Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity

PARTIAL EXACTCONTROLLABILITY

ANDEXPONENTIAL STABILITY IN

HIGHER-DIMENSIONAL LINEAR THERMOELASTICITY

WEIJIU LIU

Abstract. The problem of partial exact boundary controllability and exponential stability for the higher-dimensional linear system of ther- moelasticity is considered. By introducing a velocity feedback on part of the boundary of the thermoelastic body, which is clamped along the rest of its boundary, to increase the loss of energy, we prove that the en- ergy in the system of thermoelasticity decays to zero exponentially. We also give a positive answer to a related open question raised by Alabau and Komornik for the Lame system. Via Russell's \Controllability via Stabilizability" principle, we then prove that the thermoelastic sys- tem is partially controllable with boundary controls without smallness restrictions on the coupling parameters.

1. Introduction

Let be a bounded domain in ^Rn with smooth boundary ; =

@

of class

C

², and consider a

n

-dimensional linear, homogeneous, isotropic, and ther- moelastic body occupying in its non-deformed state. For a material point with conguration

x

= (

x

¹

x

ⁿ) at time

t

, let

u

(

xt

) = (

u

¹(

xt

)

u

ⁿ(

xt

)) and

(

xt

) denote the displacement and temperature deviation, re- spectively, from the natural state of the reference conguration. Then

u

and

satisfy the system of thermoelastic equations

8

<

:

u

^00;

u

^;(

+

)^rdiv

u

+

^r

= 0 in (0

¹)

^0;

+

div

u

⁰= 0 in (0

¹)

u

(0) =

u

⁰

u

⁰(0) =

u

¹ (0) =

⁰ in

⁽¹

:

1) in the absence of external forces and heat sources, where

,

>

0 are Lame's constants and

,

>

0 the coupling parameters. By⁰we denote the derivative with respect to the time variable.

^r

div denote the Laplace, gradient, and divergence operators in the space variables, respectively.

u

(0),

u

⁰(0) and

(0) denote the functions

x

^!

u

(

x

0),

x

^!

u

⁰(

x

0) and

x

^!

(

x

0), respectively. For the derivation of (1.1), we refer to 17] and 40].

The goal of this paper is to study the exponential stability and partial exact controllability of system (1.1). Our main results will be presented in Section 2 and proved in Section 3.

School of Mathematics and Applied Statistics, University of Wollongong, North elds Avenue, Wollongong, NSW 2522, Australia. E-mail: weijiu [email protected]

Submitted April 15, 1997. Revised October 13, 1997. Accepted for publication De- cember 10, 1997.

c

Societe de Mathematiques Appliquees et Industrielles. Typeset by TEX.

Before we give our main results, let us briey describe the existing liter- ature.

Under the Dirichlet-Dirichlet boundary conditions

u

= 0 = 0 on ;(0

¹)

(1

:

2) the thermoelastic energy of (1.1) can be dened as

E

(

t

_{) = 12}^Z

h

j

u

⁰(

xt

)^j2+

^jr

u

(

xt

)^j2 + (

+

)^jdiv

u

(

xt

)^j2+

^j

(

xt

)^j2i

dx:

⁽¹

:

3) Here we have used the notation

jr

u

(

xt

)^j2= ^Xn

ij=1 j

@u

ⁱ

@x

^jj2

:

It is easy to verify that the energy

E

(

t

) decreases on (0

¹), but, in general, does not tend to zero as

t

^!1. In fact, it has been shown by Dafermos in his pioneering work 9] that the energy of every solution of (1.1) and (1.2) converges to zero as

t

^!1if and only if satises the following condition:

(

H

) There is no non-trivial eigenfunction

=

(

x

) of the Lame system

;

^;(

+

)^rdiv

=

²

in

= 0 on

@

such that div

= 0 in .

In 9], it was pointed out that (

H

) holds \generically" for smooth do- mains. It was also shown that (

H

) fails when is a ball (see 32]).

Subsequently, the attention was paid to the problem of the uniform ex- ponential decay rate of energy. In 1992, Hansen 12] made the rst attempt on this problem by considering the one dimensional linear thermoelastic system subject to Dirichlet-Neumann or Neumann-Dirichlet boundary con- ditions. Applying the analysis of nonharmonic Fourier series, he succeeded in establishing the uniform exponential decay rate.

Hansen's results rely signicantly on the boundary conditions of Dirichlet- Neumann or Neumann-Dirichlet type. The case of Dirichlet-Dirichlet type was left as an open problem. Later, by using dierent methods, Kim 14]

and Liu and Zheng 35] independently solved this problem, showing the ex- ponential decay under Dirichlet-Dirichlet boundary conditions. While Kim used the energy method, multiplier techniques and compactness properties, Liu and Zheng's method was based on an abstract theorem about the expo- nential stability of semigroups.

