On Spectrum and Riesz Basis Assignment of Infinite-Dimensional Linear Systems by Bounded Linear...

SIAM J. CONTROL AND OPTIMIZATIONVol. 34, No. 2, pp. 521-541, March 1996

() 1996 Society for Industrial and Applied Mathematics0O7

ON SPECTRUM AND RIESZ BASIS ASSIGNMENT OFINFINITE-DIMENSIONAL LINEAR SYSTEMS BY BOUNDED

LINEAR FEEDBACKS*

CHENG-ZHONG XU AND GAUTHIER SALLETtAbstract. This paper deals with spectrum and Riesz basis assignability of infinite-dimensional

linear systems via bounded linear feedbacks. The necessary and sufficient condition of Sun [SIAM J.Control Optim., 19 (1981), pp. 730-743] is generalized to a large class of boundary control systems.Two typical examples are presented to illustrate the application of our results. The results obtainedin this paper may have potential applications in nondissipative spectral systems.

Key words, distributed parameter systems, bounded linear feedback control perturbation,spectral determination, stability, flexible structures

AMS subject classifications. 93C20, 93D15, 93B60, 93B55

1. Introduction. In this paper, we consider directly the following linear evolu-tion systems on a separable Hilbert space H (the inner product and induced norm inS are denoted by (., .) and I1" II, respectively):

(Eo) (t) AX(t) + bu(t),

where A is the infinitesimal generator of a C0-semigroup on H and the input elementb is not necessarily admissible in the sense of [7]. Throughout the paper, A and b areassumed to satisfy the conditions H1, H2, and H3.

Hypothesis H1. The operator A has compact resolvent. We suppose that thespectrum a(A) {n, n E IN} is simple.

Hypothesis H2. The domain 7:)(A*) of the adjoint operator A* is a Hilbert spacewith the graph norm. I)’(A*) is the topological dual of :D(A*). We suppose that bbelongs to T)’ (A*).

Hypothesis H3. The eigenvectors {k; k E IN} of A form a Riesz basis in H.The biorthogonal sequence correspondingto the eigenvectors of A* is denoted by{k; k IN} and is also a Riesz basis of H [5, p. 310]. We set bk (k,b), where(., .) is the classical duality product on (A*) x 7)’(A*). Here, it is defined as thecontinuous extension of the inner product on H (H is dense in 7:)’(A*)). We supposethat bk = 0 for all k IN. Let dn be the distance of {An} to the rest of the spectruma(A). We consider the set of disks Dn, n lN centered at {An} with radius 1/2dn. Nowsuppose that there exists a positive constant M such that for all (Jje Dj andall m IN,

+ bn 2

(1) E ) /nn----1

and

n=l,nm

bn < M < +c.

Received by the editors June 23, 1993; accepted for publication (in revised form) November 15,1994.

Institut National de Recherche en Informatique et en Automatique-Lorraine (Projet CONGE),Centre National de la Recherche Scientifique Unit de Recherche Associe 399, CESCOM, 4 rueMarconi, 57070 Metz, France (xu@+/-lm. loria, fr, salletilm, loria, fr).

521

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

522 CHENG-ZHONG XU AND GAUTHIER SALLET

The assumptions H2 and H3 allow us to take into account the cases of boundarycontrol because many linear distributed parameter systems can be formulated in theform of (Eo) (see [7] and also [19], [15], [16]). In particular, the cases of [21], thecantilever beam equation with lateral force control [15], [24], or moment force control[15], heat conduction equations [7], and the wave equation [18] enter the class of thesystems considered here. For each fixed A E p(A*), the resolvent set of the adjointoperator A*, and all r E H, it follows from the hypothesis H3 that

The condition (1) forces the constant K to be bounded uniformly with respect to/k [Jye D (see 3). In certain cases, the condition (2) of H3 implies the condition(1). See the examples in 2.

A closed linear operator A (A) H is called regular spectrM if its resolventis compact and its eigenvectors form a Riesz basis of H [20]. Sun has proved in [21]that under the hypothesis H1, with b H, and a stronger hypothesis than H3, thefollowing condition is necessary and sufficient for the operator A + b(., h} (h H) tobe regular spectral and to have the spectrum {v; k } assigned:

(3) v-bk

More results on spectrum assignment via linear feedback at the boundary have beenobtained in [6], [12], [11], [15], and [16]. Notably, Rebarber has shown that for somecases, it is possible to assign uniformly an infinite number of eigenvalues by unboundedbut admissible linear feedback at the boundary [16]. In [11], Lasiec and iggianihave given fine sufficient conditions on a(A) and b such that the operator A+ b(., h)Uis regular spectral. In [12], Liu has generalized Sun’s condition to the class of systemsfor which the hypotheses H1-H3 are satisfied, with the following condition replacingthose of (1) and (2):

