RD-Ri42 888 FINITE DIMENSIONAL COMPENSATORS FOR INFINITE /

UIMENSIONAL SYSTEMS WITH UNB..(U) WISCONSINUNIV-MRDISON MATHEMATICS RESEARCH CENTERSIID RF CURTAIN ET AL. MAY 84 MRC-TSR-2684 F/G 12/1i N

USEhhshEmhEEEEEEEEEEEEEEmhhEEIEu'."ommso

III EMll 332

NAI L BU3 O STNAD-96-

lu1 ,."

til 1.25 111111.4 1 1.6liiWi.- = u

MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF STANDARDS- 1963-A

* WTFNITE DIMENSIONAL COM4PENSATORS

L 0

* Mathematics Research CenterUniversity of Wisconsin-Madison610 Walnut StreetMadison, Wisconsin 53705

May 1984 .

(Received March 5, 1984) -- "

0..! b JULl 1 O19N4.

LA.Approved for public release

* Distribution unlimited

Sponsored by

U.S. Arm Research OfieNational Science Fo~undationl

P.O. Box 12211 Washington, D.C. 20550

Research Triangle ParkNorth Carolina 27709

84 6 01

UNIVERSITY OF WISCONSIN-MADISONMATHEMATICS RESEARCH CENTER

FINITE DIMENSIONAL COMPENSATORS FOR INFINITE DIMENSIONALSYSTEIS WITH UNBOUNDED CONTROL ACTION

R.F. Curtain* and D. Salamon**

Technical Summary Report #2684May 1984

ABSTRACT

This paper contains a design procedure for constructing finite

dimensional compensators for a class of infinite dimensional systems with

unbounded input operators. Applications to retarded functional differential

systems with delays in the input or the output variable and to partial

differential equations with boundary input operators are discussed. 1 /

AMS (MO) Subject Classifications: 34K20, 93C15, 93C25, 93DI1

Mey Words: Infinite Dimensional Systems, Unbounded Input Operators,'Compensator, Retarded Functional Differential Equations,

Partial Differential Equations, Boundary Control

Work Unit Number 5 (Optimization and large Scale Systems)

*Mathematisch Instituut, Rijksuniversiteit Groningen, The Netherlands

Sponsored by the U.S. Army under Contract No. DAAG29-80-C-0041.**This material is based upon work supported by the National Science

Foundation under Grant No. MCS-8210950.

A . ._-

SIGNIFICANCE AND EXPLANATION

For a large number of control and observation processes appropriate

mathematical models incorporate partial or functional differential equations

and therefore give rise to differential equations in an infinite dimensional

state space. In order to improve the stability properties of the underlying

process, dynamic compensators have to be designed. A straight forward

generalization of the finite dimensional theory leads to compensators which

themselves are infinite dimensional systems and therefore cannot be

implemented directly. Hence the question arises whether it is possible to

stabilize an infinite dimensional system through a finite dimensional

compensator. This problem is not new, however previous results are restricted

either to a special class of infinite dimensional systems or to systems with

bounded input/output-operators. But for many control and observation

processes there are severe limitations such as delays in the control actuator

and measurement devices and restrictions of controls and sensors to a few

points of the boundary of the spatial domain. Modelling such limitations

results in unbounded input/output-operators.

It is the purpose of this paper to present a general approach for the

design of finite dimensional compensators for infinite dimensional systems

V which allows for unbounded input/output-operators and can be applied to

r parabolic and hyperbolic PDEs as well as retarded and neutral FDEs.

The responsibility for the wording and views expressed in this descriptivesumary lies with MRC, and not with the authors of this report.

.A, %

FINITE DIMENSIONAL C0HPENSATORS FOR INFINITE DIMENSIONALSYSTEMS WITH UNBOUNDED CONTROL ACTION

R.F. Curtain* and D. Salamon**

1 . INTRODUCTION

In (15], (16), [17] SCHUMACHER presented a design procedure for constructing

stabilizing compensators for a class of infinite dimensional systems. The novel feature

was that the compensators were finite dimensional and that they could be readily

numerically calculated from finitely many system parameters. The class included those

systems described by retarded functional differential and partial differential equations

provided that the eigenvectors of the system operators were complete and provided that the

input and output operators were bounded. In (3] CURTAIN presented an alternative

compensator design which applied to essentially the same class of systems, except that for

the special case of parabolic systems unbounded input and output operators were allowed.

By means of enlarging the state space of the given distributed boundary control system,

CURAIN in 12] essentially transformed the original problem with unbounded control into

one with bounded control action so that the techniques of either (161 or (31 could be

applied. The resulting control, however, was of integral type. Neither of these two

compensator designs are applicable to retarded system with delays in the control or the

observation.

In the present paper we make use of the abstract approach developed by SALAMON [141

to extend the results of SCHUMACHER [16] to allow for unbounded control action. This is

dome in a direct way without reformulating the original problem into one with a bounded

input operator. In section 2 we outline the abstract formulation and prove a theorem on

the existence of a finite dimensional compensator paralleling the development in (16]. In

section 3 the general approach is than applied to retarded functional differential systems

Nathematisch Instituut, Rijksuniversiteit Groningen, The Netherlands

Sponsored by the U.S. Army under Contract No. DAAG29-80-C-0041.*-This material is based upon work supported by the National Science Foundation underGrant No. NC-8210950.

... ..... %........................................... . ...........................

S%.

with delays in either the input or the output variables. The conditions are easy to check

and they are quite reasonable, except for the assumption of completeness of the

eigenfunctions which seems to be too strong. In a special case we are able to weaken this

assumption.

Finally, in section 4, we show how boundary control systems fit into the abstract

framwork of section 2 and give an example. The results are compared with the approach in

[31.

2. A GIUWERL RESULT

We consider the abstract Cauchy problem

12.111) xt) = x(t) + Bu(t), x(O) - e X,, (2.1;2) y(t) - (:x(t),

on the real, reflexive Banach space X where A s P(A) + X is the infinitesimal

generator of a strongly continuous amigroup 8(t) on X and C 0 L(xR3

).

In order to give a precise definition of what we mean by an unbounded input operator

we need an extended state space Z D X . For this purpose let us first introduce the

subspace

z- V,(A) CX

X

endowed with the graph norm of A*. Then Ze becomes a real, reflexive Danach space and

the injection of Z* into X* is continuous and dense. Defining Z to be the dual

space of Z , we obtain by duality that

XC Z

with a continuous dense injection.

-2-

%4

44-%

% %

4IN:

REMARKS 2.1

Ui) A can be regarded as a bounded operator from Z into X and S (t) restricts

to a strongly continuous semigroup on Z B y duality, A extends to a bounded operator

from X into Z. This extension, regarded as an unbounded operator on Z, is the

infinitesimal generator of the extended semigroup S(t) e L(z) (see 114, Isa 1.3.21).

(ii) If Pi 9 U(A) - OWA), then the operator U1 - A :X + Z is bijective.

Furthermore, this operator commutes with the somigroup SC t) so that it provides a

similarity action between the semigroups s(t) e LCX) and 8(t) e LUz.

Ciii) It follows from (ii) that the exponential growth rate of the semigroup 8(t) is

the same on the state spaces X and Z, i.e.

W- liz t logIS~t)E liz t logIS~t)El~)

(iv) It also follows from (ii) that the spectrum of A on the state space X coincides

awith the spectrum of A on z (see ( 14, Lema& 1.3.21). Furthermore, the generalized

siqenvoctors for both operators are the same, since the eigenvectors of A on Z are

contained in DZ(A) - X. Finally, the (generalized) eigenvectors of A are complete in

X if and only if they are complete in Z.

I

We will always assume that B is a bounded, linear operator from R into Z.

4However, we want the solutions of (2.111) to be in the smaller space X on which the

output operator is defined. Therefore we need the following hypothesis.