Applying the abstract exponential stability theorem mentioned above, Burns, Liu and Zheng 5] further considered all other possible boundary conditions for the 1-D system of thermoelasticity (such as stress free at both ends and stress free at one end) and showed that the semigroups associated with these boundary conditions are also exponentially stable.

In summary, the problem of exponential decay rate of energy for the one- dimensional linear thermoelastic system has been completely solved by now.

For the higher-dimensional linear thermoelastic system, the problem is much more complicated. Since the total energy in higher-dimensional linear thermoelasticity does not always decay to zero, one has to try to partition the total energy into a dissipative part and a conservative part.

The question of partition of the energy was rst studied by Lax and Phillips 23] for a classical solution of the wave equation. The rst work about the partition of the energy in the linear thermoelasticity is due to Dassios and Grillakis 10], who studied how the energy associated with the longitudinal and thermal wave is divided into kinetic, strain, and thermal energy in the case = ^R3. They concluded that all three parts of the energy decay to zero as

t

^!+¹ at a polynomial rate. Further, Rivera 43]

studied the decomposition of the displacement vector eld in^Rn(

n >

1) into two parts. One of them is the solenoidal part, namely, the nondissipative component that conserves its energy and the other the dissipative component that decays to zero as fast as

t

^;n=2 when

t

approaches innity.

For bounded domains, Chirita 8] proved that the mean thermal energy tends to zero as time goes to innity and that the asymptotic equiparti- tion occurs between the Ces aro means of the kinetic and strain energies.

This shows that thermal eects do not inuence explicitly the asymptotic equipartition of the mean kinetic and strain energies. Chirita's method relies on the Lagrange-Brun identities.

In special situations where the restoring force is proportional to the vector velocity of the displacement vector eld, Pereira and Menzala 42] proved that in a bounded domain the kinetic, strain and thermal energies tend to zero exponentially as

t

^!+¹.

Recently, Lebeau and Zuazua 25] gave a su!cient and necessary condi- tion ensuring that the energy tends to zero exponentially as

t

^! +¹ in a bounded multi-dimensional smooth domain . This condition is written in terms of the dynamics of the rays of geometric optics. As a consequence of the result of 25], it follows that when is a bounded smooth convex open set, the energy does not decay exponentially to zero.

In addition, there has been a lot of work on von Karman's system of thermoelastic plates. For details, we refer to 3], 6], 20], 22], 34], 36], 37], 38].

While there has been extensive work on the stabilization of the linear thermoelasticity, relatively little is known about the controllability.

The earliest results appear to be in the paper 40] of Narukawa, who proved the partial exact boundary controllability for the general form of the thermoelastic system on a bounded domain in^Rn. Later, this result was improved by Lions 30, p. 32-60] by introducing the Hilbert Uniqueness Method. In both Narukawa and Lions' results, only the displacement is controlled, disregarding the values of the temperature. This is the so-called partial controllability property. Such a partial controllability property was also proved for von Karman's system of thermoelastic plates (see 20], 21], 22]). This drawback (i.e., lack of information on the controllability of the temperature) was avoided by Hansen 13], who showed that, for at least

the one-dimensional thermoelastic system, exact controllability of both the displacement and temperature is possible by only controlling the thermal or mechanical component on the boundary in the case where

u

and

satisfy the Dirichlet-Neumann or Neumann-Dirichlet boundary conditions and the coupling parameters satisfy some further restrictions. Hansen's results were proved by making use of the moment problems and the theory of nonhar- monic Fourier series. It seems that Hansen's method is not applicable to the multi-dimensional space case. Thus, the problem of exact controllabil- ity of both the displacement and temperature is much more complicated in this case. In order to attack this problem, Zuazua 48] recently intro- duced the concept of exact-approximate controllability and made signicant progress. He proved that, if

T

is large enough, then the thermoelastic system is exact-approximately controllable with a control supported in a neighbor- hood of the boundary of , i.e., the displacement is shown to be exactly con- trollable and the temperature approximately controllable. The method of Zuazua combines multiplier techniques, compactness arguments and Holm- gren's Uniqueness Theorem among other tools. More recently, Teresa and Zuazua 47] proved that the same kind of results hold for thermoelastic plates. In addition, Lebeau and Zuazua 26] proved that the system of lin- ear thermoelasticity with periodic boundary conditions is null controllable with a volume force located in a subset satisfying the geometric control con- dition of 4]. Their method of proof is based on a spectral decomposition of the system and its adjoint on the basis generated by the eigenfunctions of the Laplacian. The spectrum is split into a parabolic and a hyperbolic part, and then the techniques of 4] and 24] are combined.

