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On exponential stability of infinite dimensionallinear systems with bounded or unboundedperturbationsPeng Li a & N.U. Ahmed ba Department of Electrical Engineering, University of Ottawa, Ottawa, Ontario,Canadab Department of Electrical Engineering and Department of Mathematics, Universityof Ottawa, Ottawa, Ontario, CanadaVersion of record first published: 02 May 2007.

To cite this article: Peng Li & N.U. Ahmed (1988): On exponential stability of infinite dimensional linear systems withbounded or unbounded perturbations, Applicable Analysis: An International Journal, 30:1-3, 175-187

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dpplrtohii .Anoh i i . ~ Lo1 30. pp 175- I87 Rcprlnts d \a~iahle d~rectly from the pubilsher Photocopj~ng pernm~ttcd hq l~ ienre on15

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On Exponential Stability of infi- nite Dimensional Linear Systems with Bounded or Unbounded Perturbations Communicated by Z. Nashed

KEY WORDS: Infinite dimensional systems. stabilizability. exponential stability. semigroup, relative bounded perturbations, m- dissipativity, feedback control

( R e w i w / / b r Publication 14 April 1988)

Peng Li Department of Electlical Erlginee~ing, University of Ottawa, Ottawa, On- tario, Canada I i l N G N j

IU.U.11llrned Dcpartnlent of Electrical Engineering and Department of hlathematics, University of O t t ana , Ottawa. Ontalio, Canada I i l K 6x5

Al~st ract In thi, paper. n e stutlj the qi~estion of stal~ilization of uncertain systems governed 11) e ~ o l u t ~ o n ecjuat~ons in IIilbert space. It is shown that exponential stabllit?; can be acliieved h?; choice of suitable state feedback controls even in the presence of l~ountled or relatively bounded(uncertain) perturbations. Our proof is l~ascd on perturbation theory of semigroups. The results are illustrated 115 two examples involving heat equation and wave equation.

This work was supported in part I]?; tlw Natural Science and Engineering Research Council of Canada under Giant No.X7109.

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176 PENG LI AND N. U. AHMED

1. INTRODUCTION

The question of stabilization of uncertain linear systems described by ordinary differential equations,

i ( t ) = ( A + B ( r ) ) x ( t ) + Cu( t )

x (0) = so.

with deterministic feedback control has received considerable attention in cur- rent literature[l-61 where the state r € Rn, A is an n x i z matrix, B ( r ) is the

system uncertainty, C is a matrix of suitable dimension and u is the control. In many scientific and engineering problems the system is modelled by par-

tial differential equations, integral equations or coupled ordinary and partial

differential equations. \I7e know that a large class of linear delay differen- tial equations and partial differential equations can be modelled as abstract evolution equations of the form

k ( t ) = Ax( t ) + Cu( t )

.c (O) = ro

in an appropriate Banach space X.The system operator A may be bounded or

unbounded, the control operator C is bounded and u ( t ) is the control force. The uncertainty considered here is the perturbation of operator governing the

dynamics of the system. That is,

?( t ) = (il + B ( r ) ) z ( t ) + Cu( t )

x ( 0 ) = 5 0

where Bjr ) is a relatively bounded perturbation of A for each r in a suitable

set R; B ( r ) may be bounded or unbounded. The parameter r is not known but

the set R is known. The problem is to design a state feedback linear control

law that assures exponential stability of the zero state uniformly with respect

to r E R. To our knowledge, this problem, involving infinite dimensional

systems with bounded or unbounded perturbation, has not been considered in the literature before. D

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INFINITE DIMENSIONAL SYSTEM STABILITY

2. PROBLEM STATEMENT AND ASSUMPTIONS

Consider an abstract system

on a real Hilbert space X where A: D(A) + X is the infinitesimal generator of - a strongly continuous semigroup(Co- semigroup) T( t ) , t 2 0, on X, D(A) = X. C is a bounded linear operator from CJ into X, where U is another real Hilbert space (with dimension(U)finite or infinite), and { B ( r ) , r E R) is a family

of linear operators perturbing the generator A. We introduce the following assumptions for the operators A, B(r) , r E R and C.

Assum~tion 1

The pair {A, C ) is exponentially stabilizable; that is, there exists a Do E

L(X, U) such that 2 - A+ CDo generates an exponentially stable contraction semigroup Tf in X satisfying \/Tf)I < edwt, for some w > 0.

Assumpt.ion 3,

B(r) Bl(r) + B2(r) , where D(A) C D(Bl( r ) ) C X for all r E R, and for the given R, there exist constants CY E [O,1) and y 2 0 such that, for all r E R,

(x, BI(T)x) L a(x , Ax), for x E D(A)

Assum~tion 3 The linear operator CC* is coercive; that is,

( I , CC*x) 2 x ~ [ x I ( ~ , for some X > 0 and x E X,

where C* is the dual of the operator C.