(HI) For every T > 0 there exists a constant bT >0 such that I 8(T-s)Bu(s)ds e x I4 0

andT

If S(T-s)Bu(s)dol X b T*uC.)I X

for every u(*) e [PO,TIR I where 1 4 p <

-3-

10 e 0 % . .. . . .

" * .7 k : ;.

Zn the following we collect some important consequences of (HI) which have been

established in [14, section 1.3].

REMARKS 2.2

(i) (HI) is satisifed if and only if the inequality

IB 8 ( )x I L[0 ,TIR i b T Ix Il

holds for every x e z and every T > 0 where 1/p + I/q 1.

(ii) If (HI) is satisfied, then

t(2.2) x(t) - S(t)x0 + j S(t-s)Du(s)ds e x

0

in the unique strong solution of (2.101) for every x0 e x and every

u(-) e LP[o,TR i]. More precisely x(t) is continuous in X on the interval (0,T]

and satisifles

tx(t) - X0 + J (Ax(s) + Bu(s)]ds, 0 4 t 4 T,

0

where the integral has to be understood is the state space Z. Thus equation (2.111) is

satisfied in the space Z for almost every t e [0,T].

(iii) If (31) is satisifed, then for every w(-) e CEO,TX] there exists a unique

x(*) e CEOTrX] satisfying the equation

tx(t) - w(t) + ) S(t-s)BFx(s)ds, t • 0.

0

This solution x(•) depends continuously on w(•).

-4-

F rLr -L / * .%. .r.. F.%.%...'P

• - .- - . _ . . . . . . .

4.

7-

Moreover hypothesis (HI) implies the following important perturbation result.

-/THEOREM 2.3

Let F e L(XifI ) be given. Then the following statements hold.

(i) here exists a unique strongly continuous semigroup SF(t) on X satisfying

a.. t(2.3) S(t)x - S(t)x + J S(t-s)FSF (s)xds

0

for every x e X and every t ; 0. Its infinitesimal generator is given by

D(A,) - {x e xAx + BFx e x),

AFx - Ax + BFx

(ii) A F - A + F :V(A z X

(iii) S (t) extends to a strongly continuous semigroup on Z and the infinitesimal

generator of the extended semigroup is given by A + BF : X + Z.

(iv) Let x0 e O and v() e LP[0,T;RI] be given and let x(-) e C[0,T;X] be the unique

solution of

t(2.4) x(t) - S(t)x 0 + J S(t-s)B[Fx(s) + v(s)lds, 0 4 t C T

0

Then

t(2.5) x(t) - SF (t)x 0 + J SF(t-s)Bv(s)ds, 0 f t 4 T

0

(v) Hypothesis (HI) is satisfied with S(t) replaced by Sp(t) and S,(t) satisfies

t(2.6) S F(t)x - S(t)x + j S (t-5B)FS(s)xds

0*0

%. . .. .~~ *.*. . . I** * * * *4 .. .* .9.- . 0 . .. :.

,%, -.-.L,..,3%;_,. %;:,,,: .- .--. :......... .....- %%...... .............-......... .. .. ,.. ,,,

for every x 43X and every t O 0

(vi) let xO 6 and f(-) e Lpj0,?uXJ be given and define

Vt

(2.7) x(t) - S1.(t)xo + j 8 tv(t-s)f(s)ds. 0 4 t 4 T

Then

t(2.8) X~t) - S(t)x 0 + j S(t-a)[Drx(s) + f(s)]du, 0 < t 4 T

0

Psoor Statement Mi has been shown in [14, theorem 1.3.7] and (ii) follows from (14,

theorem 1.3.91 since the input space is finite dimensional. By (ii), S (t) restricts

to a strongly continuous semtigroup on Z D (A; ) and hence Sr(t) extends to a

semigroup on Z. Dy remark 2.1(i), the extended semigroup is generated by the adjoint

operator of A 1 where A; is regarded as a bounded operator from Z into X This

proves statement (iii).

In order to establish statement (i10, let us first assume that v(-) 6 C I 10TfE I and

let x(-) e C(O.Tjj be the unique solution of (2.4). Then it follows from Remark 2.2

(ii) that xC*) e C ICO.TIZ) and

d/dt x(t) -(A + B)x(t) + Bv(t), 0 4t <

Hence it follows from (iii) and a classical result in senigroup theory that x(o) is

given by (2.5). In general, statement (iv) follows from the fact that the unique

solutions of both (2.4) and (2.5), regarded as continuous functions with values in Z,

depend continuously on v(*) I L (0,TjR 1 .

It follows imediately from (iv) that (I) is satisfied with S(t) replaced by

Sr(t). Now let x(t), t )0, be defined by the RHS of (2.6). Then it follows from (iv)

that

N V -6-

t5x(t) - SWtX + jS(t-s)D[F) S F(a-flBFS(r)xd'r + FS(s)x]da

0 0

t- SWtX + j S(t-8)DPx(s)d8

0

for t 0 ,and hence x(t) - S7Ct)x, by the definition of Sl,(t). This prove.

statement (v).

Statement (vi) can be established straight forwardly by inserting (2.3) into (2.7)

~. , and interchanging integrals..

The aim of this section is to give sufficient conditions under which system (2.1) can

be stabilized by a finite dimensional compensator of the form

(2.901) ;(t) - w(t) - Ny~t), v(0)-

where N e It , H e3 R K 6 a are suitably chosen matrices. To this end we need

the following well posedness result for the connected system (2.1), (2.9).

PROPOSITION 2.4

let (Hi) be satisfied. Then for all x06 X, w 0 6 R , v(*) e L1 (Oc,-;3 I there

exists a unique solution pair, x(t), w(t) of (2.1) and (2.9). This means that x(t) is

continuous in X and absolutely continuous in 2, that (2.111) is satisfied for almost

every t '.1 0 where u(t) is given by (2.912) and that w(t) e 6 is continuously

differentiable and satisfies (2.911) where y(t) is given by (2.1,2).

PROOr Lt us introduce the spaces X .Xx RZe 0 - U R x Rm and the

operators 5e Ct) 9 L(xe), Se *L(ueze), 6e L( XeUe) by

-7-

4 -,o

A,

S (t) - [(t) 0] _1, e B b - 0i e 0 e4t e 0 H

Then hypothesis (Hi) in satisfied with X, Z, S(t), B replaced by Xe , Ze , Se(t), Be'

respectively. Moreover x(t) e X and w(t) e N satisfy (2.1) and (2.9) in the above

sense if and only if the following equation holds for every t ) 0

x t(X(t)) Set(O) + IS(t-s)B [F (x(s)) + (v(s)),]w(t) ee e e W

This proves the statement of the Proposition.

* 0The following hypothesis together with (HI) will turn out to be sufficient for the

existence of a stabilizing, finite dimensional compensator for system (2.1). It

generalizes the approach of OCHuIACHER [15], [16], [171 to systems with unbounded input

operators.

(N2) Suppose that there exist operators F e L(X,§), G e L(In,X) and a finite

dimensional subspace W C X such that the following conditions are satisfied.

1. The feedback sefiqroup SF(t) e L(x), defined by (2.3), is exponentially stable.

2. The observer semigroup SG(t) e L(X), generated by A + OC is exponentially

stable.

3. S(t)W C V for all t > 0.

4. Range G C w.

If (f2) is satisfied and N - dim W, then there exist linear maps I RN +X,

N: X R satisfying

(2.10) i - id, twx - x, x e w

Al~

~ .N~ -.5, % ~ * .

7-v ..- 771

Moreover, w C V(AF) and hence wA Fiis a weil defined linear map on R.We will show

that the system

;(t) - ,(CA F+ GC)iw(t) - lrGy(t), w(0) w.

u(t) -Flw(t),

defines a stabilizing compensator for the Cauchy problem (2.1).

THEOREM4 2.5

if (H11), (H12) and (2.10) are satisfied, then the closed loop system (2.1), (2.11) is

exponentially stable.