(4) minf >_ I1 and

for some N and > 0. However the latter condition is restrictive in the sensethat it singles out the one-dimensional wave equation and cantilever beam equationwith moment control. The aim of this paper is to expand the result of Sun to a moregeneral class of systems satisfying our conditions HI-Ha. It is clear that the condition(4) implies the conditions (1) and (2) of the sumption

In this paper, we do not restrict our study to he ce of admissible input elements[6], but consider input elements in ’(A*). However, we do restrict our study to thecase of bounded linear feedbacks (BLF): (t) {z(t),h} with h e H. We provethat under the hypotheses HI-Hg, the condition (a) is also a necessary and sucientcondition for spectrum and Ries basis assignment. On he one hand, after each BL,the controlled operator A + b{., h} is still regular spectral. On the other hand, givena set of points satisfying the condition (a), we can compute explicitly the BL whichrealizes he spectrum assignment. The main difference between the work of this paperand that of [6], [12], and [16] is that our condition (heorem 1) is not only sumcient,but also necessary. The necessary part of the condition may find applications incontroller design for infinite-dimensional linear systems (cf., [14] and [9]). Our results

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

SPECTRUM AND RIESZ BASIS ASSIGNMENT 523

also allow us to explain why the system (Eo) cannot be exponentially stabilized bythe BLF laws when it has an infinite number of eigenvalues in e(s) >_ 0 and itsinput vector b is admissible in the sense of [7]. However we prove that in some cases,the uniform assignment of the spectrum can be achieved by BLFs. We should pointout that the construction of BLF laws is simple and systematic as illustrated by ourexamples.

The paper is organized as follows. In 2, we present our main result and twotypical examples. Section 3 is devoted to the proof of our main theorem. The lastsection contains our conclusions.

2. Main results. As only BLF laws are considered in the paper, the closed-loopsystem is governed by the evolution equation (Ec) in the phase space H

(Ec)" (t) AX(t) + b(X(t), hI.The linear operator A" I)(A) ---. H C :D’(A*) admits the unique extension

e (H, T’(A*))

by continuity because T(A) is dense in H. Accordingly, the linear operator

Ah A + b(., hl I)(A --, )’(A*)

admits a unique extension from H to :D(A*), still denoted by Ah, and for all x E H,Ahx x + b(x, h).

Define now I)(Ah) {x H;x+ b(x, h) H}. We use here the same definitionfor the unbounded linear operator Ah l)(Ah) ---, H as that of [16]. In the following,instead of directly dealing with Ah, we study the unbounded linear operator LhA* + h(., b) because the infinitesimal generation property is equivalent between Ahand Lh if they are adjoint w.r.t, each other. It is easy to see that (Lh) :D(A*)from the hypothesis H2 and that Lh is closed because h(., b) is A*-compact [8, p. 194].

LEMMA 1. The unbounded linear operator Ah T)(Ah) H is the adjointoperator of Lh with the inner product (.,.I on H.

Proof. First, let us prove that for all x H and y :D(A*2),

(5) (y, x) (A’y, x).

Given all x e T(A) and y e :D(A*2), we have

(y, fix) (y, Ax) (y, Ax) (A’y, x) (A’y, x).

Since the domain :D(A) is dense in H and both the injection H T’(A*) and theoperator are continuous from H to T’(A*), the equality (5) is true for all x Hand y T(A*).

Now, for all x e :D(L,) and y e T(A*),

(y, Ahx) (y,x + b(x, hl) (y,x) + (h,x)(y,b)(y, fix) + (h(y, b), x (A*y + h(y, b), x)(Lhy, x) (Lhy, x) (y, nx) (y, nx).

With )(A*) dense in (A*), Ahx L,x. This means that Ah D L.

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


On the other hand, it follows from the equality (5) that for all x E 7)(Ah) andall y e T)(A*2), (y, Ahx) (y, Ahx) (Lhy, x). Since A* is the generator of a C0-semigroup on H, for each y e :D(A*), there is a sequence Yn T)(A.2) such that

Yn -- Y and Lhyn -- Lhy in H as n -- +co. This means exactly that x T(L) andAhx Lx. Hence Ah C L. Therefore Ah L.

THEOREM 1. Assume that the hypotheses H1-H3 are satisfied. Then,1. for every h H, the feedback controlled operator Ah is regular spectral and

the spectrum a(Ah) {vk, k IN} satisfies the condition (3);2. given a set A {vk, k IN} such that vj vk for j k, there exists an

h H for the operator Ah to have a(Ah) A if and only if the set satisfiesthe condition (3). Moreover the feedback is given by

/ 1-In=l,nCj

where hj denotes the complex conjugate of hj for j IN.We should understand that the infinite product in the theorem is the limit of the

sequence in 12

n---1,njj--l,2

that is,

lim EIh-J12=0"N--*/x

We will remark that for any set A {vk,k IN} assignable by bounded linearfeedback, there exists necessarily some integer N such that for all k, j > N andk j, vk vj. The detailed proof and discussion of this result will be given in thenext section. The main idea is to prove that Lh, considered perturbation of A*, isregular spectral, and that from some rank, the eigenvalues of Lh cn be located in thedisks centered at the eigenvMues of A* with radius 6]b,hn]. Moreover we show thatthe corresponding eigenvectors of Lh form a Riesz bis in H. Then the same resultis true for the adjoint operator Ah. It follows from [3] that the controlled operatorsAh and Lh are the generators of C0-semigroups on H. Here we give only two typicalexamples to illustrate the application of Theorem 1.

