Approximate observability of abstract evolution equation with unbounded observation operator

Mathematics

and Computer Modelling 36 (ZOOZ) 729-736 www.elsevier.c~m/locat~/mcm

Approximate Observability of Abstract Evolution Equation with Un~ou~~e~ Observation Operator B. SHKLYAR Department of Exact Sciences Halon Academic Institute of Technology Holon 58102, Israel

Abstract-Propetiic Cz(t)

of the null set of the evolution equation S = Ax(t), z(0) = x0, g(t) = (A generates a strongly continuous semigroup {S(t))t>~ on a Banach space X, C is a linear

unbounded operator) are investigated. Conditions for approGmate observability of such system are obtained. @ 2002 Elsevier Science Ltd. Al3 rights reserved.

Keywords-&olution

systems,Strongly ~untinuous semigroup, Unbounded observation operator, Unobservable set, Minimal sequences of functions, Approximate observability.

1. INTR~~~CTI~N Consider the linear abstract differential equation

(1) (21 (31

c?(t) = Az(t),

z./(t)= Wt), 40) = 20,

OIt<+cq

where X,Y are Banach spaces, z(t) E X is a current state, 10 E X is an initial state, y(t) E Y, y( .) E &([O, ti], Y) is an output, A is a linear operator generating a strongly continuous semigroup {S(t)}t2e of operators in the class CO [l]; the operator C : X -+ Y is a linear possibly unbounded operator with dense S-invariant domain D(C) c X. The operator C is called the observation operator of equation (l),(2). One of the most popular examples of equations with unbounded observation operator is a system of linear partial differential equations which describes boundary or point observation. Equations having unbounded observation operator were widely studied in the literature (see [21% If z f X and f E X*, we will write {zt f) instead of Sfzrf. As usual N is the set of natural and lR is the set of real numbers. We assume (I) WA)

E WC);

0895-7177/C@% - see front matter @ 2002 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(02)00170-X

I’ypmet

by &4+W

730

B.

(II) the operator

(a(t) : D(C)

SHKLYAR

--+ Ls( [0, t], Y) defined

by the formula

@{t)x = Gs(t)x has a continuous The operator

extension

C having

to all of X for each t > 0.

abov~mention~

is called the a~~~ss~ble obse~at~o~ opera-

properties

tor [2,10].

2. NULL DEFINITION

1. A state

DEFINITION

2. The set N(t)

equation Thus,

SET

2 E X is said to be unobser~b~e

at time t, ifCS(7)s

of all states zc E X unobservable

= 0, 7 E [O, t].

at time t is said to be null set for

(2). the null set ~(~)

for equation

Iv(t) = It follows from Definition The natural question arises: a sufficiently large time t?

(1) is defined

by

{x E x : CS(T)X

= 0,

E

7

[O,t]} .

(4)

2, that the set N(t) decreases, i.e., N(ts) 2 N(ti), when ti < ts. does the decreasing process continue endlessly or does it stop after

EXAMPLE. Let X = .&[O,+oo). Denote by S(t) the left shift by t on X. limt_++a ((S(t)f - f)/t) in X, hence, Dom (A) = H’[O, +cQ). Define C on Dom (A) by C_f = f(O). For any ti > 0

N(t) = (f

E

Af

As usual,

X : f(7-) = 0 a.e., on [O,ti]).

=

(5)

If ts > ti, then the function f(t) belongs constant

= {

1,

tr < t < tar

0,

otherwise,

to N(ti) but does not belong to N(ts), set for all t > T, does not exist.

i.e., the number

T 2 0, such that

N(t)

is a

Our main task here consists of establishing conditions for independence of N(t) at least for sufficiently large t. These conditions are of importance in the theory of observation and control. 2.1.

Conditions

The operator

for the

Independence

A is assumed

of Null

to have the following

Set of i; for Sufficiently

Large

t

properties.

