July 28, 2012 9:56 WSPC/S0218-1274 1250176

International Journal of Bifurcation and Chaos, Vol. 22, No. 7 (2012) 1250176 (12 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218127412501763

PINNING IMPULSIVE STABILIZATIONOF NONLINEAR DYNAMICAL NETWORKS

WITH TIME-VARYING DELAY

JIANQUAN LUDepartment of Mathematics, Southeast University,

Nanjing 210096, P. R. ChinaPotsdam Institute for Climate Impact Research,

14412 Potsdam, Germanyjqluma@seu.edu.cn

ZIDONG WANGDepartment of Information Systems and Computing,

Brunel University, Uxbridge, Middlesex, UB8 3PH, UKZidong.Wang@brunel.ac.uk

JINDE CAODepartment of Mathematics, Southeast University,

Nanjing 210096, P. R. Chinajdcao@seu.edu.cn

DANIEL W.C. HODepartment of Mathematics,

City University of Hong Kong, Hong Kongmadaniel@cityu.edu.hk

JURGEN KURTHSPotsdam Institute for Climate Impact Research,

14412 Potsdam, GermanyDepartment of Physics, Humboldt University Berlin,

12489 Berlin, GermanyJuergen.Kurths@pik-potsdam.de

Received October 31, 2011; Revised March 8, 2012

In this paper, a new impulsive control strategy, namely pinning impulsive control, is proposedfor the stabilization problem of nonlinear dynamical networks with time-varying delay. In thisstrategy, only a small fraction of nodes is impulsively controlled to globally exponentially sta-bilize the whole dynamical network. By employing the Lyapunov method combined with themathematical analysis approach as well as the comparison principle for impulsive systems, somecriteria are obtained to guarantee the success of the global exponential stabilization process.The obtained criteria are closely related to the proportion of the controlled nodes, the impulsivestrength, the impulsive interval and the time-delay. Numerical examples are given to demonstratethe effectiveness of the designed pinning impulsive controllers.

Keywords : Pinning impulsive control; global exponential synchronization; complex networks;time-varying delay.

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1. Introduction

Recently, increasing interest has been devoted tothe study of collective behaviors in complex dynam-ical networks due to its frequently occurrence innatural systems and widely potential applications[Strogatz, 2001; Zhou et al., 2006; Gao et al., 2006;Wang et al., 2010; Motter et al., 2005; Wang &Chen, 2002]. Some recent advances on network syn-chronization have been reported in [Arenas et al.,2008].

A complex dynamical network is a large set ofinterconnected nodes, and each node can be a non-linear dynamical system with chaotic, periodic orstable behavior. Synchronization, which means thatthe state variables of the individual nodes convergetowards each other, can be realized by node infor-mation being exchanged via inter-connections [Wu,2007; Chen et al., 2004]. However, the final syn-chronous state of the network, which is achievedby the interconnecting coupling of the nodes, isvery difficult to estimate and predict due to theindividuals’ active dynamics. There is a commonneed to regulate the behavior of large ensembles ofinteracting individuals by external forces for manybiological, physical and social dynamical networks[Mazenc et al., 2008; Lu et al., 2008]. For example,many regulatory mechanisms have been uncoveredin the context of physiological, biological and cellu-lar processes to control the dynamical processes soas to evolve towards a desired structure, which arefundamental to guarantee the correct functioningof the whole network [Mazenc et al., 2008]. Con-trol methods have also been regarded as one of thefour key features to deeply understand biologicalsystems at a system level [Kitano, 2002]. The issueof consensus of multiagent systems [Olfati-Saber &Murray, 2004; Lu et al., 2009a] is closely relatedto the synchronization problem on dynamical net-works. This is also another very interesting collec-tive behavior that is of great interest to researchersin computer science, biological systems, ecosystemsand several other areas [Sole & Bascompte, 2006].