Example 1. The wave equation

utt(x, t) ux(x, t), u(0, t) 0, ux(1, t) r(t)

with the boundary control F(t). Define the Hilbert spaces H W x L2[0, 1] andW {f; f, f L2[0, 1], f(0) 0} with the inner product

[f2]fl [gl ]} :1 [flx(X)glx(x)Tf2(x)g2(x)]dx.g2 H

Using the techniques of [7], we can formally write the control system on the Hilbertspace H

(t) A(t) + br(t).

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


The semigroup generator A T(A) H is skew-adjoint with- 0

and

:D(A) { (fl, f2) e L2[0, i] x L2[0, 1], fl, flz, fizz e L2[0, 1], f2 e W,f(0) =/1() 0}

and b(x) [0, 6(1 x)]’. By direct computation one can find the following resultsLh=A=-A+h(.,b).The spectrum a(-A) { i(k + 1/2),k 0, 1,...} and the corre-sponding eigenvectors

[ 1 ]isin[(k+l/2)x]k(x)= --A 2(k+1/2)k=0,1,

Since the operator A is skew-adjoint and its resolvents are compact, the elements{} form an orthogonal basis of H. It is evident that bk (--1)k/2, k 0, 1,For all j 0, we have

+ ,1 12 1 2 + 1 + 1j ’"A’n + Aj-A-n (j-n)2r2+

n=O,nej n=l n=O,nej(j + n + 1)2r2

+ 1j--1

1 + 1 + 1 1

(j-n)22+ + =-.n=j+l

(J- n)22 (J + n + 1)2x23 k= k2x2 2

The sme result is true for j 0. Hence the condition (2) is satisfied. We showthat in this ce the condition (2) does imply the condition (1). Indeed for each+A Oj=o Dj, there is an integer mo such that

() e [(o),(o+)]because the distance dj {m(Aj)- m(Aj+l)l is greater than some constant. Thenfor j mo + 2,

I- 1 ()+ [m() -(o + + /e) + (o + + /)- (j + /e)]e() + (j -mo 1) e() + I o+1.

Forj mo- 1,- ()+ [m()- (too + 1/e) + (o + 1/)- (j + 1/e)]> e() + (o -j) e() +o 1.

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


According to Theorem 1, the spectrum assignable by continuous feedback mustsatisfy the condition

I ( + /)1 + I- +( +/) < +.n=0

Therefore the best stability result achievable by continuous feedback is strong stabil-ity. However we know that the unbounded feedback F(t) -ut(1, t) exponemiallystabilizes the system [10], [18]. Moreover the resulting operator is still regular spectrM

Example 2. Consider the nilbert spce H W[0, 1] n2[0, 1] with

and the inner product

g2 H

The cantilever beam equation with the moment force control can be formally writtenfollows [15]

(t) A(t)+ r(t),where the operator A (A) H is skew-adjoint with

f]=[O 1- o I]and

/)(A) { (fl, f2); fl e W[0, 1], f2 e W[0, 1], flxxx(1) flxx(1) 0},and b(x) [0, ’(1 x)]’.

The spectrum of-A is

a(-A) {A+k +i[kr + r/2 + O(e-k)] 2, k 1,...}.One may find two positive constants M1 and M2 such that bmn= mn with Mm M2 for n e (see [15], [19], and also [24]). The eigenvectors of A form aRiesz basis of H. Without loss of generality, we consider only the case where j 1.Then we have

I .1 [(J n)r + O(e-j) + O(e-n)]2 x [(j + n) + O(e-’) + O(-n’)]2

and

]Aj A_I2 {[jr + /2 + O(e-J)]2 + [n + /2 + O(e-n)]2}2

Then we get the following inequality for some fixed number 2:

j--1 n2 +m 1 / /17/2rr2n= (J-n)2(n+j)2+nl-Y/ < ’2

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


As in Example 1, the reader can verify that in this case the condition (2) also impliesthe condition (1). Following Theorem 1, all set A satisfying

n----c,nO

2

can be assigned for the spectrum of the operator Ah by continuous feedback. This iswhy it is possible to assign the spectrum uniformly by continuous feedback [15]. Herethe feedback is simple as given in Theorem 1. For instance, the point set A {v+-np + A+,p < 1/2,n 1,2,...} can be assigned for the spectrum of the controlledoperator Ah via the continuous feedback of Theorem 1. The resulting semigroup etAb

is exponentially stable. In particular, taking A {v+n -o2 -[-- ):t:n, 0 O E ]P, n1, 2,...}, we get the controlled semigroup etAh satisfying

lietA II (H) _< Me-ta2

for some positive constant Ms (depending on the constant a), where the decay rate isarbitrarily fast by increasing the number c2. The feedback that realizes the spectrumassignment is