In majority of engineering problems the stronger form of asymptotic sta- bility(i.e exponential asymptotic stability), is most desirable. In this paper D

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178 PENG LI AND N . U . AHMED

we are mainly concerned with this problem. In other words, we wish to find

an operator Dl E C ( X , U ) , giving the feedback control u = D l z , such that

( A + B ( r ) + CD,) is the generator of an exponmtially stable Co-se~nigrorlp in

X for all r E 0.

3. h GFXERAL STABILIZATION RESULT

Our main stability rcqult is given in the follo~r-ing theorem

Theorem 1.

conszder the system (1) tczth lznear feedback control gzven b y

I f the assurnpt~ons 1-3 are satzsfied, then for any znltcal condztzon TO E D ( A ) C X , the null-stclte x = 0 is e.rponent~ully stable unrformly wzth rfcpect to r E R

Some definit~ons and Lemmas useful for the p ~ o o f of the Thtorem are glvcn

below:

Definition 1.

A dissipatice operator 11 for which R(I - A) = S is called in-dissipntice.

It is clear that if A is dissipative so is pr l for all p > 0, therefore if A is

m-dissipatiae then R(XI - A ) = X for every X > 0.

Lemma l .(Paq[7; Theorem 3.2, p.811)

Let A and B be lznear operators In S such that D ( B ) > D ( A ) nnd A + tB 2s d~ss~patzve for 0 5 t 1 1 . I f IIBzIl I. a l /Ax / l + Jllxll for x E D ( A ) where

0 < cr < 1, ,O 2 0 a n d f o r some to E [0 , 11. A + t o B 1s m-dmzpatzce then :I+ t B

zs m-dzsszpat~ve for all t 6 [O, 11.

Lemma 2.

Let T and A be linear operators in X , D ( A ) C D ( T ) C S and D ( A ) = .Y. If ( x , T x ) 1 ( x , A x ) , x E D ( A ) , then l\Txll 1 llAxll, x E D ( A ) . D

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INFINITE DIMENSIONAL SYSTEM STABILITY 179

- Since D ( A ) = X it follows that for any 0' E -Y with l I r l t i l = 1, one can fiild a

sequence { T ~ ) c D ( A ) with llsnll = I , n = 1 , 2 . ... such that 77, 4 T ~ ' . S ~ n c e D ( A ) c D ( T ) C X , T x E X for x E D ( A ) and ( v n , T x ) 5 llAx11. taking the

limit we have

( ? ' , T x ) JIAzJl for x E D ( A ) .

Since 0' is an arbitrary element of X satisfying i/r1'1/ = 1, this inequality is

satisfied for all such 7'. Hence

/ ( T x j ( = sup ( q ' , T x ) 5 llAx// for x E D ( A ) . ll7'll=l

Thus if T is form-relatively hounded with respect to '1, that is

- ( x , T x ) < a ( x , A x ) + ?(a , x ) for x E D(A), U(A) = S

then T is operator-relatively bounded with respect to A , tha t is.

Therefore, assumption 2 implies that B ( r ) is relati~ely bounded with re-

spect to A for all r E 0.

Lemma 3. Le t A be t h e genera tor o f C o - s e m l g r o u p Tt. t 2 0, zn X . Then T t , t 2 0. r r

a n exponen tml l y stable contraction scrnzyroup rf and o n l y 1f zts generato? I S

s trzc t ly dzsszpahue, t ha t zs, (x, As) 5 - t ~ j j - c j ( ~ f o r s E D ( A ) . a n d R((1- r c ) I - A ) = X for s o m e w > 0 .

Proof: Applying Lumer-Phillips Theorem (Pazy[7; Theorem 4.3, p.141) to the linear D

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180 PENG LI AND N. U. AHMED

operator B = A + w I , the result follows upon noting that B generates the

Co-semigtuup St = eu"Tt of contractions given that {Tt , t 2 0) is the Co- semigroup generated by A if and only if B is rn-dissipatiue. rn

IVitll feedback control (2) and subject to the assumptions 1-3, the system

(1) becomes

nllerc Ck B l ( r ) + n C D o . C l f B2(r ) - (*I/X)CC* ale linear operators in S. LVith these preparations we can now give a proof of our main result.

Proof of Theorem 1:

In ordel to prove the Theorem, one must show that (a) gi\en any initial condition x(0) = s o E D(A), there is a t least one classical

solution of (3).

(b) A + C, is the generator of an exponentially stable contraction semigroup with exponent independent of r E R. For (a) i t suffices to show that A -+ C, generates a Co-semigroup for every

r E n. This, however, follows from (b). Hence we concentrate on proving (b).