PROOF By Proposition 2.4, the system (2.1), (2. 11) is a well posed Cauchy problem. Now

let x~t) e x, w(t) e le be any solution pair of (2.1), (2.11) and define'-I'T

*(2.12) z(t) i w(t) - x(t) e x, t o

Then

(2.13) ;(t) - 7rAFiw(t) + irGCz(t)

wA itand hence, using wS F(t~ti e and Theorem 2.3 (vi) with f(t) =GCz(t), we get

tZ(t) IN tSF(t)1w 0 + J ITSF(t-s) iiGCz(s)ds - x(t)

0

t

% %S 1 ~l 0 + j S,(t-s)GCz(s)ds -x(t)

0

t=S(t)1lv + S(t-s)(BFiw(s) + GCz(s)]ds

-9-

% .1

0:!

t.. S(t)x .. S(t-.).u(s)ds

0

t= S(t)z(O) + J S(t-s)GCz(o)ds, t ) 0

0

This implies z(t) - sG(t)z(O) and hence, by (2.13), stability of the pair z(t), w(t).

Now the stability of the pair x(t), w(t) follows from (2.12).o

Clearly, the hypothesis (R2) is not very useful in the present form since it is rather

difficult to check in concrete examples. Pollowing the ideas of SCHUNACHZR [161, we

transform (R2) into an easily verifiable criterion. The basic idea is to approximate G

by generalized eigenvectors of AF and to show that, if A has a complete set of

generalized eigenvectors and is stabilizable through B, then there exists a stabilizing

feedback operator r which does not destroy the completeness property of A.

More precisely, we need the following assumptions on A.

(W3) The resolvent operator of A is compact and the set A - {A 6 P0(A) j ReA • -

is finite for some w > 0

If (R3) is satisifed, then we may introduce the projection operator

1-

r

where r is a simple rectifiable curve surrounding A but no other eigenvalue of A.

Clearly, PA is a projection operator on both X and Z. Correspondingly we obtain the

decomposition

A AX "XA X, Z XA Z

-10-

%,,

XW -0 -A~5 .5%* .. ~*.~ .' .*.

where x A rage P., Z =ker PA' x Z r) Xl are invariant subspaces under S(t). If

AA

NNA A

IA R *XAI TA R

* - with the properties

(2.14) itAt A - id, A IFA - PA

Then the projection xAt M wAe(t) of a solution to (2.1) satisfies the finite

dimensional ODE

(2.1) XAt)W AAxA(t) + BAUMt) KA(0) T AxO

yA(t) C AxA(t)

(2.16) A A WA AIAV SA W WA B C A -1 CA

Now we can replace (H2) by the following stronger conditions which can in many cases

be easily verified. The result has been proved by SCHU.MACHER [161 for the case that

range B C X (bounded input operator).

PROPOSITION 2.6

Let the operator A satisfy (H3), assume that the exponential estimate

(2.17) ISMtI x A'LxAV 4 e t ;0 0

-11- . . . . . . . . .

S.%

L?. "...4 . . .

holds for some M I 1 and that the reduced finite dimensional system (2.15) is

controllable and observable. Furthermore, assume that the generalized eigenvectors of

A are complete in X. Then (142) is satisfied.

PROOF Since (2.15) is controllable, there exists a matrix FA e R such that the

matrix AA + BAFA is stable. Furthermore, the estimate (2.17) implies that

ISW z(t) AnL(zA) A Me t 0 0

(Remark 2.1(11i1)). It is a well known result in infinite dimensional systems theory (see

e.g. [5] or (16]) that under these assumptions the closed loop semigroup S F(t) e Lz),generated by A + BF : X + Z with F - FAI A : Z + R is exponentially stable. By theorem

2.3, SP(t) restricts to a strongly continuous semigroup on X and the operator

VI - A - BF : X + Z provides a similarity action between both semigroups (Remark

2.1(11)). Hence the restricted semigroup SF(t) e Lx) is still exponentially stable

(Remark 2.1(i111)).

By assumption, the generalized eigenvectors of A are complete in X and hence they

.4 are complete in Z (Remark 2.1(iv)). Since F is of finite rank, this implies that the

' generalized eigenvectors of A + BF : X + Z are complete in Z (SCHUMACHER [16]). It

follows again from the above similarity argument, that this completeness property carries

over to the restricted operator AF : V(AF) + X introduced in Theorem 2.3 i).NAxm

Now choose GA e R such that AA + GAC A is stable and define

G A AG A F + X Then it is again a well known fact from infinite dimensional linear

systems theory that A + GC . V(A) + X generates an exponentially stable semigroup on

X (see (5) or [161). It is also well known that A + GC still generates an exponentially

stable semigroup on X whenever IG - GI is sufficiently small. Now we make useL(Rm,x)

of the fact that the generalized eigenvectors of A. are complete in X. This implies

that G Rm + X can be approximated arbitrarily close by an operator G m + X whose

-12-

.. . 4 - * 'JV %.. % ... . ,, . ,,,. .,.

- ,"% -.% ." . .. .' .'*. '..%. . ,

.

range is spanned by finitely many generalized eigenvectors of A.. We choose G in such

4A

a way that A + GC generates a stable semigroup and denote by W the finite dimensional

subspace of X which is invariant under A. and generated by those generalized

eigenvectors which span the range of G. Since W is a finite dimensional subspace

contained in NA) and invariant under A., the restriction of A to W is a

bounded, linear operator generating a semigroup Sw(t) on W. Since d/dt SW(t)x -

AvSW(t)x, the semigroup Sw(t) coincides with SF(t) on W. Hence W is also

invariant under the semigroup S,(t). We conclude that the operators F : Z + it and

ItG : X satisfy hypothesis (H2).s

Combining Proposition 2.6 with Theorem 2.5, we obtain a constructive procedure for

designing a finite dimensional compensator for the Cauchy problem (2.1). The construction

is based on the knowledge of the finite dimensional reduced system (2.15) and on the

knowledge of sufficiently many eigenvalues and eigenvectors of the operator A.. For the

case of bounded input operators (range B C 1) the procedure has been described in detail

by SCHUKACHER [16]. Precisely the same algorithm applies to the case where range B i X.

3. RETARDED SYSTMS

In this section we apply the abstract result of the previous section to retarded

functional differential equations (RFDE) with delays either in the input or in the output

variable. If delays occur in the input and output variables at the same time, the RFDE

can still be reformulated as an abstract Cauchy problem (see e.g. PRITCHARD-SALAMON [12])

however, the completeness assumption will no longer be satisfied.

3.1 RETARDED SYSTE1D4 WITH OUTPUT DILAYS

We consider the linear RFDE

-13-

% % %

4*. e - F -e

xt)- Lxt + Boult)

(3.1)

y(t) - Cx

where x(t) 0 IP, u(t) e RI, y(t) e It and xt is defined by xtlT) - x(t+T) for

-h T < 0, h > 0 Correspondingly B0 is a real n x t- matrix and L and C are

bounded linear functionals on C- C(-h,O13n] with values in 3I and IF,

respectively. These can be written in the form

0 0- J dn(T)#(T) , C4 - J dY()#(r) e e C,

-h -h

where n(T) and y(T) are normalized functions of bounded variation, i.e. they vanish

for T • 0, are constant for T 4 -h and left continuous for -h < T < 0

It is well know that equation (3.1) admits a unique solution

p _n Ip n• (" • "loco,'," for every inputu(, .,x(O) 6 E-h,(**h I () foWvr nu u(-) e 1[0,-Ifa I and every

initial condition of the form

(3.2) x(O) -0

, x(T) 4 1t(T) , -h T t C 0

where ( 1, *I ) I L (-h,Ogl ] - NIP . Moreover, in these spaces the solution

x(e) of (3.1) and (3.2) depends continuously on # and u(.) . This has motivated the

definition of the state of system (3.1) at time t ) 0 to be the pair

tt(3-3) (t) - (x(t),x t ) e Hp

•

The evolution of x(t) can be described by the variation-of-constant. formula

t(3.4) x(t) - S(t)# + J 8(t-m)Bu(s)ds , t > 0

0

where 3 6 L( 1 ,AP ) maps u * 3 into the pair Ou - (B0u,0) and s(t) e L(M3 ) is the %

-14-

.1

b

.. U F..5P;'-°Fp ¢ *L'.- : * * ' 5' ' .- . .... ,. .