In this paper, we establish a su!cient condition which guarantees the exponential decay rate of the energy by means of an additional boundary damping. It is well known that the reason why the energy

E

(

t

) does not tend to zero as

t

^!1 is that the total energy is not dissipated completely in the form of thermal energy (see 25] for more details). Thus we introduce here a velocity feedback on part of the boundary of the thermoelastic body, which is clamped along the rest of the boundary, to increase the loss of energy.

In order to state the boundary velocity feedback, we set

;¹ =^f

x

²; :

m

(

x

)

(

x

)0^g

(1

:

4)

;² =^f

x

²; :

m

(

x

)

(

x

)

>

0^g

(1

:

5) where

m

(

x

) =

x

^;

x

⁰ = (

x

^{1 ;}

x

⁰¹

x

^n;

x

⁰ⁿ) for some

x

^{0 2 Rn},

= (

^{1 n}) denotes the unit normal on ; directed towards the exterior of and

m

=

m

(

x

)

(

x

) =^Xn

i=1

(

x

^i;

x

⁰ⁱ)

ⁱ

:

;¹ is assumed either to be empty or to have a nonempty interior relative to

;. Note that assumptions (1.4) and (1.5) imply that the domain is simply connected and star-shaped with respect to

x

^{0 2} or = ^{1 ;2}, both ¹ and ² being star-shaped with respect to

x

⁰. Especially, the domain

can be a bounded smooth convex open set. As mentioned before, for the convex domain, Lebeau and Zuazua 25] has showed that the energy does not decay exponentially to zero in general. Thus, the feedback is necessary for this case.

The boundary velocity feedback can be stated as follows

8

>

>

>

<

>

>

>

:

= 0 on ;(0

¹)

u

= 0 on ;¹(0

¹)

@u @

^{+ (}

+

)div(

u

)

+

am u

+

m u

⁰ = 0 on ;²(0

¹)

(1

:

6) where

a

=

a

(

x

) is a given nonegative function on ;² with

a

(

x

)²

C

¹(;²)

:

(1

:

7) It is clear that if

m

(

x

)

(

x

)

on ;² for some

>

0 then

a

(

x

)

m

(

x

)

(

x

) can be any nonnegative function as we can take

a

(

x

) =

f

(

x

)

=

(

m

(

x

)

(

x

)),

f

(

x

) being any nonnegative function. It is well known that the boundary velocity feedback is an eective mechanism to increase the loss of energy, and has been extensively used for the wave equation 7], 16], 18], elastodynamic systems 2], 19] and viscoelasticity 27], 28].

We will prove that the energy of the solutions of (1.1) and (1.6) decays to zero exponentially as

t

^!1 if

A

= max

x2;2

a

(

x

) is small enough (see Theorem 2.1 below).

In the special case of the Lame system

8

>

>

>

>

>

<

>

>

>

>

>

:

u

^00;

u

^;(

+

)^rdiv

u

= 0 in (0

¹)

u

= 0 on ;¹(0

¹)

@u @

^{+ (}

+

)div(

u

)

+

am u

+

m u

⁰= 0 on ;²(0

¹)

u

(0) =

u

⁰

u

⁰(0) =

u

¹ in

(1

:

8)

the exponential stability still holds even though

A

is large. This result answers an open question raised by Alabau and Komornik 2].

On the other hand, as the consequence of the uniform stabilization, we use the \Controllability via Stabilizability" principle to prove the partial ex- act boundary controllability for the thermoelastic system without smallness restrictions on the coupling parameters

and

.

The main results of this paper are presented in Section 2 and proved in Section 3. The methods of our proofs are based on multiplier techniques, the asymptotic property of the semigroups and Russell's \Controllability via Stabilizability" principle.

2. Main results

In what follows,

H

^s() denotes the usual Sobolev space (see 1]) and^{k ks} denotes its norm for any

s

^2R. For

s

0,

H

^0s() denotes the completion of

C

⁰¹() in

H

^s(), where

C

⁰¹() denotes the space of all innitely dieren- tiable functions on with compact supports in . Let

X

be a Banach space.

We denote by

C

^k(0

T

]

X

) the space of all

k

times continuously dieren- tiable functions dened on 0

T

] with values in

X

, and write

C

(0

T

]

X

) for

C

⁰(0

T

]

X

).