3. Proof of Theorem 1. To simplify the presentation, we introduce the fol-lowing notation. Define the bounded linear functional 9v E :Dt(A*) such that forall g e T(A*), 9V(g) (g, b). For all e p(A*), the resolvent set of the operatorA*, define the characteristic function Fh()) 1- (R(, A*)h), where h H andR(A,A*) (- A*) -1. For each A0 e p(A*), the linear functional $" o R(A0, A*) e(H, ). The complex function Fh()) is analytic on the resolvent set p(A*) (see [6]for a proof). We set the perturbation operator T h9v which is A* compact [8,p. 194]. Then the operator A* + T is closed with T(A* + T) :D(A*). The followingresult can be proved by direct computation.

LEMMA 2. For all ik p(A*) such that Fh()) O, we have

R(A,A* + T) R(A,A*) + R(I,A*)TR(A,A*)/Fh(A)

and p(A* + T). Moreover the perturbed operator A* + T has compact resolvents.From this lemma, we know that the spectrum a(A* + T) consists entirely of

isolated eigenvalues with finite multiplicity [8, p. 187]. Now consider the set of disksbj j IN, centered at {j} with radius -d. It is evident that /j /t 0 for3 3"

j 1. Technically we suppose that [.Jje Dj implies that its complex conjugate- jIN Dj. This assumption is minor because in applications the spectrum a(A)is usually symmetric with respect to the real axis in the complex plane. Otherwise itis sufficient to rewrite the condition (1).

LEMMA 3. For each h =1 hyCj, there exists a positive number R1 > 0 suchthat the subset in the complex plane

,s’ < R,} U Uj=l

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


contains all the spectrum points a(A* + T).Proof. To prove this lemma, it is sufficient to verify that Fh(A) 7 0 for all A S

(see Lemma 2). From the hypotheses H1 and H2, for each h E H and all

+ hnbn(8) JZ(R(A’A*)h) Z -n"n--1

Since (n +)n=l is a Riesz basis of H, the following is true for some numbers M1,M>O:(9) M2 Z IhJl -< Ilhll/-< M22 Z

jElN

It follows directly from (8) that for all A I,Jj=

(10) IU(R(A’A*)h)I -- 2 ]-_ nl-t- 2 Ihnl 2 - ’"’n=l n>_N1+l n>_N1+l

Using the condition (1) of the hypothesis H3, we can choose a large integer N1 suchthat

(11) [hnl2 Z bn 2 1

nN+l nN+l -- n nN+l

(where we have used the fact that A Uje implies that Uye J) and thenchoose a positive number R1 large enough so that for all

N IhbnI < 1(12) IX-n=l

It follows from the conditions (10)-(12) that for all A S,1

](R(A, A*)h)[ <

that is, Ih()l This proves that a(A* + T) C S.We let u(A, A*) and u(A, A* + T) denote the algebraic multiplicities of A eigen-

value of A* and A* + T, respectively, and n(A) denote the order of A zero of thecharacteristic function Fh(A). (The order of A as pole of the characteristic functionFh(A) counts as negative and u(A,A*) counts zero if A e p(A*).) In [12], Liu hproved the following result.

PROPOSITION 1. For all A in the complex plane,

(a) (,A* + T) (a,A*) + ().LEMMA 4. For each h H, there ezists an integer N such that the infinite

pa of the spectm points {, n N1 } of A* + T are simple and the coespondingeigen+ectors are given by

(14) Cn bnFh(-’)

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


R1

FIG. 1. Illustrative distribution of the spectrum.

Moreover the whole spectrum of A* + T satisfies the condition

Vn nb"n < -00.

n--1

Proof. Consider the disk D(0, R1) centered at zero with the radius R1 defined inLemma 3 such that A E D(0, R). For all A E Dj,

-Xlland

Since lim#+ JAy +, only a finite number No of disks y intersect the disk

D(0, R1). Take the boundary C {; IA- l d/3} of the disk D. Define theclosed curve 0 by the boundary of the union

j=l

(as indicated in Fig. 1). om the proof of Lemma 3, we know that

1sup I(R(A,A*)h)I .D

ownl

oade

d 04

/23/

13 to

142

.51.

1.21

2. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


From the hypothesis H1, the function Fh(A) has at most No poles in the domainApplying the Rouch theorem on the functions g() 1 and Fh(A) allows us to saythat Fh(A) has the same number of zeros as poles in , for 1 --Fh(A) .T’(R(A,A*)h).Then from the identity (13),

E (’ A* + T) No.

