

∫ t

t0−τ

maxt0−τ≤σ≤t0+a

[D[U(σ), V (σ)]e−λσ ]eλσdσ

≤ D1[U, V ]

∫ t

t0−τ

eλσdσ

= D1[U, V ]1

λ[eλt − eλ(t0−τ)]

=1

λD1[U, V ]eλt.

Thus we obtain

K

∫ t

t0

D0[(Uξ, Vξ)]dξ ≤K

λD1[U, V ]eλt,

which implies that, on J0

e−λtD[TU(t), TV (t)] ≤K

λD1[U, V ].

Choosing λ = 2K and maximum over t, on J0, we have

D1[TU, TV ] ≤1

2D1[U, V ]

which means that the operator T on C[J0, Kc(RN )] is a contraction. Thus

there exists a unique fixed point U ∈ C[J0, Kc(RN )] of T from the contraction

principle. Hence U(t) = U(t0, Φ0)(t) is the unique solution of the IVP (2.1).

We now prove a comparison theorem in this context, which is a usefultool in proving the global existence theorem.


Theorem 3.2. Assume thatF ∈ C[R+ × C0, Kc(R

N )] and D[F (t, Φ), F (t, Ψ)] ≤ g[t, D0[Φ, Ψ]] fort ∈ R+, Φ0, Ψ0 ∈ C0, where g ∈ C[R2

+, R+]. Let r(t) = r(t, t0, w0) be themaximal solution of

w′ = g(t, w)

w(t0) = w0 ≥ 0 existing for t ≥ t0

Then D0[Φ0, Ψ0] ≤ w0 implies

D[U(t), V (t)] ≤ r(t), t ≥ t0,

where U(t) = U(t0, Φ0)(t) and V (t) = V (t0, Ψ0)(t) are the solutions of (3.1).

Proof. Since U(t), V (t) are solutions of (3.1) for small h > 0, the differ-ences U(t + h) − U(t), V (t + h) − V (t) exist. Now for t ∈ R+, set m(t) =D[U(t), V (t)]. Then using the properties of Hausdorff metric, (2.2) and (2.4),we have

m(t + h) − m(t) = D[U(t + h), V (t + h)] − D[U(t), V (t)]

≤ D[U(t + h), U(t) + hF (t, Ut)]

+ D[U(t) + hF (t, Ut), V (t) + hF (t, Vt)]

+ D[V (t) + hF (t, Vt), V (t + h)] − D[U(t), V (t)]

≤ D[U(t + h), U(t) + hF (t, Ut)] + D[V (t) + hF (t, Vt), V (t + h)]

+ hD[F (t, Ut), F (t, Vt)]

from which we get, using (2.2) and (2.4) again,

m(t + h) − m(t)

h≤ D[

U(t + h) − U(t)

h, F (t, Ut)]

+ D[F (t, Vt),V (t + h) − V (t)

h] + D[F (t, Ut), F (t, Vt)] .

Taking limit supremum as h → 0+, gives,

D+m(t) = limh→0

m(t + h) − m(t)

h

≤ D[F (t, Ut), F (t, Vt)]

≤ g(t, D0[Ut, Vt])

= g(t, |mt|0) .

The above inequality, along with the fact that |mt|0 = D0[Φ0, Ψ0] ≤ w0,implies from the comparison theorem for ordinary delay differential equations[9], that

D[U(t), V (t)] ≤ r(t), t ≥ t0 .


We are now ready to prove the global existence theorem.

Theorem 3.3. Let F ∈ C[R+ × C0, Kc(RN )] and for (t, Φ) ∈ R+ × C0,

D[F (t, Φ), θ] ≤ g(t, D0[Φ, θ]),

where g ∈ C[R2+, R+], g(t, w) is nondecreasing in w for each t ∈ R+. Assume

that the solutions w(t, t0, w0) of w′ = g(t, w), w(t0) ≥ w0 exist for t ≥ t0,

and F is smooth enough to assume local existence. Then the largest intervalof existence of any solution U(t0, Φ0)(t) of (3.1) is [t0,∞).

Proof. Suppose that U(t0, Φ0)(t) is a solution of (3.1) existing on someinterval [t0 − τ, β), where t0 < β < ∞. Assume that β cannot be increased.Define for t ∈ [t0 − τ, β),

m(t) = D[U(t0, Φ0)(t), θ]

mt = D[Ut(t0, Φ0), θ] and |mt|0 = D0[Ut(t0, Φ), θ]

Then reasoning and working as in the comparison theorem, we get the dif-ferential inequalities

D+m(t) ≤ g(t, |mt|0), t0 ≤ t < β .

Choosing |mt|0 = D0[Φ0, θ] ≤ w0, we arrive at

D[U(t0, Φ0)(t), θ] ≤ r(t, t0, w0), t0 ≤ t < β .

Now g(t, w) ≥ 0 implies that r(t, t0, w0) is nondecreasing in t, which furtheryields

D[Ut(t0, Φ0), θ] ≤ r(t, t0, w0), t0 ≤ t < β . (3.5)

Consider t1, t2 such that t0 < t1 < t2 < β, then using (2.2)–(2.4), we get

D[U(t0, Φ0)(t1), U(t0, Φ0)(t2)]

= D[U(t0, Φ0)(t1), U(t0, Φ0)(t1) +

∫ t2

t1

F (s, Us)ds]

= D[θ,

∫ t2

t1

F (s, Us)ds]

≤

∫ t2

t1

D[F (s, Us), θ]ds

≤

∫ t2

t1

g(s, D0[Us(t0, Φ0), θ])ds .

Next, using the fact that g is monotonically nondecreasing in w and therelation (3.5) in the above inequality, we obtain

D[U(t0, Φ0)(t1), U(t0, Φ0)(t2)] ≤

∫ t2

t1

g(s, r(s, t0, w0))ds

= r(t2, t0, w0) − r(t1, t0, w0) .

(3.6)


If we let t1, t2 → β in the above relation (3.6), then limt→β− U(t0, Φ0)(t)exists, because of Cauchy’s criterion for convergence. We now defineU(t0, Φ0)(β) = limt→β− U(t0, Φ0)(t) and consider Ψ0 = Uβ(t0, Φ0) as thenew initial function at t = β. The assumption of local existence implies thatthere exists a solution U(β, Ψ0)(t) of (3.1) on [β − τ, β + α], α > 0. Thismeans that the solution U(t0, Φ0)(t) can be continued beyond β, which iscontrary to our assumption that the value of β cannot be increased. Hencethe theorem.

4 Practical Stability

In this section, we present a result on nonuniform practical stability of (3.1)using perturbing Lyapunov functions. See [10] for details. Before proceedingfurther, we need the following classes of functions

K = a ∈ C[[0, A], R+], a(0) = 0 and a(u) is strictly increasing

CK = σ ∈ C[R+ × [0, A], R+] : σ(t, u) ∈ K for each t ∈ R+

We now define practical stability in this context.

Definition 4.1. The system (3.1) is practically stable, given (λ, A) with0 < λ < A, we have D0[Φ, θ] < λ implies D[U, θ] < A, t ≥ t0 for somet0 ∈ R+.

We setS(A) = [ U ∈ Kc(R

n) : D[U(t), θ] < A]

andΩ(A) = Φ ∈ C0 : D0[Φ, θ] < A .

We are now in a position to prove the following result on practical stability.

Theorem 4.1. Assume that(i) 0 < λ < A ;(ii) V1 ∈ C[ R+ × S(A) × Ω(A), R+],for (t, U1, Φ), (t, U2, Φ) ∈ R+ × S(A) × Ω(A)

|V1(t, U1, Φ) − V1(t, U2, Φ)| ≤ L1 D[U1, U2], L1 > 0;

for each (t, U, Φ) ∈ R+ × S(A) × Ω(A),

V1(t, U, Φ) ≤ a1(t, D0[Φ, θ]), a1 ∈ CK,

and

D+V1(t, U, Φ) ≡ lim suph→0+

1

h[V1(t + h, U + hF (t, Ut), Ut+h) − V1(t, U, Ut)]

≤ g1(t, V1(t, U, Φ))


where g1 ∈ C[R2+, R];

(iii) V2 ∈ C[ R+ × S(A) × Ω(A), R+],for (t, U1, Φ), (t, U2, Φ) ∈ R+ × S(A) × Ω(A),

|V2(t, U1, Φ) − V1(t, U2, Φ)| ≤ L2 D[U1, U2], L2 > 0;

for each (t, U, Φ) ∈ R+ × S(A) × Ω(A),

b(D[U, θ]) ≤ V2(t, U, Φ) ≤ a2([D0[Φ, θ]]),

D+V1(t, U, Φ) + D+V2(t, U, Φ) ≤ g2(t, V1(t, U, Φ) + V2(t, U, Φ))

where a2, b ∈ K and g2 ∈ C[R2+, R];

(iv) a1(t0, λ) + a2(λ) < b(A) for some t0 ∈ R+;(v) u0 < a1(t0, λ) implies u(t, t0, u0) < a1(t0, λ) for t ≥ t0 where u(t, t0, u0)is any solution of

u′ = g1(t, u), u(t0) = u0 (4.1)

and v0 < a1(t0, λ) + a2(λ) implies

v(t, t0, v0) < b(A), t ≥ t0,

for every t0 ∈ R+, where v(t, t0, v0) is any solution of

v′ = g2(t, v), v(t0) = v0 ≥ 0 (4.2)

Then the system (3.1) is practically stable.

Proof. We have to prove that given 0 < λ < A, D0[Φ, θ] < λ thenD[U(t), θ] < A where U(t) = U(t0, Φ0)(t) is any solution of (3.1), fort ≥ t0. Suppose it is not true, then there exists t2 > t1 > t0 and a solutionU(t0, Φ0)(t) of (3.1) such that

D[U(t1), θ] = λ and D[U(t2), θ] = A (4.3)

and λ ≤ D[U(t), θ] ≤ A for t1 ≤ t ≤ t2. Now using the hypothesis (iii) and(v) and the standard arguments, we get

V1(t, U(t0, Φ0)(t), Ut(t0, Φ0)) + V2(t, U(t0, Φ0)(t), Ut(t0, Φ0))

≤ r2(t, t1, V1(t, U(t0, Φ0)(t1), Ut1(t0, Φ0))

+V2(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0)))

(4.4)

for t1 ≤ t ≤ t2, where r2(t, t1, v0) is the maximal solution of (4.2) through(t1, v0), and v0 = V1(t1, U(t0, Φ0)(t1), Ut(t0, Φ0))+V2(t1, U(t0, Φ0)(t1), Ut(t0, Φ0)).Similarly, condition (ii) gives the estimate

V1(t, U(t0, Φ0)(t), Ut(t0, Φ0)) ≤ r1(t, t0, V1(t0, Φ(0), Φ0)), t0 ≤ t ≤ t1


where r1(t, t0, u0) is the maximal solution of (4.1) with u0 = V1(t0, Φ(0), Φ0).Since D0[Φ0, θ] < λ, using hypothesis (ii)

V1(t0, Φ(0), Φ0) ≤ a1(t0, D0[Φ0, θ])

≤ a1(t0, λ) .

Also, we have

V2(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0)) ≤ a2(D0[Ut1 , θ]) .

≤ a2(λ) .

Thus we get

V1(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0)) + V2(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0))

≤ a1(t0, λ) + a2(λ) .

Now using the relation (4.4) and the hypothesis (v), we obtain


≤ r2(t2, t1, a1(t0, λ) + a2(λ))

≤ b(A) .

(4.5)

However, using the relation (4.3) and the hypothesis (ii) and (iii), we get


≥ V2(t2, U(t0, Φ0)(t2), Ut2(t0, Φ0))

≥ b[D[U(t2), θ]] = b(A)

which contradicts (4.5). Thus the proof of our claim.

5 Boundedness

Similar to the earlier section, in this section, we study the nonuniform bound-edness property for the system (3.1). We define the concept of boundednessas follows.

Definition 5.1. The differential system (3.1) is said to be

(1) Equibounded, if for any α > 0 and t0 ∈ R+, there exists a β whereβ = β(t0, α) > 0 such that

D0[Φ, θ] < α implies D[U(t), θ] < β, t ≥ t0,

where U(t) = U(t0, Φ0)(t) is any solution of (3.1).

(2) Uniform Bounded, if β in (1) does not depend on t0.


The following theorem uses the method of perturbing Lyapunov functionsto obtain nonuniform boundedness property. For that purpose, setS(ρ) = U ∈ Kc(R

n) : D[U, θ] < ρ and S(ρ) = Φ ∈ C0 : D0[Φ, θ] < ρ .

Theorem 5.1. Assume that

(i) ρ > 0, V1 ∈ C[R+ × S(ρ) × S(ρ), R+], V1 is bounded

for (t, U, Φ) ∈ R+ × ∂S(ρ) × ∂S(ρ);

|V1(t, U1, Φ) − V1(t, U2, Φ)| ≤ L1D[U1, U2], L1 > 0

and for (t, U, Φ) ∈ R+ × Sc(ρ) × Sc(ρ)

D+V1(t, U, Φ) ≡ lim suph→0+

1

h[V1(t + h, U + hF (t, Ut), Ut+h) − V1(t, U, Ut)]

≤ g1(t, V1(t, U, Φ))

where g1 ∈ C[R2+, R];

(ii) V2 ∈ C[R+ × Sc(ρ) × Sc(ρ), R+],

b(D[U, θ]) ≤ V2(t, U, Φ) ≤ a(D0[Φ, θ])

D+V1(t, U, Φ) + D+V2(t, U, Φ) ≤ g2(t, V1(t, U, Φ) + V2(t, U, Φ))

where a, b ∈ K and g2 ∈ C[R2+, R];

(iii) The scalar differential equations

w′1 = g1(t, w1), w1(t0) = w10 ≥ 0 (5.1)

w′2 = g2(t, w2), w2(t0) = w20 ≥ 0 (5.2)

are equibounded and uniformly bounded respectively.

Then the system (3.1) is equibounded.

Proof. Let B1 > ρ and t0 ∈ R+ be given.Let α0 = maxV1(t0, U0, Φ0) : U0 = Φ0(0) ∈ clS(B1) ∩ Sc(ρ),

Φ0 ∈ clS(B1) ∩ Sc(ρ)

α∗ ≥ V1(t, U, Φ) for (t, U, Φ) ∈ R+ × ∂S(ρ) × ∂S(ρ),and set α1 = α1(t0, B1) = max(α0, α

∗).Since the scalar differential equation (4.1) is equibounded, given α1 > 0

and t0 ∈ R+, there exists a β0 = β0(t0, α1) such that

w1(t, t0, w0) < β0, t ≥ t0 , (5.3)

provided w10 < α1, where w1(t, t0, w10) is any solution of (5.1).Let α2 = a(B1)+β0. Then the uniform boundedness of the equation (5.2)

yieldsw2(t, t0, w20) < β1(α2), t ≥ t0 (5.4)


provided w20 < α2, where w2(t, t0, w20) is any solution of (5.2).Choose B2 satisfying

b(B2) > β1(α2) . (5.5)

We now claim that Φ0 ∈ S(B1) implies that U(t) ∈ S(B2) for t ≥ t0,

where U(t) = U(t0, Φ0)(t) is any solution of (3.1).If it is not true, there exists a solution U(t0, Φ0)(t) of (3.1) with Φ0 ∈

S(B1), such that, for some t∗ > t0, D[U(t0, Φ0)(t∗), θ] = B2. Since B1 > ρ,

there are two possibilities to consider,

(i) U(t0, Φ0)(t) ∈ Sc(ρ) for t ∈ [t0, t∗],

(ii) there exists a t ≥ t0 such that U(t0, Φ0)(t) ∈ ∂S(ρ) andU(t0, Φ0)(t) ∈ Sc(ρ) for t ∈ [t, t∗] .

If (i) holds, we can find a t1 > t0 such that

U(t0, Φ0)(t1) ∈ ∂S(B1)

U(t0, Φ0)(t∗) ∈ ∂S(B2)

U(t0, Φ0)(t) ∈ Sc(B1), t ∈ [t1, t∗]

(5.6)

Setting m(t) = V1(t, U(t0, Φ0)(t), Ut(t0, Φ0))+V2(t, U(t0, Φ0)(t), Ut(t0, Φ0))for t ∈ [t1, t

∗] and using the standard arguments, we obtain the differentialinequality [15]

D+m(t) ≤ g2(t, m(t)), t ∈ [t, t∗] ,

It then follows from the Comparison Theorem 1.4.1 of [9]

m(t) ≤ r2(t, t1, m(t1)), t ∈ [t1, t∗]

where r2(t, t1, V0) is the maximal solution of (5.2) with

r2(t1, t1, V0) = V0 =V1(t, U(t0, Φ0)(t1), Ut1(t, Φ0))

+ V2(t, U(t0, Φ0)(t1), Ut1(t, Φ0)) .

Thus,

V1(t∗, U(t0, Φ0)(t

∗), Ut∗(t, Φ0)) + V2(t∗, U(t0, Φ0)(t

∗), Ut∗(t, Φ0))

≤ r2(t∗, t1, V1(t1, U(t0, Φ0)(t1), Ut1(t, Φ0)))

+r2(t∗, t1, V2(t1, U(t0, Φ0)(t1), Ut1(t, Φ0))) .

(5.7)

Similarly, we get

V1(t, U(t0, Φ0)(t), Ut(t0, Φ0)) ≤ r1(t1, t0, V1(t0, U(t0, Φ0)(t0)), Ut0(t0, Φ0))(5.8)

where r1(t, t0, u0) is the maximal solution of (5.1) with

u0 = V1(t0, U(t0, Φ0)(t0), Ut0(t0, Φ0)) = V1(t0, Φ0(0), Φ0) .


Setting w10 = V1(t0, Φ0(0), Φ0) < α1, and using the relation (5.3) we get,

V1(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0)) ≤ r1(t1, t0, w10) ≤ β0.

Furthermore, V2(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0)) ≤ a(B1)and from (5.6), we have

w20 = V1(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0)) + V2(t1, U(t0, Φ0)(t1), Ut1(t0, Φ0)) .

which gives w20 ≤ β0 + a(B1) = α2 .Now combinging (5.4), (5.5),(5.6) we have

b(B2) ≤ m(t∗) ≤ r(t∗) ≤ β1(α2) < b(B2) (5.9)

which is a contradiction.If, case (ii) holds, we also come up with the inequality (5.7), where t1 > t

satisfies (5.6). We then obtain, in place of (5.8) the relation

V1(t1, U(t0, Φ0)(t1), Φt1) ≤ r1(t1, t, V1(t, U(t0, Φ0)(t)), Φt) ,

since U(t0, Φ0)(t) ∈ ∂S(ρ) and

V1(t, U(t0, Φ0)(t), Φt) ≤ α∗ ≤ α1 ,

arguing as before, we get a contradiction.This proves that, for any given B1 > ρ, t0 > 0 there exists a B2 such

that Φ0 ∈ S(B1) implies U(t0, Φ0)(t) ∈ S(B2), t ≥ 0. For the case B1 < ρ,we get B2(t0, B1) = B2(t0, ρ) and hence the proof.

6 References

[1] Brandao Lopes Pinto, A. J., De Blasi, F. S. and Irevolino, F., Uniqueness andexistence theorems for differential equations with compact convexvalued solutions,Boll. Unione. Mat. Italy, 3, (1970) 47-54.

[2] Gnana Bhaskar, T. and Lakshmikantham, V., Set differential equations and flowinvariance, Applicable Analysis 82, (2003) 357-368.

[3] Gnana Bhaskar, T. and Lakshmikantham, V., Generalized flow invariance for dif-ferential equations, (to appear).

[4] Gnana Bhaskar, T. and Lakshmikantham, V., Lyapunov stability for set differentialequations,(to appear).

[5] Gnana Bhaskar, T. and Lakshmikantham, V., Generalized proximal normals andflow invariance for differential inclusions, Nonlinear Studies, (to appear)

[6] Gnana Bhaskar, T. and Lakshmikantham, V., Generalized proximal normals andLypaunov stability, (to appear)

[7] Lakshmikantham, V., Set differential equations versus fuzzy differential equations,(to appear)

[8] Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S., Theory of Impulsive Dif-

ferential Equations, World Scientific, Singapore, 1989 .


[9] Lakshmikantham, V., Leela S., Differential and Integral Inequalities, Vol. I and II,Academic Press, New York 1969.

[10] Lakshmikantham, V., Leela S., and Martynyuk, A.A., Practical Stability of Nonlin-ear Systems, World Scientific, 1990

[11] Lakshmikantham, V., Leela S., and Sivasundaram S., Lyapunov Functions on Prod-uct Spaces and Stability Theory of Delay Differential Equations, Journal of Math-

ematical Analysis and Application, 154, (1991) 391-402.

[12] Lakshmikantham, V., Leela S. and Vatsala, A.S., Set-valued Hybrid DifferentialEquations and Stability in Terms of Two Measures, J. Hybrid Systems 2 (2002),169-187.

[13] Lakshmikantham, V., Leela S. and Vatsala, A.S., Interconnectedness Between Setand Fuzzy Differential Equations, Nonlinear Analysis, 54, (2003) 351-360

[14] Lakshmikantham, V., and Tolstonogov, A., Existence and Interrelation Between Setand Fuzzy Differential Equations, Nonlinear Analysis, (to appear)

[15] Lakshmikantham, V., and Zhang, Y., Strict Practical Stability of Delay DifferentialEquations, Applied Mathematics and Computation 118 (2001), 275-285.

[16] Tolstonogov, A., Differential Inclusions in a Banach Space, Kluwer Academic Pub-lishers, Dordrecht 2000.

[17] Vasundhara Devi J., Basic Properties in Impulsive Set Differential Equations , Non-

linear Studies (to appear)


OVERLAPPING QUADRATIC OPTIMAL CONTROLOF TIME-VARYING DISCRETE-TIME SYSTEMS

L. Bakule 1 and J. Rodellar 2 and J.M. Rossell 3

1Institute of Information Theory and Automation,Academy of Sciences of the Czech Republic,182 08 Prague 8, Czech Republic

2Department of Applied Mathematics IIIUniversitat Politecnica de Catalunya (UPC), Campus Nord, C-2, 08034-Barcelona, Spain

3Department of Applied Mathematics III

Universitat Politecnica de Catalunya (UPC), Campus Manresa, 08240-Manresa, Spain

Abstract. Overlapping quadratic optimal control of linear time-invariant systems and a

commutative class of continuous-time time-varying systems has been recently developed

by using a generalized structure of complementary matrices. It has been shown that these

structures offer a powerful and effective mean for decentralized control design for these

classes of systems. In this paper, these structures are presented for general linear time-

varying discrete-time systems. The results presented here concern the transition matrices

and explicit conditions on complementary matrices for this class of systems. It essentially

differs from continuous-time linear time-varying systems, where general results hold only

for aggregations and restrictions. A guideline for their selection is given. The effectiveness

of this generalized structure is illustrated by a numerical example of overlapping decen-

tralized control design.

Keywords. Decentralized control, large-scale systems, linear systems, discrete–time sys-

tems, time–varying systems, overlapping decompositions, optimal control.

AMS (MOS) subject classification: 93A14, 93A15, 93B17, 93C05, 93C35, 93C50

1 Introduction

Whenever possible, a model simplification in large-scale systems modelling,which results in smaller dimensional models keeping the most important fea-tures of the original system, is always a preferable and desirable procedure.One way to follow this line is a mathematical framework for comparing twodynamic systems known as Inclusion Principle. It states relations betweentwo dynamic systems with different dimensions under which solutions of thesystem with larger dimension include solutions of the system with smallerdimension. An original system with common shared parts (overlapping) maybe expanded into a larger space in order to become disjoint. Thereby, well-established decentralized control methods for disjoint subsystems may beused in the expanded space. Subsequently, such designed controllers arecontracted for implementation into the original space.

304 L. Bakule, J. Rodellar and J.M. Rossell

1.1 Relevant references

The inclusion principle has been developed by Siljak and his co-workers [6],[7], [12]. The relation between both systems is constructed usually on thebase of appropriate linear transformations between the corresponding sys-tems in the original and expanded spaces, where a key role in the selection ofappropriate structure of all matrices in the expanded space is played by theso called complementary matrices [6], [12]. The conditions on complemen-tary matrices, which are presented there, have implicit character in the sensethat it is difficult to select their specific values. In fact only two particularforms of aggregations and restrictions have been used for the control designand numerical computations [12]. Recently, a new generalized structure ofcomplementary matrices has been used as a mean to overcome this drawback[2], [3]. The validity of the general selection of complementary matrices isrestricted on the class of continuous-time LTV commutative systems and itsextension to discrete-time LTV systems is not at all straightforward [4]. Theimportance of the inclusion principle is underlined by various applications,for instance in applied mathematics [11], automated highway systems [13],flexible structures [1] or electric power systems [12].

One of the open research issues within the inclusion principle is the exten-sion of the results available for LTI systems to LTV systems. The availableresults in this direction for continuous-time LTV systems are in [4], [8] and[14]. To the authors knowledge, discrete-time LTV systems are consideredonly in [5], which serves as a preliminary version of this paper.

1.2 Outline of the paper

The strategy of generalized selection of complementary matrices has beenused as an effective tool to find both structure and optimal values of freeelements of complementary matrices for LTI systems. These free elementsplay an important role when considering suboptimality. The main part of thispaper concerns an extension of this strategy for overlapping state LQ controlfor discrete-time LTV systems including the contractibility conditions.

The paper is organized as follows. The problem is formulated in Sec-tion 2. The main results are presented in Section 3, identifying a new blockstructure of the complementary matrices that for the first time present ina compact form results for expansion-contraction of pairs of systems andoptimal control criteria. Subsection 3.1 presents general conditions on thecomplementary matrices in the LQ control of discrete-time LTV systems. Itincludes aggregations and restrictions as important particular cases. Subsec-tion 3.2 describes expansion-contraction process for the considered problemat a general level by using a new basis and including contractibility condi-tions. Subsection 3.3 outlines a guideline for selection of these matrices. InSection 4, a numerical example of the overlapping state LQ optimal controldesign is supplied to illustrate advantages of this procedure.

Time-Varying Discrete-Time Systems 305

2 Problem formulation

2.1 Preliminaries

Consider the optimal control problems as follows:

minu(k)

J(

x(k0), u(k)

)

= xT (kf)Πx(k

f)+

+

kf−1∑

k=k0

[

xT (k)Q∗(k)x(k) + uT (k)R∗(k)u(k)]

,

s.t.S : x(k + 1) = A(k)x(k) + B(k)u(k) (1)

and

minu(k)

J(

x(k0), u(k)

)

= xT (kf)Π x(k

f)+

+

kf−1∑

k=k0

[

xT (k)Q∗(k)x(k) + uT (k)R∗(k)u(k)]

,

s.t. S : x(k + 1) = A(k)x(k) + B(k)u(k), (2)

where k0

is the initial time, kf

is the final time and integers k∈[k0, k

0+

1, ..., kf]. The vectors x(k)∈R

n

, u(k)∈Rm

and x(k)∈Rn

, u(k)∈Rm

are the

states and inputs of S and S, respectively, at time k for k∈[k0, k

f]. The

matrices A(k), B(k) and A(k), B(k) have n×n, n×m and n×n, n×m di-mensions, respectively. Q∗(k), Q∗(k) are symmetric, nonnegative definitematrices of dimensions n×n, n×n, respectively. R∗(k), R∗(k) are symmet-ric, positive definite matrices of dimensions m×m, m×m, respectively. Π, Πare constant, symmetric, nonnegative definite matrices of dimensions n×n,n×n, respectively. In problems (1) and (2) the final time kf is fixed andx(kf ) is free. The minimization of J(x(k

0), u(k)) searches for a control u(k)

without an excessive effort able to maintain the state vector x(k) close to thezero required state at any time k∈[k0, kf ], with particular emphasis at theterminal time kf as weighted by matrix Π. It is well known that the solutionof the problem (1) exists, is unique and given in the form u(k)=−K(k)x(k).It is based on the solution of the corresponding Riccati equation [9]. If kf

is finite, this control law ensures a bounded state and the stability issuesare absent. If kf is infinite (with Π=0), the question of stability becomesimportant. There are results ensuring that this control guarantees that theclosed-loop system is exponentially stable under certain conditions related tocontrollability and observability [9]. We assume that the system S satisfiessuch conditions. Similar comments hold for problem (2). Suppose that thedimensions of the state and input vectors x(k), u(k) of S are smaller than (or


at most equal to) those of x(k), u(k) of S. Denote x(k)=x (k; x(k0), u(k)),

x(k)=x (k; x(k0), u(k)) the solutions of (1), (2) for given inputs u(k), u(k),

respectively. Given an initial time k0, an initial state x(k

0) and an input

signal u(k) defined for all k∈[k0, k

f], it is well known that

x(k) = Φ(k, k0)x(k

0) +

k−1∑

j=k0

Φ(k, j + 1)B(j)u(j),

x(k) = Φ(k, k0) x(k

0) +

k−1∑

j=k0

Φ(k, j + 1)B(j)u(j)

(3)

are the unique solutions of the systems (1) and (2), respectively. Φ, Φ arediscrete-time transition matrices [10]. Suppose that these sums are zero ifk=k0. Denote

Φ(k, j) = A(k − 1)A(k − 2) · · · A(j), k > j (4)

by adopting the convention that an empty product is the identity, Φ(k, j)=I,if k=j.

Let us consider the following linear transformations:

V : Rn

−→ Rn

, U : Rn

−→ Rn

,

R : Rm

−→ Rm

, Q : Rm

−→ Rm (5)

with rank(V )=n, rank(R)=m satisfying UV =In, QR=Im, where In, Im areidentity matrices of indicated dimensions. The systems S and S are re-lated through the transformations x(k)=V x(k), x(k)=Ux(k), u(k)=Ru(k),u(k)=Qu(k), where V , U , R and Q are constant matrices of appropriatedimensions and full ranks. Time-varying transformation matrices have beenconsidered in [14].

