Dynamical stabilizers and coupled systems

Huitiemes Entretiens du Centre Jacques Cartier URL:http://www.emath.fr/proc/Vol.2/

DYNAMICAL STABILIZERS AND COUPLED SYSTEMS

FARID AMMAR KHODJA, ASSIA BENABDALLAH, AND DJAMEL TENIOU

Abstract. Given a linear (unbounded) operatorAin a Hilbert space,

we describe classes of operators B andC which allow strong or

expo-nential stability of the system u 0 =

Au+Bv ====v i = ;Bu+Cv.

Applications to mathematical models of physics are given which illus-trate our results.

Key words:

Mathematics subject classication:

1. Introduction

Let's consider the equation

u 0

(t) =Au(t) +Bv(t)

where (AD(A)) is a linear operator on a Hilbert spaceXwhich is an

inni-tesimal generator of a C 0

;semigroup andB is a closed linear operator from

a Hilbert space Y toX. Our problem is to nd, among linear (unbounded)

operators (C D(C)) which are generators of C 0

;semigroups on Y , those

which insure the exponential or strong decay of the energy of the system

8 > > > > < > > > > :

u 0(

t) =Au(t) +Bv(t)

v 0(

t) =;B

u 0(

t) +Cv(t)

u(0) =u 0

v(0) =v 0

(1)

astgoes to innity. A pair (BC) which satises this property will be called

a dynamical stabilizer ofA. In the nite dimensional case, this problem has

been intensively studied and we refer the interested reader to 8].

This paper is organized as follows. In the second section, we study the eect of the pair (BC) whenever the semigroup S

A(

t) generated by A is

initially uniformly stable: sucient conditions on (BC) are given which

allow conservation of the uniform stability property. The third section is devoted to the case where S

A(

t) is not uniformly stable. In the rst

subsec-tion, the strong stability is studied while the second subsection deals with uniform stabilization. Each of our theoretical results is applied to mathe-matical models of physics (linear thermoelasticity and thermoplate systems or closely related systems). The proofs of the results are omitted but we indicate references where they may be found.

2. Coupling two uniformly stable equations

In this section, we study the eect of the coupling operator B on two

uniformly stable dynamics. More precisely, we set the following assumption:

Assumption (E.S)

1)- (AD(A)) is a generator of an exponentially stable semigroup of

con-tractions (S A(

t))

t on the Hilbert space

X: there exists! A

>0 andM A

>0

such that

kS A(

t)k X

M A

e ;t!

A

8t0

2)-(C D(C)) is self-adjoint on the Hilbert space Y and generator of an

exponentially stable semigroup (S C(

t)) t.

3)-(BD(B)) is an operator from Y to X such that D(A)D(B ) and

it is (;C) 1

2- bounded, i.e:

D((;C) 1 2)

D(B) and

B(;C) ;

1 2

2L(YX)

4)- The operator

L=

A B

;B

C

with D(L) = D(A)D(C) is closed and there exists 0

> 0 such that

( 0

I;L) is into.

Proposition 1. Under assumption (E.S), the operatorLis an innitesimal

generator of a C 0

;semigroup S L(

t) of contractions.

Theorem 2. The semigroup S L(

t) generated by (LD(L)) is exponentially

stable: there exist !>0 and M >0 such that

kS L(

t)k XY

Me ;! t

8t0

For the proof, see 5]. An example where C is nonselfadjoint is given in

6]

Example 1.

Let's consider the system

8 > > > > > > > > < > > > > > > > > :

u

00 =

u;a(x)u 0+

bw

w

0 =

;b

u 0+

w

u =w= 0 @

u(0) =u 0

u 0(0) =

u 1

w(0) =w 1

(2)

where is an open bounded subset of R

n with a smooth boundary, a 2 L

1( R

+) and (

bD(b))is a linear operator onL

2() which will be precised

later. System (2) can be written formally:

8 > > < > > :

Y 0=

LY

Y(0) =Y 0 =

2 4

u 0 u

1 w

0 3

5 (3)

with:

L= 0 B B @

0 I

;a

0

b

;