Various control strategies including adaptivecontrol [Lu et al., 2008; Zhang et al., 2008; Wanget al., 2008; Gleiser & Zanette, 2006], impulsivecontrol [Liu et al., 2005; Guan et al., 2010; Zhanget al., 2010; Tang et al., 2010] and pinning state-feedback control [Li et al., 2004; Chen et al., 2007;Lu et al., 2009b; Yu et al., 2009; Sorrentino et al.,2007; Lu & Ho, 2011] have been proposed to regu-late different kinds of complex dynamical networks

with time-delay or noise. Pinning state-feedbackcontrol here means that only a small fraction ofnodes is selected to be controlled to stabilize thewhole dynamical network. A great number of resultshave been published in the recent literature on thegeneral topic of pinning state-feedback control fornetwork synchronization, most of which are essen-tially obtained based on the fact that the largesteigenvalue of the coupling matrix can be changedfrom zero to be negative via state-feedback pin-ning [Chen et al., 2007; Lu et al., 2009c]. Adaptivecontrollers and impulsive controllers are simultane-ously used to synchronize a stochastic coupled net-work in [Tang et al., 2011]. Pinning control makesthe network control more convenient to implementsince only a small fraction of nodes is directly con-trolled. On the other hand, it is well-known that theimpulsive control allows the stabilization of dynam-ical networks and chaotic systems using only smallcontrol impulses, it has been widely used to stabi-lize dynamical networks and chaotic systems [Yang,2001; Liu & Zhao, 2011]. However, impulsive controlis not easy to implement in a complex network envi-ronment if all nodes have to be controlled as in [Liuet al., 2005; Guan et al., 2010; Zhang et al., 2010;Tang et al., 2010]. Unfortunately, to the best of ourknowledge, pinning impulsive control for networkstabilization, though vitally important for under-standing network stabilization processes and impul-sive feature, has not yet been studied primarily dueto the mathematical difficulties.

It is noted that time delays are often encoun-tered in real world [Cao et al., 2008; Ponce et al.,2009], and the delays are usually time-varying inelectronic implementation of analog networks dueto the finite switching speed of amplifiers. In manycases, only partial information about the time-varying delay is known such as the upper bound,and the time-varying delay can be even nonsmooth.Design flaws and incorrect analytical conclusionscan be obtained if time-varying delays are not con-sidered or not well described in system modeling.Therefore, it is of great importance to consider theeffects of time-varying delay for the pinning impul-sive stabilization of dynamical networks.

Motivated by the above discussions, we aim tostudy the pinning stabilization problem of nonlin-ear complex dynamical networks with time-varyingdelay. Instead of state-feedback controllers, impul-sive controllers will be used to pin a small fractionof nodes for successful control, and the states of

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Impulsive Stabilization of Nonlinear Dynamical Networks with Time-Varying Delay

the whole dynamical network can be globally expo-nentially forced to the objective state. The nodesto be impulsively controlled should be different atdistinct control instants. Some easily-verified crite-ria will be presented by employing some analyticalmethods, Lyapunov method and comparison princi-ple for impulsive systems. The relationship amongthe proportion of the controlled nodes, an upperbound of time-varying delay, impulsive strengths,and impulsive interval of the dynamical networks iswell explained in the obtained criteria. Simulationexamples are given to demonstrate the effectivenessof the proposed pinning impulsive method.

The rest of this paper is organized as follows. InSec. 2, some preliminaries are presented. In Sec. 3,pinning impulsive control strategy is given and somestabilization criteria are derived. In Sec. 4, numeri-cal examples are given to demonstrate the effective-ness of the results obtained. Concluding remarks aredrawn in Sec. 5.

Notations. The standard notations will be usedin this paper. In is the identity matrix of order n.We use λmax(·) to denote the maximum eigen-value of a real symmetric matrix. R

n denotesthe n-dimensional Euclidean space. 0 denotesn-dimensional zero vector. R

n×n are n × n realmatrices. The superscript “T” represents thetranspose. diag{· · ·} stands for a block-diagonalmatrix. Matrices, if not explicitly stated, areassumed to have compatible dimensions. Let ‖x‖be the Euclid vector norm of x ∈ R

n, and PC(m)denote the class of piecewise right continuous func-tion ϕ : [t0 − τ,+∞) → R

m(m ∈ N) with the normdefined by ‖ϕ(t)‖τ = sup−τ≤s≤0 ‖ϕ(t + s)‖. Forϕ : R → R, denote ϕ(t+) = lims→0+ ϕ(t + s) andϕ(t−) = lims→0− ϕ(t + s). #G denotes the numberof elements of a finite set G. The Dini derivative ofϕ(t) is defined as D+ϕ(t) = lim sups→0+

ϕ(t+s)−ϕ(t)s .

2. Some Preliminaries

In this section, some preliminaries including modelformulation and some lemmas are presented. Thebasic nonlinear dynamical network model will beintroduced in various stages starting with a triv-ial network of decoupled nodes governed by delayeddifferential equations.