Since C and Do are bounded operators, ~ ( 2 ) = D ( A $ CDo) = D(A) and - ~ ( 2 ) are both in X and D ( A ) = X. By assumption 1 and Lemma 3, we know

that 2 is strictly dissipative satisfying

Define A, = A + w I , then by definition A, is rn-dissipatice and D(C,) > D(A,). LVe prove that A, + tCr is dissipative for 0 5 t 5 1. Indeed by assumptions 2 and 3

< t a ( x , Ax) + tcu(x, CDox) t ty(x, x) - t(y/X)(x, CC*x)

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INFINITE DIMENSIONAL SYSTEM STABILITY

Hence A, + C, is dissipative and A + Cr is strictly dissipative.

Utilizing assumption 2, we obtain

= a ( x , Ax)

5 a ( x , A , x ) for x E D ( A , ) = D ( A ) ,

and hence it follows from Lemma 2 that

Since C E L(U, X) it follows from assumption 2 that there exists a constant

P = P(7, A, IlCll) > 0 such that

As A, is rn-dissipative and CI is relatively bounded, it follows from Lemma

1 that A, + Cr is m-dissipative uniformly with respect to r 6 R. Thus by

Lemma 3 A + C, is the generator of an exponentially stable contration semi-

group with exponent w which is indepent of r € 0. This completes the proof

of Theorem 1. W

We have shown in Theorem 1 that if the perturbing operator B ( r ) , r E

R, in system (I), is uniformly form-relatively bounded then the system is

exponentially stable. Next we show that if B ( r ) , r E R, is operator-relatively

bounded then the same result follows under certain additional assumptions.

For this purpose we shall use the following Lemma.

Lemma 4.(I<ato[9; Theorem 4.12, p.2921) Let A be selfadjoint and nonnegative and T symmetn'c with D(T) > D(A) and

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182 PENG L1 AND N. U. AHMED

Then

( T u , u ) < ( A u , u ) for u E D ( A )

iVe replace the assumption 2 by

Assumption 2'

B ( r ) r B l ( r ) + B z ( r ) , where D ( A ) C D ( B l ( r ) ) C X for all r E a, B l ( r ) is

symmetric and for the given R, there exist a E [ O , l ) and y 2 0 such that

IIB(r)x1I < CY\IAXI( + Y / ( x ~ / for x E D ( A ) and for all r E 0.

Under the assumptions 1, 2' and 3 it is easy to prove the following result.

Corollan 1.

Conszder the system (1) where A zs a selfadjoznt posztzve definzte h e a r operator

wzth feedback control zi = D l x where D l zs as dejined 211 equatzon (2). If the

assumptions 1 , 2 ' and 3 are satzsjied then x = 0 zs exponentzally stable.

In case of bounded perturbation, B ( r ) = B 2 ( r ) , we have the following

result.

Corollarv 2.

Conszder the system (1) and suppose the assumptrons 1 and 3 hold wzth the

ptrturbzng operator B(r) bounded such that lJB(r) j j 5 7 for a l l r E R. Then

wzth control u = ( D O - ( y /X )C t ) x , the system (1 ) is exponentrally stable wzth

respect t o the null state.

4. FINITE DIhIEIiSIOn'.4L COKTROL

In practical situations, one usually considers the system

i ( t ) = ( A + B ( r ) ) x ( t ) + C u ( t )

= ( A + B ( r ) ) x ( t ) + C:,c,u,(t), x ( 0 ) = xo E D ( A ) ( 5 )

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INFINITE DIMENSIONAL SYSTEM STABILITY 183

where c, E X, u, E L?'(O, w). This is a special case of (1) corresponding

to finite dimensional control action, u E Rm. Under the assumptions 1-3,

Theorem 1 is also true for the system ( 5 ) . In this case, C : Rm i X. Let us

consider the system with a different constraint on C. Instead of CC' being a

coercive operator we may assume that the following condition holds:

Assumption 2"

B ( r ) = B1 ( r ) + B % ( r ) , where D ( A ) C D ( B ( r ) ) c X for all r E 0, and there

exist a E [0, l) , yl > 0 such that for all r E f2

(x, B1 ( r ) ~ ) 5 Q ( X , A x ) for x E D ( A ) ,

and Ba(r ) _< l l C C i for some yl > 0 and all r E R.

Theorem 2.

Consider the system (5) with linear feedback control given by

and D o zs as gzaen zn ccssumptzon 1. If the assumptzons 1 and 2 " are sa t~$ied ,

then for any admzsszble uncertaznty r E Cl , and any znitzal condztzon xo E

D ( A ) c X , the null state x = 0 zs exponentzally stable.