Definition 1 A system S includes the system S, that is S⊃S, if there existsa quadruplet of constant matrices (U, V, Q, R) such that UV = In, QR = Im

and for any initial state x(k0) and any fixed u(k) of S, the choice x(k

0) =

V x(k0) and u(k) = Ru(k) implies x(k; x(k

0), u(k)) = Ux(k; V x(k

0), Ru(k))

for all k∈[k0, k

f].

Definition 2 A pair (S, J) includes the pair (S, J), that is (S, J)⊃(S, J), ifS⊃S and J(x(k

0), u(k)) = J(V x(k

0), Ru(k)) for all k∈[k

0, k

f].

Definition 3 If (S, J)⊃(S, J), then (S, J) is called an expansion of (S, J)and (S, J) is a contraction of (S, J).

There exist two important cases satisfying S⊃S. They are called restric-tions and aggregations. They are given by the following definitions.


Definition 4 A system S is a restriction of S if there exists a pair (V, R)such that UV = In, QR = Im for some U , Q, and such that for any initialstate x(k

0) and any fixed input u(k) of S, the choice x(k

0) = V x(k

0) and

u(k) = Ru(k) implies x(k; x(k0), u(k)) = V x(k; x(k

0), u(k)) for all k∈[k

0, k

f].

Definition 5 A system S is an aggregation of S if there exists a pair (U, Q)such that UV = In, QR = Im for some V , R, and such that for any initialstate x(k

0) and any fixed input u(k) of S, the choice x(k

0) = Ux(k

0) and

u(k) = Qu(k) implies x(k; x(k0), u(k)) = Ux(k; x(k

0), u(k)) for all k∈[k

0, k

f].

Definition 6 A control law u(k) = −K(k) x(k) for S is contractible to thecontrol law u(k) = −K(k)x(k) for S if the choice x(k

0) = V x(k

0) and

u(k) = Ru(k) implies K(k)x(k; x(k0), u(k)) = QK(k)x(k; V x(k

0), Ru(k))

for all k∈[k0, k

f], any initial state x(k

0) and any fixed input u(k) of S.

We may also say that the gain matrix K(k) is contractible to the gainmatrix K(k). Contractibility implies that the expanded closed loop sys-tem x(k + 1)=[A(k)− B(k)K(k)]x(k) includes the closed loop system x(k +1)=[A(k) − B(k)K(k)]x(k).

2.2 The problem

Suppose given the pairs of matrices (U, V ) and (Q, R). Then the matricesA(k), B(k), Π, Q∗(k), R∗(k) and K(k) can be described as

A(k) = V A(k)U + M(k), B(k) = V B(k)Q + N(k),

Π = UT ΠU + MΠ, Q∗(k) = UT Q∗(k)U + MQ∗(k),

R∗(k) = QT R∗(k)Q + NR∗(k), K(k) = RK(k)U + F (k)

(6)

where M(k), N(k), MΠ, MQ∗(k), NR∗(k) and F (k) are called complementarymatrices.

The motivation of this work is to obtain a general structure of complemen-tary matrices for overlapping state linear quadratic optimal control problemfor discrete-time LTV systems. The specific goals are the following:• To derive explicit conditions on complementary matrices for discrete-timeLTV systems such that (S, J)⊃(S, J), including contractibility conditions.• To present a guideline for their selections.• To illustrate the derived results by using a numerical example.

3 Main results

This section gives the results covering mainly the expansion-contraction pro-cess for discrete-time LTV systems. Subsection 3.1 includes the general re-sults. Subsection 3.2 characterises the expansion-contraction process pre-sented at a general level by using a new basis and including contractibility


conditions. Subsection 3.3 offers a guideline for selection of complementarymatrices.

3.1 General discrete-time LTV systems

It is possible to give another definitions, equivalent to Definitions 1 and 6,respectively, in terms of discrete-time transition matrices.

Definition 7 Consider the systems S and S given by (1) and (2), respec-tively. A system S includes the system S if

U[

Φ(k, k0) x(k

0) +

k−1∑

j=k0

Φ(k, j + 1)B(j)u(j)]

=

= Φ(k, k0)x(k

0) +

k−1∑

j=k0

Φ(k, j + 1)B(j)u(j) (7)

for all k∈[k0, k

f].

Definition 8 A control law u(k) = −K(k) x(k) for S is contractible to thecontrol law u(k) = −K(k)x(k) for S if

QF (k)[

Φ(k, k0)V x(k

0) +

k−1∑

j=k0

Φ(k, j + 1)B(j)u(j)]

= 0 (8)

hold for all fixed k∈[k0, k

f].

Previously, define for integers r, s∈[k0, k

f] and any square matrix M the

matrix product M[s, r] as follows:

M[s, r] = M(s)M(s − 1) · · · M(r), s > r,

M[s, r] = M(s), s = r.(9)

M[s, r] is undefined for s < r.Now, we are ready to formulate the Inclusion Principle for discrete-time

LTV systems in terms of complementary matrices. Some proofs are omittedbecause they can be obtained easily.

Theorem 1 Consider (1) and (2). A pair (S, J)⊃(S, J) if and only if

UM [s, r]V = 0, UN(s)R = 0,

UM [q, p]N(p− 1)R = 0, V T MΠV = 0,

V T MQ∗(k)V = 0, RT NR∗(k)R = 0

(10)


f], all r, s such that k

06r6s6k− 1 and all p, q such

that k0+ 16p6q6k − 1.


Proof. The necessary and sufficient condition are proved simultaneously.Definition 7 is equivalent to the following conditions:1) U Φ(k, k

0)V x(k

0)=Φ(k, k

0)x(k

0), and

2)∑k−1

j=k0

U Φ(k, j+1)B(j)u(j)=∑k−1

j=k0

Φ(k, j+1)B(j)u(j) for all k∈[k0, k

f].

Substitute Φ(k, k0) = A(k−1)A(k−2)···A(k

0) into 1) where A(k)=V A(k)U+

M(k). Substitute Φ(k, j+1)=A(k−1)A(k−2)···A(j+1) and B(j)=V B(j)Q+N(j) into 2). Then, if k=k

0, 1) and 2) hold directly. From 1), for all

fixed k∈[k0

+ 1, kf] and comparing both sides, we obtain UM [s, r]V =0 for

all s∈[k0, k − 1] and all r∈[k

0, s], i.e. UM [s, r]V =0 for all r,s such that

k06r6s6k−1. By comparing both sides of 2) for all fixed k∈[k

0+1, k

f], the

equation is equivalent to UN(s)R=0, UM [q, p]N(p−1)R=0, for all s∈[k0, k−

1] and all p, q such that k0

+ 16p6q6k − 1. The remaining conditions areobtained readily when considering J(x(k

0), u(k))=J(V x(k

0), Ru(k)).

Note. It is important to recognize that it is not necessary to know thetransition matrices explicitly in order to select the complementary matricessatisfying the required conditions.

Proposition 1 Consider S and S given by (1) and (2), respectively. A pair(S, J)⊃(S, J) if V T MΠV = 0, V T MQ∗(k)V = 0, RT NQ∗(k)R = 0 andeither

a) UM [s, r] = 0, UN(s) = 0 or

b) M [s, r]V = 0, N(s)R = 0(11)


f] and all r, s such that k

06r6s6k − 1.

Proof. The proof follows immediately from Theorem 1.

The conditions a) and b) are two independent sets of sufficient conditionsfor (S, J) to be an expansion of (S, J).

Proposition 2 The system S is a restriction of the system S if and only if

M [k − 1, r]V = 0, UM [n, m]V = 0,

UN(n)R = 0, N(k − 1)R = 0,

UM [u, t]N(t − 1)R = 0, M [k − 1, p]N(p − 1)R = 0

(12)


f], all r such that k

06r6k − 1, all m, n such that

k06m6n6k − 2, all t, u such that k

0+ 16t6u6k − 2, and all p such that

k0

+ 16p6k − 1.

Proof. Imposing the conditions x(k; x(k0), u(k)) = V x(k; x(k

0), u(k)) given

by Definition 4 and following an analogous process as Theorem 1, the proofis concluded.

Proposition 3 The system S is an aggregation of the system S if and onlyif

UM [s, k0] = 0, UM [q, p]V = 0,

UN(s) = 0, UM [q, p]N(p− 1) = 0(13)



f], all s∈[k

0, k − 1] and all p, q such that k

0+

16p6q6k − 1.

Proof. Imposing the conditions x(k; x(k0), u(k)=Ux(k; x(k

0), u(k)) given

by Definition 5 and following an analogous process as that one in Theorem1 the proof is concluded.

Proposition 4 The system S is a restriction of the system S if M(s)V = 0and N(s)R = 0 for all s∈[k

0, k − 1].

Proposition 5 The system S is an aggregation of the system S if UM(s) =0 and UN(s) = 0 for all s∈[k

0, k − 1].

The sufficient conditions given by Propositions 4 and 5 in order that S⊃S

correspond with usual conditions on complementary matrices when consid-ering restrictions and aggregations.

Note. In (12), the conditions M [k − 1, r]V =0 and UM [n, m]V =0 im-ply UM [s, r]V =0 in (10). The relations UN(n)R=0, N(k − 1)R=0 implyUN(s)R=0. Finally, UM [u, t]N(t − 1)R=0, M [k − 1, p]N(p − 1)R=0 implyUM [q, p]N(p − 1)R=0 for all fixed k and all m, n, p, q, r, s, t, u belonging tothe corresponding intervals. Obviously, it is a consequence of the fact thata restriction is a particular case of expansion. A similar note can be makewhen considering aggregations.

Proposition 6 A control law u(k) = −K(k) x(k) for S is contractible to thecontrol law u(k) = −K(k)x(k) for S if

QF (k)V = 0, QF (k)M [k − 1, r]V = 0,

QF (k)N(k − 1)R = 0, QF (k)M [k − 1, p]N(p− 1)R = 0(14)


f], all r∈[k

0, k − 1] and all p∈[k

0+ 1, k − 1].

Proof. Consider the following three conditions:

1) QF (k)Φ(k, k0)V =0,

2) QF (k)∑k−1

j=k0

Φ(k, j + 1)V B(j)=0 and

3) QF (k)∑k−1

j=k0

Φ(k, j + 1)N(j)R=0

for all fixed k∈[k0, k

f], which imply Definition 8. Substitute Φ(k, k

0) = A(k−

1)A(k − 2) · · · A(k0) into 1) and Φ(k, j + 1)=A(k − 1)A(k − 2) · · · A(j + 1)

into 2) and 3) where A(k)=V A(k)U +M(k). When k=k0, only the equation

QF (k0)V =0 is defined. Suppose that QF (k)V =0 and QF (k)M [k−1, r]V =0

hold for all fixed k∈[k0+1, k

f] and all r∈[k

0, k−1] which imply 1). Moreover,

1) implies 2). Suppose that QF (k)N(k−1)R=0 and QF (k)M [k−1, p]N(p−1)R=0 for all p∈[k

0+ 1, k − 1] which imply 3) and the proof is concluded.


3.2 Expansion–contraction process

Change of basis

The change of basis in the expansion-contraction process introduced in [7],[12] represents S in a canonical form. Since the inclusion principle does notdepend on the specific basis used in the state, input and output spaces, wemay introduce convenient changes of basis in S for a prespecified purpose [2].The expansion-contraction process between S and S can be illustrated in theform

S −→ S −−→˜S −−→ S −→ S,

Rn V

−→ Rn T

−1

A−−−→ R

n TA

−−→ Rn U

−→ Rn

,

Rm R

−→ Rm T

−1

B−−−→ R

m TB

−−→ Rm Q

−→ Rm

,

(15)

where ˜S denotes the expanded system in the new basis. Given full row-rank matrices V and R, define U=(V T V )−1V T , Q=(RT R)−1RT as theirpseudoinverses, respectively. Let us consider

TA

= (V WA) , T

B= (R W

B) , (16)

where WA, W

Bare chosen such that Im W

A=Ker U , Im W

B=Ker Q. By us-

ing these transformations, the conditions U V =In, V U=(

In 00 0

)

and QR=Im,

RQ=(

Im 00 0

)

can be easily verified, where V =T−1A

V =(

In

0

)

, U=UTA=( In 0 )

and R=T−1B

R=(

Im

0

)

, Q = QTB=( Im 0 ). Note that the motivating factor for

defining TA

and TB

is the fulfillment of these conditions. They play a crucialrole in deriving explicit block structured complementary matrices (with zeroblocks) including a general strategy for their selection.

Expansion-contraction in the new basis

Consider the system S partitioned into 3×3 blocks, where the diagonal sub-matrices Aii(k), Bii(k), i=1, 2, 3 are ni×ni, ni×mi matrices, respectively.This structure has been adopted as a prototype structure for overlappingdecompositions [7], [8], [12].

Define (˜S, ˜J) as follows:

min˜u(k)

˜J(

˜x(k0), ˜u(k)

)

= ˜xT (kf) ˜Π ˜x(k

f)+

+

kf−1∑

k=k0

[

˜xT (k) ˜Q∗(k)˜x(k) + ˜uT (k) ˜R∗(k)˜u(k)]

,

s.t. ˜S : ˙x(k + 1) = ˜A(k) ˜x(k) + ˜B(k) ˜u(k) (17)

for all k∈[k0, k

f], where ˜A(k), ˜B(k), ˜Π, ˜Q∗(k) and ˜R∗(k) are matrices of

appropriate dimensions. The vectors ˜x(k) and ˜u(k) are defined as

˜x(k) = T−1A

V x(k) = V x(k), ˜u(k) = T−1B

Ru(k) = Ru(k). (18)


Now, the relations between ˜S and S are defined as

˜A(k) = V A(k)U + M(k), ˜B(k) = V B(k)Q + N(k),

˜Π = UT ΠU + MΠ, ˜Q∗(k) = UT Q∗(k)U + MQ∗(k),

˜R∗(k) = QT R∗(k)Q + NR∗(k),

(19)

where new complementary matrices are

M(k) = T−1A

M(k)TA, N(k) = T−1

AN(k)T

B,

MΠ = T TA

MΠTA, MQ∗(k) = T T

AMQ∗(k)T

A,

NR∗(k) = T TB

NR∗(k)TB.

(20)

First, we analyze the structure of the matrices M(k), N(k), MΠ, MQ∗(k)

and NR∗(k) in the expanded system. Consider the matrices of S in theform M(k)=(Mij(k)), N(k)=(Nij(k)), MΠ=(MΠij ), MQ∗(k)=(MQ∗

ij(k)),

NR∗(k)=(NR∗

ij(k)) for i, j=1, ..., 4, where MΠij

=MTΠji

, MQ∗

ij(k)=MT

Q∗

ji(k),

NR∗

ij(k)=NT

R∗

ji(k) and all matrices have corresponding dimensions. Con-

sider M(k)=(

M11(k) M12(k)

M21(k) M22(k)

)

, N(k)=(

N11(k) N12(k)

N21(k) N22(k)

)

, MΠ=(

MΠ11MΠ12

MTΠ12

MΠ22

)

,

MQ∗(k)=

(

MQ∗

11(k) MQ∗

12(k)

MTQ∗

12

(k) MQ∗

22(k)

)

, NR∗(k)=

(

NR∗

11(k) NR∗

12(k)

NTR∗

12

(k) NR∗

22(k)

)

where M11(k),

M22(k) are n×n, n2×n2 matrices, respectively. N11(k), N22(k) are n×m,n2×m2 matrices, respectively. MΠ11

, MΠ22are n×n, n2×n2 matrices, respec-

tively. MQ∗

11(k), MQ∗

22(k) are n×n, n2×n2 matrices, respectively. NR∗

11(k),

NR∗

22(k) are m×m, m2×m2 matrices, respectively. We need to know the

conditions that these submatrices must verify.

Theorem 2 A system ˜S includes the system S if and only if M(s) and N(s)

have the structure M(s) =(

0 M12(s)

M21(s) M22(s)

)

, N(s) =(

0 N12(s)

N21(s) N22(s)

)

and

M12(p)M21(p − 1) = 0, M12(p)M22[p − 1, j]M21(j − 1) = 0,

M12(p)N21(p − 1) = 0, M12(p)M22[p − 1, j]N21(j − 1) = 0(21)


f], all s∈[k

0, k − 1], all p∈[k

0+ 1, k − 1] and all

j∈[k0

+ 1, p − 1].

Proof. Denote M(k)=(

M11(k) M12(k)

M21(k) M22(k)

)

. Consider UM [s, r]V =0, for k06

r6 s6 k − 1 given by Theorem 1. UM [k0, k

0]V =0 implies M11(k0

)=0 fork=k

0+1. We obtain M11(k0

)=0, M11(k0+1)=0 and M12(k0

+1)M21(k0)=0

for k=k0+2. We get M11(s)=0 together with the conditions M12(p)M21(p−

1)=0, M12(p)M22[p − 1, j]M21(j − 1)=0 for all s∈[k0, k − 1], all p∈[k

0+

1, k − 1] and all j∈[k0

+ 1, p − 1] when repeating this process. Similarly, theconditions M12(p)N21(p−1)=0, M12(p)M22[p−1, j]N21(j−1)=0 are obtainedby using the relations U N(s)R=0, UM [q, p]N(p−1)R=0, respectively, givenby Theorem 1.


Proposition 7 Consider S and ˜S given in (1) and (17), respectively. A

pair (˜S, ˜J)⊃(S, J) if MΠ =(

0 MΠ12

MTΠ12

MΠ22

)

, MQ∗(k) =

(

0 MQ∗

12(k)

MTQ∗

12

(k) MQ∗

22(k)

)

,

NR∗(k) =

(

0 NR∗

12(k)

NTR∗

12

(k) NR∗

22(k)

)

and either

a) M(p) =(

0 0M21(p) M22(p)

)

, N(p) =(

0 0N21(p) N22(p)

)

or

b) M(p) =(

0 M12(p)

0 M22(p)

)

, N(p) =(

0 N12(p)

0 N22(p)

)

(22)


f] and all p∈[k

0+ 1, k − 1].

Proof. Considering the conditions a) and b) given by Proposition 1, respec-

tively, in the new expanded system ˜S, the proof is concluded.

Contractibility

The idea is to design a control law for S so that it can be contracted andimplemented into S. Now, we want to determine the conditions under which a

control law designed in ˜S can be contracted into S in terms of complementarymatrices.

Denote matrices appearing in the contractibility process as follows. Thecomplementary matrix F (k) has the form F (k)=(Fij(k)), i, j=1, ..., 4, whereF11(k), F22(k), F33(k) and F44(k) are m1×n1, m2×n2, m2×n2 and m3×

n3 matrices, respectively. Define F (k)=(

F11(k) F12(k)

F21(k) F22(k)

)

, where F11(k) and

F22(k) are m×n and m2×n2 matrices, respectively. Similarly, denote thegain matrix K(k)=(Kij(k)), i, j=1, 2, 3, where Kii(k) are mi×ni matrices,

respectively. The gain matrix ˜K(k) for ˜S has the form ˜K(k)=RK(k)U+F (k),

where ˜K(k)=T−1B

K(k)TA

and

F (k) = T−1B

F (k)TA. (23)

So far we do not know the form of the complementary matrix F (k) andthe corresponding contractibility conditions. The following theorem solvesthe problem.

Proposition 8 A control law ˜u(k) = −˜K(k)˜x(k) for ˜S is contractible to the

control law u(k) = −K(k)x(k) of S if F (k) =(

0 F12(k)

F21(k) F22(k)

)

and

F12(k)M21(k − 1) = 0, F12(k)N21(k − 1) = 0,

F12(k)M22[k − 1, p]M21(p − 1) = 0, F12(k)M22[k − 1, p]N21(p − 1) = 0(24)


f] and all p∈[k

0+ 1, k − 1].


Proof. Suppose F (k)=(

F11(k) F12(k)

F21(k) F22(k)

)

. Consider the conditions by Proposi-

tion 6. Only the relation QF (k)V =0 is defined for k=k0, and then F11(k0

)=0is obtained. From QF (k)V =0, QF (k)M [k − 1, r]V =0 and QF (k)N(k −

1)R=0 we get F11(k0+1)=0, F12(k0

+1)M21(k0)=0 and F12(k0

+1)N21(k0)=0

for k=k0

+ 1, respectively. Repeating this process for all k, F11(k)=0,F12(k)M21(k− 1)=0, F12(k)N21(k− 1)=0, F12(k)M22[k− 1, p]M21(p− 1)=0and F12(k)M22[k − 1, p]N21(p − 1)=0 for all fixed k∈[k

0, k

f] and all p∈[k

0+

1, k − 1] are obtained and the proof is concluded.

3.3 Guideline for selection of complementary matrices

The above results give a formal block structure of the complementary ma-trices M(k), N(k), MΠ, MQ∗(k), NR∗(k) and F (k) in the new basis. Theyare general, in the sense that the obtained structure does not depend on anyparticular selection of the transformation matrices V and R. Therefore, theyhold for any expansion-contraction process.

However, an important practical question arises when the goal is to selectspecific numerical values of the complementary matrices for a given problem.In this case, it is necessary to translate the structure of the complementarymatrices M(k), . . . , NR∗(k) and F (k) into the corresponding structure of thecomplementary matrices M(k), . . . , NR∗(k) and F (k) in the original basis.To do this, specific matrices V and R have to be selected to expand thegiven problem, such that the changes of basis TA and TB in (16) are defined,which are needed to relate the matrices M(k), . . . , NR∗(k) and F (k) withthose M(k), . . . , NR∗(k) and F (k) as given in (20) and (23). The choice ofthe expansion matrices is limited by the information structure constraintsrequiring the preservation of the integrity of the local feedback and subsys-tems in overlapping decentralized control. Although the resulting structureof the complementary matrices M(k), . . . , NR∗(k) and F (k) is then prob-lem dependent, it is sufficiently general to give a rich margin for selectingnumerical values. For this selection, the designer can follow different speci-fications. It can be advisable to simplify the structure as much as possiblewithin given constraints by fixing a possible maximum of zero blocks andleaving nonzero blocks with some free parameters for optimization. The au-thors believe that it is not possible to give an explicit numerical algorithmfor a general scenario. The above recommendations can serve as a guidelineto build a selection procedure for any given problem.

In the remaining of this section, the structure of the complementary ma-trices in the initial basis is derived for a specific expansion scheme. Further,the next section gives a numerical example.

We select the following expansion transformation matrices, which are the


most frequently used in the literature [12]:

V =

In10 0

0 In20

0 In20

0 0 In3

, R =

Im10 0

0 Im20

0 Im20

0 0 Im3

. (25)

By using these transformations, x2(k) and u2(k) appear in a repeated form in

x(k)=(

xT1 (k), xT

2 (k), xT2 (k), xT

3 (k))T

and u(k)=(

uT1 (k), uT

2 (k), uT2 (k), uT

3 (k))T

by using (25), respectively. The change of basis results in

TA

=

In10 0 0

0 In20 In2

0 In20 −In2

0 0 In30

(26)

and its corresponding inverse T−1A

. The matrices TB, T−1

Bhave an analogous

structure. The following theorems present the structure of the complemen-tary matrices M(k), N(k), MΠ, MQ∗(k), NR∗(k) and F (k) in the initialbasis.

Theorem 3 Consider S and S given by (1) and (2), respectively. A systemS includes the system S if and only if M(s) has the following structure

M(s) =

(

0 M12 −M12 0M21 M22 M23 M24

−M21 −(M22+M23+M33) M33 −M24

0 M42 −M42 0

)

(s) (27)

and their blocks satisfy(

M12

M23+M33

M42

)

(p)

(

M21 M22+M23 M24

)

(p − 1) = 0,

(

M12

M23+M33

M42

)

(p)

(

M22+M33

)

[

p − 1, j]

(

M21 M22+M23 M24

)

(j − 1) = 0,

(

M12

M23+M33

M42

)

(p)

(

N21 N22+N23 N24

)

(p − 1) = 0,

(

M12

M23+M33

M42

)

(p)

(

M22+M33

)

[

p − 1, j]

(

N21 N22+N23 N24

)

(j − 1) = 0

(28)for all fixed k∈[k

0, k

f], all s∈[k

0, k − 1], all p∈[k

0+ 1, k − 1] and all j∈[k

0+

1, p − 1], where the matrix N(s) has the same structure as M(s).

Proof. Consider M(s)=T−1A

M(s)TA. From Theorem 2, M11(s)=0 and

the remaining matrix blocks Mij(s), i, j=1, 2, can be identified and M(s)is obtained. An analogous procedure holds for the matrix N(s) when ap-plying Theorem 2. Now, (28) is obtained imposing M12(p)M21(p − 1)=0,M12(p)M22[p− 1, j]M21(j − 1)=0, M12(p)N21(p− 1)=0 and M12(p)M22[p−1, j]N21(j − 1)=0 for all fixed k∈[k

0, k

f], all p∈[k

0+ 1, k − 1] and all j∈[k

0+

1, p − 1] given by Theorem 2.


We may identify two important cases from (28):

a) M12(p) = 0,(

M23 + M33

)

(p) = 0, M42(p) = 0,

b) M21(p − 1) = 0,(

M22 + M23

)

(p − 1) = 0, M24(p − 1) = 0,

N21(p − 1) = 0,(

N22 + N23

)

(p − 1) = 0, N24(p − 1) = 0

(29)

for all fixed k∈[k0, k

f] and all p∈[k

0+1, k−1]. The above case b) corresponds

with restrictions.

Proposition 9 Consider S and S given by (1) and (2). A pair (S, J)⊃(S, J)if the matrices MΠ, MQ∗(k), NR∗(k) have the structure

MΠ =

0 MΠ12−MΠ12

0

MTΠ12

−MΠ23−MT

Π23−MΠ33

MΠ23MΠ24

−MTΠ12

MTΠ23

MΠ33−MΠ24

0 MTΠ24

−MTΠ24

0

,

MQ∗(k) =

0 MQ∗

12−MQ∗

120

MTQ∗

12

−MQ∗

23−MT

Q∗

23

−MQ∗

33MQ∗

23MQ∗

24

−MTQ∗

12

MTQ∗

23

MQ∗

33−MQ∗

24

0 MTQ∗

24

−MTQ∗

24

0

(k),

NR∗(k) =

0 NR∗

12−NR∗

120

NTR∗

12

−NR∗

23−NT

R∗

23

−NR∗

33NR∗

23NR∗

24

−NTR∗

12

NTR∗

23

NR∗

33−NR∗

24

0 NTR∗

24

−NTR∗

24

0

(k),

(30)

and either

a) M(p) =

( 0 0 0 0M21 M22 M23 M24

−M21 −M22 −M23 −M24

0 0 0 0

)

(p),

N(p) =

(

0 N12 −N12 0N21 N22 N23 N24

−N21 −(N22+N23+N33) N33 −N24

0 N42 −N42 0

)

(p)

or

b) M(p) =

(

0 M12 −M12 00 M22 −M22 00 M32 −M32 00 M42 −M42 0

)

(p), N(p) =

(

0 N12 −N12 00 N22 −N22 00 N32 −N32 00 N42 −N42 0

)

(p)

(31)


f] and all p∈[k

0+ 1, k − 1].

Proof. The proof of the assertion is straightforward when applying Propo-sition 7.

Proposition 10 A control law u(k) = −K(k)x(k) for S is contractible to

u(k) = −K(k)x(k) of S if F (k) =

(

0 F12 −F12 0F21 F22 F23 F24

−F21 −(F22+F23+F33) F33 −F24

0 F42 −F42 0

)

(k) and


the equations

(

F12

F23+F33

F42

)

(k)

(

M21 M22+M23 M24

)

(k − 1) = 0,

(

F12

F23+F33

F42

)

(k)

(

N21 N22+N23 N24

)

(k − 1) = 0,

(

F12

F23+F33

F42

)

(k)

(

M22+M33

)

[

k − 1, p]

(

M21 M22+M23 M24

)

(p − 1) = 0,

(

F12

F23+F33

F42

)

(k)

(

M22+M33

)

[

k − 1, p]

(

N21 N22+N23 N24

)

(p − 1) = 0

(32)hold for all fixed k∈[k

0, k

f] and all p∈[k

0+ 1, k − 1].

Proof. Consider F (k)=T−1B

F (k)TA

with T−1B

, TA

given by (26). F11(k)=0and the other matrix blocks Fij(k), i, j=1, 2 can be identified from Proposi-tion 8. Thus, the structure of F (k) is obtained. Further, we get (32) imposing(24).

From Proposition 10 by choosing F12(k)=0,(

F23+F33

)

(k)=0 and F42(k)=0

for all k∈[k0, k

f], any control law designed for S is contractible to S.

4 An example

Objective

Consider the problem (1) with the specific matrices

A(k) =

1 0 0 p

p−1

− − − −

kp

p−1 −1 p

p0

0 p

p0 0 p

pk

− − − − −

1 p

p0 −2 0

, B(k) =

1 0 p

p−1

− − −

0 p

p0 p

pk

1 p

p−1 p

p0

− − − −

0 p

p−1 0

, (33)

Π=Q∗=diag(1, 1, 1, 1), R∗=diag(1, 1, 1) and initial state x(k0)=(1, 1, 1, 1)

T.

The overlapping decomposition is determined by dashed lines. Consider theinitial and the final time as k

0=0 and kf=3, respectively. It is well known

that a discrete-time time-varying system is controllable if and only if the

matrixk

f∑

k=k0

Φ(kf , k)B(k)BT (k)Φ(kf , k) is invertible [9]. The system (33) is

controllable.The objective is to show the potential advantages that offer the charac-

terization of the presented complementary matrices for an overlapping de-centralized state LQ optimal control design.

We consider the following steps:


1) The pair (S, J) in (1) is expanded to (S, J) by using the transformationsV and R given in (25). The system S can be represented as:

S : x(k + 1) = AD

(k) x(k) + BD

(k) u(k) + AC(k) x(k) + B

C(k) u(k) (34)

with AD

(k), BD

(k) block diagonal matrices and AC(k), B

C(k) the corre-

sponding interconnection matrices. Check the controllability of the pair(A

D(k), B

D(k)).