A neural network is a mathematical model con-sisting of an interconnected group of artificial neu-rons, which has potential applications in securecommunication and pattern recognition [Fischer

et al., 2000]. In this paper, the following neural net-work with time-varying delay [Cao & Ho, 2005] istaken as a single node of the nonlinear dynamicalnetwork:

y(t) = −Cy(t) +Bf1(y(t))

+Df2(y(t− τ(t))) (1)

where y(t) = (y1(t), y2(t), . . . , yn(t))T ∈ Rn is the

state vector associated with the neurons; C =diag(c1, c2, . . . , cn) > 0 (a positive-definite diago-nal matrix), B = (bij)n×n and D = (dij)n×n arethe connection weight matrix and the delayed con-nection matrix, respectively; τ(t) is the transmis-sion time-varying delay satisfying 0 < τ(t) ≤ τ∗;f1(y(t)) = (f11(y1(t)), . . . , f1n(yn(t)))T ∈ R

n andf2(y(t − τ(t))) = (f21(y1(t − τ(t)), . . . , f2n(yn(t −τ(t))))T ∈ R

n denote the activation functions ofthe neurons satisfying f1(0) = 0 and f2(0) = 0.

By nonlinear coupling of N nodes in a com-plex network, we can obtain the following nonlineardynamical network model:

xi(t) = −Cxi(t) +Bf1(xi(t)) +Df2(xi(t− τ(t)))

+ cN∑

j=1

aijΓH(xj(t)), i = 1, 2, . . . , N,

(2)

where xi(t) ∈ Rn is the state vector of the ith

node; Γ = diag{γ1, γ2, . . . , γn} > 0 is the innercoupling matrix between two connected nodes;c > 0 is the coupling strength of the dynam-ical network; the nonlinear function is definedas H(xj(t)) = (h(xj1(t)), h(xj2(t)), . . . , h(xjn(t)))T

and h(·) is a monotonically increasing function sat-isfying [(h(u)−h(v))/(u−v)] ≥ ϑ > 0 for any u, v ∈R; A = (aij)N×N is the Laplacian coupling matrixrepresenting the coupling topology of the dynami-cal network [Chung, 1997]. The elements aij of thematrix A are defined as follows: if there is a connec-tion between nodes i and j (i �= j), aij = aji > 0;otherwise aij = aji = 0 (i �= j), and the diffusivitycoupling condition aii = −∑N

j=1,j �=i aij is satisfiedfor the diagonal elements aii (i = 1, 2, . . . , N). TheLaplacian matrix A is assumed to be irreduciblethroughout this paper. This diffusivity conditionimplies that the nonlinearly coupled dynamical net-work would be decoupled when the array of nodesis synchronized.

In this paper, the objective trajectory that thenonlinear dynamical network (2) will be forced to,

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is selected as any solution s(t) of the system (1) sat-isfying s(t) = −Cs(t)+Bf1(s(t))+Df2(s(t−τ(t))).Let ei(t) = xi(t) − s(t) be the error state of nodei between the current state xi(t) and the objectivestate s(t). Then, the following error dynamical net-work can be obtained by considering the fact thatc∑N

j=1 aijΓH(s(t)) = 0:

ei(t) = −Cei(t) +Bf1(ei(t)) +Df2(ei(t− τ(t)))

+ c

N∑j=1

aijΓH(ej(t)), i = 1, 2, . . . , N,

(3)

where f1(ei(t)) = f1(xi(t)) − f1(s(t)), f2(ei(t)) =f2(xi(t)) − f2(s(t)), and H(ej(t)) = H(xi(t)) −H(s(t)) := (h(ej1(t)), h(ej2(t)), . . . , h(ejn(t)))T ).By referring to the property of the function H(·),we have [(h(u) − h(v))/(u − v)] ≥ ϑ > 0.

The following assumption is needed for theproof of the main results in the next section.

Assumption 1. The neuron activation functionsf1(·) and f2(·) of the neural network satisfy thefollowing Lipschitz condition:

‖f1(x) − f1(y)‖ ≤ l1‖x− y‖,‖f2(x) − f2(y)‖ ≤ l2‖x− y‖,

∀x, y ∈ Rn, (4)

where l1 and l2 are positive constants.