The domain D(A*) is dense in X*. The operator A has a purely point spectrum ~7which is either finite or has no finite limit points and each X E (T is of a finite multiplicity. (V) There exists a time moment T 2 0 such that for all z E X and t > T the weak so(l),(2) is expanded in a series of generalized eigenveclution z(t) = S(t) x of equation tors of the operator A, converging uniformly with respect to t on an arbitrary interval [?I;, Ts] (Ti > T)i in the strong topology of X.

(III) (IV)

LetnumbersXjEo, j=l,Z ,..., be enumerated let cyj be a multiplicity of Xj E u, and let

(Pjkl

and

@jk’jkt,,

j=1,2

3.s.;

k=l,.s.,mj;

in the order of nondecreasing

t=1,2,..v,@jk;

absolute

Ffijjb=a3,

k==l

lPossibly

for a certain

grouping of terms.

values,

Unbounded Observation Operator be generalized

731

eigenvectors of the operators A and A*, respectively, such that (‘pjp&-l+1:

j,kEF$

p=l,...,

Tnj;

7&g) = ~jlc&&

1=1 )“‘I

pjp;

s=l,...,

??zrc; q=l

I.“., &.

(6)

Denote

The vectors

rpjkl, j = 1,2,. . . , k = 1,2, . . . , rnj, are eigenvectors of the operator A.

DEFINITION 3. A sequence {Q}~EH of functions from Lp[O, +co) is called minimal on [0,v] (v > 0) if there exists a sequence {gj}?en of functions from Lz[O, v] such that

s

0vMt)9i

i,j = 1,2 ,...,

(t>>d-J= &j,

where bij is the Kronecker symbol. the sequence {z:j}j,==, on [0, ~1.

The sequence {Yj}JEn is called the sequence biorthogonal to

THEOREM 1. If tl 5 t2? then N(t2)

c N(tl).

If the sequence j = 1,2,. . .;

fjk(t) = t’eXjt,

k = 1,. ..,aj;

t E [0,+m),

is minimal on [0, Y], then N(tr) = N(t2) for each tl,t2 > T + ZI. PROOF. The inclusion N(t2) C N(tr) for ti < t2 is evident. Now we will prove the inclusion N(ti) C N(t2) for all 7’ + TJ< tr < t2. Let x E N(tr), tr > T + U. By (4) CS(r)z

= 0,

7 (5 [O,ill.

(8)

In accordance with Property (V) we have S(t)x =

2 *jexp (A$)

(z, QjjT :

T < t < t1,

j=l

where C,“=,

is considered

in the topology

of X.

Let

j = 1,2,. . . ,

be diagonal block matrices, where

hjl =

are

x 8,Jordan

blocks.

I xj

1

0

...

0

xj

1

...

0

0

4

...

.,. 0 0

.,, 0 0

... 0 0

... ... ...

= 1.2.. . . .m;.

0 0 0 Xj

. 0. .

0 0

0 1 .%. . 1

(9)

B. SNKLYAR

732

If C is bounded operator, then one can easily show, that t > T,

CS(t)x = j=l

but formula (10) cannot be directly applied for unbounded operator C. Let Tl = tl - 2), Denote

By (V) and Tl r T we obtain

By virtue of (II) and (12)

in &([O, ~3,Y), It is easy to prove that (13) implies

s V

V

CS(T)X,U(T)

lim n-+03

dt =

0

/

0

CS(+z

47) dr,

for any function a(~> from &[O, w]_ Substituting (11) into (14) we obtain

2 C@j sv exp (A,r> exp (A,Tl> (x, sJT u(r) j=l

dr =

0

s”CS(T)XT,U(T) dT. 0

Let JEjl

.O .

j+

1

i

0

be diagonal block matrices, where

0

...

EjZ

*..

0 0

‘i 3

j = 1,2,. . I)

Unbounded

are &l x &l-matrices

Observation

Operator

733

with unit above the diagonal and with zeroes otherwise, 1 = 1,2, . . . , mj.