2) A decentralized control law uD(k)=−K

D(k) x(k) is designed for the de-

coupled expanded system SD

: x(k + 1)=AD(k) x(k) + B

D(k) u(k), where

KD

(k) satisfies the equation

KD

(k) =

R∗+ BTD

(k)[Q∗+ PD

(k +1)]BD(k)−1

BTD

(k)[Q∗+PD

(k+1)]AD

(k)(35)

together with

PD

(k) = ATD

(k)[Q∗ + PD

(k + 1)][AD

(k) − BD

(k)KD(k)] (36)

for all k=k0, ..., k

f−1, with the boundary condition P

D(kf )=Π

D. Select F (k)

satisfying the contractibility conditions given by Proposition 10.

3) Contract uD(k)=−QK

D(k)V x(k)=−K

D(k)x(k) to be implemented into

S. The value of the performance index (1) obtained by uD

(k) is not optimal,in general. We can compare J(x(k

0)) of the centralized optimal control law

that minimizes (1) with the value obtained by the implementation of thecontrol u

D(k) into S, which is denoted by J⊕(x(k

0)).

In this example we consider the matrices N(k)=0, MΠ=0, MQ∗(k)=0,

NR∗(k)=0 and F (k)=0, which are particular simple cases satisfying Propo-sitions 9 and 10. Consequently, because F (k)=0, the contraction of thecontrol law designed in the expanded space is possible. The complementarymatrix M(k) is selected within the cases described in the previous section andJ⊕(x(k

0)) is computed. To evaluate and compare the results obtained by the

proposed method we will also consider the selection of M(k) correspondingto the case of restrictions.

Results

Overlapping decomposition by using a restriction. Choose the typical com-plementary matrix M(k) used in [8], [12]. Then, M(k) and A(k) are givenby

M(k) =

0 0 0 0 0 00 −0.5 −0.5 0.5 0.5 00 0 0 0 0 00 0.5 0.5 −0.5 −0.5 00 0 0 0 0 00 0 1 0 −1 0

, A(k) =

1 0 0 0 0 −1k −1 −1 0 0 00 0 0 0 0 kk 0 0 −1 −1 00 0 0 0 0 k1 0 0 0 −2 0

. (37)


The decoupled matrices AD

(k) and BD

(k) are in this situation given by

AD

(k) =

1 0 0 0 0 0k −1 −1 0 0 00 0 0 0 0 00 0 0 −1 −1 00 0 0 0 0 k0 0 0 0 −2 0

, B

D(k) =

1 0 0 00 0 0 01 −0.5 0 00 0 0 k0 0 −0.5 00 0 −0.5 0

. (38)

The contracted gain matrices, KD

(k), k=0, 1, 2, computed from step 3),are the following:

KD(0) =

(

0.56 −0.09 −0.09 0.000.28 0.28 0.87 0.000.00 0.00 0.00 0.00

)

, KD(1) =

(

0.64 0.20 0.20 0.000.19 0.26 1.04 −0.470.00 −0.41 −0.32 0.06

)

,

KD(2) =

(

0.54 0.00 0.00 0.000.18 0.00 0.80 −0.400.00 −0.40 −0.40 0.00

)

.

(39)

Overlapping decomposition by using the proposed method. Suppose the se-lection of the transformation matrices according to the guideline in Subsec-tion 3.3. Then, the zero blocks of M(k) have been selected in such a waythat M12(p)=0,

(

M23 + M33

)

(p)=0,(

M22 + M23

)

(p − 1)=0, M24(p − 1)=0

and(

M22 + M33

)

[p − 1, j]=0 for all fixed k∈[k0, k

f], all s∈[k

0, k − 1], all

p∈[k0+1, k− 1] and all j∈[k

0+1, p− 1]. The remaining nonzero blocks have

been considered with a fixed submatrix M42(p)=(

0 1)

and M21(p−1)=(

−k0

)

with a free parameter k for optimization. It satisfies the necessary constraintM42(p)M21(p− 1)=0. Such particular selection satisfies the constraints (28).M(k) and A(k) are given by

M(k) =

0 0 0 0 0 0−k 0 0 0 0 0

0 0 0 0 0 0k 0 0 0 0 00 0 0 0 0 00 0 1 0 −1 0

, A(k) =

1 0 0 0 0 −10 −0.5 −0.5 −0.5 −0.5 00 0 0 0 0 k2k −0.5 −0.5 −0.5 −0.5 00 0 0 0 0 k1 0 0 0 −2 0

. (40)

The decoupled matrices AD

(k) and BD

(k) are

AD

(k) =

1 0 0 0 0 00 −0.5 −0.5 0 0 00 0 0 0 0 00 0 0 −0.5 −0.5 00 0 0 0 0 k0 0 0 0 −2 0

, BD(k) =

1 0 0 00 0 0 01 −0.5 0 00 0 0 k0 0 −0.5 00 0 −0.5 0

. (41)

The matrix M(k) satisfies the structural requirements given by Theorem 3.The corresponding contracted gain matrices K

D(k) are given by

KD(0) =

( 0.55 −0.03 −0.03 0.000.13 0.03 0.63 0.000.00 0.00 0.00 0.00

)

, KD(1) =

( 0.55 −0.03 −0.03 0.000.15 0.04 0.83 −0.470.00 −0.18 −0.15 0.02

)

,

KD(2) =

(

0.55 0.00 0.00 0.000.18 0.00 0.80 −0.400.00 −0.20 −0.20 0.00

)

.

(42)

Following the steps given above, we can summarize the obtained resultsas follows:

proposed method restriction centralized case

J⊕ =12.8 J⊕ = 38.3 J = 6.7


Note that J is the cost for the centralized optimal control solving (1). Sincethe goal of a decentralized control is to drive the system as close as possible tothe (ideal) centralized control, we may observe essentially better performancewhen using the proposed method for the selection of the complementarymatrices. Thereby, we have reduced the cost approximately to one thirdcompared with the cost in the case of restriction. This method illustratesthe freedom introduced by this approach to select the convenient structure ofcomplementary matrices to reduce the cost in overlapping quadratic optimalcontrol.

By comparing (33) and (37) (restriction), it can be seen that the expandedsystem preserves the identity of the initial overlapping structure, as the orig-inal overlapping subsystems can be recognized within the matrix A(k). Thisidentity is not maintained in the case of the proposed expansion procedure(40). However this is not a basic matter of concern. The expanded sys-tem is only an artificial mathematical construction and the crucial issue isthat, with the proposed complementary matrices, the inclusion principle isguaranteed. Moreover, the expanded system (40) has a structure with sub-systems of equal dimensions as the original overlapping subsystems. This is astructural constraint that is preserved in the proposed approach. The benefitof this approach may be recognized in a significant reduction of the perfor-mance index satisfying simultaneously conditions on the original informationstructure constraints of controllers.

5 Conclusion

The Inclusion Principle has been specialized for a quadratic optimal controldesign for general discrete-time LTV systems. The generalized structure ofcomplementary matrices has been presented for this class of systems. Ex-plicit conditions on their structure have been proved, including conditions oncontractibility of controllers. In the case of continuous-time LTV systems,these structures have been recently available only for the specific class ofcommutative systems. In contrast to this case, it has been shown that thegeneralized structure of complementary matrices can be obtained withoutany restrictions for general discrete-time LTV systems. A guideline for selec-tion of complementary matrices is included when considering a particular setof expansion/contraction transformation matrices. The effectiveness of theuse of these generalized structures has been demonstrated on an illustrativeexample.

6 Acknowledgements

This work was supported in part by the DURSI under Grant PIV, by theASCR under Grant A2075304 and by the CICYT under Grant DPI2002-04018-C02-01.


7 References

[1] L. Bakule, and J. Rodellar, Decentralized control and overlapping decomposition ofmechanical systems. Part 1: System decomposition. Part 2: Decentralized stabiliza-tion, Internat. J. of Control, 61, (1995) 559-587.

[2] L. Bakule, J. Rodellar, and J.M. Rossell, Structure of expansion-contraction matricesin the inclusion principle for dynamic systems, SIAM J. Matrix Anal. Appl., 21(4),(2000) 1136-1155.

[3] L. Bakule, J. Rodellar, and J.M. Rossell, Generalized selection of complementarymatrices in the inclusion principle, IEEE Trans. Automat. Control, 45(6), (2000)1237-1243.

[4] L. Bakule, J. Rodellar, and J.M. Rossell, Overlapping quadratic optimal control oflinear time-varying commutative systems, SIAM J. Control Optim., 40(5), (2002)1611-1627.

[5] L. Bakule, J. Rodellar, and J.M. Rossell, Overlapping LQ control of discrete-timetime-varying systems, Large Scale Systems: Theory and Applications 2001, Proc.IFAC, Pergamon, Oxford, (2002) 425-430.

[6] M. Ikeda, and D.D. Siljak, Overlapping decompositions, expansions and contractionsof dynamic systems, Large Scale Systems, 1, (1980) 29-38.

[7] M. Ikeda, D.D. Siljak, and D.E. White, An inclusion principle for dynamic systems,IEEE Trans. Automat. Control, 29, (1984) 244-249.

[8] M. Ikeda, D.D. Siljak, and D.E. White, Overlapping decentralized control of lineartime-varying systems, Advances in Large Scale Systems, JAI Press, 1 (1984) 93-116.

[9] H. Kwakernaak, and R.Sivan, Linear Optimal Control Systems, Wiley, New York,1972.

[10] W.J. Rugh, Linear System Theory (2nd ed.), Prentice Hall, Upper Saddle River, NewJersey, 1996.

[11] M.E. Sezer, and D.D. Siljak , Nested epsilon decomposition of linear systems: weaklycoupled and overlapping blocks, SIAM J. Matrix Anal. Appl., 12(3), (1991) 521-533.

[12] D.D. Siljak, Decentralized Control of Complex Systems, Academic Press, New York,1991.

[13] D.D. Siljak, S.M. Mladenovic, and S. Stankovic, Overlapping decentralized observa-tion and control of a platoon of vehicles, Proc. of the American Control Conference,

San Diego, California, (1999) 4522-4526.

[14] S.S. Stankovic, and D.D. Siljak, Inclusion principle for time-varying systems, SIAM

J. Control Optim., 42(1), (2003) 321-341.

email:[email protected]://monotone.uwaterloo.ca/∼journal/


INCLUSION PRINCIPLE FOR DISCRETE-TIMETIME-VARYING SYSTEMS

Srdjan S. Stankovic

Faculty of Electrical EngineeringUniversity of Belgrade, Belgrade, Serbia and Montenegro

Abstract. In this paper the concept of inclusion is applied to dynamic linear discrete-time

time-varying dynamic systems. Starting from general definitions, which include restriction

and aggregation as special cases, inclusion conditions are derived. It is proved that any

inclusion can be considered as a specific composition of sequences of restriction/ aggre-

gation pairs. The paper contains also a formulation of the contractibility conditions for

implementation of time-varying state-feedback controllers.

Keywords. Inclusion principle, restriction, aggregation, discrete-time time-varying sys-

tems

1 Introduction

A considerable attention among researchers in the field of complex and large-scale systems has been paid to the inclusion principle, which has been foundto provide a convenient framework for diverse approaches to both analysisand design. The basic idea, applied to linear time-invariant systems, hasbeen presented in [16], starting from the fundamental geometric arguments[1,31]. It has been found that restriction and aggregation represent specialcases of inclusion, having fundamental theoretical and practical importance.Numerous successful applications of the inclusion principle to control design,including decentralized overlapping control and reduced order design, havebeen reported, e.g. [13,16,22,23,26,27]. Considerable attention is paid to thepractically important problem of controller contractibility (expandability), inthe sense of ensuring contraction/expansion of controllers from one statespace to another [2,3,9,10,11,12,23,28]. Applications of the inclusion con-cept to performance indices, closed-loop systems with dynamic controllers,stochastic systems and nonlinear systems have been described, for exam-ple, in [7,9,10,19,24]. However, the application of the inclusion principle todynamic time-varying systems has remained insufficiently clarified. In [15],this problem has been considered for the first time from the point of viewof time-invariant expansions/contractions applied to the state inclusion, inthe context of restriction and aggregation. In [7] the focus is on the con-tractibility of dynamic controllers for discrete-time time-varying systems with

324 S. S. Stankovic

time-invariant expansions/contractions. In [4,5] the attention is still paid totime-invariant expansions/contractions. In [30] a general treatment of theproblem has been provided for linear continuous-time time-varying systemsand time-varying exoansions/contractions.

In this paper, general aspects of the inclusion principle applied to lineardiscrete-time time-varying dynamic systems are presented. Starting froma general formulation of the principle, inclusion conditions are derived interms of system characteristics under time-varying input/state/output ex-pansion/contraction relationships. Further, attention is paid to the input/state /output restrictions and aggregations. Their properties are discussedfrom the geometric point of view, including the definition of a variety ofrestriction/aggregation types resulting from different expansion/contractionrelationships between the system variables. It is also proved that the com-

position property holds in the discrete-time time-varying case, in the sensethat any input/state/output inclusion on a given time interval can be de-composed into a sequence of time-varying input/state/output aggregationand input/state/output restriction pairs. A treatment of state feedback con-tractibility (expandibility) is also presented, including an elaboration of di-verse restriction/aggregation types.

In spite of obvious direct analogies between the continuous-time anddiscrete-time cases, specific properties of discrete-time systems deserve a par-ticular attention and a self-contained treatment within the inclusion frame-work. This is especially pronounced in relation with the composition prop-erty. Moreover, to the author’s knowledge, no systematic presentation of theinclusion concept applied to discrete-time systems has been presented yet.

2 Inclusion Principle for Discrete-Time Time-

Varying Systems

Consider a pair of linear discrete-time time-varying systems representedby

S : x(k + 1) = A(k)x(k) + B(k)u(k) (x(k0) = x0),

y(k) = C(k)x(k), (1)

S : x(k + 1) = A(k)x(k) + B(k)u(k) (x(k0) = x0),

y(k) = C(k)x(k), (2)

where k ∈ I = . . . ,−1, 0, 1, . . . is the set of integers, x ∈ X , u ∈ U and y ∈Y are n−, m− and l− vectors, and x ∈ X , u ∈ U and y ∈ Y n−, m− and l−vectors, respectively, satisfying n ≤ n, m ≤ m and l ≤ l. For k ≥ k0 the inputfunctions u[k0,k] : [k0, k] → U and u[k0,k] : [k0, k] → U belong to the sets Uf

and Uf , respectively. The elements of the matrices in S = (A(k), B(k), C(k))

Discrete-Time Time-Varying Inclusion 325

and S = (A(k), B(k), C(k)) functions of k. Denote by x(k; k0, x0, u[k0,k])and x(k; k0, x0, u[k0,k]) the unique solutions of (1) and (2) for the initial timek0, initial states x0 and x0, and fixed control functions u[k0,k] ∈ Uf and

u[k0,k] ∈ Uf , respectively. It is further assumed that the inputs, states and

outputs of S and S are related by linear time-varying expansion/contractiontransformations defined by the following monomorphisms: V (k) : X → X ,R(k) : U → U and T (k) : Y → Y and the following epimorphisms: U(k) :X → X , Q(k) : U → U and S(k) : Y → Y, so that

x(k) = V (k)x(k) or x(k) = U(k)x(k),

u(k) = R(k)u(k) or u(k) = Q(k)u(k),

andy(k) = T (k)y(k) or y(k) = S(k)y(k).

We also assume that U(k) = V (k)L, Q(k) = R(k)L and S(k) = T (k)L, wherethe superscript L denotes the left inverse.

Definition 1. The system S includes the system S on [ka, kb], i.e., S ⊂ S

(S is an expansion of S and, vice versa, S is a contraction of S) if thereexists a quadruplet of full rank matrices (U(k), V (k), R(k), S(k)) such thatfor any k0 ∈ [ka, kb], x0 ∈ X and u[k0,k] ∈ Uf , the choice x0 = V (k0)x0 andu(k) = R(k)u(k) implies x(k; k0, x0, u[k0,k]) = U(k)x(k; k0, x0, u[k0,k]) andy[x(k)] = S(k)y[x(k)], for all k ∈ [k0, kb].

Introduce, as in [26,29,30], two ordered pairs of variables, composed ofindependent variables in the above expansion/contraction relationships. Thepair P1 corresponds to the variables x0, x0 and u, u that can be chosen exoge-nously, irrespective of the system dynamics, while the pair P2 correspondsto the variables x, x and y, y, reflecting the system behavior. For example,P1 = x0, u stands for x0 = V (k0)x0 and u(k) = Q(k)u(k), P1 = x0, u forx0 = U(k0)x0 and u(k) = R(k)u(k), P2 = x, y for x(k) = U(k)x(k) andy(k) = T (k)y(k), etc. The form P1 ⇒ P2 is a shorthand notation for differenttypes of inclusion in the sense of Definition 1. If, for example, P1 = x0, uand P2 = x, y, the form x0, u ⇒ x, y will be given the following mean-ing: conditions x0 = V (k0)x0 and u(k) = R(k)u(k) imply that, for anyx0 ∈ X and any u[k0,k] ∈ Uf , x(k; k0, x0, u[k0,k]) = V (k)x(k; k0, x0, u[k0,k])and y[x(k)] = S(k)y[x(k)], ∀k ≥ k0. We have now for Definition 1 the fol-lowing compact formulation:

Definition 1. S ⊂ S on [ka, kb] if

∃(U(k), V (k), R(k), S(k)) x0, u ⇒ x, y,

∀k0 ∈ [ka, kb].

326 S. S. Stankovic

Denote by Φ(k, j) and Φ(k, j) (k ≥ j) the transition matrices of S and S,respectively, given by

Φ(k, j) =

A(k − 1) · · ·A(j), k ≥ j + 1I, k = j

Φ(k, j) =

A(k − 1) · · · A(j), k ≥ j + 1I, k = j

Inclusion conditions, expressed in terms of the characteristics of S and S,are then:

Theorem 1. S ⊂ S on [ka, kb] iff ∀k0, k, j ∈ [ka, kb], k0 ≤ j ≤ k,

Φ(k, k0) = U(k)Φ(k, k0)V (k0),

Φ(k, j)B(j) = U(k)Φ(k, j)B(j)R(j),

C(t)Φ(k, k0) = S(k)C(k)Φ(k, k0)V (k0),

C(k)Φ(k, j)B(j) = S(k)C(k)Φ(k, j)B(j)R(j). (3)

Proof: According to Definition 1,

Φ(k, k0)x0 +

k−1∑

j=k0

Φ(k, j)B(j)u(j)

= U(k)Φ(k, k0)V (k0)x0 + U(k)

k−1∑

j=k0

Φ(k, j)B(j)R(j)u(j) (4)

wherefrom two first relations in (3) follow.The remaining relations are ob-tained similarly.

In the time-varying case it is not possible to express the inclusion condi-tions through direct and simple relations between the system matrices of S

and S as in the time-invariant case [14,16,23]. For example, the first relationin (3) gives for i = j + 1 and i = j + 2

A(j) = U(j + 1)A(j)V (j),

A(j + 1)A(j) = U(j + 2)A(j + 1)A(j)V (j).

The remaining relations in (3) give for i = j and i = j + 1

B(j) = U(j)B(j)R(j), C(j) = S(j)C(j)V (j),

A(j)B(j) = U(j + 1)A(j)B(j)R(j)

C(j + 1)A(j) = S(j + 1)C(j + 1)V (j)


C(j)B(j) = S(j)C(j)B(j)R(j),

C(j + 1)A(j)B(j) = S(j + 1)C(j + 1)A(j)B(j)R(j),

C(j + 1)A(j)B(j) = SC(j + 1)A(j)B(j)R.

When the expansions/contractions are time invariant, one obtains from (3)

A(j) = UA(j)V, A(j + 1)A(j) = UA(j + 1)A(j)V,

B(j) = UB(j)R, A(j)B(j) = UA(j)B(j)R,

C(j) = SC(j)V, C(j + 1)A(j) = SC(j + 1)V,

C(j)B(j) = SC(j)B(j)R,

C(j + 1)A(j)B(j) = SC(j + 1)A(j)B(j)R.

More compact formulations of (3) can be obtained in the case of time-invariant systems:

Ai = UAiV, AiB = UAiBR,

CAi = SCAiV, CAiB = SCAiBR, (i = 0, 1, . . .).

The last conditions are formally identical to those obtained in the continuous-time case [13,23,30].

Example. Let

S : x(k + 1) = kx(k) + 2u(k), y(k) = x(k), (5)

i.e., in terms of (1), A(k) = k, B(k) = 2 and C(k) = 1. Let V (k)T =[v1(k) v2(k)] = [1 k]. It is easy to verify that

A(k)V (k) − kV (k + 1) = 0, (6)

where

A(k) =

[

0 1k2 1

]

.

Multiplying (6) successively with A(k−1), . . . , A(k0), and applying V (k)A(k−1) = A(k − 1)V (k − 1), one obtains V (k)Φ(k, k0) = Φ(k, k0). As V (k) ismonic, there exists U(k) such that U(k)V (k) = 1 for all k (for example,U(k) =

[

12

12k

]

), and the first condition for inclusion in (3) is satisfied.Similarly, if

B(j) =

[

22j

]

= V (j)B(j) = 2V (j),

one obtains, after multiplying both sides of (6) successively with A(j +1), . . . , A(k), that V (k)Φ(k, j)B(k) = Φ(k, j)B(j), which gives the secondrelation in (3) (assuming that R(j) = 1).

328 S. S. Stankovic

The third and the fourth relation in (3) are verifiable analogously if oneassumes that S(k) = 1 and

C(k) = 1 = [1 0]V (k) = C(k)V (k).

Consequently, the system

S : x(k + 1) =

[

0 1k2 k

]

x(k) +

[

22k

]

u(k),

y(k) = [1 0]x(k) (7)

includes S in the sense of Theorem 1.

3 Restriction and Aggregation

Restriction and aggregation represent the most important special cases of in-clusion in both time-invariant and time-varying cases [14,16,23,30]. We shallgive here general definitions for the discrete-time case, covering a variety ofinput/output expansion/contraction combinations [27,26,30].

Definition 2. S is a restriction of S if one of the following conditions issatisfied on [ka, kb], ∀k0, k ∈ [ka, kb], k0 ≤ k:

R(a) : ∃(V (k), R(k), T (k)) x0, u ⇒ x, y,R(b) : ∃(V (k), R(k), S(k)) x0, u ⇒ x, y,R(c) : ∃(V (k), Q(k), T (k)) x0, u ⇒ x, y,R(d) : ∃(V (k), Q(k), S(k)) x0, u ⇒ x, y.

Consequently, we have restriction (a), denoted as SR(a)⊂ S or R(a), restric-

tion (b), denoted as SR(b)⊂ S or R(b), etc.

Definition 3. S is an aggregation of S if one of the following conditionsis satisfied on [ka, kb], ∀k0, k ∈ [ka, kb], k0 ≤ k:

A(a) : ∃(U(k), Q(k), S(k)) x0, u ⇒ x, y,A(b) : ∃(U(k), R(k), S(k)) x0, u ⇒ x, y,A(c) : ∃(U(k), Q(k), T (k)) x0, u ⇒ x, y,A(d) : ∃(U(k), R(k), T (k)) x0, u ⇒ x, y.

We have four aggregation types: aggregation (a), denoted as SA(a)⊂ S or

A(a), etc.


Theorem 2. Let

(R1) : V (k)Φ(k, k0) = Φ(k, k0)V (k0),

(R2) : V (t)Φ(k, j)B(j) = Φ(k, j)B(j)R(j),

(R3) : V (t)Φ(k, j)B(j)Q(j) = Φ(k, j)B(j),

(R4) : T (k)C(k)Φ(k, j) = C(k)Φ(k, k0)V (k0),

(R5) : C(k)Φ(k, j) = S(k)C(k)Φ(k, k0)V (k0),

for all k0, j, k ∈ [ka, kb], k0 ≤ j ≤ k. Then

SR(a)⊂ S ⇔ (R1) ∧ (R2) ∧ (R4),

SR(b)⊂ S ⇔ (R1) ∧ (R2) ∧ (R5),

SR(c)⊂ S ⇔ (R1) ∧ (R3) ∧ (R4),

SR(d)⊂ S ⇔ (R1) ∧ (R3) ∧ (R5).

Proof: According to Definition 2, condition (R1) holds for all the restric-tion types. The remaining conditions follow directly from the correspondingdefinitions.

Theorem 3. Let

(A1) : U(k)Φ(k, k0) = Φ(k, k0)U(k0),

(A2) : Φ(k, j)B(j)Q(j) = U(k)Φ(k, j)B(j),

(A3) : Φ(k, j)B(j) = U(k)Φ(k, j)B(j)R(j),

(A4) : C(k)Φ(k, k0)U(k0) = S(k)C(k)Φ(k, k0),

(A5) : T (k)C(k)Φ(k, k0)U(k0) = C(k)Φ(k, k0),

for all k0, j, k ∈ [ka, kb], k0 ≤ j ≤ k. Then

SA(a)⊂ S ⇔ (A1) ∧ (A2) ∧ (A4),

SA(b)⊂ S ⇔ (A1) ∧ (A3) ∧ (A4),

SA(c)⊂ S ⇔ (A1) ∧ (A2) ∧ (A5),

SA(d)⊂ S ⇔ (A1) ∧ (A3) ∧ (A5).

Proof: The condition (A1) follows directly from Definition 3. The remainingconditions are obtained similarly as in Theorem 2.

Let, in general, the matrices in S be represented as A(k) = V (k)A(k)U(k)+M(k), B(k) = V (k)B(k)Q(k) + N(k), C(k) = T (k)C(k)U(k) + L(k), whereM(k), N(k) and L(k) are complementary matrices.

330 S. S. Stankovic

Theorem 4.

SR(a)⊂ S if (M(k)V (k) = [V (k + 1) − V (k)]A(k))∧

(N(k)R(k) = 0) ∧ (L(k)V (k) = 0),

SR(b)⊂ S if (M(k)V (k) = [V (k + 1) − V (k)]A(k)))∧

(N(k)R(k) = 0) ∧ (S(k)L(k)V (k) = 0),

SR(c)⊂ S if (M(k)V (k) = [V (k + 1) − V (k)]A(k)))∧

(N(k) = 0) ∧ (L(k)V (k) = 0),

SR(d)⊂ S if (M(k)V (k) = [V (k + 1) − V (k)]A(k)))∧

(N(k) = 0) ∧ (S(k)L(k)V (k) = 0),(∀k ∈ [ka, kb])

Proof: Taking k = j + 1 in (R1) and replacing j with k one obtains

V (j + 1)A(j) = A(j)V (j),

which is equivalent to

M(k)V (k) = [V (k + 1) − V (k)]A(k).

For R(a) and R(b) we have (R2), which gives

V (k)B(k) = B(k)R(k) ⇔ N(k)R(k) = 0.

Similarly, (R4) provides

T (k)C(k) = C(k)A(k) ⇔ L(k)V (k) = 0.

The remaining relations can be derived analogously.

Theorem 5.

SA(a)⊂ S iff (U(k)M(k) = U(k)A(k) − A(k)U(k))∧

(U(k)N(k) = 0) ∧ (S(k)L(k) = 0),

SA(b)⊂ S iff (U(k)M(k) = U(k)A(k) − A(k)U(k))∧

(U(k)N(k)R(k) = 0) ∧ (S(k)L(k) = 0),

SA(c)⊂ S iff (U(k)M(k) = U(k)A(k) − A(k)U(k))∧

(U(k)N(k) = 0) ∧ (L(k) = 0),

SA(d)⊂ S iff (U(k)M(k) = U(k)A(k) − A(k)U(k))∧

(U(k)N(k)R(k) = 0) ∧ (L(k) = 0),(∀k ∈ [ka, kb])

Proof: Condition (A1) yields U(k + 1)A(k) = A(k)U(k) which is equivalent


to the first condition for all aggregation types. The remaining conditions canbe derived as in Theorem 3.

It is easy to verify that the all restriction and aggregation conditionssatisfy the inclusion conditions from Theorem 1. For example, conditionsV (k)Φ(k, k0) = Φ(k, k0)V (k0) and U(k)Φ(k, k0) = Φ(k, k0)U(k0) imply thatthere exist a left inverse U(k) = V (k)L and a right inverse V (k0) = U(k0)

R,respectively, so that Φ(k, j) = U(k)Φ(k, k0)V (k0).

Restriction/aggregation conditions for the time-invariant case can be eas-ily derived as special cases. For example,

SR(a)⊂ S iff (AV = V A) ∧ (BR = V B) ∧ (CV = TC)

⇔ (MV = 0) ∧ (NR = 0) ∧ (LV = 0),

or

SA(a)⊂ S iff (UA = AU) ∧ (UB = BQ) ∧ (SC = CU)

⇔ (UM = 0) ∧ (UN = 0) ∧ (SL = 0).

Notice that R(c) implies both R(a) and R(d), and that both R(a) andR(d) imply R(b); analogously, A(c) implies both A(a) and A(d), and bothA(a) and A(d) imply A(b).

4 Composition Property

The importance of restriction and aggregation for the inclusion relationshipsbetween two dynamic systems is illustrated by the following theorem, statingthat any input/state/output inclusion can be considered as a specific compo-sition of one input/state/output restriction/aggregation pairs. This propertyhas been shown for the state inclusion of linear time-invariant systems in [16]and for linear continuous-time time-varying systems in [30].

Let

S : x(k + 1) = A(k)x(k) + B(k)u(k) (x(k0) = x0),

y(k) = C(k)x(k),

where x(k) ∈ X , u(k) ∈ U and y(k) ∈ Y with n ≤ d(X ) ≤ n, m ≤ d(U) ≤ m

and l ≤ d(Y) ≤ l (d(.) denotes the dimension of the indicated space).

Theorem 6. S ⊂ S on [ka, kb] iff for each k ∈ [ka, kb] there exist bothan interval [ks, kf ] containing k, and a system S defined on the same inter-

val, such that SA(b)⊂ S

R(b)⊂ S on [ks, kf ], ka ≤ ks ≤ kf ≤ kb.