3. Main Results

In this section, pinning impulsive controllers willbe designed to globally and exponentially stabilizethe dynamical network with nonlinear coupling. Inorder to drive the nonlinear dynamical network intothe objective state s(t), the following impulsive con-trollers are constructed for l nodes (l N):

ui(t) =+∞∑k=1

µei(t)δ(t − tk), i ∈ D(t), (5)

where µ ∈ (−2,−1) ∪ (−1, 0) is a constant, whichmeans that the impulsive effects can be used to sta-bilize the nonlinear dynamical network; the timeseries {t1, t2, t3, . . .} is a sequence of strictly increas-ing impulsive instants satisfying limk→∞ tk = ∞,and the index set of l nodes D(t) which should beimpulsively controlled is defined as follows: At timeinstant t, for the error states e1(t), e2(t), . . . , eN (t),

one can reorder the vectors such that ‖ep1(t)‖ ≥‖ep2(t)‖ ≥ · · · ≥ ‖epl(t)‖ ≥ · · · ≥ ‖epN (t)‖. Then,the index set of l controlled nodes D(t) is definedas D(t) = {p1, p2, . . . , pl}.Remark 3.1. If there are more than l states withthe same norm (i.e. ‖epl(t)‖ = ‖ep,l+1(t)‖ = · · · =‖ep,l+�(t)‖), any node with the same norm can beselected into the set D(t) for satisfying #D(t) = l.

After adding the pinning impulsive con-trollers (5) to the nodes D(t), the controlled dynam-ical network can be rewritten as follows:

ei(t) = −Cei(t) + Bf 1(ei(t))

+ Df 2(ei(t− τ(t))) + c

N∑j=1

aijΓH(ej(t)),

t ≥ 0, t �= tk, k ∈ N,

ei(t+k ) = ei(t−k ) + µei(t−k ), i ∈ D(tk).

(6)

Remark 3.2. From the second equation of (6), wecan see that only l (l N) nodes are impulsivelycontrolled. This means that we do not need to addcontrollers to each node, and hence the control costcan be substantially lower and our strategy can beeasier to implement. It is noted that the controllednodes can be distinct at different instants.

The initial conditions of dynamical network (6)are given by

ei(t) = φi(t), −τ∗ ≤ t ≤ 0, i = 1, 2, . . . , N,(7)

where φi(t) ∈ C([−τ∗, 0],Rn) is the set of continu-ous functions from [−τ∗, 0] to R

n.Throughout this paper, we always assume that

ei(t) is right continuous at t = tk, i.e. e(tk) = e(t+k ).Therefore, the solutions of (6) are piecewise right-hand continuous functions with discontinuities att = tk for k ∈ N.

Definition 3.1. The nonlinear dynamical net-work (2) is said to be globally exponentially stabi-lized to the objective state s(t) if there exist λ > 0,T0 > 0 and θ > 0 such that for any initial valuesφi(·) (i = 1, 2, . . . , N),

‖ei(t)‖ = ‖xi(t) − s(t)‖ ≤ θe−λt

hold for all t > T0, and for any i = 1, 2, . . . , N .

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Impulsive Stabilization of Nonlinear Dynamical Networks with Time-Varying Delay

Lemma 1 [Yang & Xu, 2007]. Let 0 ≤ τi(t) ≤ τ,

F (t, u, u1, . . . , um) : R+ ×

m+1︷ ︸︸ ︷R × · · · × R → R

be nondecreasing in ui for each fixed(t, u, u1, . . . , ui−1, ui+1, . . . , um), i = 1, 2, . . . ,m,and Ik(u) : R → R be nondecreasing in u. Sup-pose that u(t), v(t) ∈ PC(1) satisfy

D+u(t) ≤ F (t, u(t), u(t − τ1(t)), . . . ,

u(t− τm(t))), t �= tk, t ≥ 0

u(tk) ≤ Ik(u(t−k )), k ∈ N

andD+v(t) > F (t, v(t), v(t − τ1(t)), . . . ,

v(t− τm(t))), t �= tk, t ≥ 0

v(tk) ≥ Ik(v(t−k )), k ∈ N.

Then u(t) ≤ v(t), for −τ ≤ t ≤ 0 implies u(t) ≤v(t), for t ≥ 0.

Remark 3.3. Lemma 1 is an extended compari-son theorem for the impulsive system with time-delay. By using the comparison principle and theLyapunov stability method [Lu et al., 2011; Luet al., 2010; Yang & Xu, 2007], the main result onpinning impulsive stabilization of nonlinear dynam-ical networks will be derived.