- - - % .- " . ' " . "

1- FMCw TZ T. v. - . . .- T

strongly continuous seuigroup generated by

1,pl I

D(A) e 6 MPI1 W , 1(0) 400A#- (L 1,

Here W1 'P denotes the Sobolev space V lP[-h,0,1n].

We vil consider the evolution of the state (3.3) of system (3.1) in the dense

subepace 1(#(0),4)I4 e 1"p C Np which we shall identifiy with W1 .p . Then 3 becomes

an 'unbounded' operator ranging in the larger space P. iowever, it follows from the

uniqueness and continuous dependence result for the solutions of (3.1) and (3.2) that the

state x(t) of (3.1), (3.2) defines a continuous function in W1*p provided that

* 6 1,p and u(*) e LPo(O,.R He " snoe, the operators P and B satisfy hypothesis

(NI) with Z - Np and X - .l ep This implies that the state ^(t) 6 W1 " p of (3.1),

(3.2) with * e Wl1 p satisfies the Cauchy problem

d/dt xlt) - A;(t) + Bu(t) . x(0) - e 1 .p(3.5)

y(t) - Cx(t)

in the sense of Remark 2.2(11). Of course, the output operator C t W'1 P IF is given

by

0co - dy(T), (r) 6 6

-h

On the state space W1 p this operator is bounded.

UIARMU 3.1

(L) If the equation

(3.6) n(T) - n(-h) + A1 -h < T ( e-h

-15-

4e -. e

"~ ~ -- - - "- 77 -. ."--7777L. i. "' ".,'

holds for some £ > 0 , then the generalized eigenfunctions of A are complete in MP

and in WpIP if an only if

(3.7) det A1 * 0

(see MANITIUS (91, SALANGI (14, chapter 31).

(ii) It is well known that the operator A satisifes (H3).A

(iii) The exponential growth of the semigroup S(t) on the complementary subspace X

corresponding to A - JA e O(A)IRA ) 0} is determined by

supIReXIx e *(A), Rex < o} < 0.

(iv) Let (2.15) denote the reduced finite dimensional system obtained by spectral

projection of the solutions of (3.5) on the generalized eigenspace XA . Then (2.15) is

controllable iff

(3.8) rank[)I-L(ex°),B 0 - n VA e A

(PINDOLFZ (10]) and observable iff

XI-L(e )(3.9) rank [ n VX e A

(OURT-ADIVO [1), &AUMON [131, (141).

Combining these facts with Proposition 2.6 and Theorem 2.5, we obtain the following

existence result for a finite dimensional compensator for system (3.1).

3.2

If (3.6-9) are satisfied, then there exists a finite dimensional compensator of the

form (2.9) such that the closed loop system (3.1), (2.9) is exponentially stable.

-16-

.................................................." " .......... "

% mm+ '

*l'~ l - . ... , t ++

' "- + - * r.. * p*e'""" "*, "

• - -. .. •.

A7

3.2 RETARDED SYSTES WITH INPUT DELAYS

In this section we consider the RFDE

x(t) - Lxt + BUt ,

(3.10)y(t) - C x(t)

0

with general delays in the state and input and no delays in the output variable. This

time C0 is a real m x n - matrix and 8 a bounded linear functional on C[-h,0,3 I

nwith values in R given by

0E - J dG()'r), C[-h,OR I

-h

where O(M) is an n x I - matrix valued, normalized function of bounded variation. Of

course, we can immediately get an existence result for a finite dimensional compensator

for system (3.10) by dualising Theorem 3.2. However, for reasons to become clear later,

we make use of a direct approach for system (3.10), following the ideas of VINTER and

KNOG (18] (see also DELFOUR (61, SALAMON (14]).

First note that (3.10) admits a unique solution x(.) e Lo[-h,m Rn] w [0,-,R n ]

loc loc

for every input u(.) e _o[0,- I and every initial condition of the form

x(0) - x(T) - * 1Cr) ,

u(-r) - (T), -h 4 T < 0

where * pI and E e Lp[-h,0; I I In order to reformulate system (3.10) as an

evolution equations in a product space, we rewrite (3.10-11) as

0 0;(t) dn(T)x(t+T) + j d(T)u(t+T) + f (-t),

-t -t(3.12)

y(t) - C0x(t) , x(O) - fO

-17-

.. o . . , . " ..-.. .' .d ".- .. . . .NL :%

Swhere the pair f -(f 0, f1) e 14p given by

- fO .r. rn 0. ,

0 1

(3.13) a a

f f1 0) -JdinT)* 1 (r-c) + Jd9(T)C(T-0) ,-h 4C a 4. 0

-h -h

is regarded as the initial state of system (3.12). The corresponding state at time

t 0 is given by

x(t) - (x(t),x) t 6 H

(3.14)a

x t(a) - f dn(T)x(t+t[-O) + f d(T)u(t+T-O) + f lO-t)O-t O-t

It has been shown in (61, (141, (16] that the evolution of this state can be described by

the variation-of-constants formula

t - *

(3.15) x(t) - S? (t)f + ! (t-s) u(s)ds, t ) 00

Here 8 (t) L( p) is the adjoint smigroup of sT(t) 6 L(N , 1/p + 1/q - I , which

corresponds to the transposed equation x(t) - LTt in the sense of section 3.1. Since8 5T(t) restricts to a samigroup on the dense subspace V 1

q C K

q , the adaont

-s- emiqrOup ST

(t) extends to the dual spaes W"1,

p -(Wl'q)

* which contains H

P an a

*.1, dons" subpace in a natural way. The input operator B e L(R *W 'p) is the adjoint

of 1,qoperator of e L(W ,qR ) given by

0" BT* 0 dO(')* l(V

) e R it e) V 1,q C H q •

Since the infinitesimal generator AT of ST t) and the input operator BT satisfy

the hypothesis (MI) of section 2 with X - NP and Z - WV1 1p (see SAL4ON [14]). the state

*x(t) 6 HP of system (3.12), given by (3.14), defines the unique solution of the abstract

-19-%

14.-1

i"~

Cauchy problem

d/dt X(t) - A x(t) + B .(t), ;(o) - e 6

(3.16)

y(t) - Cx(t)

in the sense of Remark 2.2(11). Of course, the output operator C t H *R is now given

by

Cf - C f O

e jr , f e Hp

•0

In order to make the results of this section more precise, we briefly outline the

construction of the reduced system (2.15). For this purpose let XA C W1 ' nd

xC denote the generalized sigenspaces of A and AT , respectively,

corresponding to A - (X a(A)IR*A ) 01 . Since A is a symetric set, we can choose

real bases {*1,.,.,*wA of IA and } of X s uch that the matrices

nXNA'..- [,1...W I • v1 'q[-hoiaR I

N l nxNA6 w e v'[-h,0aRt

satisfy

0 0IT101400 + I JOY T(a)d~jlT)4IT-a~do" I e R A X

"A

•

-h T

:A p AThen A sR A and A Mp R may be defined by

lAXA 1 0 *(O)XA , AxA - d(l)*(tO)XA , -h 4 a c 0 ,-h

-19-

,.,

.---

7~ 777

A - yT(o)f0 + JyT(O)f I()do

-hN AN xNAN A X MX NAfor xA R and f e Hp and the matrices A A AXNA B A e R A e R

are given by

0

A((0)0)- (0(o),#)AA & B - 0 (T)dB(T) , CA - C04(O)-h

(see SAL140H [14, section 2.4]).