Proof: The sufficiency follows from the basic fact that restrictions and ag-gregations are special cases of inclusion, and that (S ⊂ S) ∧ (S ⊂ S) ⇒

332 S. S. Stankovic

S ⊂ S. Namely, any [ka, kb] can be represented as a union of consecutivenonoverlapping intervals:

K1 = [k1s = ka, k1

f ], K2 = [k2s = k1

f + 1, k2f ] . . .Ki = [ki

s, kif ], . . . ,

so that [ka, kb] =⋃

i Ki (for finite intervals [ka, kb] the number of the in-tervals Ki is finite). Take one set Ki covering [ka, kb]. By assumption,

on each Ki there exists a system S which satisfies SA(b)⊂ S

R(b)⊂ S. As-

sign quadruplets (U ri (k), V r

i (k), Rri (k), T r

i (k)) to S ⊂ S and quadruplets(Ua

i (k), V ai (k), Ra

i (k), T ai (k)) to S ⊂ S on Ki, i = 1, 2, . . .. Therefore, the

choice(x(tis) = V r

i (tis)x(tis)) ∧ (u(k) = Rri (k)u(k))

implies

(x(k) = U ri (k)x(k; ki

s, x(kis), u[ki

s,k])) ∧ y[x(k)] = Sr

i (k)y[x(k)],

as well as the choice

(x(tis) = V ai (tis)x(tis)) ∧ (u(k) = Ra

i (k)u(k))

implies

(x(k) = Uai (k)x(k; ki

s, x(kis), u[ki

s,k])) ∧ (y[x(k)] = Sa

i (k)y[x(k)]),

∀k ∈ Ki = [kis, k

if ], i = 1, 2, . . .. Consequently, the choice

(x(tis) = V ri (tis)V

ai (tis)x(tis)) ∧ (u(k) = Rr

i (k)Rai (k)u(k))

implies

(x(k; kis, x(ki

s), u[kis,k]) = Ua

i (k)U ri (k)x(k; ki

s, x(kis), u[ki

s,k]))

∧(y[x(k)] = Sai (k)Sr

i (k)y[x(k)]),

and, therefore, S ⊂ S on each Ki, i = 1, 2, . . .. Thus, S ⊂ S on [ta, tb],with the quadruplet (U(k) = Ua

i (k)U ri (k), V (k) = V r

i (k)V ai (k), R(k) =

Rri (k)Ra

i (k), S(k) = Sai (k)Sr

i (k)) ∀k ∈ [ka, kb].The necessity will be proven constructively, similarly as in [30], taking

into account specific properties of discrete-time systems.Assume that S ⊂ S on [ka, kb], with the assigned quadruplet (U(k), V (k),

R(k), S(k)).

Choose V r(k) = [V′

(k)...V (k)], where V

′

(k) = (I − V (k)U(k))W (k), andW (k) is any basis matrix for

W(k) = [R(〈A(k)|V (k); ka〉) + R(〈A(k)|N(k)R(k); ka〉)] ∩R(V (k))⊥,

where

〈A(k)|V (k); ka〉 = [V (k)...A(k − 1)V (k − 1))

...


. . ....A(k − 1)A(k − 2) . . . A(ka)V (ka)],

〈A(k)|N(k)R(k); ka〉 = [N(k)R(k)...A(k − 1)N(k − 1)R(k − 1))

...

. . ....A(k − 1)A(k − 2) · · · A(ka)N(ka)R(ka))]

and (.)⊥ denotes the annihilator of the indicated map. Notice that thedefined matrices represent reachability matrices for the time-varying pairs(A(k), V (k + 1)) and (A(k), N(k + 1)R(k + 1)) on the interval [ka, k], sinceA(j − 1) · · · A(j0) = Φ(j, j0) [20,21]. Having in mind that

[V′

(k)...V (k)] = [W (k)

...V (k)]

[

I 0

−U(k)W (k) I

]

and U(k)V (k) = I, we conclude that V r(k) is monic, as well as that

R(V r(k)) = R(〈A(k)|V (k); ka〉) + R(〈A|(N(k)R(k); ka〉)

and N (U(k)) = R(V′

(k)), implying the existence of the left inverse V′

(k)L

such that the choice U r(k) =

[

V′

(k)L

U(k)

]

implies U r(k)V r(k) = I.

However, the rank of both 〈A(k)|V (k); ka〉 and 〈A(k)|N(k)R(k); ka〉 is,in general, time varying (depending on both k and ka), and, consequently,dimensions of both V r(k) and U r(k) are, in general, time-varying. Prop-erties of these matrices can be analyzed by using the symmetric positivesemidefinite matrices

〈A(k)|V (k); ka〉〈A(k)|V (k); ka〉T

and

〈A(k)|N(k)R(k); ka〉〈A(k)|N(k)R(k); ka〉T .

Their elements, together with their nonnegative real eigenvalues λi(k) andµi(k), (i = 1, . . . , n), respectively, are functions of time (see, for example, [8]).These eigenvalues can be equal to zero either on intervals of nonzero length, oron a set of isolated discrete-time instants. Therefore, rank(〈A(k)|V (k); ka〉)and rank(〈A(k)|N(k)R(k); ka〉) are piecewise constant functions of time, im-plying that for each k ∈ [ka, kb] there exists an interval [ks, kf ] ⊆ [ka, kb]containing k where both functions have constant values (these intervals canbe reduced to single points, when ks = kf ).

Choose also Rr(k) = [R′

(k)...R(k)], where R

′

(k) is any basis matrix forU ∩ U⊥ ⊂ N (N(k))∩U⊥ (if N (N(k))∩R(R(k))⊥ = ∅, then Rr(k) = R(k)),

and T r(k) = [T′

(k)...T (k)], where T

′

(k) is any basis matrix for Y ∩ Y⊥ .Therefore, there exist left inverses R

′

(k)L and T′

(k)L such that the choice

334 S. S. Stankovic

Qr(k) =

[

R′

(k)L

Q(k)

]

and Sr(k) =

[

T′

(k)L

S(k)

]

implies both Qr(k)Rr(k) = I

and Sr(k)T r(k) = I.Let the matrices in S be

A(k) = U r(k + 1)A(k)V r(k),

B(k) = U r(k)B(k)Rr(k),

C(k) = Sr(k)C(k)V r(k).

¿From the general expression

A(k) = V r(k)A(k)U r(k) + M r(k)

we obtainM r(k)V r(k) − [V r(k + 1) − V r(k)]A(k)

= [I − V r(k + 1)U r(k + 1)]A(k)V r(k) = 0,

after replacing the expression for A(k) and using the fact that A(k)Vr(k) ⊂R(V r(k + 1)) by definition.

Note also that

R(B(k)Rr(k)) ⊂ R(V (k)) + R(N(k)Rr(k)) ⊂ R(V r(k)),

having in mind that B(k) = V (k)B(k)Q(k) + N(k) and R(N(k)Rr(k)) =R(N(k)R(k)), according to the definition of Rr(k). From

B(k) = V r(k)B(k)Qr(k) + N r(k)

we conclude, consequently, that

N r(k)Rr(k) = [B(k) − V r(k)B(k)Qr(k)]Rr(k)

= [I − V r(k)U r(k)]B(k)Rr(k) = 0.

Also, from C(k) = T r(k)C(k)U r(k)+Lr(k) we obtain Sr(k)Lr(k)V r(k) =0.

Therefore, according to Theorem 2, SR(b)⊂ S on [ks, kf ] with the quadru-

plet (U r(k), V r(k), Rr(k), Sr(k)).

Choose now V a(k) =

[

0I

]

, Ua(k) = [0 I], Ra(k) =

[

0I

]

, Qa(k) =

[0 I], T a(k) =

[

0I

]

and Sa(k) = [0 I]. Obviously, Ua(k)U r(k) = U(k),

V r(k)V a(k) = V (k), Rr(k)Ra(k) = R(k) and Sa(k)Sr(k) = S(k).¿From Theorem 5 one obtains that for any type of aggregation

U(k + 1)M(k) = [I − U(k + 1)V (k)]A(k)U(k),


where (U(k), V (k)) is any expansion/contraction pair. Therefore, using

A(k) = V a(k)A(k)Ua(k) + Ma(k)

one obtains

Ua(k + 1)Ma(k) − [I − Ua(k + 1)V a(k)]A(k)Ua(k)

= U(k + 1)A(k)[V′

(k)...V (k)] − A(k)[0

...I].

By assumption, V′

(k) = (I−V (k)U(k))W (k), where W (k) ∈ W(k). Further,one concludes that

U(k + 1)A(k)(I − V (k)U(k))A(k − 1) · · · A(k − j + 1)V (k − j) = 0,

implying that U(k+1)A(k)(I−V (k)U(k))〈 A(k)|V (k); k0〉 = 0. Analogously,one can show that

R(〈A(k)|B(k)R(k); k0〉) ⊂ N (U(k))

using the second relation in (3). As S ⊂ S implies

U(k)Φ(k, j)B(j) = U(k)Φ(k, j)V (j)U(j)B(j)R(j) = Φ(k, j)B(j),

one obtains also that

R(〈A(k)|V (k)U(k)B(k)R(k); k0〉) ⊂ N (U(k)),

implyingR(〈A(k)|N(k)R(k); k0〉) ⊂ N (U(k)).

Therefore, according to the definition of W(k), we have Ua(k)Ma(k) = 0.We have, also,

B(k) = V a(k)B(k)Qa(k) + Na(k)

andUa(k)Na(k)Ra(k) = 0

using the inclusion relation U(k)B(k)R(k) = B(k).Reasoning similarly, we start from C(k) = T (k)C(k)U(k) + L(k) and the

inclusion relations related to C(k) in (3), and obtain

S(k)L(k)〈A(k)|V (k); k0〉) = 0

andS(k)L(k)〈A(k)|B(k)R(k); k0〉) = 0.

Consequently,

N (S(k)L(k)) ⊃ R(〈A(k)|V (k); k0〉) + R(〈A(k)|N(k)R(k); k0〉)

336 S. S. Stankovic

and

S(k)L(k)V′

(k) = 0.

Therefore, we get

Sa(k)La(k) = Sa(k)[C(k) − T a(k)C(k)Ua(k)] = S(k)(T (k)C(k)U(k)

+L(k))[V′

(k)...V (k)] − C(k)[0

...I] = 0

having in mind that C(k) = T a(k)C(k)Ua(k) + La(k).

Therefore, according to Theorem 3, SA(b)⊂ S on [ks, kf ] with the quadru-

plet (Ua(k), V a(k), Ra(k), Sa(k)).

5 State Feedback Contractibility

The problem of inclusion of systems with feedback structure is importantbecause in decentralized control applications it is necessary to carry out feed-back control design in the expanded space, contract the obtained control law,and implement it in the original system. This problem has been discussed inthe time-invariant case in numerous papers, especially from the point of viewof designing decentralized control law for systems with the overlapping struc-ture, e.g. [9,10,11,12,23,25,26,30]. Alternatively, a model reduction schemecan be used to simplify the feedback design; then the obtained control lawmust be expanded before implementation [6,22].

Consider the systems S and S under the time-varying state feedbackdefined by

C : u(k) = K(k)x(k) + ν(k), C : u(k) = K(k)x(k) + ν(k), (8)

where ν(k) and ν(k) are reference inputs. Define the closed loop systemsby the corresponding triplets of matrices: Sf = (Af (k), B(k), C(k)) and

S = (Af (k), B(k), C(k)), where Af (k) = A(k) + B(k)K(k) and Af (k) =

A(k) + B(k)K(k). Obviously, Sf ⊂ Sf on [ka, kb] iff the conditions of The-

orem 1 are satisfied, with the transition matrices Φf (k, j) and Φf (k, j) for

Sf and Sf , respectively. This formulation does not allow, however, getting

explicit formulations of the conditions for K(k) and K(k) to satisfy Sf ⊂ Sf ,suitable for design purposes. Sufficient conditions for controller contractibil-ity (expandibility) can be formulated starting from:

Definition 4. Controller C is contractible to the controller C (or C isexpandable to C) if the condition (C1) ∨ (C2) holds on [ka, kb], where

(C1) : ∃(U(k), V (k), R(k), S(k))

x0, u ⇒ x, y ∧ (R(k)K(k)x(k) = K(k)x(k)),


(C2) : ∃(U(k), V (k), Q(k), S(k))

x0, u ⇒ x, y ∧ (K(k)x(k) = Q(k)K(k)x(k)).

Theorem 7. C is contractible to C on [ka, kb] iff the condition (D1)∨ (D2)holds ∀k0, j, k ∈ [ka, kb], k0 ≤ j ≤ k:

(D1) : R(k)K(k)Φ(k, k0) = K(k)Φ(k, k0)V (k0),

R(k)K(k)Φ(k, j)B(j) = K(k)Φ(k, j)B(j)R(j) (9)

(D2) : K(k)Φ(k, j) = Q(k)K(k)Φ(k, j)V (k0),

K(k)Φ(k, j)B(j)Q(j) = Q(k)K(k)Φ(k, j)B(j). (10)

Proof: The proof follows directly from the proof of Theorem 1.

It is very important to notice that the contractibility in the sense of Def-inition 4 and Theorem 7 implies the inclusion of the corresponding closedloop systems. Indeed, condition (C2), for example, ensures that, for anyx0 and u[k0,k], x(k; k0, x0, u[k0,k]) = U(k)x(k; k0, x0, u[k0,k]), where u[k0,k] isgenerated by u(k) = Q(k)u(k) (∀k ∈ [k0, k]) and x0 = V (k0)x0. Therefore,the input u(k) = K(k)x(k) is just one particular choice satisfying the con-ditions for Sf ⊂ S, provided u(k) = K(k)x(k) = Q(k)K(k)x(k). We have a

completely analogous situation with (C1). In both cases S ⊂ S, while theadditional conditions for K(k) and K(k) can be considered as the regula-tor output inclusion conditions (if K(k) and K(k) are considered as outputmatrices, and Q(k) and R(k)) as the corresponding expansion/contractionmaps). Consequently, one of the important implications of contractibility isthe preservation of stability after contraction. It follows from (3) that, ingeneral,

Sf ⊂ Sf ⇔ ‖Φf (k, k0)‖ ≤ ‖U(k)‖ ‖Φf(k, k0)‖ ‖V (k0)‖, (11)

so that ‖Φf(k, k0)‖≤ M < ∞ ⇒ ‖Φf (k, k0)‖ ≤ M < ∞, that is, stability of

Sf implies stability of Sf , provided (‖U(k)‖ ≤ Mu < ∞) ∧(‖V (k0)‖ ≤ Mv <

∞), ∀k0, k ∈ [ka, kb], k ≥ k0.

In the case of restrictions and aggregations contractibility conditions be-come:

Theorem 8. C is contractible to C if one of the following conditions is

338 S. S. Stankovic

satisfied ∀k ∈ [ka, kb]:

(R1) : (SR(a)⊂ S) ∧ (R(k)K(k) = K(k)V (k)),

(R2) : (SR(b)⊂ S) ∧ (R(k)K(k) = K(k)V (k)),

(R3) : (SR(c)⊂ S) ∧ (K(k) = Q(k)K(k)V (k)),

(R4) : (SR(d)⊂ S) ∧ (K(k) = Q(k)K(k)V (k)).

Theorem 9. C is contractible to C if one of the following conditions issatisfied:

(A1) : (SA(a)⊂ S) ∧ (K(k)U(k) = Q(k)K(k)),

(A2) : (SA(b)⊂ S) ∧ (R(k)K(k)U(k) = K(k)),

(A3) : (SA(c)⊂ S) ∧ (K(k)U(k) = Q(k)K(k)),

(A4) : (SA(d)⊂ S) ∧ (R(k)K(k)U(k) = K(k)).

The inclusion conditions obtained in Theorems 8 and 9 are, in fact, com-pletely analogous to the conditions derived in [30] for the continuous-timecase, and result, essentially, from different combinations of input/output ex-pansions/contractions. It should be also noted that all the conditions fromTheorems 8 and 9 satisfy the contractibility conditions from Theorem 7.

6 Conclusion

This paper gives basic aspects of the inclusion principle applied to lineartime-varying discrete-time systems. A set of necessary inclusion conditionsis derived in terms of time-varying system matrices. Restrictions and aggre-gations of different types have been introduced covering different combina-tions of time-varying input/state/output expansions/contractions. The maincontribution of the paper is contained in the theorem dealing with the com-position property of the time-varying discrete-time inclusion. This theoremrepresents the discrete-time counterpart of the the basic theorem dealing withthe composition property in the continuous-time case [30]. It has been provedthat any input/state/output inclusion relation on a given time-interval canbe decomposed into a specific sequence of pairs of input/state/output restric-tions and input/state/output aggregations. The paper encompasses the prac-tically important problem of contractibility (expandibility) of time-varyingstate feedback controllers. Starting from a general definition, sufficient con-tractibility conditions are derived.

Further efforts could be oriented towards the application of the above de-veloped methodology to the inclusion of time-varying observers and dynamiccontrollers, in the context of the inclusion of performance indices.


7 Acknowledgements

This paper has been inspired by the results of a long and fruitful collaborationwith Professor Dragoslav D. Siljak from Santa Clara University related todifferent aspects of the inclusion principle.

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HYPERGRAPH PARTITIONING TECHNIQUES

Dorothy Kucar, Shawki Areibi2 and Anthony Vannelli

Department of Electrical and Computer EngineeringUniversity of Waterloo, Waterloo, Ontario N2L 3G1

2School of Engineering

Guelph University, Guelph, Ontario N1G 2W1

Abstract. Graph and hypergraph partitioning are important problems with applications

in several areas including parallel processing, VLSI design automation, physical mapping of

chromosomes and data classification. The problem is to divide the vertices of a hypergraph

into k similar–sized blocks such that a given cost function defined over the hypergraph is

optimized. The intent of this paper is to expose our readers to the theoretical and imple-

mentation issues of hypergraph partitioning techniques.

Keywords. Hypergraph Partitioning, Iterative Improvement, Multilevel Techniques, ILP

Model, Eigenvector Methods, Simulated Annealing, Genetic Algorithms, Tabu Search

1 Introduction

Partitioning is used to divide a hypergraph into smaller, more manageableblocks while minimizing the number of connections between the blocks, calledcutnets. Vertices that are strongly “related” are assigned to the same block;vertices that are weakly “related” are assigned to different blocks. The rela-tionship between vertices is determined by the number of, or the weight ofedges connecting the vertices. From this description, it is evident that parti-tioning is a generic tool applicable to problems in a wide variety of scientificdisciplines:

Large–Scale Dynamical Systems: Given a large dynamical systemof the form

z′ = Az + Bu

where z represents the state of the system and u the input to thesystem: the goal is to decouple the directed graph associated with thesystem into smaller, highly–connected subgraphs. The system takes ona block–triangular structure where the blocks correspond to strongly–connected structures. It is then computationally easy to determinewhether the blocks are individually stable, hence whether the overallsystem is stable [52, 51].

342 D. Kucar, S. Areibi, A. Vannelli

Large–scale matrix computations: The goal is to minimize com-munication (edges) between processors (blocks) so that computationson each processor are nearly independent of computations on otherprocessors. This computational model is used in the solution of dis-crete partial differential equations arising from a variety of scientificapplications.

Linear systems of the type Ax = b are solved by factoring A into aproduct of lower and upper triangular matrices; the system LUx = bis solved relatively easily. The order in which L and U are formeddetermines the sparsity of L and U. Partitioning can be used to reorderA so as to minimize the non–zero entries of L and U.

VLSI Physical Design: In the placement stage, logic gates (mod-ules) are grouped according to net connectivity and highly–connectedmodules are placed on a two–dimensional grid in proximity to one an-other so as to minimize overall net usage and reduce wiring congestion.The minimization of net usage and wiring congestion is accomplishedthrough a combination of netlist partitioning, terminal propagation andthe judicious use of quadratic optimization solvers. More thorough de-scriptions of VLSI placement techniques are found in [14, 39]

Due to the ever–increasing sizes of data sets involved in the above areas,partitioning will continue to retain its importance of the tools used in com-puter science. In this work, we are interested primarily in netlist partitioningapplied to problems from VLSI design, although the problem formulationscan be easily adapted to formats present in other research areas. The intentof this paper is to expose our readers to the theoretical and implementationissues of hypergraph partitioning techniques.

The rest of this paper is structured as follows: Section 2 defines parti-tioning and introduces relevant notation. Section 3 discuses iterative im-provement methods that move individual vertices to different blocks if thecost function improves. Section 4 covers advanced search heuristics such asGRASP, Simulated Annealing, Tabu Search and Genetic Algorithms. Sec-tion 5 discuses relevant developments in multilevel methods in which clus-tering is combined with iterative improvement. Section 6 discuses mathe-matical programming formulations of hypergraph partitioning starting witha general integer linear programming formulation, which forms the basis ofapproaches that use eigenvectors. Section 7 describes testing methodolo-gies and compares the various methods with respect to run–time and solutionquality.

2 Preliminaries

Although this paper deals with the general problem of hypergraph partition-ing, we will use examples from the Very Large Scale Integrated (VLSI) circuit

Hypergraph Partitioning Techniques 343

a

b

h

d

i

g

fe

c1

2

3

4

5

6

7 j

terminal

net

Figure 1: Graphical Representation of a Netlist

domain. A circuit is a collection of logic gates connected to one another bywires at the same electrical potential called signal nets. The points at whichsignal nets come into contact with logic gates are called pins. If a pin con-nects gates to areas outside circuit block, it is referred to as a terminal. Gateconnectivity information is provided in the form of a netlist which containsnet names followed by the names of gates they are connected to. A schematicidentifying the various terminology is shown in Figure 1. Once the physicsof circuit operation is abstracted away, a circuit becomes a purely combi-natorial structure. The combinatorial and highly “visual” nature of VLSIphysical design problems lends itself quite nicely to formulations involvinggraphs or matrices. Most often, a circuit is represented as a hypergraphwhere logic gates are weighted vertices (the weight is typically proportionalto the number of ports) and nets are hyperedges. A hypergraph is the mostnatural representation of a circuit because, in a circuit, more than two ver-tices may be connected by the same signal net. In this section, we assume ahypergraph comprises of |V | vertices, |E| hyperedges, P pins and K blocks.Although a hypergraph is the natural representation of a circuit, it is verydifficult to work with. Often, a hypergraph is represented as clique graph.

Definition 1 A clique is a subgraph of graph G(V, E) in which every vertexis adjacent to every other vertex.

In the clique representation of a hypergraph, each vertex induces an edge,

thus a net on |Vi| vertices will be represented by

(

|Vi|2

)

edges. In the

generic clique model, the weight of each edge is

1

|Vi| − 1

although, many other models have been proposed [2], but none model the cutproperties of a net exactly [32] in the sense that regardless of how vertices


in the clique are partitioned, the number of cuts should add up to 1. Forexample: a net on 4 vertices is represented by an equivalent 6–edge completegraph. To cut off one vertex from the rest requires cutting 3 edges, so theweight should be 1/3 (for a total edge weight of 1). However, to cut offtwo vertices from the rest requires cutting four edges (i.e. the edge weightshould be 1/4) so there is clearly an inconsistency in this weighting scheme.Hypergraph and clique graph representations of the hypergraph in Figure 1are depicted in Figure 2

b

a

c

d

eg

fi

jh

2

1

3

4

6

72

5

(a) Hypergraph

2

5

1

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7

1/3

3/2

1/2

1

1/21/2

1

1/2

1/3 4/3

1/3

1/3

1/2

1/21/2

1/2

1/2

1/2

1/2

1

1

(b) Clique Graph

Figure 2: Netlist Representations

Given that a circuit is a fairly simple combinatorial object, it can also berepresented by different types of matrices:

Definition 2 Given a graph on |V | vertices, the adjacency matrix, A, isdefined as

aij =

wij if vertices i and j are adjacent i 6= j0 otherwise

Definition 3 Given a vector containing the degrees of |V | vertices, the di-agonal degree matrix, D, is defined as:

dii =

|V |∑

j=1

aij

Definition 4 Given a graph on |V | vertices, the corresponding |V | × |V |Laplacian matrix, L is given by L = D − A

If the graph is connected then one of the eigenvalues is 0 and the rest lie inthe interval (0, 1). In general the number of 0 eigenvalues equals the numberof connected components in the graph. Lastly, we introduce some definitionswhich come in useful when we discuss hypergraph partitioning:


Definition 5 A partitioning, f(PK) of |V | vertices into K blocks, V1, . . . , VKis given by V1 ∪ V2 ∪ · · · ∪ VK = V where Vi ∩ Vj = ∅ and α|V | ≤ |Vi| ≤ β|V |for 1 ≤ i < j ≤ K and 0 ≤ α, β ≤ 1.

Definition 6 Edges which have vertices in multiple blocks belong to the cut-set of the hypergraph. Reducing the size of the cutset, f(PK) is one of theprincipal objective of partitioning.

2.1 Solving the Partitioning Problem

One of the most interesting features of partitioning is that even the simplestcase—graph bisection—is NP–complete [20] thus hypergraph partitioning isat least as hard. Some practical consequences of this fact are that the num-ber of problem solutions grows exponentially with the problem size and thatthere may be multiple optimal solutions (although the ratio of optimal so-lutions to all solutions is nearly zero). The reality that problem sizes areprohibitively large imposes fairly strict criteria on possible approximationalgorithms: the run–time complexity should be linear or log–linear with thesize of the problem.

Currently, the most popular partitioning techniques are multilevel tech-niques that use interchange heuristics introduced by Kernighan/Lin [37]and Fiduccia/Mattheyses [19] or eigenvector–based partitioners [46, 48, 27] .These algorithms are included in partitioning packages such as hMetis [35],Chaco [27] and MLPart, [3]. These techniques have been studied and improvedextensively over the past three decades. Current implementations have verymodest run–time complexities and produce optimal or near–optimal results.

2.2 Partitioning Objectives and Constraints

In partitioning, we typically want to minimize the number of cutnets. Moreformally, if E(Ci) represents hyperedges that belong to the set of cutnets,then the objective is:

minimize f(PK) =K

∑

i=1

|E(Ci)|

Two types of constraints may be present: the first constraint ensures thateach block has approximately the same number of vertices, or else, thatit occupies approximately the same area. A typical balance constraint forbisection is:

b =|V1|

|V |

with

0.45 ≤ b ≤ 0.55


For K–way partitioning, some researchers [36] use

|V |

100K≤ |Vi| ≤

100|V |

K

The second type of constraint takes into account that some vertices maybe fixed in a specific partition. The presence of fixed terminals actuallyadds some convexity to this otherwise highly non–convex problem, renderingit significantly easier (and faster) to solve [16]. In [1], the authors pointout that the presence of fixed vertices requires fewer starts in an iterativeimprovement–based partitioner to approach a “good” solution.

3 Iterative Improvement Methods

In this section, we will outline the most significant developments in the fieldof iterative improvement partitioners. Iterative improvement partitioners ini-tially fix nodes to arbitrary blocks. This is followed by a process of assigningnodes to different blocks if the connectivity between blocks is reduced. Thegoal in the end is to find an assignment of nodes such that the connectivitybetween blocks is globally minimal.

3.1 Kernighan–Lin

Kernighan and Lin’s work was the earliest attempt at moving away fromexhaustive search in determining the optimal cutset size subject to balanceconstraints. In their paper [37], they proposed a O(|V |2 log |V |) heuristic forgraph bisection based on exchanging pairs of vertices with the highest gainbetween two blocks, V1 and V2. They define a gain of a pair of vertices as thenumber of edges by which a cutset would decrease if vertices x and y were tobe exchanged between blocks. Assuming aij are entries of a graph adjacencymatrix, the gain is given by the formula:

g(vx, vy) =

∑

vj 6∈V1

axj −∑

vj∈V1

axj

+

∑

vj 6∈V2

ayj −∑

vj∈V2

ayj

− 2axy

The terms in parentheses count the number of vertices which have edgesentirely within one block minus the number of vertices which have edgesconnecting vertices in the complementary block.

The procedure works as follows: vertices are initially divided into two sets,V1 and V2. A gain is computed for all pairs of vertices (vi, vj) with vi ∈ V1

and vj ∈ V2; the pair of vertices (vx, vy) with the highest gain are selected forexchanged; vx and vy are then removed from the list of exchange candidates.The gains for all pairs of vertices vi ∈ (V1 − vx) and vj ∈ (V2 − vy)are recomputed and the pair of vertices resulting in the highest gain areexchanged. The process continues until g(vx, vy) = 0, at which point, the


algorithm will have found a local minimum. The algorithm can be repeatedto improve upon the current local minimum. Kernighan and Lin observe thattwo to four passes are necessary to obtain a locally optimal solution.

Kernighan and Lin also make the observation that using exhaustive search,

partitioning |V | vertices into 2 blocks of size |V |2 results in

1

2

(

|V ||V |2

)

configurations, which is a very large number indeed. As an example, theyshow that a specific 32–vertex graph has 3–5 global optima, thus a randomconfiguration has a 1 in 107 chance of being an optimal solution.

In [50], Schweikert and Kernighan introduce a model which deals with hy-pergraphs directly. They point out that the major flaw of the (clique) graphmodel is to exaggerate the importance of nets with more than two connec-tions. After partitioning, vertices connecting large nets tend to end up in thesame block, whereas vertices connected to two point nets are split up betweenthe blocks. They combine their hypergraph model in the Kernighan–Linpartitioning heuristic and obtain much better results on circuit partitioningproblems.

3.2 Fiduccia–Mattheyses

The Fiduccia–Mattheyses [19] method is a linear–time (O(P )), per pass,hypergraph bisection heuristic. Its impressive run–time is due to an cleverway of determining which vertex to move based on its gain and to an efficientdata structure called a bucket list

The gain, g(i), of a vertex is defined as the number of nets by which acutset would decrease if vertex i were to be moved from the current blockto the complementary block. Fiduccia and Mattheyses also define a balancecriterion given as

b =|V1|

|V1| + |V2|

Once a vertex is moved—subject to satisfying the balance criterion—it islocked in the complementary block for the remainder of the pass. Verticescan be moved until they all become locked or until the balance criterionprevents further moves. Once a vertex i is moved, the gains of a specific setof its free neighbors are recomputed. This set is composed of vertices incidenton nets which are either entirely within the starting block, or nets which haveonly vertex in the starting block (and other vertices in the complementaryblock).