Theorem 1. Let ρ = N+lµ(µ+2)N ∈ (0, 1) and T =

maxk∈N{tk+1 − tk} <∞. Suppose that

λmax(−C − CT ) + 2l1√λmax(BTB)

+ 2l2

√λmax(DTD)

ρ+

ln ρT

< 0, (8)

then the complex dynamical network (6) with pin-ning impulsive controllers is globally exponentiallystable in the following sense:

‖ei(t)‖ ≤ ρ−1 sup−τ∗≤s≤0

{N∑

i=1

‖φi(s)‖2

}e−λt (9)

where λ > 0 is a unique solution of

λ− β + ρ−12 d · eλτ∗

= 0, t > 0 (10)

in which

β = −(λmax(−C − CT )

+ 2l1√λmax(BTB) +

d√ρ

+ln ρT

)

and d = l2√λmax(DTD). This means that the non-

linear dynamical network (2) can be exponentiallystabilized to the objective state s(t) by the pinningimpulsive controllers (5) if the inequality (8) issatisfied.

Remark 3.4. In Theorem 1, T = maxk∈N{tk+1 −tk} is the maximum impulsive interval whichmeans the maximum time difference between twoimpulses. Normally, this concept is used, whenthe impulses are synchronizing (or stabilizing), inorder to characterize the frequency of impulsesand to guarantee that the frequency of impulsesshould not be too low. By revising the proofof the main results, a newly proposed conceptof “average impulsive interval” [Lu et al., 2010]can be used to make the obtained result lessconservative.

Remark 3.5. Although the coupling matrix A in thispaper is assumed to be irreducible, the results inTheorem 1 can be easily extended to the case of areducible coupling matrix [Lu et al., 2009b] by uti-lizing the normalized left eigenvector of the couplingmatrix with respect to the eigenvalue zero and usingthe fact that the maximum value of the real part ofthe eigenvalue is zero. For directed networks witha rooted spanning tree, it would be much easier tocontrol the whole network by impulsively control-ling a small fraction of nodes in the group of rootednodes and the nonrooted nodes can be virtuallycontrolled.

For linearly coupled dynamical networks with-out delay term (i.e. h(xj(t)) = xj(t) and D = 0),one can easily derive the following corollary. Con-sider the following linearly coupled dynamicalnetworks:

xi(t) = −Cxi(t) +Bf1(xi(t))

+ cN∑

j=1

aijΓxj(t), i = 1, 2, . . . , N. (11)

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Corollary 3.1. Let ρ = N+lµ(µ+2)N ∈ (0, 1) and

T = maxk∈N{tk − tk−1} <∞. Suppose that

λmax(−C − CT ) + 2l1√λmax(BTB) +

ln ρT

< 0,

(12)

then the linearly coupled dynamical network (11)can be exponentially forced onto the objective tra-jectory s(t) by pinning impulsive controllers (5)with the convergence rate λ = −(λmax(−C −CT ) +2l1

√λmax(BTB) + ln ρ

T ).

Remark 3.6. It can be observed from Theorem 1 andCorollary 3.1 that whether the pinning impulsivecontrol and the stabilizing convergent rate can besuccessful depends on the proportion of the con-trolled nodes, the upper bound of the time-varyingdelay, the impulsive strengths and the impulsiveinterval of the dynamical networks. The higher theproportion of the controlled nodes is, the easier thedynamical network can be efficiently controlled.

4. Numerical Examples

In this section, two examples are shown to illustratethe results obtained in the previous section. Con-sider the following system as the individual node inthe dynamical network:

y(t) = −Cy(t) +Bf1(y(t))

+Df2(y(t− τ(t))), (13)

with C = I2, B =(

2.0 −0.11−5.0 3.2

), D =

(−1.6 −0.1−0.18 −2.4

),

and τ(t) = et

1+et ≤ 1, where y(t) = (y1(t), y2(t))T

is the state vector of the single neural network,and f1(y(t)) = f2(y(t)) = (tanh(y1), tanh(y2))T .Then, one can easily obtain l1 = l2 = 1 for satisfy-ing Assumption 1. The single individual (13) has achaotic attractor [Lu, 2002] (as shown in Fig. 1)for the initial values y1(�) = 0.2, y2(�) = 0.5,∀ � ∈ [−1, 0].

As examples, two practical network modelsincluding BA (Barabasi–Albert) scale-free network[Barabasi & Albert, 1999] and NW (Newman–Watts) small-world network [Newman & Watts,1999] are considered. A 100 nodes’ scale-free net-work is generated by taking m = m0 = 5. A 100nodes’ small-world network is generated by takinginitial neighboring nodes k = 4 and the edge addingprobability p = 0.1. The weight of the couplingmatrix A is defined as follows: if there is a con-nection between nodes i and j, then aij = aji = 1,otherwise aij = aji = 0.