0 ;b

1 C C A=

0 @

A B

;B

dened on H = H 0()

L ()L () (the energy space equipped with

the product norm denoted by k:k) and

D(L) = D(A)D(C)

= (H 2()

\H 1 0())

H 1 0()

(H 2()

\H 1 0())

Then, as a direct consequence of the previous result (Theorem 2):

Theorem 3. If :

aa 0

>0 a:e on 0

(4)

where 0 is open and satises the geometrical control property and b is

(;) 1

2-bounded, then ( S

L( t))

t0 is exponentially stable: there exist ! > 0

and M >0 such that :

kS L(

t)kMe ;! t

for al l t0

Proof:

semi-group. The other assumptions in (E.S) may be easily veried. Let's now recall that the type of the semigroup S

L(

t) is given by

!(L) := lim t!+1

lnkS L(

t)k t

Proposition 4. Ifa>0 and bare constants, then the eigenvalues( 3k +j)

of L satisfy the following properties :

(i)

3k

> ; k

k1

lim

k !+1 Re(

3k +j) = ;

a

2 j= 12 (5)

(ii) For all b2R

there existsa 0 =

a 0(

b)2]0 p

2

1 such that !(L) = !(A) if a<a

0

!(A) if a>a

0 (6)

where the

k are the eigenvalues of (

;) (with Dirichlet boundary

con-dition) and

!(A) = 8 > < > :

; a 2

if a p

2 1

; a 2 +

p a

2 ;4

1 2

if a> p

2 1

(7)

Remark 5. This last proposition proves that one cannot expect uniform stability whenevera= 0 since, in this case, (5) and the Hille-Yosida theorem

imply that !(L) = 0 (see 1] and 2]). Let's also point out that, in the

situation described by the proposition (aandbconstants), the type of S L(

t)

is larger than the one ofS A(

t):It means that, when bis bounded, the type of S

A(

t) is not in general improved once the corresponding equation is coupled

3. Dynamical stabilization

This section is devoted to the situation where S A(

t)

is not initially

stable

course, we will study separately strong and uniform stability. It will appear, in the second subsection, that uniform stabilization for system (1) requires unboundedness for the operatorB.

3.1. Strong stabilization Definition 6. A C

0

;semigroup (S(t))

t in a Banach space

E is strongly

stable if

lim

t!+1

S(t)x= 0 8x2E

In this section, assumption (E.S) is replaced by:

Assumption (A.S)

1)- (AD(A)) has a compact resolvent and is a generator of contraction

semigroup (S A(

t))

ton the Hilbert space X, .

2)-(C D(C)) has a compact resolvent, is self-adjoint on the Hilbert space Y and generator of an exponentially stable C

0

;semigroup (S C(

t)) t.

3)-i) D(C)D(B) and there exist two constants

C and

C with

0 C

<1 C

0

such that:

kBvk X

C

kCvk Y +

C

kvk Y

-ii)

D(A)D(B )

and there exist two constants A and

A with:

0 A

<1 A

0

such that:

kB

uk Y

A

kAuk X +

A

kuk X

Remark 7. Assumption (A.S3) is natural because the operator

L can be

written:

L=D+E

with

D=

A 0

0 C

E =

0 B ;B

0

Clearly Dis a generator of contraction semigroup and sinceE is dissipative, L is also a generator of contraction semigroup

Theorem 8. Under assumption (A-S), if the unique solution of the system

u

0 =

Au t0

B

u = 0 t0

is u 0, then S L(

t) is asymptotically stable.

Remark 9. In fact, one can relax the hypothesis on C . More precisely,

one may drop the selfadjointness of C and suppose Re(Cvv) = 0)v= 0

(see 6]).

Example 2.

1/ First, we consider system (2). One has:

Proposition 10. If b is (;) 1

2-bounded and

a 0 a.e. in , then for all Y

0 2X:

kS L(

t)Y 0

k !0as t!+1

provided that the unique solution of

8 > > > > < > > > > :

u =u in

bu = 0 in

u = 0 on@

is u 0 for all 2R:

This result is in particular true if b is a bounded operator in L

2(). In

this case, one cannot expect the exponential stability for if a 0 and bis a

constant, this is not true (see Remark 1).