The gains are maintained in a [−Pmax . . . Pmax] bucket array whose ithentry contains a doubly–linked list of free vertices with gains currently equalto i. The maximum gain, Pmax, is obtained in the case of a vertex of degree


CELL

2 i N... ...

...

1

max gain

−Pmax

Pmax

cell # cell #

Figure 3: Bucket List

Pmax (i.e. a vertex which is incident on Pmax nets) that is moved across theblock boundary and all of its incident nets are removed from the cutset.

The free vertices in this list are linked to vertices in the main vertexarray so that during a gain update, vertices are removed from the bucket listin constant time (per vertex). Superior results are obtained if vertices areremoved and inserted from the bucket list using a Last–In–First–Out (LIFO)scheme (over First–In–First–Out or random schemes) [24]. The authors of[24] speculate that a LIFO implementation is better because vertices thatnaturally cluster together will tend to be listed sequentially in the netlist.

3.3 Improvements on FM

In this section, we discuss four improvements to the original FM implemen-tation which have helped increase the acceptance of FM, until recently, asthe most widely–used partitioning technique. One improvement to standardFM is Krishnamurthy’s lookahead scheme [41], in which the future implica-tions of moving a vertex that is currently in the cutset, are considered inthe gain computation. Sanchis [49] extends the FM concept to deal withmulti–way partitioning. Her method computes multi–way partitionings di-rectly by constructing K(K − 1) bucket lists (because vertex 1 can move toblocks 2, 3, . . .K, vertex 2 can move to blocks 1, 3, . . . , K, etc..). One pass ofthe algorithm consists of examining moves which result in the highest amongall K(K − 1) bucket lists and the move with the highest gain and one thatpreserves the balance criterion is selected. After a move, all K(K−1) bucketlists are updated. Dutt and Deng [15] observe that iterative improvementengines such as FM or Look–Ahead do not identify clusters adequately, andconsequently, miss local optimal with respect to the number of cut hyper-edges. Their method, called CLIP, identifies and moves entire clusters ofvertices across the cutline. Cong and Lim [13] envisage K–way partitioningas concurrent bipartitioning problems where vertices are moved between twospecific clusters of vertices (each cluster contains a handful of vertices). This


is different from traditional FM partitioning where candidate vertices areinitially selected from the entire set of vertices.

4 Advanced Search Heuristics

Advanced search heuristics comprises techniques that simulate naturally–occurring process including GRASP, Simulated Annealing, Genetic Algo-rithms and Tabu Search. These techniques have been used extensively tosolve difficult combinatorial problems thanks to their uphill–climbing capa-bilities.

The motivation for the Simulated Annealing [38] algorithm comes froman analogy between the physical annealing of solids and combinatorial opti-mization problems. Simulated Annealing is widely recognized as a methodfor determining the global minima of combinatorial optimization problems.Tabu Search finds some of its motivation in attempts to imitate “intelligent”processes [47], by providing heuristic search with a facility that implements akind of “memory”. Tabu search has been applied across a wide range of prob-lem settings in which it consistently has found better solutions than othermethods. GRASP is a random adaptive simple heuristic that intelligentlyconstructs good initial solutions in an efficient manner. Genetic Algorithmson the other hand manipulate possible solutions to the decision problem insuch a way that resembles the mechanics of natural selection and offers anumber of advantages, the most important being the capability of exploringthe parameter space.

4.1 GRASP

GRASP is a greedy randomized adaptive search procedure that has beensuccessful in solving many combinatorial optimization problems efficiently[18, 9]. The GRASP methodology was developed in the late 1980s, and theacronym was coined by Feo [17]. Each iteration consists of a constructionphase and a local optimization phase. The key to success for local search al-gorithms consists of the suitable choice of a neighborhood structure, efficientneighborhood search technique, and the starting solution. The GRASP con-struction phase plays an important role with respect to this last point, sinceit produces good starting solutions for local search. The construction phaseintelligently constructs an initial solution via an adaptive randomized greedyfunction. Further improvement in the solution produced by the construc-tion phase may be possible by using either a simple local improvement phaseor a more sophisticated procedure in the form of Tabu Search or SimulatedAnnealing.

The GRASP algorithm starts with a construction phase followed by alocal improvement phase. The implementation terminates after a certainnumber of phases or runs have passed. The construction phase is iterative,


greedy and adaptive in nature. It is iterative because the initial solution isbuilt by considering one element at a time. The choice of the next element tobe added is determined by ordering all elements in a list. The list of the bestcandidates is called the restricted candidate list (RCL). It is greedy becausethe addition of each element is guided by a greedy function. The constructionphase is randomized by allowing the selection of the next element added tothe solution to be any element in the RCL. Finally, it is adaptive becausethe element chosen at any iteration in a construction is a function of thosepreviously chosen.

4.2 Simulated Annealing

Simulated Annealing (SA) searches the solution space of a combinatorialoptimization problem, with the goal of finding a solution of minimum costvalue. The motivation for the Simulated Annealing algorithm comes froman analogy between the physical annealing of solids and combinatorial opti-mization problems [38]. Physical annealing refers to the process of findinglow energy states of a solid by initially melting the substance, and then low-ering the temperature slowly, spending a long time at temperatures close tothe freezing point. In the analogy, the different states of the substance cor-respond to the different feasible solutions of the combinatorial optimizationproblem, and the energy of the system corresponds to the function to beminimized.

A simple descent algorithm is analogous to reducing temperature is re-duced quickly so that only moves which result in a reduction of the energy ofthe system are accepted. In Simulated Annealing [8], the algorithm alterna-tively attempts to avoid becoming trapped in a local optimum by acceptingneighborhood moves which increase the value of (f) with probability e(−δ/T )

(called the acceptance function) where T is a control temperature in theanalogy with physical annealing. The acceptance function implies that smallincreases in f are more likely to be accepted than large increases, and thatwhen T is high, most moves will be accepted, but as T approaches zero mostuphill moves will be rejected. The algorithm is depicted in Figure 4

In the body of the algorithm, the outer loop adjusts the temperatureand the inner loop determines how many neighborhood moves are to be at-tempted at each temperature. The determination of the initial temperature,the rate at which the temperature is reduced, the number of iterations ateach temperature and the criterion used for stopping is known as the anneal-ing schedule [30]. The choice of annealing schedule has an important bearingon the performance of the algorithm [45].

4.2.1 Annealing Schedule

The Simulated Annealing algorithm, in its original formulation convergeswith probability one to a globally minimal configuration in either one of the


SIMULATED ANNEALINGcurrent solution ← initial solutioncurrent cost ← evaluate(current solution)T ← TinitialWhile (T ≥ Tfinal)

for i = 1 to iteration(T) /* neighborhood moves */new solution ← move(current solution)new cost ← evaluate(new solution)∆cost← new cost - current cost

if(∆cost ≤ 0 OR e−∆cost/T > random())

/* accept new solution */current solution ← new solutioncurrent cost ← new cost

EndIfEndForT ← next temp(T)

EndWhile

Figure 4: A Simulated Annealing Algorithm

following cases [42]:

➊ for each value of the control parameter, Tk, an infinite number of tran-sitions is generated and limk→∞ Tk = 0. (the homogeneous algorithm);

➋ for each value Tk one transition is generated and Tk goes to zero notfaster than O([log k]−1) (the inhomogeneous algorithm).

In any implementation of the algorithm, asymptotic convergence can onlybe approximated. The number of transitions for each value Tk, for example,must be finite and limk→∞ Tk = 0 can only be approximated in a finitenumber of values for Tk. Due to these approximations, the algorithm is nolonger guaranteed to find a global minimum with probability one.

Initial Value of the Control Parameter

The initial value of T is determined in such a way that virtually all transitionsare accepted, i.e., T0 is such that exp(−δcostij/T0) ≃ 1 for almost all i andj. The following empirical rule is proposed: choose a large value for T0

and perform a number of transitions. If the acceptance ratio χ, definedas the number of accepted transitions divided by the number of proposedtransitions, is less than a given value χ0 (in [38] χ0 = 0.8), double the currentvalue of T0. Continue this procedure until the observed acceptance ratioexceeds χ0.

Final Value of the Control Parameter

A stopping criterion, determining the final value of the control parameter, iseither determined by fixing the number of values Tk, for which the algorithmis to be executed, or by terminating execution of the algorithm if the lastconfiguration of consecutive Markov chains are identical for a number ofchains.


Number of Iteration per Temperature

The simplest choice for Lk, the length of the kth Markov chain, is a valuedepending (polynomial) on the size of the problem.

Decrement of the Control Parameter

A frequently used decrement rule is given by ck+1 = α × ck, k = 0, 1, 2, ....where α is a constant close to 1.

4.3 Genetic Algorithms

Genetic Algorithms (GA’s) are a class of optimization algorithms that seekimproved performance by sampling areas of the parameter space that havea high probability for leading to good solutions [53]. The algorithms arecalled genetic because the manipulation of possible solutions resembles themechanics of natural selection. These algorithms which were introduced byHolland [29] in 1975 are based on the notion of propagating new solutionsfrom parent solutions, employing mechanisms modeled after those currentlybelieved to apply in genetics. The best offspring of the parent solutions areretained for a next generation of mating, thereby proceeding in an evolution-ary fashion that encourages the survival of the fittest. As an optimizationtechnique, Genetic Algorithms simultaneously examine and manipulate a setof possible solutions. Each candidate solution is represented by a string ofsymbols called a chromosome. The set of solutions Pj , is referred to as thepopulation of the jth generation. The population evolves for a prespecifiedtotal number of generations under the application of evolutionary rules calledGenetic Operators.

4.3.1 Characteristics of Genetic Search

There are many characteristics of Genetic Algorithms which qualify them tobe a robust based search procedure [4]. The first feature of Genetic Algo-rithms is that they are characterized to climb many peaks in parallel. Thus,the probability of finding a false peak is reduced over methods that proceedform point to point in the decision space. Secondly, the operators make useof a coding of the parameter space rather than the parameters themselves.Only objective function information is used, this results in a simpler imple-mentation. Finally, although the approach has a stochastic flavor, it makesuse of all the information that has been obtained during the search, andpermits the structured exchange of that information [23].

4.3.2 Main Components of Genetic Search

There are essentially four basic components necessary for the successful im-plementation of a Genetic Algorithm. At the outset, there must be a code or


M7M6M5M3M8M4M2M1

BLOCK 1BLOCK 0

(b) Permutation with Separator Encoding.

(a) Group Number Encoding

01110100

M8M7M6M5M4M3M2M1

Figure 5: Representation schemes

scheme that allows for a bit string representation of possible solutions to theproblem. Next, a suitable function must be devised that allows for a rankingor fitness assessment of any solution. The third component, contains trans-formation functions that create new individuals from existing solutions in apopulation. Finally, techniques for selecting parents for mating, and deletionmethods to create new generations are required.

Representation Module

In the original GA’s of Holland [29], each solution may be represented as astring of bits, where the interpretation of the meaning of the string is problemspecific. As can be seen in Figure 5a, one way to represent the partitioningproblem is to use group-number encoding where the jth integer ij ∈ 1, . . . , kindicates the group number assigned to object j. This representation schemecreates a possibility of applying standard operators [44]. However an offspringmay contain less than k groups; more-over, an offspring of two parents, bothrepresenting feasible solutions may be infeasible, since the constraint of hav-ing equal number of modules in each partition is not met. In this case eitherspecial repair heuristics are used to modify chromosomes to become feasible,or penalty functions that penalize infeasible solutions, are used to eliminatethe problem. The second representation scheme is shown in Figure 5b. Here,the solution of the partitioning problem is encoded as n + k − 1 strings ofdistinct integer numbers. Integers from the range 1, .., n represent the ob-jects, and integers from the range n+1, . . . , n+k−1 represent separators;this is a permutation with separators encoding. This representation schemeleads to 100% feasible solutions [44], but requires more computation timedue to the complexity of the unary operator involved.

Evaluation Module

Genetic Algorithms work by assigning a value to each string in the popula-tion according to a problem-specific fitness function. It is worth noting thatnowhere except in the evaluation function is there any information (in theGenetic Algorithm) about the problem to be solved. For the circuit par-


titioning problem, the evaluation function measures the worth (number ofcuts) of any chromosome (partition) for the circuit to be solved.

Reproduction Module

This module is perhaps the most significant component in the Genetic Algo-rithm. Operators in the reproduction module, mimic the biological evolutionprocess, by using unary (mutation type) and higher order (crossover type)transformation to create new individuals. Mutation is simply the introduc-tion of a random element, that creates new individuals by a small change ina single individual. When mutation is applied to a bit string, it sweeps downthe list of bits, replacing each by a randomly selected bit, if a probabilitytest is passed. On the other hand, crossover recombines the genetic materialin two parent chromosomes to make two children. It is the structured yetrandom way that information from a pair of strings is combined to form anoffspring.

4.3.3 Population Module

This module contains techniques for population initialization, generation re-placement, and parent selection techniques. The initialization techniquesgenerally used are based on pseudo-random methods. The algorithm willcreate its starting population by filling it with pseudo-randomly generatedbit strings. Strings are selected for mating based on their fitness, those withgreater fitness are awarded more offspring than those with lesser fitness.Parent selection techniques that are used, vary from stochastic to determin-istic methods. The probability that a string i is selected for mating is pi,“the ratio of the fitness of string i to the sum of all string fitness values”,pi = fitnessi

∑

j fitnessj. The ratio of individual fitness to the fitness sum denotes

a ranking of that string in the population. The Roulette Wheel Selectionmethod is conceptually the simplest stochastic selection technique used. Theratio pi is used to construct a weighted roulette wheel, with each string oc-cupying an area on the wheel in proportion to this ratio. The wheel is thenemployed to determine the string that participates in the reproduction. Arandom number generator is invoked to determine the location of the spinon the roulette wheel.

4.3.4 GA Implementation

Figure 6 shows a simple Genetic Algorithm. The algorithm begins with anencoding and initialization phase during which each string in the populationis assigned a uniformly distributed random point in the solution space. Eachiteration of the genetic algorithm begins by evaluating the fitness of thecurrent generation of strings. A new generation of offspring is created byapplying crossover and mutation to pairs of parents who have been selected


based on their fitness. The algorithm terminates after some fixed number ofiterations.

GENETIC ALGORITHM1. Encode Solution Space2.(a) set pop size, max gen, gen=0;

(b) set cross rate, mutate rate;3. Initialize Population.4. While max gen ≥ gen

Evaluate FitnessFor (i=1 to pop size)

Select (mate1,mate2)if (rnd(0,1) ≤ cross rate)

child = Crossover(mate1,mate2);if (rnd(0,1) ≤ mutate rate)

child = Mutation();Repair child if necessary

End For

Add offsprings to New Generation.gen = gen + 1

End While

5. Return best chromosomes.

Figure 6: A generic Genetic Algorithm

4.4 Tabu Search

The Tabu Search technique was originally proposed in 1990 by Glover [21]as an optimization tool to solve nonlinear covering problems. Tabu Searchhas recently been applied to problems such as integer programming, schedul-ing [31], circuit partitioning [10] and graph coloring [28]. In general terms,Tabu Search is an iterative improvement procedure that starts from someinitial feasible solution (i.e., assignment of cells to blocks for the partition-ing problem) and attempts to determine a better solution in the manner ofa steepest descent algorithm. However, Tabu Search is characterized by anability to escape local optima which usually cause simple descent algorithmsto terminate by using a short term memory of recent solutions. Moves thatreturn a previously generated partition are considered tabu and are not con-sidered. Moreover, Tabu Search permits back-tracking to previous solutions,which may ultimately lead, via a different direction, to partitions with fewercut–nets.

4.4.1 Tabu Move

There are different attributes that can be used in creating the short termmemory of Tabu lists for the circuit partitioning problem. One possibility isto identify attributes of a move based on the module value to be swappedfrom one block to another. Another way of identifying attributes of a moveis to introduce additional information, referring not only to the modules tobe moved but to positions (blocks) occupied by these modules. The recordedmove attributes are used in Tabu Search to impose the constraints, calledTabu restrictions, that prevent moves from being chosen.


4.4.2 Tabu List

Tabu list management concerns updating the Tabu list; i.e., deciding on howmany and which moves have to be made Tabu within any iteration of thesearch. The size of the Tabu list can noticeably affect the final results; a longlist may give a lower overall minimum cost, but is usually obtained in a longtime. Further, a shorter list may trap the routine in a cyclic search pattern.Empirical results show that Tabu list sizes that provide good results, oftengrow with the size of the problem. An appropriate list size depends on thestrength of the Tabu restrictions employed.

4.4.3 Aspiration Criteria

Aspiration is a heuristic rule based on cost values that can temporarily releasea solution from its Tabu status. The purpose of Aspiration is to increase theflexibility of the algorithm while preserving the basic features that allow thealgorithm to escape from local optima and avoid cyclic behavioral. At alltimes during the search procedure, each cost c has an Aspiration value thatis the current lowest cost of all solutions arrived at from solutions with valuec. The actual Aspiration rule is that, if the cost associated with a Tabusolution is less than the Aspiration value associated with the cost of thecurrent solution, then the Tabu status of the Tabu solution is temporarilyignored.

4.4.4 A Tabu Search Implementation for Circuit Partitioning

A Tabu Search implementation for circuit partitioning requires an initialfeasible solution (partition), an associated cost —in this case, the net cut,a list of Tabu solutions and a maximum number of moves, (max num iter).The stopping criteria for the current implementation is described in detail inSection 4.4.5. The Tabu list and Aspiration levels corresponds to short termmemory and the core search routine corresponds to intermediate and longterm memory. Intermediate and long term memory functions are employed toachieve regional intensification and global diversification of the search [21, 22]of the search landscape

4.4.5 List Management & Stopping Criteria

Our Tabu list management techniques are based on static and dynamic ap-proaches. In the static approach, the size of the Tabu lists remains fixedfor the entire search period. The size of the Tabu list is chosen to be afunction of the number of nodes within the circuit to be partitioned. Ourexperimentation with the Tabu Search algorithm indicates that choosing atabu list size = α × nodes (where α ranges from 0.1 to 0.2) yields good re-sults in most cases. The dynamic implementation allows the size of the Tabulist to oscillate between two ranges. The first range is determined when


TABU SEARCHInput:

The netlist or the Graph G= (V,E)K = number of partitions; | T | = size of TabuListmax num iter = maximum number of iterations

Initialization:

Initial Partition = Generate a random solutions = (V1 , V2.., Vk)num iter = 0; bestpart = s; bestcut = f(s);

Main Loop:

While num iter < max num iterPick module mi at random and compute

∆C for the interchange with mjPick best module associated with best gainIf (move not in TabuList) then

accept the moveUpdate TabuList;Update the Aspiration Level;

End IfIf (move in TabuList) then

If (Cost(s) < Aspiration(s0) thenOverride the TabuList Status and accept the moveUpdate TabuList;Update the Aspiration Level;

End WhileOutPut:

Best Partition = bestpart; Best Cut = bestcut

Figure 7: A Tabu Search Algorithm With Aspiration

cycling occurs (Tabu lists are too short). The second range is determinedwhen deterioration in solution quality occurs, which is caused by forbiddingtoo many moves (Tabu lists are too large). Best sizes lie in an intermediaterange between these extremes.

The stopping conditions used in this implementation are based on thefollowing: (i) The search will terminate when num iter is greater than themaximum number of iterations allowed max num iter, (ii) The search will ter-minate if no improvement on the best solution found so far can be made fora given number max num iter of consecutive iterations. The maximum num-ber of iterations after which the routine terminates, depends on whether theroutine starts from a random starting point or a good initial starting point.Experiments performed show the following: For random starting points, thealgorithm requires more iterations to converge to good final solutions, sothe maximum number of allowable iterations is set to “max num iter =100×nodes”, whereas for good starting points “max num iter = 20×nodes”.The final solution gives the overall best partition and best cut after the spec-ified maximum number of moves.

5 Multilevel Methods

The gist of multilevel partitioning is to construct a sequence of successivelycoarser graphs, to partition the coarsest graph (subject to a balance con-straint) and to project the partitioning onto the next level finer graph whileperforming FM–type iterative improvement to further improve the partitionThe power of multilevel partitioning is evidenced during the iterative im-provement phase, where moving one vertex across the partition boundary


corresponds to moving an entire group of vertices.

5.1 Clustering Phase

The purpose of clustering is to create a small network such that a goodpartition of the small network is not significantly worse than a partition ofthe flattened network. Ideally, large nets are contracted to nets with a fewvertices so that the iterative improvement algorithm, which is very good atdealing with small nets (and not so good at dealing with large nets) can beused to its full potential. Several authors [26, 34, 3, 35] suggest clusteringbased on vertex matchings.

Definition 7 A matching of G = (V, E) is a subset of the edges with theproperty that no two edges share the same vertex. A heavy edge match-ing means edges with the heaviest weights are selected first. A maximummatching means every vertex has been matched.

Candidates for clustering are selected based on the results of vertex matchings[3, 33]; matched vertices are then merged into “super–vertices”. Matchingsmay be selected randomly or by decreasing net–weight (i.e. using heavy edgematchings).

Vertex matchings are used extensively because they tend to locate inde-pendent logical groupings of vertices. Independence is important in that itprevents the accumulation of many vertices in one super–vertex. Karypisand Kumar [33] use the following clustering schemes:

➊ select pairs of vertices that are present in same hyperedges by findingmaximal matching of vertices based on a clique–graph representation(edge coarsening)

➋ find heavy–edge matching of vertices by non–increasing hyperedge size;after all hyperedges have been visited, collapse matched vertices (hy-peredge coarsening)

➌ after hyperedges have been selected for matching, for each hyperedgethat has not been contracted, its (unmatched) vertices are contractedtogether (modified hyperedge coarsening)

➍ for each vertex v ∈ V , consider all vertices that belong to hyperedgesincident on v, whether they are matched or not

In [33], Karypis points out that there is no consistently better coarseningscheme for all netlists. Examples can be constructed for any of the abovecoarsening methods which fail to determine the correct partitioning. Karypisalso suggests that a good stopping point for coarsening is when there are 30Kvertices where K indicates the desired number of partitions.

After the coarsening phase, an initial bisection which satisfies the balanceconstraint is performed. It is not necessary at this point to produce an


“optimal” bisection since that is ultimately the purpose of the refinementphase.

5.2 Refinement Phase

The refinement process consists of successively applying an iterative improve-ment phase to successively finer hypergraphs, while unclustering after eachpass of the interchange heuristic. Due to the space complexity of Sanchis’K–way FM algorithm and because vertices are clustered into logical super–vertices, Karypis et al. [36] use a downhill–only variant of FM which doesnot require the use of a bucket list. Their refinement method visits vertices inrandom order and moves them if they result in a positive gain (and preservethe balance criterion). If a vertex v is internal to the block being considered,then it is not moved; if v is a boundary vertex, it can be moved to a blockwhich houses v’s neighbors. The move which generates the highest gain iseffectuated. In experiments, Karypis’ et al. refinement method converges inonly a few passes.

6 Mathematical Methods

Mathematical methods use enumerative or numerical techniques to determineoptimal block assignments. Enumerative techniques such as branch–and–cutsolve partitioning as an integer linear programming problem to optimality,but take a long time to do so. Numerical techniques construct vertex order-ings using eigenvectors and split them.

6.1 An INP Formulation of Hypergraph Partitioning

The K–way partitioning problem can be formulated as a 0–1 Integer Non–linear Program (INP) [7]. Assume there are j = 1 . . .K blocks, i = 1 . . . |V |vertices, k = 1 . . . |E| hyperedges and i′ = 1 . . . pk vertices per hyperedge k;define xij = 1 if vertex i is in block j and 0 otherwise. Thus, if a specific netconsists of vertices 1 through 4, then it will be uncut if

xi1xi2xi3xi4 = 1


The non–linear programming problem consists of maximizing the sum ofuncut nets (hence minimizing the sum of cut–nets)

max

K∑

j=1

|E|∏

i=1

xij (1)

s.t.

K∑

j=1

xij = 1 ∀ i (2)

lj ≤

|V |∑

i=1

xij ≤ uj ∀ j (3)

xij = 0, 1 (4)

ykj = 0, 1 (5)

The objective function maximizes the sum of uncut nets where k(i′) findsthe coordinate of i′th vertex of edge k in V . Constraints (2) stipulate thateach vertex be assigned to exactly one block. Constraints (3) impose blocksize limits. Finally, the last two sets of constraints specify all variables in afeasible solution must take on values 0 or 1.

6.2 An ILP Formulation of Hypergraph Partitioning

We can rewrite the INP as an Integer Linear Program (ILP) by introducingindicator variables ykj = 1 if net k is entirely in block j and 0 otherwise. Thelinear programming problem is nearly identical to the non–linear program-ming formulation except that objective is

maxK

∑

j=1

|E|∑

k=1

ykj

and the net constraints are replaced by ykj ≤ xk(i′)j ∀ i′, j, k. These con-straints indicate whether vertex k(i′) is incident on hyperedge j; they effec-tively remove the nonlinearity in the objective function. They make sense be-cause, for example if a net consists of vertices 1 and 2, and if x1j , x2j ∈ 0, 1then x1jx2j ≤ x1j and x1jx2j ≤ x2j . The remaining constraints are identicalto their counterparts in the INP model.

In the case where there are two blocks, the difficulty is found in the blocksize constraint. Without this constraint, the constraints matrix is unimodularand if the constraints matrix is unimodular, the restriction xij = 0, 1 can berelaxed into xij = [0, 1]. The resulting linear program can be solved in poly-nomial time using interior point methods. However, block size constraintsare useful in order to avoid situations where a disproportionate number ofvertices are assigned to one block. In the following subsections, we derive awell–studied eigenvector technique from the ILP formulation.


6.3 An Eigenvector Method

If we examine the exact 0–1 formulation of graph partitioning, and set K = 2,l1 = u1 = l2 = u2 = |V |/2, we obtain the following problem:

max

|E|∑

k=1

yk1 +

|E|∑

k=1

yk2 (6)

s.t. yk1 ≤ xk(i′)1 (7)

yk2 ≤ xk(i′)2 (8)

xi1 + xi2 = 1 ∀i (9)

|V |∑

i=1

xi1 =|V |

2(10)

|V |∑

i=1

xi2 =|V |

2(11)

xij = 0, 1 and ykj = 0, 1 (12)

If the variables in Equations (9) and (12) are rescaled so that xi1 + xi2 = 0i.e.: xi1 = −xi2 or xi = −1, 1 then constraints (9) through (12) can berewritten as:

|V |∑

i=1

xi = 0

xi = ±1 i = 1 . . . |V |

Equations (6) through (8), on the other hand need first to be combined intoa quadratic cost function in the following manner: since the xi are all binaryvariables

yk1 ≤ xk(i′)1 for i′ = 1, 2 is equivalent to

xk(1)1xk(2)1 ≤ xk(i′)1 for i′ = 1, 2

With some algebra, the cost function can be identified as (−) the off–diagonalterms of the product of xTLx where L is the Laplacian matrix of the graphinduced by the ILP. Recall, the Laplacian is defined as

ℓij =

∑

aij if i = j−aij if i, j connected, but i 6= j0 otherwise

The diagonal terms of L are always greater than zero so the minimizationproblem is equivalent to optimizing the off–diagonal elements. This estab-lishes the equivalence between Equations (6)-(11) and the standard graph


bisection formulation [25, 12]:

min xT Lx (13)

s.t. xi = ±1 (14)

|V |∑

i

xi = 0 (15)

Due to its discrete nature, this problem is very difficult to solve exactlyexcept for very small instances. A reasonable approximation is to relax thefirst constraint to

xT x = 1 (16)

which essentially replaces the solution space hypercube with a sphere androunds the solution of the continuous problem to the nearest feasible dis-crete point. The continuous problem is normally solved by minimizing thecorresponding Lagrangian:

L(x; λ) = xTLx − λ(xT x − 1)

The solution of which isλx = Lx

i.e.: λ is an eigenvalue of the L. The maximum is given by

xTLx = λxT x = λ

by constraint (16) The minimum is given by λ2 because λ1 = 0 gives x = 1which violates constraint (15).

The components of the second eigenvector which are < 0 represent thecoordinates of vertices in the first block; components of the second eigenvectorwhich are ≥ 0 represent the coordinates of vertices in the second block.The effect is that the eigenvector components of vertices which are strongly(weakly) connected are close (far away), thus strongly connected vertices aremore likely to be assigned to the same block.

7 Testing and Results

In this section, we compare a variety of partitioning techniques includingbranch–and–cut on the ILP formulation of partitioning, multi–level inter-change via hMetis, Tabu Search, Simulated Annealing, Genetic Algorithmand the eigenvector technique. Test problem statistics are provided in Table1. Using the test problems outlined in Table 1, we make assumptions thatthe vertices are partitioned into two blocks, vertices are weighted accordingto the number of incident nets and each block is allowed to have between asclose to 50% of total vertex weight as possible.


Table 1: Test Problem Data from [40]

Circuit Cells Nets Pins Pads Cell Degree Net SizeMAX x σ MAX x σ

Fract 125 147 462 24 7 3.1 1.6 17 3.1 2.2Prim1 833 902 2908 81 9 3.4 1.2 18 3.2 2.5Struct 1888 1920 5471 64 4 2.8 0.6 17 2.8 1.9

Ind1 2271 2192 7743 814 10 3.4 1.1 318 3.5 9.0Prim2 2907 3029 18407 107 9 3.7 1.5 37 3.7 3.8Bio 6417 5742 21040 97 6 3.2 1.0 861 3.6 20.8

Ind2 12142 13419 48158 495 12 3.8 1.8 585 3.5 10.9Ind3 15057 21808 65416 374 12 4.3 1.4 325 2.9 3.2Avq.s 21854 22124 76231 64 7 3.4 1.4 4042 3.4 53.3Avq.l 25144 25384 82751 64 7 3.2 1.2 4042 3.2 49.8

7.1 Using hMetis

hMetis is a hypergraph partitioning package based on the multilevel inter-change paradigm 1. To obtain our results, we used the following parametersettings

hMetis Parameter Settings

Nparts 2 partition into 2 blocksUBFactor 1 balance factor of 49:51Nruns 20 20 runsCType 1 coarsen type: hybrid hypergraph first–choiceRType 1 refine type: Fiduccia–MattheysesVCycle 1 V–cycle refinement on final solution of each bisectionReconst 0 nodbglvl 0 don’t print intermediate output

7.2 Using Advanced Search Heuristics

This section summarizes some of the parameters selected for the differentadvanced search techniques and a comparison in terms of solution qualityand CPU time.