Proof: Identifying A = A 4 C D o and C: = B l ( r ) + a C D o it is clear from the proof of

Theorem 1 that A+c: is strictly dissipative satisfying ( ( ~ + C : ) X , s) .< - w ~ l z ~ ~ 2

for all r E fl and x E D ( A ) . Therefore it suffices to verify that the remaining

part C: r B Z ( r ) - r l C C i i s bounded and dissipative. Clearly C: is bounded

and it follows from assumption 2" that C: is dissipative. . In fact, Theorem 2 can also be applied to infinite dimensional control

problem where C E L ( U , X ) and u E U . See example 2.

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PENG LI AND N. U. AHMED

E x a m ~ l e 1.

Consider the controlled diffusion equation

z(0 , t ) = Z(1,t) = 0,

z(x, 0) = sin x x .

LVhere r is the uncertain parameter taking values from Q {[ E R : - N < - E 5 w.

iVe write this in the canonical form as follows:

t = Az + B ( r ) z + CU, z(0) = sin x x and r 6 R

with the state space X = L2(0, 1). The operator A is given by

a? a2 z A = - D(A) = {z E L2(0, 1); - E L2(0, 1) and z(0 , t ) = z (1 , t ) = 01,

8x2 ' 8x2

the operator B ( r ) is given by

B( r )z = r 2 z and C = Identity.

It is easy to verify that the assumptions 1-3 of Theorem 1 are satisfied. In

fact, the Co-semigroup T ( t ) generated by A satisfies

thus is exponentially stable (Curtain and Pritchard[lO; p.2151). Hence, in this

case we can take Do = 0. The simulation result is shown when N = 111 =3.3 for R.

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INFINITE DIMENSIONAL SYSTEM STABILITY

CONTROL

FIGURE 1 Diffusion equation (example 1)

Example 2. While considering distributed controls for vibrating systems it is required that every initial state of finite energy be transferred to a position of rest (zero state).

Consider the scalar wave equation [ll]

with initial state given by

2 z (s , 0) = x - x, x E (0 , l ) and zi(z,O) = s insx , x E (0 , l ) .

We may associate with this equation an abstract evolution equation of the form

t = Az + B(r)z + Cu, z(0) =

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186 PENG LI AND N . U. AHMED

where r E fl = {r E R : -N 5 r 5 M}, 0 5 ,V, 12f < w. In this case the state

space is given by X = lI1(O. 1) x L2(Ol l) , with D ( I ) = H2(0, 1) x H i ( 0 . 1 ) ~

A = (A A) , B(r) = (: , and C = (0, l ) ' .

The operator A generates a Co-semigroup T ( t ) and there exists Do =

(-1, -5) such that A + CDo generates an exponentially stable contraction

semigroup. Further B( r ) = CE(r)C* where E( r ) = r 2 , C * = (0, l ) . IIence,

the assumptions 1 and 2" are satisfied. For r2 5 8.05, the simulation result is

shown in the Fig.2. for N = 2 . 3 and ,I1 =2.5.

For numerical computation we have used finite difference method and the IMSL library subroutine.

FIGURE 2 Wave equation (example 2) Dow

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INFINITE DIMENSIONAL SYSTEM STABILITY

REFERENCES

[ l ] G.Leitmann, Guaranteed asymptotic stability for some linear systems with bounded uncertainties ,.Journal of Dynaniic Systems, Measurement and Control, VOL.lO1, 1\70.3,1979.

[2] G.Leitmann, On the efficacy of nonlinear control in uncertain linear sys- tems, Journal of Dynamic Systems, i\leasureinent and Control, VOL.102, N0.2,lgSl.

[3] B.R.Barmish and G.Leitmann, On ultimate boundedness control of unccr- tain systems in the absence of matching conditions,IE'EE Transactions on Automatic Control, VOL.AC-27,1982.

[4] B.R.Barmis11,I.R.Petersen and AFeuer, Linear ultimate boundedness con- trol of uncertain dynamical systems. Autornatica, L70L.19,19S3.

[J] J.R.Petersen, Structural stabilization of uncertain systems: necessity of the matching condition, SMI\I J. Confrol and Optim., L'OL.23,N0.2,1983.

(61 P.Li and N.U.Ahmed, Ol~servcr based stabilizing controllers for uncertain system with incomplete state information, Submitted.

[7] A.Pazy, Semigroups of Linear Opernfors and Applications to Partial Dif- ferential Equations, Springer-I,'erlag, 1983.

[8] G.R.Buis, Lyapunov stability for partial differential equations, Part I, N A S A Contractor Report, CR-1100, \Vashington, D.C., 1968.

[9] T.Kato, Perturbation Theory for Linear Operuiors, Springer -Lrerlag, 1980. [lo] R.F.Curtain and X.J.Pritchard, Functional Annlysis in Aloderii Applied

Mafhematics, Academic Press, 1977. [ l l ] A.T.Bharucha-Reid, Randorrl Integrnl Equations, Acadernic Press, 1972.

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