RU(Mt 3.3*

(i) If (3.6) is satisfied for some C > 0 , then the eigenfunctions of AT are

complete in N and in W- l p if and only if (3.7) holds (See MIANTIUS [9], SALAMON [14,

chapter 31).

(ii) if AA#DA*CA are defined an above, then system (2.15) is controllable iff

(3.17) rank[AI - L(e X),n(e')] - n V x e A

and observable iff

--(3.19) rank[I -Me] -n V A 6A

(see AL, OM [141)-

-20-

, - -- o. . . . . . . -.%

.F

Theorem 3.4

If (3.6-7) and (3.17-18) are satisfied, then there exists a finite dimensional

compensator of the form (2.9), such that the closed loop system (3.10), (2.9) is

exponentially stable.

Remark 3.1(1) and Remark 3.3(i) show that the completeness property of A and AT4...

No can be destroyed by arbitrarily small perturbations in the delays (compare MANITIUS

(9]). However such perturbations would not affect the stability of the closed loop system

(3.1), (2.9) respectively (3.10), (2.9). This indicates that the completeness assumption

is somewhat artifical for the purpose of stabilization by a finite dimensional

compensator. In the next section we will show that the completeness condition can in fact,...

be weakenend in the case of systems with state delays only.

3.3 A SPECIAL CASE

*- In this section we consider the special case of the RFDE

x(t) - A 0x(t) + AIx(t-h) + 0u(t)

(3.19)

y(t) - C0x(t),

with a single point delay in the sate variables and no delays in control and

observation. Following the ideas presented in the previous section we define the state of

system (3.19) to be the pair

x(t) - (x(t),xt) e mP

(3.20)

xt (a) A1x(t-h-a), -h 4 a C 0

This time the range of the input operator BT B is contained in MP , since

T T I TO01,a " (0) "BTP for e 6 w 'qC Mq Therefore x(t) satisfies the Cauchy

-21-

W .

%J*. % . . . . . . . *.% ..-

problem

d/dt x(t) - A x(t) + Bu(t),

(3.21)

y(t) - Cx(t)

We viii regard this as an equation in the closed subspace

x - (f e 4PIf(T) e range A1 , -h - T - 0}

of HP. Of course, this means that AT has to be restricted to the domain

Dx (AT V (A ) nx.

RNAM3.5

(i) it has been shown by JlANITIUS [9) that the elgenvectors of AT are complete in X

if and only if

IAX A](3.22) rank j - n + rank A

for ome A 6 This condition is obviously weaker than (3.7).

(ii) Clearly, the reduced finite dimensional system (2.15) remains the same when the

Cauchy problem (3.21) is restricted to the closed subspace X C HP . Therefore system

(2.15) in controllable and observable if and only if

rank[X -A -a -Ah AI,0 ]

(3-23) -Ahrankl -A 0 a A vXeA(3.23)- rank[ - A 0 - e " hAj - n V A

4O

' -22-

.i

where A {X e Tdet(XI - A0 - e A 0 *Re > 0)

"' (iii) It is easy to see that the stability of the abstract closed loop system (3.21),

(2.9) in the state space X x RN implies the stability of the closed loop delay equation

(3.19), (2.9).

Again combining these facts with the results of section 2, we obtain the following

existence result for a finite dimensional compensator. For system (3.19) this is a

stronger result than one would obtain by applying the theory of [16].

THEORSK 3.6

If (3.22) and (3.23) are satisfied then there exists a finite dimensional compensator

of the form (2.9) such that the closed loop system (3.19), (3.20) is exponentially stable.

RUEARK 3.7

This result suggests that it should be possible to weaken the completeness assumption

for the general RDE which would be an important improvement. Another extension would be

-u an existence result for RFDEs with delays in simultaneously the control and the

observation. However, it is not obvious how this can be achieved with the present

approach, the main difficulty being the completeness property.

RMARK 3.8

Although the main results of this section, Theorems 3.2, 3.4 and 3.6 are stated as

existence results, we emphasize that the stabilizing compensator can in fact be

constructed using exactly the same procedure as it is explained in detail in [16] for

1WDgs without delays in the external variables. The construction, as outlined in the

proof of Proposition 2.6, involves calculating finitely many eigenvalues and eigenvectors

of A and hence the projected system AA,BACA as it is described in section 3.2. Then

the matrices rA and GA can be calculated by standard finite dimensional procedures.

-23-

UP°' , € . . '- . ." . o" ." ....... . . " ' , . . - .. ' . . ., .... ,• . .

U -".. - " . . " " . . - - ' " " . .

-,, , " " " " " • . ." """ " " " .. . "•"•"• •" -"• - -"- •" - . •" -. . • . • .• .

- .....- **-I . V

The most difficult part of the design lies in finding eigenvectors of A + BF to generate

the subspace W . The approximation of G - iAGA by an operator G with range in W

then reduces to a finite dimensional linear optimization procedure. This procedure has to

be repeated - while increasing W - until ; is close enough to G . The numerical

example for a retarded system examined in [161 gives insight into the details of the

design procedure.

4. BOUNDARY CONTROL SYSTES

The purpose of this section is to show how abstract boundary control systems in

Hilbert spaces fit into the framework of section 2. When these results are applied to

obtain finite dimensional compensators for particular classes of partial differential

equations (PDR), there is a considerable overlap with results of CURTAIN in (2], (3),

[4]. The relation between both approaches will be discussed in detail at the end of this

section.

Let W,X,U,Y be Hilbert spaces and suppose that

4~ WC X

with a continuous, dense injection. Furthermore, let A 6 L(W,X), r e L(W,U), C 6 L(X,y)

be given. Then we consider the boundary control system

d/dt x(t) - A(t), x(O) - x 0 e w

(4.1)rx(t) - u(t), t ; 0

with the output

(4.2) y(t) -Cx(t) , t > 0

-24-

%* J.,

%.

,... - - . ,- - - - , .. -

DEFINITION 4.1 (strong solution, wellposedness)

(i) Let u(*) e C[0,T;U] and x0 e w satisfy rx0 - u(o) . Then a function

x(-) e C[OTiW] is said to be a (strong) solution of (4.1) if x(*) e C [0,T;X] and if

(4.1) is satisfied for every t e (0,T].

(ii) The boundary control system (4.1) is said to be well posed if the subspace

%. 1,24{ Lx e wlrx - 01 is dense in x , and if for all x0 e W and u(-) e W 20,TIU] with' .

rx0 - u(0) there exists a unique solution x(.) e C[O,t;W] n C [O,TX) of (4.1)

depending continously on x0 and u(C) • This means that there exists a constant

K > 0 such that the inequality

:, T . 2sup Ix(t)n + sup IxCt),x • K{<x0 1w + [I 1u(t)Iudt] 2}

-., LO~t4T Ot4T 0

%.* .~,holds for every solution x(t) of (4.1).

R DARKS 4.2

Let system (4.1) be well posed.

-" Ci) Taking u(t) E 0 , it follows from a classical result in semigroup theory (PHILLIPS

* [(111) that the operator

(4.3) Ax , Ax , P(A) { {x e Wlrx ,0

Vr.: is the infinitesimal generator of a strongly continous semigroup S(t) on X and that,

for every x0 e V(A) , the function x(t) - S(t)x0 is the solution of (4.1) with

%''- u(t) 3 0 •

"% (ii) As in section 2 we introduce the dense subspace Z = * ,(A*) C X* ThenX

XC Z

S-25-

. % .'.

4.-."- 5

%,,'.4.

4. .% - ...... , ,. , ,..4 ..,-.. . ,... ,;.. . ..... ,.. . . . ..*.. . ...