7.2.1 Tabu Search Parameter Setting

As explained in Section 4.4 there are a number of parameters that need to betuned for the Tabu Search implementation. This includes the tabu criteria,tabu size, aspiration criteria and stopping criteria. The following are someof the parameters used [11]:

Tabu Search Parameter Settings

1It is available free of charge at http://www-users.cs.umn.edu/∼karypis/metis/hmetis/


Partitions 2 partition into 2 blocksTabuCriteria 1 Move Tabu if module is registeredTabuLength 1 Oscillates between 2 valuesAspiration 1 override tabu status if better than bestStopping Criteria 10 nbiter = 10 × nmods

7.2.2 Genetic Algorithms Parameter Setting

The performance of any genetic algorithm implementation depends on sev-eral parameters including the population size, generation size, crossover rate,mutation rate, selection and replacement methods. The following are someparameters used in our current implementation[6]:

Genetic Algorithms Parameter Settings

Partitions 2 partition into 2 blocksinit-tech 1 Grasp Populates 25% of initial Populationcrssrate 99% Cross Over Ratemutrate 0.1% Mutation RateSelection 3 RankingReplacemenent 3 Replace Most Inferior Parent with ChildRepair Alg 2 Simple Repair Heuristicpopsize 100 Population Sizegensize 100 Generation Size (Stopping Criteria)

7.2.3 Simulated Annealing Parameter Setting

According to Section 4.2 we have adapted an annealing schedule similar tothat proposed by White[54]. The most important parameters deal with theselection of the initial temperature, final temperature, the amount of decreaseof temperature and most importantly the number of iterations performedwithin each temperature. The following summarizes the parameter settingsused for the current implementation [5].

Simulated Annealing Parameter Settings

Partitions 2 partition into 2 blocksInitTemp T0 Temp doubled until χ ≥ 80%FinalTemp Tfin 0.01IterPerTemp Lk 5 × NumOfModsTempDecrease α 0.99

7.2.4 Comparison

Table 2 shows the results obtained for different benchmarks. The best, aver-age and standard deviation are listed for Simulated Annealing, Tabu Searchand Genetic Algorithms. The results for Simulated Annealing and Tabu


Circuit Simulated Annealing Genetic Algorithms Tabu SearchBest x σ Best x σ Best x σ

Fract 11 12.2 2.4 24 26.8 1.5 11 11 0Prim1 63 72.2 6.1 61 67.7 3.4 54 59 3.5Struct 47 54 7.5 47 61.9 6.0 47 64 12.7

Ind1 51 60.2 7.9 61 78.1 9.0 49 54 5.5Prim2 225 240 11.2 178 206 10.8 191 207 18.4Bio 126 136 10.1 88 122 16.8 119 134 12.3

Ind2 371 398 19.1 279 381 46.4 353 398 34.7Ind3 522 623 34.5 533 718 55.1 299 401 119.6Avq.s 384 507 71.8 484 622 56.4 516 613 111.2Avq.l 536 656 79.5 483 593 79.8 329 681 178.2

Table 2: Advanced Search Techniques Results, Best, Averge and StandardDeviation

Search are based on five runs whereas results for the Genetic Algorithm arebased on 100 generations. GRASP has been used to construct the initial so-lution for all heuristic techniques. The Genetic Algorithm mainly uses 25% ofthe solutions obtained by GRASP and the remaining 75% of the populationis based on random initial solutions. The table indicates that on average theTabu Search and Genetic Algorithm produce better results than the Simu-lated Annealing technique. However the Genetic Algorithms consumes onaverage 10 times slower than the Tabu Search implementation. For some ofthe large circuits the Simulated Annealing technique achieves better resultsthan Genetic Algorithms and Tabu Search.

7.3 Solving the ILP Formulation

CPLEX with the following custom parameter settings is used to solve the ILP:

CPLEX Parameter Settings

node select parameter best estimate strategyvariable select pseudo–reduced costsstart algorithm primal simplexobjective lower bound |E| − (hMetis objective value)

The rationale behind selecting the objective lower bound to be equal to |E|−hM is because the result obtained by hMetis, which is a heuristic, should be aunderestimation of the optimal number of uncut nets. Giving the CPLEX sucha tight lower bound should reduce the size of the search tree considerably.

7.4 Using the Eigenvector Technique

We constructed the Laplacian of the graph representation of the variousnetlists with net weights equal to wi = 1

(∑

i6=j aij)−1. We then found the sec-

ond smallest eigenvalue and Fiedler vector, v of L and sorted it in ascending


circuit ILP time hM time SA time GA time TS time eig time

Fract 11 4 11 0.1 11 17 24 22 11 0.3 12 0.1Prim1 53 2d 53 0.33 63 125 61 147 54 3 91 4.8Struct 34 3d 34 0.29 47 281 47 307 47 4 46 14.5

Ind1 - - 19 0.56 51 312 61 335 49 8 30 21Prim2 - - 158 2.10 225 462 178 478 191 13 284 14.5Bio - - 83 2.02 126 935 88 991 119 27 150 60

Ind2 - - 191 3.7 371 2413 279 2809 353 81 525 119Ind3 - - 282 5.5 522 4322 533 4993 299 144 475 148Avq.s - - 144 5.5 384 5086 484 4555 516 251 478 1191Avq.l - - 143 4.8 536 6930 483 5274 329 416 687 1419

Table 3: 2–Way Partitioning results: (Objective: Number of Cuts, Time:Seconds)

order of components to yield v∗. We then reordered the vertices of the netlistaccording to the coordinates of v∗ with respect to v. Finally, the number

of cuts was computed by finding the cut position between rows ⌊ |V |2 ⌋ and

⌊ |V |2 ⌋ + 1 of A.

7.5 Comparison

Table 3 introduces the results obtained using (a) ILP formulation introducedin Section 6.2 (b) hMetis (c) Simulated Annealing (d) Genetic Algorithms(e) Tabu Search and (f) Eigenvector Technique. The table is divided intothree parts, small, medium and large benchmarks. The hMetis techniqueproduces optimal results for small circuits. Exact partitions for mediumand large circuits was not possible due to the large CPU time required.It is evident from Table 3 that hMetis outperforms other heuristic searchtechniques in terms of quality of solutions and CPU time. A comparisonbetween the advanced search techniques clearly indicates that Tabu Searchproduces better results than those obtained by Simulated Annealing andproduces results comparable to those obtained by the Genetic Algorithm.Tabu Search is on average 10 to 15 faster than Simulated Annealing andGenetic Algorithms. The Eigenvector approach produces good results forsmall and medium circuits, but the solution quality deteriorates as the sizeof the benchmarks increase in size.

8 Summary

Hypergraph partitioning is a useful tool in a variety of scientific disciplines,although the examples we presented were from the VLSI CAD research area.The goal of this paper was to expose the reader to a variety of partitioningtechniques ranging from multilevel methods to advanced search heuristics tomathematical programming methods. In the results section, we comparedfive different methods using run–time and number of cuts as points of com-parison.


five different methods using run–time and number of cuts as points of com-parison.

9 Acknowledgements

This research is partially supported by a Natural Sciences and EngineeringResearch Council of Canada (NSERC) operating grant (OGP 0043417).

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STABILITY OF CONTINUOUS TIME JUMPLINEAR SYSTEMS

Kenneth A. Loparo1 and Yuguang Fang2

1Department of Electrical Engineering and Computer ScienceCase Western Reserve University, Cleveland, Ohio 44106

2Department of Electrical and Computer Engineering

University of Florida, Gainesville, Florida 32611

Abstract. In this paper, we study almost sure and moment stability of continuous time

jump linear systems with a finite state Markov jump form process. We prove that the con-

cepts of δ−moment stability, exponential δ−moment stability and stochastic δ−moment

stability are equivalent and that each of these implies almost sure (sample path) stability.

We also show that for sufficiently small δ, almost sure exponential stability and δ−moment

stability are equivalent and that the region of δ−moment stability converges monotonically

to the almost sure stability region as δ ↓ 0+. Sufficient conditions for δ-moment stability

and almost sure stability are developed.

Keywords. Jump linear systems, Almost sure stability, δ−moment stability, Lyapunov

exponent.

AMS (MOS) subject classification: This is optional. But please supply them when-

ever possible.

1 Introduction

Consider the continuous-time jump linear system of the form

x(t) = A(σt)x(t) (1.1)

where σt is a finite state random process (step process), usually a finite state,time homogeneous, Markov process. The model (1.1) can be used to ana-lyze the closed-loop stability of control systems with communication delays([22],[11]) or the stability of control systems subject to abrupt changes insystem structure such as component and interconnection failures ([8]). Thestability analysis of (1.1) is therefore very important in the design and analy-sis of control systems. Stability analysis of systems of this type can be tracedback to the work of Rosenbloom ([27]), who was interested in moment sta-bility properties. Bellman ([3]) was the first to study the moment stability of(1.2) with an iid form process using the Kronnecker matrix product. Bergen([4]) used a similar idea to study the moment stability properties of system

372 K.A. Loparo and Y. Fang

(1.1) with a piecewise constant form process σt. Later, Bhuracha ([6]) usedBellman’s idea developed in [3] to generalize Bergen’s results and studied bothasymptotic stability and exponential stability of the mean. Darkhovskii andLeibovich ([9]) investigated second moment stability of system (1.1) where σtis a step process where the time intervals between jumps are iid and the modesof the system are governed by a finite state time homogeneous Markov chain.Necessary and sufficient conditions for second moment stability in terms ofthe Kronnecker matrix product for second moment stability were obtained.

There is an alternative approach to the study of stochastic stability. Katsand Krasovskii ([20]) and Bertram and Sarachik ([5]) used a stochastic versionof Lyapunov’s second method to study almost sure stability and momentstability. For certain classes of systems, such as (1.1), it is possible to obtaintestable necessary and sufficient conditions for second moment stability. Fenget al ([14],[17]) and Ji et al ([19]) used Lyapunov’s second method to studystability when σt is a finite state Markov chain.

As Kozin ([21]) pointed out, moment stability implies almost sure stabilityunder fairly general conditions, but the converse is not true. In practicalapplications, almost sure stability is most often the more desirable propertybecause it is the sample path behavior of the system that can be observed.Further, moment stability criteria are often too conservative to be practicallyuseful.

Although the Lyapunov exponent techniques may provide necessary andsufficient conditions for almost sure stability ([14],[14],[7],[17],[24],[25]), it isvery difficult to compute or estimate the top Lyapunov exponent.

Arnold et al ([1]) studied the relationship between the top Lyapnov expo-nent and the δ−moment top Lyapunov exponent for a diffusion-type process.Using a similar idea, Leizarowitz ([23]) obtained similar results for (1.1). Ageneral conclusion was that δ−moment stability implies almost sure sta-bility. Thus sufficient conditions for almost sure stability can be obtainedusing δ−moment stability. There are many definitions for moment stabil-ity: δ−moment stability, exponential δ−moment stability and stochastic δ−moment stability. Feng et al ([14],[17]) showed that all the second momentstability concepts are equivalent for the (1.1), and also proved that for onedimensional systems, the region for δ−moment stability is monotonically con-verging to the region for almost sure stability as δ ↓ 0+. This is a significantresult because the study of almost sure stability can be reduced to the studyof δ−moment stability.

In this paper, we generalize the results reported in ([17]). We show thatfor system (1.1) with a Markov form process, all δ-moment stability conceptsare equivalent and that they imply almost sure (sample path) stability. Alsofor sufficiently small δ > 0, δ-moment stability and almost sure exponentialstability are equivalent. Henceforth, almost sure stability can be inferredfrom δ-moment stability. Sufficient conditions for δ-moment stability andalmost sure stability are developed. Similar results for discrete time systemare reported in ([13]).

Stability of Jump Linear Systems 373

2 δ−MOMENT STABILITY

Consider the continuous-time jump linear system

x(t) = A(σt)x(t), x0 = x(0), t ≥ 0 (2.1)

where σt : t ≥ 0 is a finite state time homogeneous Markov process (whichis referred to as the form process) with state space S = 1, 2, . . . , N. LetQ = (qij)N×N be the infinitesimal matrix of σt and x0 is a fixed constantvector in Rn. In this paper, we use the following definitions for stochasticstability:Definition 2.1:

For the system (2.1), let Ξ denote the collection of probability measureson S and Ψ ⊂ Ξ be a nonempty subset of Ξ. The system is said to be:(I). asymptotically δ−moment stable with respect to (w.r.t.) Ψ, if for anyx0 ∈ Rn and any initial probability distribution ψ ∈ Ψ of σt,

limt→∞

E‖x(t, x0)‖δ

= 0,

where x(t, x0) is a sample solution of (2.1) initial from x0 ∈ Rn. If δ = 2,we say that the system (2.1) is asymptotically mean square stable w.r.t. Ψ;if δ = 1, we say that the system (2.1) is asymptotically mean stable w.r.t.Ψ. If Ψ = Ξ, we say simply that (2.1) is asymptotically mean square stable.Similar statements apply to the following definitions.

(II). exponentially δ−moment stable with respect to Ψ, if for any x0 ∈ Rn

and any initial distribution ψ ∈ Ψ of σt, there exist constants α, β > 0independent of x0 and ψ such that

E‖x(t, x0)‖δ

≤ α‖x0‖δe−βt, ∀t ≥ 0.

(III). stochastically δ−moment stable with respect to Ψ, if for any x0 ∈Rn and any initial distribution ψ ∈ Ψ of σt,

∫ ∞

t=0

E‖x(t, x0)‖δ

< +∞.

(IV). almost surely (asymptotically) stable with respect to Ψ, if for anyx0 ∈ Rn and any initial distribution ψ ∈ Ψ of σt,

P

limt→∞

‖x(t, x0)‖ = 0

= 1.

(V). Almost surely exponentially stable with respect to Ψ, if for anyx0 ∈ Rn and any initial distribution ψ ∈ Ψ of σt,

limt→∞

‖x(t, x0)‖ = 0

at an exponential rate almost surely, i.e., there exist M(ω) > 0 and λ(ω) > 0such that

‖x(t, x0)‖ ≤M(ω)‖x0‖e−λ(ω)t, a.s..


In the above definitions, the initial probability distribution of σt playsa very important role. The stochastic stability definitions as given can beinterpreted as being robust against (Ψ-structured) uncertainty of the initialdistributions of the form process σt. Since (x(t), σt) is the state of the systemand in practice, the initial probability distribution of the form process σtis usually not exactly known, this is a reasonable requirement. Also, asillustrated in [14], stability with respect to a single initial distribution, say, theergodic invariant distribution π, may not be a good stability criterion becausean arbitrarily small perturbation to π can actually destroy the stability ofthe system.

The above dependence on the initial distribution of σt, necessitates deal-ing with a family of Markovian form processes parameterized by the initialdistribution ψ ∈ Ψ. To signify the dependence of a quantity Q on ψ, we oftenuse a subscript Qψ.

Some preliminaries for the finite state Markov process σt are given next.For all i, j ∈ S, define

pii = 0

qi = −qii =∑

l 6=i

qil

pij =qijqi

(i 6= j)

Let rk; k = 1, 2, · · · be the (discrete-time) Markov chain defined on thestate space S with the one-step transition matrix (pij)N×N and initial dis-tribution ψ. This chain is referred to as the embedded Markov chain ofσt. We have the following sojourn time description of the process σt([28, p.254]).

Let γk, k = 0, 1, . . . be the successive sojourn times between jumps.Let tk =

∑k−1l=0 γl for k = 1, 2, . . . be the waiting time for k−th jump and

t0 = 0. Starting in state σ0 = i, the process sojourns there for a durationof time that is exponentially distributed with parameter qi. The processthen jumps to the state j 6= i with probability pij ; the sojourn time inthe state j is exponentially distributed with parameter qj ; and so on. Thesequence of the states visited by the process σt, denoted by i1, i2, · · · arethe states of the embedded Markov chain rk; k = 1, 2, · · · . Conditioning oni1, i2, · · · , the successive sojourn times denoted by γ(ik) are independentexponentially distributed random variables with parameters qik . Clearly, thejoint process (rk, γk) : k = 0, 1, . . . is a time homogeneous Markov processand it fully describes the form process σt. The following notations will beused throughout the paper. Let Ai = A(i) and κi = ‖Ai‖ for all i ∈ S. Letκ = maxi∈S κi. Let Fn = σ(rk, γk) : 0 ≤ k ≤ n be the σ-algebra generatedby (rk, γk) : 0 ≤ k ≤ n. Let Φ(t, s) denote the transition matrix of (2.1).For each i ∈ S, let ei denote the initial distribution of σt concentrated at


i-th state. If σt has a single ergodic class, let π denote the unique invariantdistribution of σt. For a matrix B, let λi(B) denote an eigenvalue of B,letλmax(B) = maxi(Reλi(B)) and λmin(B) = mini(Reλi(B)) denote the largestand smallest real parts of eigenvalues of B, respectively.

In [17], Feng et al proved that for the system (2.1), second moment sta-bility, second moment exponential stability and second moment stochasticstability are equivalent, and all of them imply almost sure stability. Thenext theorem shows that this result can be generalized to δ−moment stabil-ity.

Theorem 2.2:

For any δ > 0, δ− moment stability, exponential δ−moment stability andstochastic δ−moment stability are equivalent for system (2.1), and they allimply almost sure stability.

Before we prove the theorem, we need the following lemma:

Lemma 2.3:

(i) Let κi = ‖Ai‖ for all i ∈ S, and let κ = maxi κi. Then, for anyt′2 ≥ t′1 ≥ 0, ‖Φ(t′2, t

′1)‖ ≤ exp(κ(t′2 − t′1)). (ii) For any δ > 0, if (2.1) is

δ-moment stable, then for any ψ ∈ Ξ and an integer k ≥ 0,

E‖Φ(t, tk)‖δ|Fk ≤ η, Pψ − a.s. ∀t ∈ [tk, tk+1]. (2.2)

Where in (2.2), η = maxi∈S ηi and

ηi = Eei‖Φ(t1, 0)‖δ =

+∞∫

0

qi‖ exp(Ait)‖δe−qitdt < +∞ (2.3)

for all i ∈ S. It follows that for any ψ ∈ Ξ and integers k ≥ i ≥ 0, ift ∈ [tk, tk+1), then

E‖Φ(t, ti)‖δ|F i ≤ η(k+1−i), Pψ − a.s. (2.4)

Proof of lemma. For any matrix A, it is easy to verify that ‖ exp(At)‖ ≤exp(‖A‖t) for all t ≥ 0. Using the sojourn time description of the form processσt, the transition matrix Φ(t′2, t

′1) can be written as a product of exponential

matrices (transition matrices) of the deterministic systems x(t) = Aix(t) fori ∈ S, the result (i) follows directly.

For (ii), note that the δ-moment stability of (2.1) implies that for eachi ∈ S, (2.3) holds. To show (2.2), by the Markovian and time homogeneousproperties of the system, it is enough to show that for any ψ = (ψ1, . . . , ψN ) ∈Ξ,

Eψ‖Φ(t, 0)‖δ ≤ η, ∀t ∈ [0, t1], ψ ∈ Ξ.


However,

Eψ‖Φ(t, 0)‖δ =

N∑

i=1

ψiEei‖Φ(t, 0)‖δ

=

N∑

i=1

ψi

t∫

0

qi‖ exp(Aiαi)‖δ exp(−qiαi)dαi

≤N∑

i=1

ψi

+∞∫

0

qi‖ exp(Aiαi)‖δ exp(−qiαi)dαi

=

N∑

i=1

ψiηi ≤ η < +∞.

This shows (2.2). To show (2.4), we apply (2.2) in the following way:

E‖Φ(t, ti)‖δ|F i ≤ E‖Φ(t, tk)‖δ . . . ‖Φ(ti+1, ti)‖δ|F i= EE. . . EE‖Φ(t, tk)‖δ|Fk‖Φ(tk, tk−1)‖δ

|Fk−1 . . . |F i+1‖Φ(ti+1, ti)‖δ|F i(2.2)

≤ ηk+1−i, Pψ − a.s.

Proof of Theorem 2.2. It is very easy to show that exponential δ-momentstability implies stochastic δ-moment stability. So it is sufficient to prove thatstochastic δ-moment stability implies δ-moment stability, that δ-moment sta-bility implies exponential δ-moment stability, and that exponential δ-momentstability implies almost sure stability.

Stochastic δ-moment stability implies δ-moment stability: If (2.1) is stochas-tically δ-moment stable, then

∫∞

0E‖x(t)‖δdt < +∞. It follows that

∫ +∞

0

E‖x(t)‖δdt =

∞∑

k=0

∫ k+1

k

E‖x(t)‖δdt =

+∞∑

k=0

1∫

0

E‖x(k + t)‖δdt < +∞

(2.5)Let βk(t) = E‖x(k + t)‖δ ≥ 0, and let β(t) =

∑+∞k=0 βk(t). From (2.5), we

obtain that λ(β(t) = +∞) = 0 where λ denotes the Lebesgue measure on[0,1]. Therefore, β(t) <∞ λ−almost everywhere on [0, 1], and there exists at∗ ∈ [0, 1] such that β(t∗) < +∞, i.e.,

∑∞0 βk(t

∗) < +∞, which implies thatlimk→∞ βk(t

∗) = 0. Now, consider that for any t large enough, we have ([t]denotes the integral part of t) from (i) of Lemma 2.2 that ‖Φ(t, [t]−1+t∗)‖δ ≤


exp(κ(t− [t] + 1 − t∗)) ≤ exp(2κ)def= M . Then

E‖x(t)‖δ = E‖Φ(t, 0)x0‖δ = E‖Φ(t, [t] − 1 + t∗)Φ([t] − 1 + t∗, 0)x0‖δ

≤ E‖Φ(t, [t]− 1 + t∗)‖δ‖Φ([t] − 1 + t∗, 0)x0‖δ≤ ME‖Φ([t] − 1 + t∗, 0)x0‖δ

= ME‖x([t] − 1 + t∗)‖δ = Mβ[t]−1(t∗) −→ 0, (t → ∞)

This means that (2.1) is δ−moment stable.δ−moment stability implies exponential δ−moment stability: Suppose that(2.1) is δ-moment stable. Then, for any ψ ∈ Ξ limt→+∞Eψ‖Φ(t, 0)‖δ = 0.This implies that there exists an m > 0 sufficiently large so that

maxi∈S

Eei‖Φ(tm, 0)‖δ ≤ ρ < 1

and it also follows that

maxζ∈Ξ

Eζ‖Φ(tm, 0)‖δ ≤ ρ < 1. (2.6)

From (2.6), we have for any ξ ∈ Ξ and l ≥ 0,

Eξ‖Φ(tl+m, tl)‖δ|F l ≤ ρ, Pξ − a.s.. (2.7)

For k large, let k = pm+ q with 0 ≤ q ≤ m− 1. For t ∈ [tk, tk+1), we have

Eψ‖Φ(t, 0)‖δ ≤ Eψ‖Φ(t, tpm)‖δp∏

i=1

‖Φ(tim, t(i−1)m)‖δ

= EψEψ . . . EψEψ‖Φ(t, tpm)‖δ|Fpm‖Φ(tpm, t(p−1)m)‖δ|F (p−1)m . . . |Fm‖Φ(tm, t0)‖δ

(2.4)

≤ ηq+1EψEψ . . . Eψ‖Φ(tpm, t(p−1)m)‖δ

|F (p−1)m . . . |Fm‖Φ(tm, t0)‖δ(2.7)

≤ ηq+1ρp = ηq+1ρ−m/q(ρ1/m)kdef= M ′ρk1 .

(2.8)

where 0 ≤ M ′ < +∞ and 0 ≤ ρ1 < 1. Since there exists constants 0 < α ≤β < +∞ such that

α ≤ limk→+∞

tkk

= limk→+∞

1

k

k−1∑

i=0

γi ≤ β, a.s.

we have from (2.8) that

1

tlogEψ‖Φ(t, 0)‖δ ≤ 1

tlogM ′ +

k

tlog ρ1 ≤ 1

tlogM ′ +

k

tklog ρ1.


Taking the limit t→ +∞, we arrive at

limt→+∞

1

tlogEψ‖Φ(t, 0)‖δ ≤ α−1 log ρ1 < 0.

This implies that the system (2.1) is exponentially δ−moment stable. Sum-marizing the above, we have established the equivalence of δ−moment sta-bility, exponentially δ−moment stability and stochastic δ−moment stability.Exponential δ-moment stability implies almost sure stability. Suppose that(2.1) is exponentially δ-moment stable, i.e., for any ψ ∈ Ξ, there existM ′′ > 0and 0 < ρ0 such that

Eψ‖Φ(t, 0)‖δ ≤M ′′ exp(−ρ0t), ∀t ≥ 0. (2.9)

To show almost sure stability, by (i) of Lemma 2.2, it is enough to show thatfor n ∈ Z+, the set of positive integers,

limn→+∞

‖Φ(n, 0)‖ = 0, Pψ − a.s.. (2.10)

Let gdef= limn→+∞ ‖Φ(n, 0)‖. For any ǫ > 0, by the Markov inequality we

have

Pψg > ǫ = Pψ⋂

m≥0

⋃

n≥m

‖Φ(n, 0)‖ > ǫ ≤ Pψ⋃

n≥m

‖Φ(n, 0)‖ > ǫ

Then,

Pψg > ǫ ≤+∞∑

n=m

Pψ‖Φ(n, 0)‖ > ǫ ≤+∞∑

n=m

1

ǫδEψ‖Φ(n, 0)‖δ

and finally

Pψg > ǫ ≤ 1

ǫδ

+∞∑

n=0

M ′′(exp(−ρ0))n+m−→0, m→ +∞.

Therefore, Pψg = 0 = 1 and (2.10) holds.This theorem is very important because moment stability of jump linear

systems with a finite state Markov form process has special features. Inparticular, define the top δ−moment Lyapunov exponent:

g(δ) = limt→+∞

1

tlogE‖Φ(t, t0)‖δ

then system (2.1) is δ−moment stable if and only if g(δ) < 0. This result isnot true in general because for certain systems, even though the system isδ-moment stable, g(δ) = 0 is possible, Theorem 2.2 excludes this case.

In [17], Feng et al studied the relationship between almost sure and δ-moment stability and showed that for one-dimensional jump linear systems


(2.1) with a Markov form process with a single ergodic class, the regions ofδ−moment stability increase monotonically and converge to the almost surestability region as δ ↓ 0+. They conjectured that this result is also true forhigh dimensional systems. Next we will verify this conjecture. In the restof this section, we assume that σt has a single ergodic class E ⊂ S, unlessmentioned otherwise. Let π ∈ Ξ denote the unique invariant (ergodic) dis-tribution of σt supported on E. Define the top (sample) Lyapunov exponentas

αψ(ω) = limt→+∞

log ‖Φ(t, 0)‖. (2.11)

Let α ∈ R be the top Oseledec exponent [26], i.e.,

α = limt→+∞

log ‖Φ(t, 0)‖, Pπ − a.s. (2.12)

which is a nonrandom constant. We have the following basic result:

Proposition 2.4:

Suppose that σt has a single ergodic class. Then, for any ψ ∈ Ξ,(i) αψ(ω) = α, Pψ-a.s..(ii) α = a−1 limk→+∞

1k log ‖Φ(tk, 0)‖ = a−1 limk→+∞

1kEπlog ‖Φ(tk, 0)‖,

Pψ-a.s. with

adef= lim

k→+∞

1

ktk > 0, Pπ − a.s.. (2.13)

Proof of proposition. (i). It is enough to show that αei(ω) = α, Pei

-a.s..for all i ∈ S. In the case when i ∈ E, the result follows from the nonrandomspectrum theorem [16]. Suppose now i ∈ S\E. Since |S| is finite and E isthe only ergodic class, the first entrance time τ of σt to E is finite Pei

-a.s..Since Φ(t, τ) is invertible, we have

αei(ω) = lim

t→+∞

1

tlog ‖Φ(t, 0)‖

= limt→+∞

1

tlog ‖Φ(t, τ)Φ(τ, 0)‖

= limt→+∞

1

tlog ‖Φ(t, τ)‖, Pei

− a.s..

(2.14)

However, by the time homogeneous property and the preceding argument

limt→+∞

log ‖Φ(t, τ)‖ = α, Pei− a.s., ∀i ∈ S (2.15)

(2.14) and (2.15) yield the result for the case i ∈ S\E.

(ii). Since σt has single ergodic class, (2.13) holds. The first equality holdstrivially, because the sojourn time τk = tk+1 − tk is almost surely bounded


and then for t ∈ [tk, tk+1),

α = limt→+∞

1

tlog ‖Φ(t, 0)‖ = lim

t→+∞

1

tlog ‖Φ(t, tk)Φ(tk, 0)‖

= limk→+∞

(tkk

)−1 tkt

1

klog ‖Φ(tk, 0)‖, Pψ − a.s..