It is easy to show, that CYj-1

exp (ANT) w = c

I-~~J~E~w,

j=l,2,...,

06)

k=O

for any w E IRS”?,so that

v

exp (Ajr) exp (AjTi) (5, Qj)T u(r) dT (17)

Denote by Ujk(r), j = 1,2 )‘..) sequence (7) on [O,v]. It follows from (15)-(17)

k = O,l)...

, aj - 1, 0 5 r I v, the sequence biorthogonal to

that

I vcs() T

xTlujk(T) dt = C@jE: exp (AjTi) (z, Qj)T,

(18)

0

Since 2 E N(ti), CajE:

k=O.1,2

,...,

crj-1.

we obtain from (8) and (18) that j=1,2,...,

exp (AjTi) (z, Qj)T = 0,

k=O,l,...,

aj-l.

(19)

O
(20)

Using in (13) T2 = t2 - v instead of Tr = ti - v, where t2 > tl, we obtain cs(T)s(??2)S

=

CS(T+t2

-tl)Z~,

03 =

c

CQj exp (Aj (T + tz - tl)) exp (AjTl) (cc,X4fj)T,

j=l

where (8), (19), (20), (V), and t2 > tl imply CS(t)z

t E

i.e., N(tl)

So, we obtain by (21) that zc E IV&), This proves the theorem.

3. APPROXIMATE

= 0,

(21)

p,t21.

= N(t2) for all tl and t:! with T+v

OBSERVABILITY

< tl < t2.

CONDITIONS

In this section, we will consider application of Theorem 1 for approximate observability of equation (l),(2). Other kinds of observability can be investigated in this way also. DEFINITION

4.

(See [2,7,12].) Equation (l),(2)

if

is said to be approximately observable on [0, tl],

t1 s0

IlWt)l12 dt > 0,

(22)

for any nontrivial initial state. Let sequence (7) be minimal on [0, v] and generalized eigenvectors of the operator A* be dense in X*.

B.

734

SHKLYAR

THEOREM 2. For equation (l),(2) to be approximately null-observable on [0,tr], it is necessary and for tl > T + v suficient that system with respect to x E DomA Cx=O

(XI - A)z = 0,

(23)

have only trivial solution. PROOF. One can easily see that condition (23) is equivalent to the condition: 1,2,. . *. The vectors Cf+?.$k, k = lY2,. . . , mj, are linearly independent.

for each j, j =

SUFFICIENCY. If (22) d oes not hold, then there exists zr # 0,s E N(tr), where tl > T + u. We have obtained above (see the proof of Theorem 1) that z E N(ti) provided that tl > T +V implies

equality

aj-vector

(19).

One can consider

Et exp(AjTr)(z,

independent,

Qj)T,j

then system

(19) as a linear

(19) has only trivial

Et exp (AjTr) (x, !Pj)T = 0, Equalities

solution, j=1,2

vectors

Cpjrk,

system

with

respect

to the

k = 1,2,. . . , nzj, are linearly

i.e.,

,...,

k=O.l,2,...,cuj-1.

(24)

(24) imply (z, @j)T = 0,

The

algebraic

= 1, ‘2,. . . . If the vectors

j = I, 2,. *. .

(25)

+!~jkt, j = 1,2,. . . , k = 1,. . . ,m, 1 = 1,2,. . . ,@jk are dense

from (25), that 2 = 0. This proves the sufficiency

in X, so it follows

of (23).

NECESSITY. If condition (23) does not hold, then there exists j E N, pj # 0 such that

so

CS(t)cpj = exjtCpj Hence, equation

(l),(2)

is not approximately

= 0,

observable

\Jt 2.0. on [0, t], \J t > 0. This proves the necessity

of (23). Let Ij be oj x aj-matrix.