. . . . % . , _ - • • • % _ % . . . . . o -. • % • * . . . • • . . . , , . • - . • . . . . ,

-*, .1

with a continuous, dense injection, A extends to a bounded operator from X to Z

and S(t) to a strongly continuous semigroup on Z

(iii) It follows from [14, Lemma 1.3.2(1)] that

(x e wjrx - o) - vXIA) = {x e X = V,(A)JAx e X)

or in other words, if x e X and Ax e X , then x e V and rx = 0 . Furthermore, the

W-norm on Dx(A) is equivalent to the graph nor= of A ((14, Remark 1.3.1(11i)]). This

means that there exists a constant K1 > 0 such that the inequality

lxI V KI[1x1x + 1AxI ]

holds for all x 0 W with rx = o .

(4v) r is onto. Hence there exists a constant K0 > 0 such that for every u 0 U

there exists a w 6 w

i (4.4) rv - U , I w 4 o0 ulo •

Let us now construct the input operator a 6 L(U,Z).

1. Given u 6 U we may choose w 6 W such that rw - u since r is onto (Remark

".' 4.2(iv)). For this w e W we define Dut- Aw - Aw e Z • This expression is well

defined since Fw - 0 if and only if Aw - Aw (Rmtrk 4.2(11i1)). Hence the ap

B U + Z satisfies, by definition, the equation

(4.5) srx - Ax - Ax , x e w

2. It is easy to see that S is a linear map.

3. Let u 6 U be given and choose w e w such that (4.4) holds. Then

-26-

t.,• .. . . % % , o - -."- " -" - '% -, * " ". " . "-S. . ° . .% " ** "*" ;*."%.*'* *** ".'....". . . . . . . . . . . . . . . . . .. . .- ..... ~* ** - "C.%*.

97- 1 -

Utulz • IA - AIl ~ lVI

IBIz 41 ANL(w, z) IW

] O -AI L(W, i) IU

and therefore 8 U * Z is bounded.

LIMUA 4. 3

Let the operators A and S be defined by (4.3) and (4.5), respectively.

ftrthermore, let x 0 X and u 0 U satisfy Ax + u e X. Then

x W , rx u , Ax Ax + Bu.

Furthermore there exists a constant K > 0 such that

ii

I lxlC KXx + NUNlu+ lAx + BI.]'

for all x 6 X, u 0 U with Ax + Bue

PROOF zet x e X and u 6 U satisfy Ax + Bu 6 X and choose w e W such that (4.4)

holds. Then

'(x-w) - A.x + Bu - (A+sr)w - Ax + Bu - Aw e X

By Remark 4.2(i1), this implies that x e w and

}-. ..1II W EvE3I + Ix - wol3

4 IwlW + K[Ix - w X + IA(x - w)IXI

-27-

%ftJ

?.i. % %,

~ I + Y uid. L(WX) + 1KA L~ L(w,x) WW

+ K Ilx + K I Ac + BUl

4 K [Iui U + 1X11x + 1hz + 3ul~ix

rinally, we obtain again from Remark 4.2(111) that rx rv - u and from (4.5) that

Ax - Ax + ONx - Ax + flu a

The above operators A and 3 allow us to reformulate the boundary control system

(4.1) as a Cauchy problem of the type (2.1). M4ore precisely, we introduce the following

concept of a weak solution for (4.1).

D3WIUI?!ON 4.4 (weak solution)

Let the operators A 6 L(X,Z) and B 8 LOA.) be defined as above. Moreover, let

X0 X and u(-) e L 2 O,TUJ be given. Then x(-) 6 CfO,T;X) W V 1 , 2[O,T;Z] is said to

be a weak solution of (4.1) if

d/dt x(t) -Ax(t) + u(t) 0 -

(4.6)

'IX(O) -=

is satisfied in Z (almost everywhere).

It follows from the definition of the operator B (Remark 4.2(iv)) that every strong

solution x(-) 6 CCO,TIW]r)C (O,TIX] of (4.1) in a weak solution in the sense of

Definition 4.4. Moreover we have the following result.

-28-

~h'~% %u~%Q*~ . . . .* ~ J.~ .* ' 5AS* **~* A

• . w y -. . . . . . . ... . . . . . , . .-. . .. • -. . .. -. ... • , ".. .- . F

PROPOSITION 4.5

Suppose that the operator A defined by (4.3) is the infinitesimal generator of a

strongly continuous semigroup S(t) on X and that r e L(V,U) is onto. Furthermore

let B e L(U,Z) be defined by (4.5). Then the following statements are equivalent.

(i) System (4.1) is well posed.

(ii.) The operators A and B satisfy hypothesis (HI) of section 2 with p - 2.

2(iii) For every x0 e x and every u(*) e L [0,TIU] there exists a unique weak

solution x(*) e C(0,TXj n WI'2 (0,TIZ] of (4.1) depending continuously on x0

and u(') •

Moreover, the weak (and in particular the strong) solutions of (4.1) are given by

t

(4.7) x(t) - S(t)x 0 + J S(t-s)Bu(s)ds e X , 0 4 t C T0

PROOF It is a well known semigroup theoretic result that the solutions of (4.6), and

therefore the weak solutions of (4.1), are given by (4.7). Furthermore, it follows from

Remark 2.2(11) that (ii) is equivalent to (iii).

In order to prove that (ii) implies (i), suppose that (HI) is satisfied and let

x(t) be given by (4.7) with x0 6 W and u(*) e C (0,TgU] satisfying rx0 - u(0)

Then it is a well known result from semigroup theory that x(*) e C[O,TX] n C I[0,TgZ]

satisfies

x(t) - Ax(t) + Bu(t)

t= S(t)[Ax0 + SulO)] + J S(t-s*);(s)ds

0

t

- S(t)Ix o + f S(t-s)Du(s)ds , 0 C t 4 T00

-29-

Wa

.1 "

,,,..,.-. -... o .. ,:..-..... ...-....-..... .-. .. ... ,.... .... ... , ..... ,,, .,,,. ..,.., . .,,%" % - "a. - .'. " " " - - . . - '

BY (RI) and Remark 2.1(11), this implies that x()e C[O,TXJ and

sup Ix(t)I x sup 18tIlL(x) 16 L~w,x I01 V + bT 2.)OftOCT Oft4T L2CO,T;UJ

Applying Lease 4.3 to the term Ax(t) +- BUMt - ;(t) e x w obtain that

Ix(O) 6 C[O,WI 11 C(O,TiXJ satisfies (4.1). Since every strong solution of (4.1) is

given by (4.7), x(t) in in fact the unique solution. Furthermore, vs obtain from Les

4.3 that

*x(t)i1 C 4 IX(t)I + lu(t)I + Ixt~x

since

sup IX(t)l C sup Ia(t)Il X14ld lx Iw

OItC 04,T 2

T L 2 1,TUIand

sup iu(t)i U c in lxU Ix 0 .1 /i111~(.)I 2OCtIC? (0 L COTjU1

for u(*) 6 C (0,'1YUJ with uCO) - rx0, # this show that system (4.1) in well posed.

Conversely, suppose that system (4.1) is weilpoised In the sense of Definition 4. 1,

let v(-) 6 C I(0,Tjlvf and define

x(t) 9J(t-*)3J V(r)drdO , U~t) -JV(T)dT , 0O4(t C T'I0 0 0

7hen x(e) e C ro,TIzI and

Ax(t) + Bu~t) - (t) 8 (t-s)Bv(s)ds ,0 4 t C T

-30-

I1

*j -: 4 e. ..6 6-k

Hence x(-) e C I[O,2',X] and we obtain from remma 4.3 that x(-) e C[O,TWI, rx(t) -u~t)-

and Ax~t) -Aic(t) + Su(t) . Hence x(t) is a strong solution of (4.1) in the sense of

Definition 4.1(i) and satisfies the inequality

Ij T-)v(s)dsIx = Ix(T)I X ( KUV(.)I 20TU

This shows that the operators A and B satisfy the hypothesis (Hi)..

Having established hypothesis (HI) we are now in a position to apply the perturbation

result of section 2 (theorem 2.2) to the boundary control system (4.1).

COROLLARY 4.6

If system (4.1) is well posed, then the following statements holds.