We show the second equality next, i.e.,

α = a−1 limk→+∞

1

kEπlog ‖Φ(tk, 0)‖, Pψ − a.s.. (2.16)

First of all, the existence of the limit in (2.16) follows from the stationarity

and the fact that akdef= Eπlog ‖Φ(tk, 0)‖ is a subadditive sequence. Let

k = pm + q with 0 ≤ q < m, then using the sojourn time description of σt,we have

1

klog ‖Φ(tk, 0)‖ =

1

klog ‖ exp(Ark

γk) · · · exp(Ar0γ0)‖

≤ 1

klog ‖ exp(Ark

γk) · · · exp(Arpmγpm)‖

1pm+q

p∑

j=1

log ‖ exp(Arjm−1γjm−1) · · · exp(Ar(j−1)mγ(j−1)m)‖

(2.17)

Since ‖ exp(Ajt)‖ ≤ exp(κt) for t ≥ 0, we have

1

klog ‖ exp(Ark

γ(rk)) · · · exp(Arpm−1γ(rpm−1))‖

≤ 1

klog ‖ exp(Ark

γ(rk))‖ · · · ‖ exp(Arpm−1γ(rpm−1))‖

≤ Mγ(rpm+q) + · · · + γ(rpm−1)

k

k→∞−→ 0 Pψ − a.s. (2.18)

Moreover, because σt has a single ergodic class, this implies that the Markovchain ζk = (rk, γk) : k ≥ 0 also has a single ergodic class, and so doesthe Markov chain (ζ(j+1)m−1, . . . , ζjm) : j ≥ 0. From the Law of LargeNumbers, we have

limp→+∞

1

p

p∑

j=1

log ‖ exp(Arjm−1γjm−1) · · · exp(Ar(j−1)mγ(j−1)m)‖

= Eπ log ‖ exp(Armγm) · · · exp(Ar0γ0)‖. (2.19)


From (2.17)-(2.19), we obtain

limk→∞

1

klog ‖ exp(Ark

γk) · · · exp(Ar0γ0)‖

≤ Eπ log ‖ exp(Arm−1γm−1) · · · exp(Ar0γ0)‖ (2.20)

Let

γ = limm→+∞

1

mEπ log ‖ exp(Arm−1γm−1) · · · exp(Ar0γ0)‖

βk =1

klog ‖ exp(Ark−1

γk−1) · · · exp(Ar0γ0)‖

wk =1

k

k−1∑

i=0

log ‖ exp(Ariγi)‖ − βk.

Then wk ≥ 0, and from the Law of Large Numbers, we obtain

limk→∞

wk = Eπ log ‖ exp(Ar0γ0)‖ − limk→∞

1

kβk

(i)= Eπ log ‖ exp(Ar0γ0)‖ − aα Pψ − a.s. (2.21)

On the other hand, by stationarity, we have

Eπwk = Eπ log ‖ exp(Ar0γ0)‖

− 1

kEπlog ‖ exp(Ark−1

γk−1) · · · exp(Ar0γ0)‖k→∞−→ Eπ log ‖ exp(Ar0γ0)‖ − γ (2.22)

From Fatou’s Lemma, (2.21) and (2.22), we arrive at aα ≥ γ. This togetherwith (2.20) implies that aα = γ. The proof of the proposition is complete.

Note that α < 0 is equivalent to almost sure exponential stability. Next,we begin to study the relationship between δ-moment stability and almostsure stability. The δ-moment stability region Σδ and the almost sure stabilityregion Σa in the parameter space of the system (2.1) are defined by

Σδ = (A(1), . . . , A(N)) : (2.1) is δ-moment stable.

andΣa = (A(1), . . . , A(N)) : (2.1) is almost surely stable.

respectively. We can decompose Σa into a disjoint union of the form

Σa = Σa− ∪ Σa0


withΣa−

def= Σa ∩ (A(1), . . . , A(N)) : α < 0

andΣa0

def= Σa ∩ (A(1), . . . , A(N)) : α = 0.

The following is one of our main results:Theorem 2.5:

For the system (2.1) with a Markov form process σt, (i.) For any 0 <δ1 ≤ δ2, Σδ2 ⊂ Σδ1 ⊂ Σa and Σδ1 is open. (ii.) If σt is irreducible, i.e.,E = S, then

Σa− = limδ↓0+

Σδ =⋃

δ>0

Σδ ⊂ Σa. (2.23)

Remark: This result states that the δ−moment stability region in theparameter space of the system increases as δ decreases and in the limit as δapproaches zero, the δ−moment stability region converges to the almost sureexponential stability region. Then potentially, almost sure stability can bestudied by investigating δ−moment stability for sufficiently small δ. Gener-ally, almost sure stability is inferred from second moment stability. However,second moment stability is often too conservative for practical stability com-putations, but it is difficult to deal with almost sure (sample) stability prop-erties directly because of the complexity in computing or estimating the (top)Lyapunov exponent α. Theorem (2.5) provides an alternative approach tothe study of almost sure stability and as we will see later, δ−moment stabilitycan be used to derive testable criteria for almost sure (sample) stability.

To prove Theorem 2.5, we need some preliminary results.Lemma 2.6: (Large Deviation Theorem)

Suppose that σt is irreducible with a unique invariant distribution π. Forany fixed m > 1, let ∆ = Eπlog ‖Φ(tm, t0)‖. Then, for any ǫ > 0, thereexists M, δ > 0 such that

Pπ(1

p

p−1∑

j=0

log ‖Φ(t(j+1)m, tjm)‖ ≥ ∆ + ǫ) ≤M exp(−δp). (2.24)

Proof of Lemma. The result is a consequence of Theorem IV.1 of [10].Lemma 2.7:

Suppose that σt is irreducible. If the top Lyapunov (Oseledec) exponentα is negative, then there exists a δ > 0, such that limt→+∞Eψ‖Φ(t, 0)‖δ = 0for any ψ ∈ ΞProof of Theorem 2.5. For (i), by the Lyapunov inequality, for 0 < δ1 < δ2,we have

(Eψ‖Φ(t, 0)‖δ1)1/δ1 ≤ (Eψ‖Φ(t, 0)‖δ2)1/δ2 .Thus, Σδ2 ⊂ Σδ1 . By Theorem 2.2, we know that Σδ1 ⊂ Σa. We need toshow that Σδ is open for any δ > 0. Suppose (A(1), . . . , A(N)) ∈ Σδ. Then,


for system (2.1) with the matrices (A(1), . . . , A(N)), the system is δ-momentstable. Then, for any 0 < ρ < 1 given, there is an integer m sufficiently largesuch that

ζi(A(1), . . . , A(N))def= Eei

‖Φ(tm, t0)‖δ < ρ, ∀1 ≤ i ≤ N. (2.25)

Since

ζi(A(1), . . . , A(N)) =∑

i1,...,im−1

pii1 . . . pim−2im−1

∫‖eAim−1

tm−1 . . . eAiti‖δµim−1(dtm−1) . . . µi(dti)

is continuous, for (A(1), . . . , A(N)) in a neighbourhood of (A(1), . . . , A(N)),(2.45) still holds (for the system (2.1) with the matrices (A(1), . . . , A(N))).Consider that for t ∈ [tk, tk+1) with k large and k = pm+ q with 0 ≤ q < m,from Lemma 2.3, we have for any ψ ∈ Ξ,

Eψ ‖Φ(t, t0)‖δ ≤ Eψ‖Φ(t, tpm)‖δp−1∏

j=0

‖Φ(t(j+1)m, tjm)‖δ

= EψEψ‖Φ(t, tpm)‖δ|Fpmp−1∏

j=0


≤ η(q+1)EψEψ‖Φ(tpm, t(p−1)m)‖δ|F (p−1)mp−2∏

j=0


≤ η(q+1)ρEψp−2∏

j=0

‖Φ(t(j+1)m, tjm)‖δ ≤ . . . ≤ η(q+1)ρp (2.26)

In arriving at (2.26), we have used the time homogeneous and Markov prop-erties and η is given as in Lemma 2.3. It follows from (2.26) that

limt→+∞

Eψ‖Φ(t, t0)|δ = 0.

This shows that (2.1) with the matrices (A(1), . . . , A(N)) is also δ-momentstable. Thus, Σδ is open.

For (ii), note that Σa− ⊂ ∪δ>0Σδ by Lemma 2.7 and ∪δ>0Σ

δ ⊂ Σa byTheorem 2.2. Hence, we have

Σa− = limδ↓0+

Σδ = ∪δ>0Σδ ⊂ Σa.

Theorem 2.5 characterizes the relationship between almost sure stabilityand δ-moment stability for the jump linear system (2.1). For an illustrativeexample, see [17].


3 Testable Criteria for Almost Sure Stability

In the above section, we have shown that almost sure stability of (2.1) can bestudied using δ−moment stability. It is expected that δ−moment stabilitycriteria may give refined almost sure stability criteria as δ > 0 becomessufficiently small. In this section, we develop some testable conditions foralmost sure stability by using the relationship between δ-moment stabilityand almost sure stability. The following theorem is the main result of thissection.Theorem 3.1:

If there exists positive definite matrices P (1), . . . , P (N) such that for anyi with 1 ≤ i ≤ N ,

sup‖x‖=1

x

′(P (i)A(i) +A′(i)P (i))x

x′P (i)x+∑

j 6=i

qij log

(x′P (j)x

x′P (i)x

) < 0, (3.1)

then there exists a δ > 0 such that (2.1) is δ-moment stable and thus, (2.1)is almost surely stable.

Proof of theorem. Let xt = x(t, x0) and σt = σ(t). The joint process(xt, σt) : t ≥ 0 is a strong Markov process with right continuous samplepaths and the (weak) infinitesimal generator

L = Q+ diagx′A′(1)∂

∂x, . . . , x′A′(N)

∂

∂x. (3.2)

Consider a Lyapunov function of the form V (x, i) = (x′P (i)x)δ/2 for x ∈ Rn

and i ∈ S. Then, we have

(LV )(x, i) =N∑

j=1

qij(x′P (i)x)δ/2 + x′A′(i)

δ

2(x′A′(i)x)δ/2−1(2P (i)x)

= (x′P (i)x)δ/2

δ2

x′(AT (i)P (i) + P (i)A(i))x

x′P (i)x+

N∑

j=1

qij

(x′P (j)x

x′P (i)x

)δ/2

def= (x′P (i)x)δ/2Θ(x, i, δ) = V (x, i)Θ(x, i, δ). (3.3)

Next, we want to prove that if (3.1) holds, then there exists a δ > 0 suchthat for any i ∈ S,

sup‖x‖=1

Θ(x, i, δ) =

= sup‖x‖=1

δ

2

x′(A′(i)P (i) + P (i)A(i))x

x′P (i)x+

N∑

j=1

qij

(x′P (j)x

x′P (i)x

)δ/2 < 0


(3.4)

For notational simplicity, let

Mi(x) =x′(A′(i)P (i) + P (i)A(i))x

x′P (i)x, Gji(x) =

x′P (j)x

x′P (i)x.

Suppose that (3.4) is not true. Then for some i ∈ S and any integer k, withδ = 1/k, there exists an xk such that ‖xk‖ = 1 and

1

2kMi(xk) +

N∑

j=1

qijGji(xk)1/(2k) ≥ 0 (3.5)

The unit sphere Sndef= x : ‖x‖ = 1 is compact, and every sequence in Sn

has a convergent subsequence. Therefore, there exists a sequence δk and asequence xk in Sn satisfying δk ↓ 0+ and xk → x0 ∈ Sn as k → ∞ (where‖xk‖ = 1) such that

δk2Mi(xk) +

N∑

j=1

qijGj(xk)δk/2 ≥ 0 (3.6)

By continuity, for any ε > 0, there exists a k0 such that for k ≥ k0,

0 ≤Mi(xk) ≤Mi(x0) + ε, 0 ≤ Gji(xk) ≤ Gji(x0) + ε.

From (3.6), we have

−qi +δk2

(Mi(x0) + ε) +∑

j 6=i

qij(Gji(x0) + ε)δk/2 ≥ 0 (3.7)

Since qi =∑

j 6=i qij , we have∑j 6=i qij/qi = 1, and with δk ↓ 0+ as k → +∞,

limk→∞

∑

j 6=i

qijqi

(Gji(x0) + ε)δk/2

2/δk

=∏

j 6=i

(Gji(x0) + ε)qij/qi .

It follows that for any ε1 > 0, there exists a k1 > k0 such that for k ≥ k1,

∑

j 6=i

qijqi

(Gji(x0) + ε)δk/2

2/δk

≤∏

j 6=i

(Gji(x0) + ε)qij/qi + ε1,

i.e.,

0 ≤∑

j 6=i

qijqi

(Gji(x0) + ε)δk/2 ≤

∏

j 6=i

(Gji(x0) + ε)qij/qi + ε1

δk/2

.


Taking this into (3.7), we obtain

δk2

(Mi(x0) + ε) − qi + qi

∏

j 6=i


δk/2

≥ 0

Rewriting this in the following form:

Mi(x0) + ε− qi

1 −(∏

j 6=i(Gji(x0) + ε)qij/qi + ε1

)δk/2

δk/2

≥ 0.

Letting k −→ ∞ and using the fact that limx→0(1 − ax)/x = − log a fora > 0, we have

Mi(x0) + ε+ qi log

∏

j 6=i


≥ 0

With ε and ε1 arbitrarily chosen, it follows from the above inequality that:

Mi(x0) +∑

j 6=i

qij logGji(x0) ≥ 0.

Therefore,

sup‖x‖=1

Mi(x) +

∑

j 6=i

qij logGji(x)

≥ 0,

and this contradicts (3.2). Therefore we have proved (3.4).Now, for the choice of δ > 0 in (3.4), let −γ = sup‖x‖=1 Θ(x, i, δ) =

supx 6=0 Θ(x, i, δ) < 0. From (3.3), we obtain

(LV )(x, i) ≤ −γ(x′P (i)x)δ/2 = −γV (x, i). (3.8)

Applying Dynkin’s formula and Fubini’s theorem, we have

EV (xt, σt)|x0 = x, σ0 = i − V (x, i)

= E∫ t

0

(LV )(xs, σs)ds|x0 = x, σ0 = i

≤ E∫ t

0

−γV (xs, σs)ds|x0 = x, σ0 = i

= −γt∫

0

EV (xs, σs)|x0 = x, σ0 = ids. (3.9)


Applying the Gronwall-Bellman Lemma, we have for any x 6= 0 and i ∈ S,

EV (xt, σt)|x0 = x, σ0 = i ≤ V (x0, i) exp(−γt), ∀t ≥ 0.

Let λmin = min1≤i≤N λmin(P (i)), we have

E ‖xt‖δ|x0 = x, σ0 = i = E(x′txt)δ/2|x0 = x, σ0 = i

≤ 1

λmin

E(x′tP (σt)xt)δ/2|x0 = x, σ0 = i ≤ V (x0, i)

λmin

e−γt.

This implies that the system (2.1) is exponentially δ−moment stable andthus almost surely stable. This completes the proof of Theorem 3.1.Corollary 3.2:

If there exists positive definite matrices P (1), . . . , P (N) such that for anyi ∈ S,

λmax((P (i)A(i) +AT (i)P (i))P (i)−1) +∑

j 6=i

qij logλmax(P (j)P (i)−1) < 0.

then, (2.1) is δ-moment stable for some δ > 0 and thus almost surely stable.

Proof of corollary. Using the fact that

sup‖x‖=1

xTQx

xTPx= λmax(QP

−1) (3.10)

where Q is symmetric and P is positive definite, the corollary follows directlyby maximizing each of the terms in (3.1).

In the proof of the theorem, we used the Lyapunov function V (x, i) =(x′P (i)x)δ/2 to obtain a sufficient condition for δ-moment stability. In [17],Feng et al established a necessary and sufficient condition for second momentstability using a quadratic Lyapunov function (V with δ = 2). It is verytempting to conjecture that the Lyapunov function V is also “necessary” forδ-moment stability, i.e., if (2.1) is δ-moment stable, there exists P (i) suchthat V (x, i) = (x′P (i)x)δ/2 is a Lyapunov function for (2.1) in the sensethat LV < 0. Equivalently, we want to claim that (3.4) is also necessary forδ-moment stability for any δ > 0. However, we have not obtained a proofof this result so far. The second moment case and the one-dimensional caseillustrate this intuitive idea.Proposition 3.3:

(i) (2.1) is second moment stable if and only if with δ = 2, (3.4) holds, i.e.,for any x ∈ Rn and i ∈ S, sup‖x‖=1 Θ(x, i, δ) < 0 for some choice of positivedefinite matrices P (i) (i = 1, 2, . . . , N). Furthermore, (3.1) is satisfied. (ii)If n = 1, (2.1) is δ-moment stable for some δ > 0 if and only if (3.4) holdsand therefore, if and only if (3.1) holds. It follows that if σt is irreducible,(A(1), . . . , A(N)) ∈ Σa− if and only if (3.1) holds.


Proof of Proposition 3.3. Sufficiency is established from Theorem 3.1 andTheorem 2.5. Necessity follows from a procedure similar to that used in [17].

From this result, we see that the testable condition (3.1) for δ-momentand almost sure stability is a stronger result than the sufficient conditionsobtained from second moment stability. For a scalar system, the conditionis necessary and sufficient for almost sure exponential stability. It seemsas if (3.1) is also necessary for almost sure stability for higher dimensionalsystems. However, so far we have not obtained a rigorous proof of this result.For scalar systems, the necessity of (3.2) for almost sure exponential stabilitycan be justified algebraically. We know from [17] that for a one-dimensionalsystem, if σt has a single ergodic class, then the top exponent is given by

α = π1A(1) + . . .+ πNA(N) = πa (3.15)

where a = (A(1), . . . , A(N))′.

Theorem 3.4:

For the one dimensional system (2.1), suppose that σt is irreducible. Thesystem is almost surely exponentially stable if and only if πa < 0, or if andonly if (3.1) holds for some positive definite matrices P (i) (i = 1, 2, . . . , N),or if and only if there exists a y ∈ RN such that a+Qy < 0.

Proof. We only need to establish the equivalence of the three algebraicconditions.

For any P (i) > 0, we have for any i and x

Mi(x) +∑

j 6=i

qij logGji(x)

= 2A(i) +∑

j 6=i

qij logP (j)

P (i)= 2ai +

∑

j 6=i

qij logP (j) − (∑

j 6=i

qij) logP (i)

= 2ai +

N∑

j=1

qij logP (j) = [2(a+Q(logP (1), . . . , logP (N))′)]i

where [·]i denotes the i−th component and ai denotes the i−th componentof a. Thus, (3.1) holds for some choice of positive definite matrices P (i) ifand only if a+Qy < 0 for some y ∈ RN . Next, we show that a+Qy < 0 forsome y ∈ RN if and only if πa < 0. If a+Qy < 0, then π(a+Qy) = πa < 0,with σt is irreducible, π > 0. On the other hand, assume πa < 0. We firstshow that Im(Q) = y ∈ RN : πy = 0. In fact, if y ∈ Im(Q), there existsz ∈ RN such that y = Qz. Since πQ = 0, we have πy = (πQ)z = 0 and thusIm(Q) ⊂ y : πy = 0. However, dim Im(Q) = rank(Q) = N − 1 due to theirreducibility (or uniqueness of π). Then, dim Im(Q) = dimy : πy = 0 =N − 1 and Im(Q) = y : πy = 0. Now, let

y = −a+πa

ππ′π′ ∈ RN ,


since πy = −πa+ πaππ′

ππ′ = 0, we have y ∈ Im(Q), thus there exists a z ∈ RN ,such that Qz = y, i.e.,

Qz + a =πa

ππ′π′ < 0,

with πa < 0 and π′ > 0. This completes the proof.From the above proof, we can observe that Im(Q) is the boundary of the

almost sure stability region in the parameter space of the system. Theorem3.1 is a very general result for almost sure stability and by specifying thepositive definite matrices P (1), . . . , P (N), we can obtain some simple testcriteria for almost sure stability.Corollary 3.5:

Define µP (A) = λmax(PAP−1 + A′)/2. Then (i) If there exists a pos-

itive definite matrix P and y ∈ RN such that u + Qy < 0 with u =(µP (A(1)), . . . , µP (A(N)))′, then (2.1) is almost surely stable. (ii) If thereexists a positive definite matrix P such that

π1µP (A(1)) + π2µP (A(2)) + . . .+ πNµP (A(N)) < 0,

then (2.1) is almost surely stable.Proof of corollary: (i). In Theorem 3.1, let P (i) = αiP for some αi > 0.Then

sup‖x‖=1

x′(P (i)A(i) +A′(i)P (i))x

x′P (i)x+∑

j 6=i

qij log

(x′P (j)x

x′P (i)x

)

= sup‖x|=1

x′(PA(i) +A′(i)P )x

x′Px+∑

j 6=i

qij logαjαi

= sup‖x|=1

x′(PA(i) +A′(i)P )x

x′Px

+

N∑

j=1

qij logαj

= λmax(PA(i)P−1 +A′(i)) +N∑

j=1

qij logαj = 2µP (A(i)) +N∑

j=1

qij logαj

From the proof of Theorem 3.4, we know that

µP (A(i)) +

N∑

j=1

qij log√αj < 0, i = 1, 2, . . . , N

if and only if there exists an y ∈ RN such that u + Qy < 0. From thisand Theorem 3.1, we obtain the proof of (i). Note also that in the proof ofTheorem 3.4, we have shown that u+Qy < 0 for some y if and only if

π1µP (A(1)) + π2µP (A(2)) + · · · + πNµP (A(N)) < 0.


Thus, (ii) follows and the proof is complete.Though Theorem 3.1 is of theoretical interest, so far we have not found

a rigorous procedure to select the positive definite matrices P (1), . . . , P (N)so that (3.1) holds. Next, we provide necessary conditions for (3.1) to hold.Define

hi(x) =x′(P (i)A(i) +A′(i)P (i))x

x′P (i)x+∑

j 6=i

qij log

(x′P (j)x

x′P (i)x

),

which is a continuous function on the unit sphere Sn for all i ∈ S. (3.1) istherefore equivalent to

x′(P (i)A(i) +A′(i)P (i))x

x′P (i)x+∑

j 6=i

qij log

(x′P (j)x

x′P (i)x

)< 0, ∀x ∈ Sn, i ∈ S.

(3.16)From this we obtain

x′(P (i)A(i) +A′(i)P (i))x

x′P (i)x+

N∑

j=1

qij log(x′P (i)x) < 0, ∀x ∈ Sn, i ∈ S.

or

x′(P (1)A(1)+A′(1)P (1))xx′P (1)x

...x′(P (N)A(N)+A′(N)P (N))x

x′P (N)x

+Q

log(x′P (1)x)...

log(x′P (N)x)

< 0 (3.17)

Multiplying both sides of (3.17) by π and using πQ = 0, we obtain thefollowing necessary condition for (3.1).Theorem 3.6:

A necessary condition for (3.1) is:

π1x′(P (1)A(1) +A′(1)P (1))x

x′P (1)x+ · · ·+πN

x′(P (N)A(N) +A′(N)P (N))x

x′P (N)x< 0

(3.18)for any x ∈ Sn.

Because second moment stability implies (3.1), we see that (3.18) is alsoa necessary condition for second moment stability.

It is tempting to conjecture that (3.18) is also a sufficient condition foralmost sure stability. Unfortunately, this is not true. A reason for this is givenby the following: Suppose the conjecture is true. Then, if A(1), . . . , A(N) arestable matrices, there exists positive definite matrices P (i) for i = 1, 2, . . . , Nsuch that AT (i)P (i) + P (i)A(i) = −I and (3.18) is satisfied. But as we willsee in the following examples, stability of A(1), . . . , A(N) does not guaranteealmost sure stability of the system (2.1), in general.Example 3.7:


(i). Consider the harmonic oscillator:

u′′t + k2(1 + bξt)ut = 0, t ≥ 0

where ξt is a random telegraph process with state space −1,+1, i.e., atwo state time-homogeneous Markov chain with the infinitesimal generator

Q =

(−q qq −q

)

From Feng and Loparo ([15]), we know that for any k > 0 and b ∈ (−1,+1),the top Lyapunov exponent exists almost surely:

α = limt→+∞

1

tlog√u2(t) + k2u2(t)

regardless of the initial distribution of ξt and the initial state (u(0), u(0)).Furthermore, the number α is strictly positive.

Now, let k = 1 and b = 0.5. We have the following state space represen-tation for the oscillator: x(t) = A0(ξt)x(t) with x1 = u, x2 = u and

A0(−1) =

(0 1

−1.5 0

), A0(+1) =

(0 1

−0.5 0

)

Define a new system x(t) = A(ξt)x(t), where A(−1) = −0.5αI+A0(−1) andA(+1) = −0.5αI +A0(+1), it is easy to show that A(1) and A(2) are stablematrices. However, the top Lyapunov exponent for this system is

α = limt→∞

1

tlog ‖x(t)‖ = −0.5α+ α = 0.5α > 0. a.s..

Therefore, the system is almost surely unstable.(ii). We give another example to show that individual mode stability does

not guarantee almost sure stability.Let

A(1) =

(−a 10 −a

), A(2) =

(−a 01 −a

), Q =

(−q qq −q

),

where a > 0 and q > 0 are to be determined. Using the same notation asbefore, we have

Φ(tk+1, 0) = exp(A(rk)τk) · · · exp(A(r0)γ0)

= exp

(−a

k∑

i=0

τi

) (1 τk0 1

)(1 0

τk−1 1

)· · ·(

1 τ00 1

)

Let M(A) denote the largest element of A, then it is easy to verify that

M(Φ(tk+1, 0)) ≥ γk · · · γ0 exp

(−a

k∑

i=0

γi

).


Thus from this, we obtain

1

klogM(Φ(tk+1, 0)) ≥ 1

k

k∑

i=0

log γi − a1

k

k∑

i=0

γi.

From the Law of Large Numbers, we have

limk→∞

1

klogM(Φ(tk+1, 0)) ≥ Eπlog γ0 − aEπγ0

where π is the unique invariant measure of the joint process (γi, ri). Com-pute

Eπlog γ0 = − log q +

∫ +∞

0

(log x)e−xdx, Eπγ0 =1

q.

It follows that

limk→∞

1

klogM(Φ(tk+1, 0)) ≥ − log q +

∫ +∞

0

(log x)e−xdx− a

q.

Choose q sufficiently small so that − log q +∫∞

0 (log x)e−xdx − 1 ≥ 0, andchoose a sufficient small so that 1 − a/q ≥ 0.5, then we finally arrive at

limk→∞

1

klogM(Φ(tk+1, 0)) ≥ 1 − a

q≥ 1

2

almost surely, i.e., limk→∞M(Φ(tk, 0)) = +∞ almost surely. Therefore, thesystem is almost surely unstable with the appropriate choices of a and q.However, A(1) and A(2) are stable matrices.

Although (3.18) is not in general a sufficient condition for almost surestability of (2.1), for certain special cases, it is. Define φ(t) = x(t)/‖x(t)‖ ∈Sn, then we haveTheorem 3.8:

Suppose that (φ(t), σt) is ergodic with a unique invariant measurep(φ, σ)dφπ(dσ). If there exists a positive definite matrix P such that

max‖x‖=1

N∑

i=1

πix′(PA(i) +A′(i)P )x

x′Px

< 0,

then the system (2.1) is almost surely stable.Proof of theorem 3.8. Let

γ = max‖x‖=1

N∑

i=1


x′Px

.

Then from the assumption, we have

N∑

i=1


x′Px≤ γ < 0, ∀x ∈ Sn. (3.19)


Define the Lyapunov function V (t) = 12x

′(t)Px(t) and let A(σt) = (A′(σt)P+PA(σt))/2. We obtain

V (t) = x′(t)A(σt)x(t) =φ′(t)A(σt)φ(t)

φ′(t)Pφ(t)V (t).

Thus, we have

V (T ) = V (0) exp

(∫ T

0

φ′(t)A(σt)φ(t)

φ′(t)Pφ(t)dt

).

This implies that

limt→∞

1

tlogV (t) = lim

T→+∞

1

T

∫ T

0

φ′(t)A(σt)φ(t)

φ′(t)Pφ(t)dt (3.20)

Because the process (φ(t), σt) is ergodic, from the Law of Large Numbersand (3.20), we have

limt→+∞

1

tlogV (t) = E

φ′(t)A(σt)φ(t)

φ′(t)Pφ(t)

=

∫

Sn×S

x′A(y)x

x′Pxp(x, y)dxπ(dy)

=

N∑

j=1

πj

∫

Sn

x′A(j)x

x′Pxp(x, j)dx

=1

2

∫

Sn

N∑

j=1

πjx′(A′(j)P + P (j)A(j))x

x′Pxp(x, j)dx

≤ 1

2γ < 0

Because P is positive definite, we can easily obtain

limt→∞

1

tlog ‖x‖ =

1

2limt→∞

1

tlogV (t) ≤ 1

4γ < 0

almost surely and this implies that (2.1) is almost surely stable.

An interesting application of the above theorem is when we have a diago-nal system. Let A(j) = diaga1(j), a2(j), . . . , an(j) for j = 1, 2, . . . , N . Let


P = I, for any ‖x‖ = 1, we then have

N∑

i=1

πix′(AT (i) +A(i))x = 2

N∑

i=1

πix′A(i)x

= 2

[(N∑

i=1

πia1(i)

)x2

1 + · · · +(

N∑

i=1

πian(i)

)x2n

]

≤ 2

maxj

(N∑

i=1

πiaj(i)

)(x2

1 + · · · + x2n) = 2 max

j

(N∑

i=1

πiaj(i)

)

Thus, if maxj

(∑Ni=1 πiaj(i)

)< 0, then (2.1) is almost surely stable. In fact,

it is easy to see that this condition is also necessary for the diagonal system(2.1) to be almost surely exponentially stable.

All conditions developed previously, except for the scalar case, are suffi-cient conditions for almost sure stability. However, for some special cases,necessary and sufficient conditions can be obtained. In particular, for diago-nal systems, a necessary and sufficient condition for almost sure exponentialstability is easy to obtain. In [12],[13], Fang et al gave some necessary andsufficient conditions for almost sure stability of discrete-time jump linear sys-tems with special structures, for example, when the system matrices pairwisecommute. In the following, we summarize without proof parallel results forthe continuous-time jump linear system (2.1).Theorem 3.9:

Suppose that A(1), . . . , A(N) are given by

A(σt) =

λ1(σt) a12(σt) . . . a1n(σt)λ2(σt) . . . a2n(σt)

. . ....