The parameters corresponding to the networkcoupling are taken as follows: the nonlinear cou-pling function h is taken as h(z) = z+tanh(z) withϑ = 1, coupling strength c = 2 and the inner cou-pling matrix Γ = I2.

In both examples, according to the algorithmdescribed before Remark 3.1, l = 30 nodes areselected to be pinning controlled at each timeinstant, and the objective trajectory s(t) is takenas s(t) = 0. Take µ = −0.9 and tk − tk−1 = 0.02,

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−5

−4

−3

−2

−1

0

1

2

3

4

5

y1

y 2

Fig. 1. Chaotic attractor of the single individual (13) in the dynamical network.

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−1 −0.5 0 0.5 1 1.5−5

−4

−3

−2

−1

0

1

2

3

4

5

t

e i(t

)

Fig. 2. The states of 100 nodes’ scale-free network are successfully forced to the objective state (s(t) = 0) by impulsivelycontrolling 30 nodes.

−1 −0.5 0 0.5 1 1.5−3

−2

−1

0

1

2

3

t

e i(t

)

Fig. 3. The 100 nodes’ states in small-world network are forced to the objective original state by impulsively controlling30 nodes.

one gets that λmax(−C −CT ) + 2l1√λmax(BTB)+

2l2√

λmax(DT D)ρ + lnρ

T = −1.4508 < 0. Accordingto Theorem 1, we can conclude that both networkscan be successfully stabilized to the objective tra-jectory s(t). Figures 2 and 3 are plotted to presentthe successful stabilization process, respectively, forscale-free and small-world networks. In the simu-lations, all initial conditions of the nodes are ran-domly selected from [−5, 5].

5. Conclusion

This paper has been devoted to studying pin-ning impulsive stabilization of complex dynamical

networks with nonlinear coupling and time-varyingdelay. Different from existing results about impul-sive control, only a small fraction of nodes hasbeen selected to be impulsively controlled to sta-bilize the whole state-coupled dynamical networks.By using an impulsive delay differential inequal-ity and an efficient algorithm, a small fraction ofnodes has been carefully chosen to be controlledby designed impulsive controllers. The derived sta-bilization criterion and the convergence rate areclosely related with the proportion of the controllednodes, the time-delay, impulsive strengths, andimpulsive interval of the dynamical networks. Thereis no more constraint on the time-varying delay

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except boundedness. Two simulation examples havebeen presented to illustrate the usefulness and effec-tiveness of the main results obtained.

Acknowledgments

The work of J. Q. Lu was supported by the NationalNatural Science Foundation of China (NSFC) underGrant 61175119, the Natural Science Foundation ofJiangsu Province of China under Grant BK2010408,Huo Ying-Dong Foundation (Grant No. 132037),and the Alexander von Humboldt Foundation ofGermany. The work of J. D. Cao was supported bythe National Natural Science Foundation of Chinaunder Grant No. 11072059, and the SpecializedResearch Fund for the Doctoral Program of HigherEducation under Grant No. 20110092110017. Thiswork of D. W. C. Ho was supported by a grantfrom CityU (7002561) and a GRF grant by HKSAR(9041432). The work of J. Kurths was supported bySUMO (EU), GSDP (EU) and IRTG 1740 (DFG).

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Appendix

Proof of Theorem 1

Proof. Construct a Lyapunov function in the formof

V (t) =N∑

i=1

eTi (t)ei(t). (A.1)

For any t ∈ [tk−1, tk), taking derivative of V (t)along the solution of (6) gives that

D+V (t) =N∑

i=1

2[−eTi (t)Cei(t) + eTi (t)Bf1(ei(t)))

+ eTi (t)Df2(ei(t− τ(t)))]

+N∑

i=1

N∑j=1

2caijeTi (t)ΓH(ej(t)). (A.2)

Letting d = l2√λmax(DTD), by Assumption 1,

we obtain

2eTi (t)Bf1(ei(t))

≤ 2‖ei(t)‖ ·√

‖Bf1(ei(t))‖2

≤ 2‖ei(t)‖ ·√λmax(BTB)l21‖ei(t)‖2

≤ 2√λmax(BTB)l1eTi (t)ei(t), (A.3)

and

2eTi (t)Df2(ei(t− τ(t)))