That means that the operator A* + T has No eigenvalues (multiplicity counted) inthe region f. By construction, it is also true that for all j > No,

1sup [9(R(A, A*)h)l <e 3"

The function Fh(A) has either one pole A--j or no pole in/)j. Applying the Rouchtheorem on the functions g(A) 1 and Fh(A) allows us to say that Fh(A) has thesame number of zeros as poles in the disk Dj. Then from the identity of Liu (13), theoperator A* / T has a simple eigenvalue in Dj for j > No. Moreover this eigenvalueis either {} or the unique zero v-j = j of Fh(A) in

Actually the simple eigenvalue is situated in a smaller disk contained in j.Because

limIhjbj O,

’72:we can take some N1 > No sufficiently large such that for any A withand any integer n,

(15) sup supAn-Aj

sup sup1 1 3),_ i< supsup 6lhbl <-"

--j>_N1 nj 1- 2

It follows from the definition (8) that for all j _> N1 and all ]A AI 61h b l,m.

Ih.b.I +I(R(A,A*)h)I<_n’- n:fij

g:[hnbnl

n=l,n#j

N 31h,b.I +n=l,nj

Take an integer N2 so large that

I_hnb,l +n>Na,j IA

31h,,b l +n>N2 ,nyj

31h b l <n>N2,nj

Then we can always choose the integer N1 > No so large that for all j >_ N1,

1

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


Therefore there is some integer N1 such that for all j >_ N1 and all [A- Ajl 61hjbjl,

2I.T’(R(,k,A*)h)I <

-3

Applying again the Rouch theorem on the functions g(A) 1 and Fh(,k) and reason-ing with the identity (13) as above allows us to prove that Ivy Ayl <- 61hjbjl for allj >_ N1. It follows from the above that

2 N1--1

n--1

Vn ,n "[2bn -- E 361hjl2

n_N

Now, we compute the corresponding eigenvectors of the operator A* + T:

(16)

Observe that for any eigenvector y, -(y) :/: 0. Suppose that 9(y) 0 for some

j. The only solution of the above eigenvalue equation is vj )j and j Cj. Thisimplies that $’(y) 0, which contradicts the hypothesis H3. Setting

(17)

m=l

h-- E hm)m,m--I

we prove that vj ,kj in the eigenvalue equation, that is, ,kj E a(A* + T) if and onlyif hy O. Substituting the expression (17) into the equation (16) allows us to obtainthe following:

(18) (j ,km)O, hm.T’(j), m 1, 2,

It is evident that hy 0 if vy Aj. Suppose that hj 0. Then the characteristicfunction

is analytic at the point A Ay. This implies that the order n(Ay) of the point A Aj aszero of the function Fh (A) is greater than or equal to zero. It follows from Proposition 1that u(Ay, A* + T) >_ 1, or j Aj. In particular, for all j >_ N1, u()j, A* + T) 1.Now we are interested in the eigenvalue equation only for j >_ N1. For hj O, weknow from the above that vj )j. Then direct computation from (18) leads to

v--) )j hm_

For hj 0 and j >_ N1, vj )y and Fh(,ky) # 0 because u()j, A* + T) 1. One canfind that

bj

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


This finishes the proof of Lemma 4. rl

LEMMA 5. For some integer N2 >_ N1, the sequence {1,..., )N2, )N2+l,"" "}forms a Riesz basis of H.

Proof. Define the linear application A H H by A(i) Cj for 1 _< j _< N2and A(i) i for j _> N2 + 1. We will prove that the application A as well as itsinverse A- are bounded. Then the above sequence is also a Riesz basis because it isequivalent to the Riesz basis {i,J e ]hi} (see [5, p. 309]).

+For all g -j= OZj)j E H, using Lemma 4, we get

i= i=N=+ j= i=N=+l m E lNmTj

g + AA(g),

hmd2m

where

vi-Ai if hi 7 0,

Fh(j)’ if h 0,

and

For hi 0, the function $’(R(A, A*)h) is analytic in/)i" Then

1sup I.(R(A,A*)h)I <_ sup I.f(R(A,A*)h)l <_ -.,xeb e0

This implies that IF ( )I >_ 2/3 for all , e bj. in particular, >_ 2/3. From(19), for all j _> N2,

(20) _<

Using the fact (9) that {i} is a Riesz basis, we can obtain the following estimates:

IlzXA( )ll _<

i=N=+l

hm

hmVj Jm

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


Let us prove that for some integer N2 > N1,

2

M j=

From the conditions (20) and (15), choose an N3 with N2 >_ N1 such that

M2

j-N2+I mj,m>N3 j>N2 ,j Crn

(22) N M-- E Ihlm>_N3-t-1 j>N2 ,jym

36)j )m

2

Since from the hypothesis (2)

lim E Ihml2 EN2----*+xm--1 j>N2,jm

bj 2

we can always choose the N2 _> N1 so large that

M22 + Na

M :i= mCj,m-I

hmvy )m Ihml2

j>N2,jCm

(1)2

Aj Am 12Vj Am

Na(23) < E Ihml2 E

m=l j :> N2 ,jCm36 (3)

2

(1)2

5 -< g

Substituting (22) and (23) into (21) proves that for some integer N2 >2/3. Therefore the linear operator A and its inverse are bounded. It follows from [5,p. 309] that the sequence {1,..., N2, j,j >_ N2 / 1} is a Riesz basis of g.