2/ The second example deals with the linear thermoelasticity system. Let an open bounded set in R

n with smooth boundary ;, 8

> > > > > > > > > > > > < > > > > > > > > > > > > :

u 00=

u+ rw R

+

w 0=

w+ r:u 0

R +

u= 0 R

+ ;

w= 0 R

+ ;

u(0) =u 0

u 0(0) =

u 1

w(0) =w 0

(8)

whereu = (u 1

:::u

n)] (resp.

w) is the displacement (resp. the

tempera-ture) of the system, >0 is the coupling parameter, (see 11]).

The energy space will be the Hilbert space:

H=H 1 0())

n (L

2

())n L

2

() with the scalar product:

< 2 4

u v w

3 5

2 4

f g h

3 5

>= Z

(ru:rf +vg+w h)dx

and we denote by k:kthe induced norm onH. j:jwill denote theL

2-norm.

Let:

L= 0 B B B B @

0 @

0 I

0

1 A

0 @

0

r 1 A

;

0 r:

1 C C C C A

=

0 @

A B

;B

D(L) =D(A)D(C)

= (H 2()

\H 1 0())

n H

1 0()

n (H

2() \H

1 0())

One has

Proposition 11. The thermoelasticity system is strongly stable if the sys-tem

8 > > > > < > > > > :

u

00 =

u

u = 0 @

r:u

0 = 0

has u 0 as a unique solution.

This result was already known (9]) and we give it for illustration.

3.2. Uniform stabilization

As we are interested by equations closely related to the wave equation, we restrict ourselves to the stabilization of second order equations. More precisely, letX andY be two Hilbert spaces andA,B andCare unbounded

operators, with dense domains, acting onX, fromY toXand onY

respec-tively. We consider the system:

8 > > > > < > > > > :

u 00(

t) +Au(t) =Bw(t) t2R +

w 0(

t) +Cw(t) =;B

u 0(

t) t2R +

u(0) =u 0

u 0(0) =

v 0

w(0) =w 0

where B

is the adjoint operator of B.

Throughout, we will set two kind of hypothesis. The rst one will in-sure the semigroup property for the system (assumption (H1)). The second will insure the uniform (exponential) stability for the semigroup solution (assumption (H2)).

(H.1)

(resp. Y), strictly positive, with dense domain D(A) (resp. D(C))and

compact resolvents (ii) B (resp. B

) is an operator from

Y to X (resp. X toY) such that: D(C)D(B) (resp. D(A)D(B

)) and is

C-bounded.

In the sequel, H = D(A 1 2)

X Y will be equipped with the inner

product:

0 @ 2 4

u v w

3 5

2 4

f g h

3 5

1 A=

A

1 2

uA 1 2 f

X

+ (vg) X+ (

w h) Y

The preceding system can be equivalently written:

8 < :

Y 0(

t) =LY(t) t2R +

L = @

0 I 0

;A 0 B

0 ;B

;C A

D(L) = D(A)(D(A 1 2)

\D(B ))

D(C)

SinceLis dissipative with dense domain inH ,Lis closable. So, the closure

of L (and its closure will be denoted by the same symbol) is a generator of

a C

0-semigroup of contractions.

(H.2)

kA ;

1 2

Bwk X

ajjC 1 2

w jj Y

8w2D(C)

(ii) B

is invertible (( B

);1

2L(X Y)) and

k(B );1

Cwk X

bjjC 1 2

w jj Y

8w2D(C) a and bbeing positive real constants.