Mi For every r 6 L(XU) the operator

(4.8) x-fAx, V(A ) {(xe wjrx -Fx}

N:. is the infinitesimal generator of a strongly continuous senigroup SF(t) on X____________________

(ii) For every x 0 6 Mid,. the function x(t) - Sp(t)xo , t ), 0 , is continuous in

V , continously differentiable in X ,and satisfies the closed loop boundary control

equations

(4.9)

where the derivative has to be understood in the space X

(iii) If U in finite dimensional, then 8 7p(t) extends to a strongly continuous

-31-

04

% % %, V .

o %%'

14 't- l

semigroup on Z whose infinitesimal generator is given by the extended operator

A + BF : X - Z

PROOF By Proposition 4.5, the operators A and B defined by (4.3) and (4.5),

respectively, satisfy hypothesis (HI) of section 2. Hence it follows from Theorem 2.3(i)

that the operator

A x-Ax +BFx , (A,) -I xexAx +BFx xj

generates a strongly continuous semigroup on X (note that the proof of this result in

14, Theorem 1.3.7] does not require U to be finite dimensional). Lemma 4.3 shows that

this operator A7 coincides with the one defined by (4.8). This proves statement i).

In order to prove statement (ii), let x0 e V(A.) be given and define u(t) =

FSF(t)x 0 8 t ) 0 • Then u(t) is continuously differentiable for t ) 0 and

satisfies u(O) - Fx0 - rx0 . Hence (4.1) admits a unique strong solution

x(t), t > 0 , which by definition of the operators A and B also satisfies (4.6) and

N-P is therefore given by (4.7). This implies

tx(t) - S(t)x 0 + J S(t - s)BFSF ()x 0ds - SF (t)x0

* a 0

by definition of the semigroup SF(t)

Statement (iii) is an immediate consequence of Theorem 2.3(iii).m

so far we have shown that the general theory of section 2 also covers abstract

boundary control systems. In particular, we have reformulated the boundary control system

-32-

'"

-% %%%- wr . . C_,.' . ", • ." ,. . . . . . *.- .- - , .- -. ... - - ... . .- . . - . . ..-.. " - , , . . .. . .. - - . .. . - -

r

,',.,:% ,, .,.. -. " . . - - . " ." .'. - -.. " 4. W ". . .

,I

(4.1) in the semigroup theoretic framework with an unbounded input operator. A veryI

similar approach has been developed by HO and RUSSELL in [8] under only slightly more

restrictive assumptions. However, (8] does not contain any feedback results and also the

above Proposition 4.5 seems to he new. Furthermore we point out that earlier results in

this direction for special classes of systems can be found e.g. in CURTAIN-PRITCHARD

(5]. Another general approach has been presented by FATTORINI (7]. In (7] the input

operator is bounded, however, there are derivatives in the input function which do not

appear in our approach.

In [2] and (3] CURTAIN has used FATTORINI'S results for the constructin of finite

dimensional compensators which leads to integral terms in the loop. These integral terms

will disappear if we apply the approach of this section to obtain existence results for

finite dimensional compensators. More precisely, we have to assume that the operators A,

B and C , introduced in this section, satisfy hypothesis (H2) of section 2, or

respectively, hypothesis (H3) and the assumptions of Proposition 2.6. Under these "

conditions it follows readily from Theorem 2.5 that there exists a finite dimensional

compensator of the form (2.9) such that the closed loop system (4.1), (4.2), (2.9) is

exponentially stable.

Starting from equations (4.1), (4.2), the following problems have to be solved for

the construction of the compensator.

1. Find the operators A and B

2. Determine the spectrum of the operator A and the reduced subsystem (2.15).

3. Find the stabilizing operators F : X * U and G : Y + X I4. Determine the eigenvalues and eigenvectors of A, to approximate G

To illustrate this procedure, we consider the heat equation with Neumann boundary

conditions and boundary control which has also been treated in [3] with different methods.

EXAMPLE 4.7

Consider the parabolic PDS

-33-

04,

% "]J6-%". .'"..

: . ; ;. .. ,,; . ; , ~~~ ~~ ~~ . ._ .: ./. . . . .J;. . .' " . . ...

r%

(4.o01) t -2z• z 0 < < I t > 0,

(4.1012) z (O0t) - u(t), z (1,t) - 0 , t > 0

(4.1013) z(E,O) - 9O(0 , 0 < 1

(4.10;4) y(t) f f c(&)X(,t)d& , t > 0 .

0

This system can be written in the abstract form (4.1) with

X -L2[10,1] , WT { e 2[o 10] .1(1) -0) ,U- R,

22 [0r

z-V( ) - V(v) - * 011(0) 0 0 -

The operator A Satisfies (W3) and has a complete set of eigenvectors

40() n(E) 2 cc* nw& , correspondinq to the eigenvalues

20 - 0, A - -n ,n e N - n order to determine the operator B : R + Z, let us

choose any * 0 W such that r# - 1 , e.g. ([) - -(E-1) 2/2 . Then, for every0

e 6 , the following equation holds

> >n - ( a *,r* , - < *,ur* ) - < *,t& - A* >

- < , > <*~$~d -, -> J()*~

- 0 W 0

= -jf($)(~)+ *(4)*(C)IdC2 0

- j- *(o);(o)]

-34-

% . . ."..... ........• , - ~~~~~* N "o•o•.~ *'°." i % ,- ".* °-

2-C

1- - - ,(o).

It has been shown in PRITCHARD-SLAMON [12] that these operators A and B satisfy

'.'hypothesis (H1) and therefore system (4.10) is ellposed in X - L2[0,11 n the sense of

Definition 4.1 (see Proposition 4.5). The spectral projection of L2[0,1] onto the

eigenspace xA - (co.ia e a) of A corresponding to the unstable part A - (0) of the

spectrum is given by

1

PA - f *(T)dr , 0 < < 10

With the choice of (*0} as a basis of XA , this operator splits into PA - A

where wA : L 2[0,1] R and IA : R + L 2[0,1] are given by

1

V = f (T)dT , lAXA( ) = XA , 0 4 -9 10

Then the reduced finite dimensional system (2.15) is described by the msmtrices"

A 0 o BsA . -2 C - fc(C)d&

0

This system is controllable and observable if and only if

1

(4.11) C J c(E)dt * 00

Stabilizing matrices are given e.g. by

F A w 2 / 4 G A C A-

so that A A + BA F= 1/4 A A + G ACA" Then the operator Ar with

U-35-

. * . t t t .e ,.....t ,. C * . . - . . . . . ..., ., u ,.. '. . . -.. ,..•".. .. . .• -.- ,,-, ,, .. .,%.-..,'. , ;q , ; ,. .,., .,.. ,, ,'. ' ...- .. . .. ; . \ N . .

%"

F AA L 2(0,1] + R given by

12*1

D(,A.) - 14 e m 1o,111;(1) 0 , ;.(o) !2 I

0

-'-,The eigenvectors and eigenvalue8 of A. coincide with those of A except for X 0 0

which is now replaced by Ap = -1/4• The corresponding normalized eigenfunction is

.1 -V sin.. 2

We will choose W = span{# F } and the maps

S.