Now let us prove Theorem 1.

Proof of Theorem 1. We take the linearly independent elements {bl,..., bg},which are the generalized eigenvectors of the operator A* / T corresponding to thefinite set {gl,..., gnu} of the spectrum a(A* + T) in the following sense. Withoutloss of generality, we suppose that in the set {gl,..., N2 } there are s distinct eigen-values {1,..., s} with the respective algebraic multiplicities {u,... ,us} such that

j= uj N2. Then

Ker(l A* T)1 Span{j, j 1, 2,..., ul}

Ker(g2 A* T)": Span{j, j =/"1 -- 1,...,/]1 --/"2},

and so on. We want to prove that the sequence {@,j E ]hi} is still a Riesz basis of H.Two sequences of vectors {gj, j E IN} and {fj, j ]hi} are said to be quadratically closeif .ier Ilg.i fj ]12 < +oc. It is obvious that the two sequences {,..., CN., j, j >

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


N2} and {j,j e ]hi} are quadratically close to each other. A sequence of almostnormalized vectors {gj } is said to be w-linearily independent if the equality

Ecjgj=O for E[c.[j=l j=l

< +oe

implies that c 0 for all j >_ 1. The Bari theorem [5, Thm. 2.3, p. 317] says that asequence of w-linearily independent vectors quadratically close to a Riesz basis is alsoa Riesz basis. Since we have already shown that the sequence {1,... ,g.,(bj,j >N2+ 1 } is a Riesz basis (Lemma 5), we now need only prove the w-linear independence.

The Laurent series of the resolvent R(A, A* + T) at {1,..., 8} takes the form[8, p. 181]

(24) R(,, A* + T) , rm=l ]D (,, A*n--1 (’ m nA-1 "4- RO "3

t- T

where the operator Pm is the projector on the subspace Ker(m A* T)/2" and Dmis the nilpotent commuting with Pm for m 1, 2,... s and R0(X, A* -4- T) is analyticat v-., j 1,..., s (see [4, p. 2292]). Let {c}__ e and ’=+~ 0. Write

Applying the resolvent operator on the two sides of the last identity, we obtain

N +x(26) R(A,A*+T) Eajy=- E ,k-j

j--1 j--N2+I

Since P,P, 5,,,Pn, PnDn DnPn D, and Ro(X,A* + T)Pr 0 for alle p(A* + T) [8, p. 1811,

N2 grn"4" E ( m)n-tl E OZjj(27) R(,, A* + T)EaOj ,k mj--1 m=l n--1 j=l

Substituting (27) into (26) leads to

(28)

Since the controlled operator A* + T has compact resolvents, its spectrum has noaccumulation point different from o, that is to say, limy_.+ Ivj] +o. So there isa positive integer 5 such that for all 1 < k < s and all j > N2,

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


On the other hand, the sequence (j,j > N2} is a part of the Riesz basis in Lemma 5.This implies that for all 1 _< k _< s,

2 +o

j>N2

Thus the right side of (28) is bounded on some domain containing 1. By multiplyingthe two sides by (- 1)j, j vl,Vl 1,..., 1 and taking the limit for A 1, weget, successively,

(29) D cuej=0, m=v-l,...,1

and

(3o)j=l

Since the elements {1,..., 1} are linearly independent, the relation (30) impliesthat cU 0, j 1,... ,1. The relations (29) and (30) imply also that for all Ep(A* + T), the right side of (28) is identically zero. Repeating the same procedure forthe other eigenvalues {2, 3,..., s} we can prove that cU 0, j 1,..., N2. Sincethe sequence {j,j _> N2 + 1} is part of the Riesz basis, the relation (25) impliesthat j 0, j _> N2 + 1. Thus we have proved w-linear independence of the sequence(j, j E IN}. Therefore, it is a Riesz basis of H. So we have proved that for everyh H, the contolled operator Ah is regular spectral. This result with Lemma 4 provesthe assertion (1) of Theorem 1 and also the necessary part of the assertion (2).

Now let us prove the sufficient part of the assertion (2). Suppose that the given setsatisfies the condition (3). Consider the Hilbert space 2. We show that the element{h- }=1,2 belongs to 12, where the h’s are given by the infinite products in (7). Forthis purpose, we consider the following sequence hN 2

bj ,,j )n

This is a Cauchy sequence in 12. In fact,

j--l,2,...

Since

(3)

1-)j n_

bnbn

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


we have (see [17, p. 291])

IInj

Given an e > 0, there is an integer such that

L2

Since the condition (31) is true, the infinite sum

Enj

1- =AJ n_

converges uniformally with respect to j _< . This implies that the infinite product

converges uniformally with respect to j <_ N. Hence for sufficiently large N and L,we have

j--1 j=l

22

Thus for sufficiently large N and L,

[[hN hLl[. < e.