(iii) There exist three positive constants c 1

c 2

c

3 such that for all Y 2 D(L)

j(Au(B

);1 w)

X j

c 1

2 jjC 1 2

w jj 2 Y +

2

; c

2 jjYjj

2

+ (G(Y)LY)

(9) for all >0, whereGis a function fromH toL(H) which satises

jjG(Y)jjc 3

jjYjj 8Y 2H (10)

The main result is then:

Theorem 12. Under assumption (H), L is the generator of a uniformly

stable C

0-semigroup ( S

L(

t)): there exist !>0 and M 1 such that: kS

L( t)k

L(H) Me

;!t

8t2R +

A particular system of this kind is obtained if one sets X = Y, A = C

and B =A

where

201]. Denoting by

L =

0 @

0 I 0

;A 0 A

0 ;A

;A 1 A

D(L ) =

D(A)

D(A )

\D(A 1 2)

D(A)

we show the following:

Theorem 13. (i) The strongly continuous semigroup of contractions(S (

t))

generated by L

is uniformly stable if and only if 2

1 2

1]. In this case, the

decay rate (or the type) is !=;supRe(L ),

(L

) being the spectrum of L

.

(ii) If 20 1 2,

S (

t) is strongly stable.

(iii) The semigroup S (

t) is analytic if and only if 3 4

1.

(iv) The semigroup is compact if and only if 1 2

< 1.

This result shows the "order" of unboundedness of B with respect to A

Example 14. We consider two models of thermoelastic plates which dier by the boundary conditions. Let be an open bounded set in R

n with

smooth boundary ;,

8 > > > > > > > > > > > > < > > > > > > > > > > > > : u 00= ; 2

u;w R

+

w 0=

w+ u 0

R +

u= u= 0 R

+ ;

w= 0 R

+ ;

u(0) =u 0

u 0(0) =

u 1

w(0) =w 0

(11)

The energy space will be the Hilbert space:

H = (H 2

()\H 1 0())

L 2

()L 2

() with the scalar product:

< 2 4 u v w 3 5 2 4 f g h 3 5 >= Z

(u:f+vg+w h)dx

and we denote by k:kthe induced norm onH. j:jwill denote theL

2-norm. Let: L= 0 B B B B @ 0 @ 0 I ; 2 0 1 A 0 @ 0 ; 1 A ; 0 1 C C C C A

D(L) =f u v w

]2H uvw2H 2

()\H 1 0()

g

This model has already been studied in 14] where uniform stability is proved. This system satises Assumption (H) with A=

2 and

D(A) =

u2H 2

()\H 1 0()

u2H 2

()\H 1 0()

C =B =A 1 2 =

; withD(C) =D(B) =H 2()

\H 1

0() and

F = 0. So

Proposition 15. The thermoelastic plates system (11) is uniformly stable. Moreover, the semigroup associated to L is compact and the decay rate of

the energy is supRe(L).

The second model is the following

8 > > > > > > > > < > > > > > > > > : u 00= ; 2

u;w R

+

w 0=

w+ u 0 R + u= @u @ =

w= 0 R

+ ;

u(0) =u 0

u 0(0) =

u 1

w(0) =w 0

(12)

The energy space will be the Hilbert space:

H=H 2 0()

L 2

()L 2

with the scalar product:

< 2 4

u v w

3 5

2 4

f g h

3 5

>= Z

(u:f+vg+w h)dx

and we denote by k:kthe induced norm onH. j:jwill denote theL

2-norm.

Let:

L= 0 B B B B @

0 @

0 I

; 2 0

1 A

0 @

0

; 1 A

;

0

1 C C C C A

D(L) = 8 < : 2 4

u v w

3 5

2Hu2H 2()

v2H 2 0()

w2(H 2() ^

E\H 1 0())

9 =

This model has also been studied in 10] where it is proved that the system is uniformly stable using an indirect method. We obtain here the same result but with an estimate of the decay rate. Let's denote byA=

2 and

D(A) = H

4() \H

2 0(),

C=B =; with D(C) =D(B) =H 2()

\H 1 0().

Proposition 16. The thermoelastic plates system (12) is uniformly stable. In these two last models, the method we use to prove uniform stability is based on the construction of Lyapunov functions (using the multipliers method). It allows us to give an estimate of the decay rate of the energy.

The reader can nd all the proofs of the results of this subsection in 3] and 4]. For more precisions on the denitions of the dierent concepts of stability, see12], 13].

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