(1 )( ) = * (EIv , 0 < & < I , w e R

F F

(i)(Ed) J F ( M1)4)d, , 0 e L 210,11

So that iF F : L2[0,1 ] W is the orthogonal projection onto W and w T = I

Let us now consider the case that c(g) - E for 0 4 E C 1 . Then CA ' /2 and we

choose GA = -ig/2d, g > 0 . With this choice the operator G R L (0,1] is given by

We replace this operator by

~-36-

II

e r t

semtigroup it satisfies the "spectrum determined growth" assumption. Furthermore, its

spectrum is given by4

CA + GC)

1_{-2Igfi - os 3]= 8K + w2 - vtiw 2w 2 ]il/,/ sinww, w ol

if g > 0 *in the case g <0 there is an additional positive eigenvalue X0 W > 0

where

e N+ e~-2. .,Nsx -,2 + /i~

We conclude that A + GC generates an exponentially stable semigroup if and only if

g > 0 . Hence the operators P and G satisfy the hypothesis CR2) with the one

dimensional subspace W =span{4 In this case the compensator (2.9) is described by

the "matrices"

M if (A +G;C) 1 1 4(

F F F 4

K F A r A Ir

1 r

(4.12)

uV

'S-37-

% -% %j

' AR 4.8 '

The results of this section show that the abstract framework of section 2 is general

enough to cover both FD-s and PDEs We mention that the approach of this section can also

be applied to damped hyperbolic systems. Hence this paper represents a complete

generalization of the compensator design of SCHU14ACHER (161 to infinite dimensional

t~r systems with unbounded control action. However, the degree of unboundedness which we can

" allow for the input/output operators is not as general as one would desire. For example,

for the parabolic PDZ (4.10) we cannot allow simultaneously Neumann boundary control and

point observation. Also we cannot allow Dirichlet boundary control when the outputoperator is an arbitrary functional on 2[0,1] A general theory which covers these

cases would require the consideration of unbounded output operators as well. The

development of such an approach seems to involve some further difficulties and would be an

interesting problem for future research.

RMARK 4.9

Using the abstract approach outlined in section 2, it is possible to directly extend

the results of oCHACHER 15a, d17e on tracking and regulation infin ite dimensions to

unbounded control action. A different approach i to useee o extended system formulation

pitscussed in 2, which results in integral control action and this can be found in

-38-

*4r

r is an

s d r e te c o

AZ dl

REFERENCES

(1] K.P.M. BRAT/H.N. KOIVO

Modal characterization of controllability and observability for time delay systems.

IEEE Trans. Autom. Control AC-21 (1976), 292-293.

v92) R.?. CURTAIN

On semigroup formulations of unbounded observations and control action for

distributed systems.

International MTNS-Symposium, 1983, Beer-Sheva, Israel.

(3] R.F. CURTAIN

-Finite dimensional compensators for parabolic distributed systems with unbounded

control and observation.

SIM J. Control Opt. 22 (1984).

(4] R.F. CURTAIN

Tracking and regulation for distributed parameter systems with unbounded observation

and control.

Matheatica Aplicada e Computacional, 2 (1983).

15] R.F. CURTAIN/A.J. PRITCHARD

Infinite Dimensional Linear Systems Theory

LNCIS 8, Springer-Verlag, Berlin 1978.

-39-

A. 1%

%

(6] M.C. DELFOUR

The linear quadratic optimal control problem with delays in state and control

variables: a state space approach.

Centre de Recherche de Mathfimateques Appliqu6es, Universit6 de Montrial, CNMA-1012,

1981.

[7] H.O. FATTORINI

Boundary Control Systems.

SIAN J. Control 6 (1968), 349-385.

[81 L.F. HO/D.L. RUSSELL

Admissable input elements for systems in Hilbert space and a Carleson measure

criterion.

SIAN J. Control Opt. 21 (1983), 614-640.

[9] A. ANITIUS

Completeness and F-completeness of eigenfunctions associate with retarded functional

differential equations.

J. Diff. Equations 35 (1980), 1-29.

[10] L. PANDOLFI

Feedback stabilization of functional differential equations.

Bollettino U.N.I. 12 (1975), 626-635.

[11] R.S. PHILLIPS

A note on the abstract Cauchy problem.

Proc. Nat. Acad. Science USA 40 (1954), 244-248.

-40-

%-,.

~~[121 A.J. PRITCHARD/D. 8SALMuON

The linear quadratic optimal control problem for infinite dimensional system with

...

unbounded input and output operators.

NRC, University of Wisconsin-Madison, TSR-2624, 1984.

(131 D. SALMNON

Observers and duality between dynamic observation and state feedback for time delay

systems.

ISE3 Trans. Autom. Control AC-25 (1980), 1187-1192.

(14] D. SAL-MON

Control and Observation of Neutral Systems INK 91, Pitman, London 1984.

(15] J.1. SCHUMACHER

% I'M Dynamic Feedback in finite- and infinite-dimensional space.

Mathematical Centre Tracts, No. 143, Mathematisch Centrum. Amsterdam 1961.

[16) J.M. SCHUMACHER

( A direct approach to compensator design for distributed parameter systems.

SIAM J. Control Opt. 21 (1983), 823-836.

(17] .7.M. SCHUMACHER*Finite dimensional regulators for a class of infinite dimensional systems.

- Systems a Control Letters 3 (1983), 7-12.

(16] R.B. Vfl4ZR/R.H. WONG

The infinite time quadratic control problem for linear systems with state and

control delays: an evolution equation approach.

SIAM J. Control Opt. 19 (1991), 139-153.

-41-

..~~~~ ~~ .J .. .oto Opt.. 1. .18) 1-5.

'6 %4 . • , . . ". . . . .; . . . . . - . , -, - , . . -. . - , , . . , . , , ,

SECURITY CLASSIFICATION OF THIS PAGE (Wheun Dais Entr.0D

* . REPORT DOCUMENTATION PAGE BFRED INSTRUCTONS~

* . REPORT NUMBER 2.GOVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER

4. TITLE (and Subtitle) S. TYPE OF REPORT A PERIOD COVERED

Finite Dimensional Compensators for Infinite Summary Report - no specificDimensional Systems with Unbounded Control reporting periodAction 6. PERFORMI1NG ORG. REPORT NUMBER

*7. AUTNOR(e) S. CONTRACT OR GRANT NUMBER(q)

R.F. Curtain and D. Salamon DAAG29-80-C-OO 41MCS-82 10950

II. PERFORMING ORGANIZATION NAME AND ADDRESS 1O.- PROGRAM ELEMENT. PROJECT, TASKMathmatcs eseach entr, Uivesit ofAREA & WORK UNIT NUMBERSMathmatcs eseach entr, Uivesit ofWork Unit Number 5 -

610 Walnut Street Wisconsin Optimization and LargeMadison, Wisconsin 53706 Scale SystemsII. CONTROLLIG OFFICE NAME AND ADDRESS 12. REPORT DATE

May, 1984(See Item 18 below) 1S. NUMBER OF PAGES

4114. MONITORING AGENCY NME & ADDRESSUf different frow Controlling Office) IS. SECURI TY CLASS. (of tsl report)

UNCLASSIFIED- - 15. DECL ASS, FICATION/ DOWNGRADING

SCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered In Sleek 20 It diffeent from Report)

IS. SUPPLEMENTARY NOTES

U. S. Army Research Office National Science Foundation-PP. 0. Box 12211 Washington, DC 20550

Research Triangle ParkNorth Carolina 27709

1S. KEY WORDS (Continuo on revere., side it necessar end Identity bl ock number)

infinite Dimensional Systems, Unbounded input Operators, Compensator,Retarded Functional Differential Equations, Partial Differential Equations,

A Boundary Control

*20. ABSTRACT (Continue an reverse aide It necessar and Identify by blook amber)

This paper contains a design procedure for constructing finitedimensional compensators for a class of infinite dimensional systems withunbounded input operators. Applications to retarded functional differentialsystems with delays in the input or the output variable and to partial

*differential equations with boundary input operators are discussed.

D F ONR7 1473 EDITION OF I NOV 65 IS OBSOLETC UNCLASSIFIEDSECURITY CLASSIFICATION Of THIS PAGE (Whent Data Ratare*

V%

w r w .]r w .r r r ' - v • - . "- -- ---.-,-"---. ..-. ,,-

rz w w . ' -" .. - " -. ' .: -,-' .-

44,.. .I . i " ... .- " :! * ." . ,, ..... .... . ,.,_ , _. ,.

.. ., .. .4 ..,

#'', 1.' - ,. . . ,,4, ;I . ., ,cJ,

'" - " I - :,. ' . .. ..4" . : "

o--