This proves that the limit h, which is the feedback, belongs to the Hilbert space H.We must still prove that the controlled operator has the spectrum assigned a(A* +T) {j}-__. With the feedback element h given in (7) and from the proof ofLemma 4, we know that the controlled operator A*+T has No eigenvalues (multiplicitycounted) in the region gt and the other eigenvalues are simple and each of them issituated in the corresponding disk. It is sufficient to verify that the finite part of thespectrum contained in t is equal to the subset {;j 1,2,...,N0} and the onlysimple eigenvalue in the disk /j is equal to j for j > No. Reorder the elements1, 2,..., No such that vj Aj if some gk E a(A*) for k _< No. From the expressionof the feedback element, hi 0 if vj Aj. So Aj E a(A* + T) with simple algebraicmultiplicity (see Proposition 1). The rest of the spectrum a(A* + T) is equal to thezero set of the function Fh(A). From Proposition 1 and the hypothesis that vjfor j - n, it is easy to see that each eigenvalue {vj } is of simple algebraic multiplicity.

We claim that imposing the function Fh(A) to be zero on the point j . a(A*)gives the following unique solution (7) in 12

+ b,h,Fh(m) 1- E m_’-Xn

n=l

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


Without loss of generality, the above equation is equivalent to

(32) hm + E vm m bnhn Vm mn:m bm "m- An b,

Define the linear operator O, T 12 -- 12 such that

Oh Vm Ambm

bnhn_ }+

"m /nm--1

and

T=I+O.

We shall prove that the operator O is compact and that the operator T is one-to-one.Then the operator T has a bounded inverse. Set g {(m "m)Ibm}m=+Cxl. Thenthe above equation has a unique solution h T-lg. Define also the sequence ofoperators Tn 12 12 by

Trr, +

imb,

1’m

bjrj for m <_ n,

for m>n+l.

Using the same argument as that used to prove that O is compact, we can see thatthis sequence of operators is bounded. Direct matrix computations (tedious but ele-mentary) allow us to show that Tn is invertible and that

Tlg_. gmH.m’_.j for rn <_ n,j#m

gm for rn _> n + 1.

As in the above we can show that

lim T- jn+

g gmjm m Aj

m--l,2

By direct matrix computations (which are also tedious) we can prove that the sequenceof operators T-I is bounded. Let us prove that

h= T-lg= lim Tlg.

Since we have the identity

Tlg T-lg TI(T Tn)T-lg,

it is sufficient to prove that for any r E/2,

(33) lim (T- Tn)r O,

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


which will be evident in the following.Now return to proving the compactness of the operator O and the one-to-one

property of the operator T. Take any weakly convergent sequence gk E 2. ThenIlgkll2 _< M and

Ogk v.bm

bngn_m n

m--1

We prove that for every e > 0, there is an N > 0 such that for all k _> N,

In fact,

Note that by hypothesis

lim EN---+cxm_Nl,mn bm

bm

Therefore there is an integer N1 such that for all m >_ N1 and n - m,

2

Then for all m >_ N1 and n m,

Since the conditions (3) and (1) are satisfied, we can choose an N > N1 such that forall m > ,

(35)

2

bn 2 +oo

l=/=m

bn

2

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


Since the point j__ p(A*), " o R(j, A*) E (H, ). It is implied that the elementsrm {bn/(m An)}+ belong to the Hilbert space 12 The weak convergence ofthe sequence gk implies that there is an N :> 0 such that for all k > N

(36)

2

The addition of the inequalities (35) and (36) implies that of (34). Therefore we haveproved that

and, as a result, the compactness of the operator O. Note that for all r E/2,

(T- Tn)r

for m <_ n,

for m_>n+l.

In fact, we have

which tends to zero for n -- +c, for the two terms

-t-o nacx

l>n m>_n+l

2

go to zero for n +cx. Suppose that Tr 0. Then r TI (Tn T)r. Taking thelimit for n --, +cx, we prove that r 0. So the operator T is one-to-one. Finally theunique solution of (32) is

h T-l {m -m }+

brn m=

lim rl{rn-’rn}n +x bm m=!

jmm=l

So we have finished the proof of Theorem 1. D

4. Conclusions. In this paper, the necessary and sufficient condition of Sun [21]has been generalized to a large class of distributed parameter systems with boundarycontrols, which allows us to exploit the limitations imposed by BLF. For example,

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


the input vector being admissible implies that {bk} e [22], [25]. In this case,BLF cannot uniformly assign the spectrum of the systems. We have proved that it ispossible to achieve exponential stabilization of some systems by means of BLF only(Example 2). For an assignable spectrum set, we have given an explicit feedback lawwhich realizes the spectrum assignment with the resulting controlled operator beingregular spectral. The paper has also given a systematic method to assign a finitenumber of spectrum points. This method could find potential applications in dampedflexible systems as illustrated by [24] (see [19] and [1] for other models). We shouldmention that the assumption that the eigenvectors of the operator A constitute a Rieszbasis practically reduces the applications to evolution systems in space-dimension one.