COROLLARY3. For equation (l),(2) to be approximately null-observable on [0, tl], it is necessary and for tl > T + u sufhcient that ran~{hj~~Aj}=aj,

j=1,2

,...,

(26)

hold.2 PROOF. We have AQj = @&, By (27), we obtain

the equivalence

of conditions

j = 1,2 ,.... (23) and (26).

DEFINITION 5. Equation (l),(2) is said to be approximately observable if

for any nontrivial initial state. *See also [7, Theorem

1.61

(27)

UnboundedObservationOperator THEOREM 4. For equation

(1) to be approximately

735

nuke-observable

it is necessary

and suficient

that, (23) hold. PROOF. If (28) does not hold,

obtain

then there exists 2 # 0, 2 E N = nEP=, N(t). By Theorem 1 we V’tl > T + Y. The proof is finished by means of the proof of Theorem 2.

N = N(tl),

Theorems

2 and 4 yield the following

COROLLARY 5. If equation (lf,f2) t2 > T + v, then it is approximately COROLLARY 6. If equation

(I),fZ) observable

then it is approximately

This holds true also for other REMARK 1. Theorems

hold. For example, observable. EXAMPLE.

parabolic

equation

with boundary

The output

is approximately observable on [0, tz] for some finite time t2, observable on [0, tl] for any finite time tl, tl > T -t ‘u. is ap~roxjrna~e~y observable (on [0, +co), see ~e~nitio~ on [0, tl] for any finite time tl, tl > T + ‘u.

a simple

31,

kinds of observability.

2 and 4 and Corollaries

equation

Consider

corollaries.

5 and 6 do not hold if Conditions

(5) is not observable illustrative

(I)-(V)

do not

on [O,tl] for any tl > 0 but it is approximately

example

of observation

problem

described

by partial

[2] af @.f -=ae”’ dt

0 I t,

f(t, 0) = 0,

f(t, r) = 0,

O<@
(29)

conditions

is defined

0 < t.

(30)

by [2]

I#) = f(t>~),

w here a E (0,~)

Let X = Lz[O,?r). D enote by A the di~erential {f E Lz[O,~] :f' E AC[O],

where AC[O, 7r] is the set of absolutely It is well known [2], that space X = Lz [O,r] and

SW =

operator

t 2 0.

(31)

Af = f” with the domain

f" E L2[O,-/r], f(o) = f(r) = 01,

continuous

the operator

is a fixed number,

functions

A generates

(32)

in Lz[O, ~1.

a compact

fJ(_f, ~n)e-n2ti0n,VfEX,

self-adjoint

S(t) in the state

ost,

(33)

?a=1

where (pn(0) = fisinn@, 0 < 8 I n; (f, cpn) = fi orthonormal base for X. Let the operator C be defined by

Cf = f(a)*

s: f(8) sin&de,

functions

CS(t)f

S(B)sinaBdB)

2 (Lr T-k=1

~~(61) yield an

Dom C = CL~ [0, n],

where CL, [O,z] is the set of continuous every t > 0 and

= fi

and vectors

in Lz[O, ~1.

e-“2tsinna

Obviously, D(A) C D(C); I\CS(t)fllz 2 Ilfl11/2dm, admissible observation operator. Since Cz.1 l/n2 < foe, the sequence e--lbzt is minimal

(34) We have S(t)_f

E DomC

for

(see [2]).

[2], so that

the operator

on [0, v] for any w > 0, 1131.

C is an

B. SHKL'IAH

736

Using Theorems I. and 2, and the density of functions we obtain the validity of the following statements:

sin n@, rt. = 1,2, . . . , 8 f [0, ;?-I, in Lz(O, x],

* the null set of equations ~29),~30) with output (31) does not depend on t for any t > 0; * for equations ~29)~~3~) with output (31) to be approximately observable on 10, TV],tit1 > 0, it is necessary

and sufficient

that sin 7zQr# 0,

hold.

Condition

coincides

Tz= lt 2,. . .

(35) holds if and only if the number

(35) LY/~ is irrational

number.

This

with the result of [2].

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