≤ 2 ·√

d√ρ‖ei(t)‖2 ·

√√ρ

d‖Df2(ei(t− τ(t)))‖2

= 2 ·√

d√ρ‖ei(t)‖2

·√√

ρ

dfT

2 (ei(t− τ(t)))DTDf2(ei(t− τ(t)))

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≤ 2 ·√

d√ρ‖ei(t)‖2

·√√

ρ

dλmax(DTD)l22‖ei(t− τ(t))‖2

= 2 ·√

d√ρ‖ei(t)‖2 ·

√d√ρ‖ei(t− τ(t))‖2

≤ d√ρ‖ei(t)‖2 + d

√ρ‖ei(t− τ(t))‖2. (A.4)

Let eι(t) = (e1ι(t), e2ι(t), . . . , eNι(t))T andH(eι(t)) = (h(e1ι(t)), h(e2ι(t)), . . . , h(eNι(t)))T .From the diffusivity property of matrix A and theassumption of [(h(u) − h(v))/(u − v)] ≥ ϑ > 0, itfollows that

N∑i=1

N∑j=1

2caijeTi (t)ΓH(ej(t))

=N∑

i=1

N∑j=1

2caij

[n∑

ι=1

eiι(t)γιH(ejι(t))

]

=n∑

ι=1

2cγι

N∑

i=1

N∑j=1

eiι(t)aijH(ejι(t))

=n∑

ι=1

2cγι(eι(t))TAH(eι(t))

= −n∑

ι=1

cγι

N∑i=1

N∑j=1,j �=i

aij(eiι(t) − ejι(t))

× (H(eiι(t)) −H(ejι(t)))

≤ −n∑

ι=1

N∑i=1

N∑j=1,j �=i

ϑcγιaij(eiι(t) − ejι(t))2

≤ 0. (A.5)

Considering the inequalities (A.3)–(A.5), itfollows from (A.2) that

D+V (t) ≤N∑

i=1

[−2eTi (t)Cei(t)

+ 2√λmax(BTB)l1eTi (t)ei(t)

]

+N∑

i=1

d√ρ‖ei(t)‖2

+N∑

i=1

d√ρ‖ei(t− τ(t))‖2

≤N∑

i=1

(α+

d√ρ

)eTi (t)ei(t)

+N∑

i=1

d√ρ‖ei(t− τ(t))‖2

=(α+

d√ρ

)V (t)

+ d√ρV (t− τ(t)), (A.6)

where α = λmax(−C − CT ) + 2√λmax(BTB)l1.

When t = tk, we have

V (t+k ) =N∑

i=1

eTi (t+k )ei(t+k )

=∑

i∈D(tk)

(1 + µ)2eTi (t−k )ei(t−k )

+∑

i�∈D(tk)

eTi (t−k )ei(t−k ). (A.7)

Let ϕ(t−k ) = min{‖ei(t−k )‖ : i ∈ D(t−k )} andψ(t−k ) = max{‖ei(t−k )‖ : i �∈ D(t−k )}, then we canget ψ(t−k ) ≤ ϕ(t−k ).

Since ρ = N+lµ(µ+2)N ∈ (0, 1), one has N − l =

[ρ−(1+µ)2]l(1−ρ) ≥ 0, which further gives that

∑i�∈D(tk)

eTi (t−k )ei(t−k )

≤ (N − l)ψ2(t−k ) ≤ (N − l)ϕ2(t−k )

≤ [ρ− (1 + µ)2]l(1 − ρ)

ϕ2(t−k )

≤ ρ− (1 + µ)2

(1 − ρ)

∑i∈D(tk)

eTi (t−k )ei(t−k ).

(A.8)

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Then by some calculations, we have

(1 + µ)2∑

i∈D(tk)

eTi (t−k )ei(t−k ) +∑

i�∈D(tk)

eTi (t−k )ei(t−k )

≤ ρ

N∑i=1

eTi (t−k )ei(t−k ) = ρV (t−k ). (A.9)

If follows from (A.7) that

V (t+k ) ≤ ρV (t−k ). (A.10)

For any ε > 0, let ν(t) be a unique solution ofthe following impulsive delayed systems (A.11):

ν(t) =(α+

d√ρ

)V (t) + d

√ρ · ν(t− τ(t))

+ ε, t ≥ 0, t �= tk, k ∈ N,

ν(t+k ) = ρν(t−k ), k ∈ N,

ν(t) =N∑

i=1

‖φi(t)‖2, −τ∗ ≤ t ≤ 0.