λn(σt)

and suppose that σt has a single ergodic class. Then the system (2.1) isalmost surely exponentially stable if and only if

Eπλi(σ0) = π1λi(1) + π2λi(2) + · · · + πNλi(N) < 0, i = 1, 2, . . . , n

where π = π1, . . . , πN is the unique invariant measure of the form processσt.Corollary 3.10. Suppose that σt has a single ergodic class. IfA(1), . . . , A(N)can be simultaneously transformed into the following upper triangular forms

λ1(i) a12(i) . . . a1n(i)λ2(i) . . . a2n(i). . .

...λn(i)

, i = 1, 2, . . . , N,


by the same similarity transformation, then the system (2.1) is almost surelyexponentially stable if and only if

Eπλi(σ0) = π1λi(1) + π2λi(2) + · · · + πNλi(N) < 0, i = 1, 2, . . . , n

where π = π1, . . . , πN is the unique invariant measure of the form processσt.

It is well-known ([18]) that if A(1), . . . , A(N) pairwise commute, thenthere exists a unitary matrix U such that U ′A(j)U for j = 1, 2, . . . , N areeach upper triangular matrices. Thus, using Theorem 3.9, we can completelysolve the almost sure exponential stability problem for the jump linear system(2.1) with pairwise commuting system matrices. We haveCorollary 3.11. If σt has a single ergodic class. Suppose thatA(1), . . . , A(N)pairwise commute, then there exists a unitary matrix U such that U ′A(j)Ufor j = 1, 2, . . . , N are of the following form:

U ′A(i)U =

λ1(i) a12(i) . . . a1n(i)λ2(i) . . . a2n(i). . .

...λn(i)

, i = 1, 2, . . . , N.

It follows that the system (2.1) is almost surely exponentially stable ifand only if

Eπλi(σ0) = π1λi(1) + π2λi(2) + · · · + πNλi(N) < 0 (i = 1, 2, . . . , n)

where π = π1, . . . , πN is the unique invariant measure of the form processσt.

4 Conclusions

We have studied the stability properties of continuous time jump linear sys-tems, which can be used to model linear systems subject to abrupt changes instructure, possibly resulting from component failure or communication delaysin networked control systems. After introducing some stability concepts, wehave shown that all δ−moment stability properties are equivalent and thateach imply almost sure (sample path) stability. It is then proved that theregion of δ−moment stability is monotonically converging to the region ofalmost sure exponential stability as δ ↓ 0+. This provides a new approachfor the study of almost sure stability and for developing testable conditionsfor almost sure stability. It can be seen from our results, that the proof ofthe equivalence between δ-moment stability for sufficiently small δ > 0 andalmost sure stability crucially depends on the large deviation principle givenin ([10]). The study of almost sure stability for a more general class of sys-tems using similar results in large deviation theory is a direction for futureresearch.


5 References

[1] L. Arnold, E. Oelieklaus and E. Pardoux, “Almost Sure and Moment Stability forLinear Ito Equations,” in Lyapunov Exponents: Proceeding, Bremen 1984 (Ed:Arnold,L. and Whistutz,V.), Lecture Notes in Mathematics 1186, Springer-Verlag,1986.

[2] L. Arnold and V. Wihstutz, Lyapunov Exponents, Lecture Notes in Math., 1186,1985.

[3] R. Bellman, “Limiting Theorems for Non-commutative operators,” Duke Math. J.,21, 456, 1960.

[4] A. R. Bergen, “Stability of Systems with Randomly Time-varying Parameters,” IRETrans., AC-5, 265, 1960.

[5] J. E. Bertram and P. E. Sarachik,“Stability of Circuits with Randomly Time-varyingParameters,”Trans. IRE, PGIT-5, Special Issue, 260, 1959.

[6] B. H. Bharucha, “On the Stability of Randomly Varying Systems,” Ph.D Thesis,Dept. Elec. Eng., Univ. Calif. ,Berkeley, July, 1961.

[7] P. Bougerol and J. Lacroix, Products of Random Matrices with Applications toSchrodinger Operators, Birkhauser, 1985.

[8] H. J. Chizeck, A. S. Willsky and D. Castanon, “Discrete-time Markovian -jumpQuadratic Optimal Control,” Int. J. Control, Vol. 43, No. 1, 213-231, 1986.

[9] B.S. Darkhovskii and V.S. Leibovich, “Statistical Stability and Output Signal Mo-ments of a Class of Systems with Random Variation of Structure,” Auto. Remot.Control, Vol.32, No.10, 1560-1567, 1971.

[10] R.S. Ellis, “ Large Deviations for A General Class of Random Vectors,” Annal.Prob., 12, 1-12, 1984.

[11] Y. Fang, K.A. Loparo and X. Feng, “Modeling Issues for the Control Systems withCommunication Delays,” SCP Research Report for Ford Motor Company (2), Oct.,1991.

[12] Y. Fang, K.A. Loparo and X. Feng, “Almost Sure Stability and δ−moment Stabilityof Jump Linear Systems” Int. J. Control, Vol. 59, No. 5, pp. 1281-1307, 1994.

[13] Y. Fang, K.A. Loparo and X. Feng, “Stability of Discrete Time Jump Linear Sys-tems,” Journal of Mathematical Systems, Estimation and Control, Vol. 5, No. 3,pp. 275-321, 1995.

[14] X. Feng, “Lyapunov Exponents and Stability of Linear Stochastic Systems ,” Ph.DThesis, Dept. Systems Eng., Case Western Reserve Univ., 1990.

[15] X. Feng and K.A. Loparo, “Almost Sure Instability of the Random Harmonic Os-cillator,” SIAM J. Appl. Math., Vol.50, No.3, pp.744-759, June, 1990.

[16] X. Feng and K.A. Loparo, “ A Nonrandom Spectrum Theorem for Products ofRandom Matrices and Linear Stochastic Systems,” J. Math. Syst., Est., Contr.,2,3, pp.323-338, 1992.

[17] X. Feng, K.A. Loparo, Y. Ji, and H. J. Chizeck, “Stochastic Stability Propertiesof Jump Linear Systems,” IEEE Trans. Automat. Control, Vol.37, No.1, pp.38-53,1992.

[18] R.A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

[19] Y. Ji, H. J. Chizeck, X. Feng and K.A. Loparo,“Stability and Control of Discrete-time Jump Linear Systems,” Control—Theory and Advanced Tech., Vol.7, No.2,247-270, 1991.

[20] I. I. Kats and N. N. Krasovskii, “On the stability of Systems with Random Param-eters,” Prkil. Met. Mek. 24, 809, 1960.


[21] F. Kozin,“A Survey of Stability of Stochastic Systems,” Automatica, Vol.5, 95-112,1969.

[22] R. Krtolica, U. Ozgunder,, H. Chan, H. Goktas, J. Winkelman and M. Liubakka,“Stability of Linear Feedback Systems with Random Communication Delays,” ACC’91, pp. 2648-2653, Boston, MA., June, 1991.

[23] A. Leizarowitz, “Estimates and Exact Expressions for Lyapunov Exponents ofStochastic Linear Differential Equations,” Stochastics, Vol.24, pp.335-356, 1988.

[24] A. Leizarowitz, “On the Lyapunov Exponet of a Harmonic Oscillator Driven by afinite-State Markov Processes,” SIAM J. Appl. Math., Vol.49, No.2, pp.404-419,1989.

[25] A. Leizarowitz, “Exact Results for the Lyapunov Exponents of Certain Linear ItoSystems,” SIAM J. Appl. Math., Vol.50, No.4, pp.1156-1165, 1990.

[26] V. I. Osecledc, “A Multiplicative Ergodic Theorem: Lyapunov Characteristic Num-bers for Dynamical Systems” Trans. Moscow Math. Soc., 19, pp.197-231, 1968.

[27] A. Rosenbloom,“Analysis of Linear Systems with Randomly Time-varying Param-eters,” Proc. Symp. Inf. Nets, Vol.III, Poly. Inst. Brooklyn, 145, 1954.

email:[email protected]://monotone.uwaterloo.ca/∼journal/


COMPUTATION OF AN OVER-APPROXIMATIONOF THE BACKWARD REACHABLE SET USING

SUBSYSTEM LEVEL SET FUNCTIONS

Dusan M. Stipanovic1, Inseok Hwang1, and Claire J. Tomlin1

1Department of Aeronautics and Astronautics

Stanford University, Stanford, CA 94305

Abstract. In this paper, we present a method to decompose the problem of computing the

backward reachable set for a dynamic system in a space of a given dimension, into a set of

computational problems involving level set functions, each defined in a lower dimensional

(subsystem) space. This allows the potential for great reduction in computation time.

The overall system is considered as an interconnection of either disjoint or overlapping

subsystems. The projection of the backward reachable set into the subsystem spaces is

over-approximated by a level set of the corresponding subsystem level set function. It is

shown how this method can be applied to two-player differential games. Finally, results

of the computation of polytopic over-approximations of the unsafe set for the two aircraft

conflict resolution problem are presented.

Keywords. Reachable sets; interconnected systems; over-approximations.

AMS (MOS) subject classification: 70H20, 49N70, 49N75, 93A14, 93A15

1 Introduction

Computation of reachable sets for dynamic systems has an important appli-cation in the automatic verification of safety properties and synthesis of safecontrollers for air traffic systems [12, 18]. The exact reachable set boundaryis known to be the zero level set of the viscosity solution [2] of a Hamilton-Jacobi type of partial differential equation (PDE) [14]. To the best of ourknowledge, Leitmann was the first to recognize the relationship between Bell-man functions and the boundaries of the reachable sets [11]. In [22] it wasshown how to approximate boundaries of reachable sets with an arbitraryaccuracy using smooth functions. In the sequence of papers by Khrustalev[4, 5], locally Lipschitz functions are used to describe, arbitrarily accurate,under- and over-approximations of reachable sets. Finally, polytopic [19]and ellipsoidal approximations [7, 8, 9, 10] were developed to approximatereachable sets for linear systems (with and without perturbations).

The numerical solutions which provide convergent approximations of reach-able sets for dynamic systems have computational complexity which is expo-

400 D.M. Stipanovic, I. Hwang, and C.J. Tomlin

nential in the continuous variable dimension [14, 13]. To overcome this prob-lem, in [3] the authors use a polytopic approximation method [19], based onoptimal control methods (e.g., [6, 20]), and polytopic level set functions, toefficiently compute approximations of forward and backward reachable setsfor linear dynamic systems. This approach is extended to feedback lineariz-able nonlinear systems, linear dynamic games, and norm-bounded nonlinearsystems in [3].

In [21], Vincent and Wu consider an approximation of the projection ofthe backward reachable (controllable) set for a class of hierarchical dynamicsystems. Mitchell and Tomlin [15] approximate the backward reachable setfor differential games by solving a set of lower dimensional Hamilton-Jacobi-Isaacs PDEs in projected spaces.

In this paper, we propose a method which decomposes the overall prob-lem of computing an over-approximation for the backward reachable set of ageneral nonlinear dynamic system, into a number of lower dimensional com-putational problems. By doing so, the computational complexity of the prob-lem is decreased: in general, we solve many reachability problems of lowerdimension than the original reachability problem. For implementation, com-putational load may be distributed over a network of parallel computers. Theoverall dynamic system is considered as a set of interconnected subsystems.Then, with each subsystem we associate a level set function of Lipschitz typeof which a particular level set is an over-approximation of the boundary ofthe backward reachable set in the subsystem state space (projection space).The interconnection of these sets is an over-approximation of the backwardreachable set. The idea for this approach has been strongly motivated by thesuccessful application of vector Liapunov functions to establish stability ofinterconnected dynamic systems [23].

The paper is organized as follows. In Section 2 we present the mathemat-ical background for backward reachable sets. The analysis of the computa-tion of over-approximations in the subsystem (projected) spaces is providedin Section 3. The method is extended to differential games in Section 4,with an application to the computation of the unsafe set in the two aircraftcollision avoidance problem. Conclusions are presented in Section 5.

2 Backward Reachable (Controllable) Sets

Let us consider the following dynamic system

x = f(t, x, u) , t ∈ T = [0, tf ] , x(tf ) ∈ Xf (1)

where 0 ≤ tf <∞ , x ∈ Rn is the state, u ∈ U ⊂ Rm is the control input withU a compact set independent of the state x. A compact set of final states isdenoted as Xf , Xf ⊂ Rn. The function f : T×Rn ×U → Rn is assumed tobe Lipschitz continuous. The control input function t→ u(t) is a measurablefunction, and the trajectory of the system t→ x(t) is an absolute continuous

Comp. of an Over-App. of the Back. Reach. Set using Sub. Level Set Func. 401

function such that equation (1) is satisfied almost everywhere. We will referto the set of final states as the target set.

The backward reachable (controllable) set (from a given target set Xf )for the system (1) is defined as follows:

Definition 1. The backward reachable set C(t) at time t (t ∈ T), of thesystem (1) from the target set Xf , is the set of all states x for which thereexists an admissible control input u(τ) (τ ∈ [t, tf ]) and a corresponding ab-solute continuous trajectory x(τ) (τ ∈ [t, tf ]) of the system (1), such thatx(tf ) ∈ Xf and x = x(t) .

We define the Hamiltonian for the backward propagation of the system(1) as

Hf (t, x, p) = minu∈U

pT f(t, x, u) (2)

with p ∈ Rn being the adjoint state vector. The inner product of two vectorsa, b ∈ Rn, is denoted as aT b. It is well known [14] that for any t ∈ T, theexact boundary of the backward reachable set can be computed as the zerolevel set of the viscosity solution v(t, x) of the Hamilton-Jacobi equation,

∂v

∂t+Hf (t, x,

∂v

∂x) = 0 (3)

with the final condition v(tf , x) having zero level set which is the boundaryof the target set Xf . Let us rewrite system (1) in decomposed form as a setof N subsystems

xi = fi(t, x, u) , i ∈ N, N = 1, ..., N (4)

where xi ∈ Rni is the i-th subsystem state such that x = [xT

1 , ..., xTN ]T ∈ R

n,that is, the subsystems are disjoint. Then,

Hf (t, x, p) = minu∈U

pTf(t, x, u)

= minu∈U

N∑

i=1

pTi fi(t, x, u) ≥

N∑

i=1

minu∈U

pTi fi(t, x, u)

(5)

By defining the subsystem Hamiltonians as

Hfi(t, x, pi) = min

u∈U

pTi fi(t, x, u) (6)

from equations (5) and (6) it follows that

Hf (t, x, p) ≥

N∑

i=1

Hfi(t, x, pi) (7)

If we assume that each subsystem has its own independent input, that is,

fi(t, x, u) = fi(t, x, ui) (8)


with ui ∈ Rmi , m =N∑

i=1

mi, (that is, decomposition of the input vector is

disjoint) then using (5), (6), and (8) we obtain the following:

Hf (t, x, p) =

N∑

i=1

Hfi

(t, x, pi). (9)

Thus, when the subsystems can be chosen to have independent inputs, in-equality (7) becomes equality (9).

To conclude this section, let us state that the motivation for derivinginequality (7) and equality (9) is the idea of converting the high dimensionalproblem of computing the backward reachable set for the overall system intoa set of lower dimensional problems involving subsystem Hamiltonians.

3 Over-Approximations of the Subsystem Pro-

jections of the Backward Reachable Sets

In order to compute an over-approximation of the backward reachable set,denoted as ℜ(t), (C(t) ⊆ ℜ(t)), to each subsystem we associate a level setfunction (t, xi) → vi(t, xi) which is assumed to be Lipschitz continuous withpositive Lipschitz constant ki. By computing its derivative along the solutionof system (1) we obtain

dvi(t, xi)

dt

∣∣∣∣(1)

=∂vi

∂t+

(∂vi

∂x

)T

f(t, x, u) =∂vi

∂t+

(∂vi

∂xi

)T

fi(t, x, u) (10)

For pi = ∂vi/∂xi, from equation (6) we obtain

(∂vi

∂xi

)T

fi(t, x, u) ≥ Hfi(t, x,

∂vi

∂xi

) (11)

Now, from equations (10) and (11) it follows:

dvi

dt

∣∣∣∣(1)

≥∂vi

∂t+Hfi

(t, x,∂vi

∂xi

) (12)

If there exists a measurable function µi(t) such that (see [4, 5, 8] for appli-cation to a single system)

∂vi

∂t+Hfi

(t, x,∂vi

∂xi

) ≥ µi(t) (13)

for all t ∈ T and x ∈ Rn, then from (12) and (13) it follows:

vi(t, xi(t)) ≤ −

tf∫

t

µi(τ)dτ + maxxi(tf )∈(Xf )i

vi(tf , xi(tf )) (14)


where (Xf )i is the projection of the target set into the subsystem space Rni ,where xi ∈ Rni . It is important to note that the choice of an appropriateµi(t) in inequality (13) is very much problem dependent, and no suggestionsof how to compute the set of measurable functions µi(t)

Ni=1 for the general

system can be given. We will address this issue in more detail in Section3.1 by considering interconnected systems as an example of systems with aspecial structure.

The over-approximation of the reachable set for the subsystem with statexi is given by the following formula [4, 5, 8]:

ℜi(t) = xi|vi(t, xi) ≤ −

tf∫

t


vi(tf , xi(tf )) (15)

Notice that from equation (15) it follows that we can compute an over-approximation of the backward reachable set for only a portion of the statespace, that is, xi. The over-approximation of the reachable set for the overallstate is given by

ℜ(t) = ℜ1(t) × ...×ℜN (t), C(t) ⊆ ℜ(t) (16)

where symbol “×” denotes the Cartesian product.

3.1 Interconnected Systems

In this section we consider a class of nonlinear systems with a special struc-ture, that is, interconnected systems, for which we can analyze the procedureof determining a bound, represented by inequality (13), in more detail. Letus assume that functions fi

Ni=1 in equation (4) are of the following form:

fi(t, x, u) = gi(t, xi, u) + hi(t, x, u) (17)

where gi : T × Rni × U → R

ni is a Lipschitz continuous function thatrepresents subsystem dynamics, and hi : T × Rn × U → Rni is a Lipschitzcontinuous function that represents interconnections between subsystems, forall i, i ∈ N.

Let us assume that for each i there is a measurable function t → µxi(t)

over the finite time horizon T, such that

∂vi

∂t+Hgi

(t, xi(t),∂vi

∂xi

) ≥ µxi(t) (18)

where the subsystem Hamiltonian Hgi(·, ·, ·) (as defined in (6)) is computed

with respect to the function gi(·, ·, ·) defined in (17). Then,

dvi(t,xi)dt

∣∣∣(1)

= ∂vi

∂t+

(∂vi

∂xi

)T

fi(t, x, u) ≥∂vi

∂t+Hfi

(t, x, ∂vi

∂xi)

≥ ∂vi

∂t+Hgi

(t, x, ∂vi

∂xi) + min

u∈U

(

∂vi

∂xi

)T

hi(t, x, u)

≥ µxi(t) + min

u∈U

(

∂vi

∂xi

)T

hi(t, x, u)

(19)


We assume that the interconnections satisfy sector bounds [23]

||hi(t, x, u)|| ≤

N∑

j=1

βijµxj(t) (20)

for some βij ’s. Since vi(t, xi) is a Lipschitz function in xi with positiveLipschitz constant ki, we define

βi(t) =

N∑

j=1

kiβijµxj(t) (21)

such that

minu∈U

(∂vi

∂xi

)T

hi(t, x, u) ≥ −βi(t) (22)

From (19) and (22) it follows that:

dvi(t, xi)

dt

∣∣∣∣(1)

≥ µxi(t) − βi(t) (23)

By equating µi(t) = µxi(t)−βi(t), and integrating (23), the over-approximation

of the backward reachable set for xi is given by


tf∫

t


vi(tf , xi(tf )) (24)

which has the same form as (15), and the over-approximation ℜ(t) of thebackward reachable set for the state x is obtained using (16).

3.2 Overlapping Over-Approximations

We have considered decomposing the overall state into disjoint subsystems.In this section, we allow partitioned subsystem states to overlap (for moredetails on the overlapping subsystems see [23]). Consider equations (14) and(15), and for simplicity of the presentation (and without loss of generality),assume that we have only two subsystems with states x1 ∈ R

n1 , and x2 ∈Rn2 , such that they overlap, that is, x1 and x2 share components, meaningx1∩x2 = x12, x12 ∈ Rn12 , n12 ≥ 1. Notice that n1 +n2−n12 = n. We defineextended over-approximations as

ℜ1(t) = ℜ1(t) × R × . . .× R︸︷︷︸

n−n1 times

ℜ2(t) = R × . . .× R︸︷︷︸

n−n2 times

×ℜ2(t)(25)


where ℜ1(t) and ℜ2(t) are computed as in (15) or, in the case of intercon-nected systems, as in (24). Then, the following is true:

(C(t) ⊆ ℜ1(t)) ∧ (C(t) ⊆ ℜ2(t)) ⇒ C(t) ⊆ ℜ1(t) ∩ ℜ2(t) (26)

where symbol “∧” denotes the “and” logic operator. In other words, wecan compute an over-approximation as an intersection of any set of extendedlower dimensional over-approximations in the subspaces that cover the wholespace. The extension to N subsystems with overlapping states is clear. Thisgives us more freedom, since we are not any more restricted only to disjointpartitions of the state space.

4 Differential Games with Application to Con-

flict Resolution

In this section, we show how the analysis presented in Section 3 carries overto accommodate the computation of the backward reachable set for the two-player differential game. Let us consider a two-player game described as thefollowing dynamic system:

xi = gi(xi) + qi1(x)u + qi2(x)d + hi(x) , t ∈ T , i ∈ N (27)

where functions u = u(·) ∈ U = measurable functions : T → U andd = d(·) ∈ D = measurable functions : T → D are the control functions ofthe first and the second player, respectively [1]. We assume that U, U ⊂ Rmu ,and D, D ⊂ Rmd are compact sets. The right-hand side of equation (27)is Lipschitz continuous, and the solution trajectories t → x(t) are assumedto be absolute continuous. The partition xi

Ni=1 of the state x is a set of

subsystem states that are either disjoint or overlapping. The target set isdenoted as Xf .

In order to define a game, let us introduce the nonanticipating strategyfor the first player as [1]:

Definition 2. A strategy for the first player, which is a map α : D → U ,is nonanticipating if for any t > 0 and d, d ∈ D, d(τ) = d(τ) for all τ ≤ timplies α[d](τ) = α[d](τ) for all τ ≤ t.

We denote the set of nonanticipating strategies for the first player as Γ,

and recall that the feedback strategy α[d](t)∆= φ(t, x(t)), where φ : T×Rn →

U such that t→ φ(t, x(t)) is a measurable function along the solutions of (27),and the solutions are unique for a given initial condition, is nonanticipating[1].

Now, we can define the backward reachable set for the system (27) asfollows [14]:


Definition 3. The backward reachable set C(t) at time t (t ∈ T), of thesystem (27) from the target set Xf , is the set of all states x for which thereexists a nonanticipating strategy α ∈ Γ for the first player such that forany admissible control input d ∈ D, an absolute continuous trajectory x(τ)(τ ∈ [t, tf ] ) of the system (27) satisfies x(tf ) ∈ Xf and x = x(t).

The interpretation of Definition 3 is that there exists a strategy for thefirst player that drives the system to Xf despite the choice of the controlinput of the second player.

To compute an approximation (at this point it is important to note thatthis is not necessarily an over-approximation for the backward reachable setas defined in Definition 3), for a given Lipschitz level set function (t, xi) →v(t, xi) we introduce

λi(t, x) = ∂vi

∂t+

(∂vi

∂x

)Tgi(xi) +

(∂vi

∂xi

)T

qi1(x)u(t, x)

+(

∂vi

∂xi

)T

qi2(x)d(t, x) + βi(t)(28)

where

u(t, x) = argminu∈U

(∂vi(t,xi)

∂xi

)T

qi1(x)u

d(t, x) = arg maxd∈D

(∂vi(t,xi)

∂xi

)T

qi2(x)d

(29)

and we assume that (∂vi/∂xi)Thi(x(t)) ≤ βi(t) holds for all t ∈ T and

x ∈ Rn. If there exist a measurable function µi(t) such that

µi(t) ≤ λi(t, x) (30)

for all t ∈ T and x ∈ Rn, then the following set is the approximation ofthe backward reachable projection into Rni , the state space of the subsystemwith state xi:


tf∫

t


vi(tf , xi(tf )) (31)

Notice that in (29), d(t, x) is a function of both independent variable t anddependent variable x and thus does not belong to the set of measurablefunctions D. In the case when d(t, x) ≡ d(t), equation (31) does representan over-approximation of the reachable set as defined in Definition 3. Ingeneral, since the set of feedback strategies is larger then the set D this mightnot be true. The motivation to compute an approximation using equations(28)-(31) is that it is computationally efficient, and that in a number ofimportant applications the maximization of the control strategies for thesecond player does depend only on t (as it is the case in which we use polytopicapproximations for the linear two-player differential games as considered inthe conflict resolution problem).


4.1 Conflict Resolution between Two Aircraft

To demonstrate the proposed procedure we consider the two aircraft collisionavoidance problem [14] which is modeled as differential game (27). In Figure1 we show the relative configuration of the two aircraft where aircraft 1 triesto avoid the collision regardless of the behavior of aircraft 2. In this problemwe wish to compute the backward reachable (unsafe) set from the target set(protected zone), that is, the set which includes all the states (in relativecoordinates) from which aircraft 2 can choose a control that will lead to lossof separation for any control strategy of aircraft 1.

Protected zone

x r

y r

ψ r Aircraft 2

Aircraft 1

Figure 1: Relative coordinate system for the two aircraft conflict resolutionproblem.

We use the planar kinematic model for each aircraft, and after dynamicextension and feedback linearization of the model [16], we obtain the followinglinear model in terms of relative coordinates of the two aircraft [17]:

x =

0 0 1 00 0 0 10 0 0 00 0 0 0

x+

0 00 01 00 1

u2 −

0 00 01 00 1

u1 (32)

where x ∈ R4 is the state vector, u1 ∈ U1 ⊂ R2, and u2 ∈ U2 ⊂ R2,are control inputs of the two aircraft, respectively. From the definition ofthe unsafe set notice that u1 corresponds to d, and u2 corresponds to u inequation (27). We decompose the state vector x as x = [xT

1 , xT2 ]T , where

x1 = [xr , yr]T , and the two remaining states form x2. In other words, the

overall state vector x is decomposed into two disjoint subsystems, each of


dimension two. Then, system (32) in a decomposed form can be written as

x1 = x2

x2 = u2 − u1(33)

with x1, x2, u1, u2 ∈ R2.

As level set functions we choose polytopic functions described as inter-sections of the supporting hyperplanes of the form [19]:

v1(t, x1) = ℓT1 (t)x1 ,v2(t, x2) = ℓT2 (t)x2

(34)

The protection zone is the target set. Using (28) and the procedure proposedin [3, 19], we compute

dv1

dt= ℓT1 x1 + ℓT1 x2

µ1(t) = maxu1∈U1, u2∈U2

ℓT1 (tf )(u2 − u1)

, for ℓ1(t) ≡ ℓ1(tf )

dv2

dt= ℓT2 x2 + ℓT2 (u2 − u1)

µ2(t) = minu2∈U2

ℓT2 (tf )u2

− min

u1∈U1

ℓT2 (tf )u1

, for ℓ2(t) ≡ ℓ2(tf )

(35)

Using equations (31), (34), and (35) we obtain the over-approximation ℜ1

for x1, that is, rectangular relative coordinates xr and yr. Notice that inthis procedure, the term ℓT1 x2 = ℓT1 (u2 − u1) is treated as an interconnection(that is, perturbation h1(x) in (27)). By computing the backward reachableapproximation ℜ2 in the x2-space, the overall over-approximation is com-puted as ℜ = ℜ1 × ℜ2. In Figure 2, the inner set is the exact unsafe set in(xr , yr, ψr)-space, ψr being the relative heading angle as in Figure 1, and theouter set is the over-approximation obtained by projecting ℜ = ℜ1 ×ℜ2 into(xr , yr, ψr)-space. The exact set is computed in [14], using the convergentapproximation designed there. This exact computation took approximately5 minutes on a Sun UltraSparc with 50 grid nodes in each dimension. Theover-approximation computation (using MATLAB on a 700 MHz PentiumIII PC) took 0.2 seconds (which includes plotting the figure).

If we choose the first three coordinates of the state space vector x asour subsystem, which turns out to be a finer decomposition, and computethe over-approximation in this three dimensional space, we obtain a betterover-approximation of the exact set, as shown in Figure 3.

The over-approximation computation plotted in Figure 3 (again, usingMATLAB on a 700 MHz Pentium III PC), took 0.31 seconds (includingplotting the figure).

Further refinements can be obtained by intersections of various disjointor overlapping over-approximations, and an efficient methodology for doingthis is the subject of our current research.


Figure 2: An over-approximation of the exact unsafe set (two 2-dimensionalsubsystems).

5 Conclusions

A method for solving the problem of computation of the backward reachableset, as proposed in this paper, decomposes the problem into a set of smallerdimensional problems on the basis of the decomposition of the overall dy-namic system into a set of subsystems. Then the analysis is carried overusing a set of subsystem level set functions of which the level sets providethe over-approximations of the projections of the reachable set into subsys-tem spaces. The method accommodates computation of the reachable setsfor two player dynamic games and the application to computation of theunsafe set for the two aircraft collision avoidance problem produced encour-aging results. Our future work is to consider computing reachable sets forhybrid systems using subsystem level set functions, and to work on higherdimensional applications.

Finally, it is important to note that the analysis presented in this papercarries over to the computation of the forward reachable set of dynamicsystems in an obvious way.

6 Acknowledgements

This research is supported by DARPA under the Software Enabled ControlProgram (AFRL contract F33615-99-C-3014), and by the DoD Multidisci-plinary University Research Initiative (MURI) program administered by theOffice of Naval Research under Grant N00014-02-1-0720.


Figure 3: An over-approximation of the exact unsafe set (one 3-dimensionalsubsystem).

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