Acknowledgments. The authors would like to thank the referees for their valu-able suggestions and constructive comments on the paper.

REFERENCES

[1] J. BAILLIEUL AND M. LEVI, Rotational elastic dynamics, Physica, 27D (1987), pp. 43-62.[2] G. CHEN, M. C. DELFOUR, A. M. KRALL, AND G. PAYIE, Modeling, stabilization, and control

of serially connected beams, SIAM J. Control Optim., 25 (1987), pp. 526-546.[3] a. F. CURTAIN, Spectral systems, Internat. J. Control, 39 (1984), pp. 657-666.[4] N. DUNFORD AND J. SCHWARTZ, Linear Operators Part III: Spectral Operators, Wiley-

Interscience, New York, 1971.[5] I. C. GOHBERG AND M. G. KREN, Introduction to the Theory of Linear Nonselfadjoint Oper-

ators, American Mathematical Society, Providence, RI, 1969.[6] L. F. Ho, Spectral assignability of systems with scalar control and application to a degenerate

hyperbolic system, SIAM J. Control Optim., 24 (1986), pp. 1212-1231.[7] L. F. HO AND L. RUSSELL, Admissible elements for systems in Hilbert space and a Carleson

measure criterion, SIAM J. Control Optim., 21 (1983), pp. 614-639.[8] T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976.[9] T. KOBAYASHI, A digital PI-controller for distributed parameter systems, SIAM J. Control

Optim., 26 (1988), pp. 1399-1414.[10] V. KOMORNIK, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29

(1991), pp. 197-208.[11] I. LASIECKA AND R. TRIGGIANI, Finite rank, relatively bounded perturbations of semigroup

generators. Part II spectrum and Riesz basis assignment with application to feedbacksystems, Ann. Mat. Pura Appl. (4), 143 (1986), pp. 47-100.

[12] J. Q. LIu, Perturbation of one rank and the pole assignment, J. Systems Sci. Math. Sci., no.2(2)(1982), pp. 81-94. (In Chinese with English abstract.)

[13] A. PAZY, Semigroup of Linear Operators and Applications to Partial Differential Equations,Springer-Verlag, New York, 1983.

[14] S. A. POHJOLAINEN, Robust multivariable PI-controller for infinite-dimensional systems, IEEETrans. Automat. Control, 27 (1982), pp. 17-30.

[15] R. L. REBARBER, Spectral determination for a cantilever beam, IEEE Trans. Automat. Control,a (s), . 0-0.

[16] -------, Spectral assignability for distributed parameter systems with unbounded scalar control,SIAM J. Control Optim., 27 (1989), pp. 148-169.

[17] W. RUDIN, Real and Complex Analysis, McGraw-Hill, New York, 1966.[18] P. RIDEAU, Contrdle d’un assemblage de poutres flexibles par des capteurs-actionneurs

ponctuels: dtude du spectre du systme, Thtse, Ecole Nationale Suprieure des Mines deParis, Sophia-Antipolis, France, 1985.

[19] D. L. RUSSELL, Mathematical models for the elastic beam and their control-theoretic impli-cations, in Autumn College on Semigroups and Applications, International Center forTheoretical Physics, Italy, 1984.

[20] J. SCHWARTZ, Perturbation of spectral operators and applications I. Bounded perturbation,Pacific J. Math., 4 (1954), pp. 415-458.

[21] S. H. SUN, On spectrum distribution of completely controllable linear systems, SIAM J. ControlOptim., 19 (1981), pp. 730-743.

[22] G. WEISS, Admissibility of input elements for diagonal semigroup on 2, Systems Control Lett.,10 (1988), pp. 79-82.

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


[23] C. Z. Xu AND J. BAILLIEUL, Stabilizability and stabilization of a rotating body-beam systemwith torque control, IEEE Trans. Automat. Control, 38 (1993), pp. 1754-1765.

[24] C. Z. Xu AND (. SALLET, Boundary stabilization of rotating flexible systems, in Lecture Notesin Control and Information Sciences 185, R.F. Curtain, A. Bensoussan, and J.L. Lions,eds., Springer-Verlag, Berlin, New York, pp. 347-365.

[25] , On spectrum assignment of infinite-dimensional linear systems by bounded linear feed-back, Rapport de Recherche, No. 1705, INRIA, 1992.

Dow

nloa

ded

04/2

3/13

to 1

42.5

1.1.

212.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

On Spectrum and Riesz Basis Assignment of Infinite-Dimensional Linear Systems by Bounded Linear...

Documents

Transcript of On Spectrum and Riesz Basis Assignment of Infinite-Dimensional Linear Systems by Bounded Linear...