(A.11)

According to Lemma 1, we have ν(t) ≥ V (t) ≥0 for any t ≥ 0.

By using the formula for the variation of param-eters [Lakshmikantham et al., 1989], we can obtainthe following integral equation for ν(t):

ν(t) = W (t, 0)ν(0)

+∫ t

0W (t, s)[d

√ρ · ν(s− τ(s)) + ε]ds,

t ≥ 0, (A.12)

where W (t, s) (t > s ≥ 0) is the Cauchy matrix ofthe following linear impulsive system:w(t) =

(α+

d√ρ

)w(t), t ≥ 0, t �= tk, k ∈ N

w(t+k ) = ρw(t−k ), k ∈ N.

(A.13)

Since 0 < ρ < 1 and T = maxk∈N{tk − tk−1},we have

W (t, s) = e(α+ d√

ρ)(t−s)

∏s<tk≤t

ρ

≤ e(α+ d√

ρ)(t−s)

ρ[ (t−s)T

−1]

= ρ−1 · e(α+ d√ρ+ ln ρ

T)(t−s)

,

t ≥ s ≥ 0. (A.14)

Letting θ = ρ−1 sup−τ∗≤s≤0{∑N

i=1 ‖φi(s)‖2}and β = −(α+ d√

ρ + lnρT ), we have

ν(t) ≤ ρ−1N∑

i=1

‖φi(0)‖2 · e(α+ d√ρ+ lnρ

T)t

+∫ t

0ρ−1e

(α+ d√ρ+ ln ρ

T)(t−s)

× [d√ρ · ν(s− τ(s)) + ε]ds

≤ θ · e−βt +∫ t

0e−β(t−s)

× [ρ−12 d · ν(s− τ(s)) + ρ−1ε]ds,

t ≥ 0. (A.15)

Let g(λ) = λ − β + ρ−12 d · eλτ∗

. Since β > 0,d > 0, 0 < ρ < 1 and α + 2d√

ρ + ln ρT < 0, we

have g(0) = −β + ρ−12 d < 0, g(∞) > 0 and

g′(λ) = 1 + λρ−12d · eλτ∗

> 0. Consequently, wecan conclude that g(λ) = 0 has a unique solutionλ > 0.

Since 0 < ρ < 1, for −τ∗ ≤ t ≤ 0, we have

ν(t) ≤ ρ−1 ·N∑

i=1

‖φi(t)‖2

< θe−λt +ε

βρ− d√ρ. (A.16)

In the following, we shall prove that, for t ≥ 0,the following is true:

ν(t) < θe−λt +ε

βρ− d√ρ. (A.17)

If (A.17) is not true, then there exists a t∗ > 0such that

ν(t∗) ≥ θe−λt∗ +ε

βρ− d√ρ, (A.18)

and

ν(t) < θe−λt +ε

βρ− d√ρ, for t < t∗. (A.19)

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By using (10), (A.15) and (A.19), some detailed calculations give that

ν(t∗) ≤ θ · e−βt∗ +∫ t∗

0e−β(t∗−s)[ρ−

12 d · ν(s− τ(s)) + ρ−1ε]ds

< e−βt∗[θ +

ε

βρ− d√ρ

+∫ t∗

0eβs

[ρ−

12 d · (θe−λ(s−τ(s))) + ρ−

12 d · ε

βρ− d√ρ

+ ρ−1ε

]ds

]

≤ e−βt∗[θ +

ε

βρ− d√ρ

+ ρ−12 dθeλτ∗

∫ t∗

0e(β−λ)sds+

βε

βρ− d√ρ

∫ t∗

0eβsds

]

= θe−βt∗ +ε

βρ− d√ρ· e−βt∗ + e−βt∗ρ−

12 dθeλτ∗ · e

(β−λ)t∗ − 1β − λ

+ e−βt∗ · βε

βρ− d√ρ· e

βt∗ − 1β

= θe−λt∗ +ε

βρ− d√ρ, (A.20)

which leads to a contradiction with (A.18). Therefore, the inequality (A.18) holds. Letting ε→ 0, for t ≥ 0,we have

ν(t) ≤ θe−λt. (A.21)

which further implies that∑N

i=1 eTi (t)ei(t) = V (t) ≤ ν(t) ≤ θe−λt. The proof is hence completed by

referring to Definition 3.1. �

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