Let

a

(

x

) be the nonnegative function given in (1.6) satisfying (1.7) and

= (

^{1 n}) the unit normal on ; directed towards the exterior of . Suppose that ;¹ and ;² are given by (1.4) and (1.5), respectively, and ;¹ either is empty or has a nonempty interior relative to ;. Set

H

^;11() =^f

u

²

H

¹() :

u

= 0 on ;^1g

H

^;21() =^f

u

²

H

²() :

u

= 0 on ;^1g

V

=^f(

uv

)²(

H

¹())ⁿ(

L

²())ⁿ :

Z

;

m ud

; +

Z

vdx

= 0^g

W

=

V

if ;¹ = and

a

(

x

)0

(

H

^;11())ⁿ(

L

²())ⁿ

otherwise,

H=

W

L

²()

:

Further, if ;¹ = and

a

(

x

)0, we set

D

^;1 =ⁿ(

uv

) :

²(

H

²()^\

H

⁰¹())

(

uv

)²((

H

²())ⁿ(

H

¹())^n\

V

@u @

^{+ (}

+

)div(

u

)

+

am u

+

m v

= 0 on ;^2o

:

Otherwise, we set

D

^;1 =ⁿ(

uv

) :

²(

H

²()^\

H

⁰¹())

(

uv

)²(

H

^;21())ⁿ(

H

^;11())ⁿ

@u @

^{+ (}

+

)div(

u

)

+

am u

+

m v

= 0 on ;^2o

:

Note that the norm on

W

k(

uv

)^kW =1 2

Z

^jr

u

^j2+ (

+

)^jdiv(

u

)^j2+^j

v

^j2]

dx

+ 12^Z;2

am

^j

u

^j2

d

;¹⁼² (2

:

1) is equivalent to the usual one induced by (

H

¹())ⁿ (

L

²())ⁿ. When proving this, the only delicate case is when ;¹ = and

a

(

x

)0. We argue by contradiction. If the norms are not equivalent, then there is a sequence

f(

u

ⁿ

v

ⁿ)^g2

V

such that

Z

^jr

u

^nj2 +^j

u

^nj2+^j

v

^nj2]

dx

= 1

(2

:

2)

and

Z

j

u

^nj2

dx n

^Z

^jr

u

^nj2+ (

+

)^jdiv(

u

ⁿ)^j2+^j

v

^nj2]

dx:

(2

:

3) Then, we have

lim

n!1 Z

jr

u

^nj2

dx

= 0

(2

:

4) lim

n!1 Z

j

v

^nj2

dx

= 0

:

(2

:

5) Moreover, by (2.2), we may assume that ^f

u

^ng converges to

u

weakly in (

H

¹())ⁿ. Since the injection of (

H

¹())ⁿ into (

L

²())ⁿ is compact, we may assume that ^f

u

^ng converges to

u

strongly in (

L

²())ⁿ. Since in the sense of distribution

lim

n!1

@u

ⁿ

@x

^{i =}

@u

@x

ⁱ

(2.4) implies that

@u

@x

^{i = 0}

:

Thus

u

=

C

(constant)

:

Because (

u

ⁿ

v

ⁿ)²

V

, we have

Z

;

m u

ⁿ

d

; +

Z

v

ⁿ

dx

= 0

:

(2

:

6) By (2.5), we deduce that

lim

n!1 Z

v

ⁿ

dx

= 0

:

It therefore follows from (2.6) that

lim

n!1 Z

;

m u

ⁿ

d

; = 0

:

On the other hand, by (2.4) and the fact that^f

u

^ngconverges to

C

strongly in (

L

²())ⁿ, we can deduce that^f

u

^ngconverges to

C

strongly in (

H

¹())ⁿ. Thus, by the trace theorem, we have

C

^Z

;

m d

; = lim

n!1 Z

;

m u

ⁿ

d

; = 0

:

This shows that

C

= 0. Consequently, it follows from (2.4) and (2.5) that lim

n!1 Z

^jr

u

^nj2+^j

u

^nj2+^j

v

^nj2]

dx

= 0

which is in contradiction to (2.2).

If

a

(

x

)⁶0, the proof is similar. If ;^{1 6}=, this is just the consequence of Poincare inequality (see 1, p. 159]).

In the sequel, we use the energy norm on ^H:

k(

uv

)^kH=1 2

Z

^jr

u

^j2+ (

+

)^jdiv(

u

)^j2+^j

v

^j2+

^j

^j2]

dx

+ 12^Z;2

am

^j

u

^j2

d

;¹⁼² (2

:

7) for (

uv

)^2H, which is equivalent to the usual one induced by (

H

¹())ⁿ (

L

²())ⁿ

L

²().

We consider the thermoelastic system with a velocity feedback:

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

u

^00;

u

^;(

+

)^rdiv

u

+

^r

= 0 in (0

¹)

^0;

+

div

u

⁰ = 0 in (0

¹)

= 0 on ;(0

¹)

u

= 0 on ;¹(0

¹)

@u @

^{+ (}

+

)div(

u

)

+

am u

+

m u

⁰= 0 on ;²(0

¹)

u

(0) =

u

⁰

u

⁰(0) =

u

¹ (0) =

⁰ in

:

(2

:

8)

It is well known that problem (2.8) is well-posed (see 40]). In fact, the system generates a strongly continuous semigroup

S

(

t

) in ^H. This has been proved in 33] for the more general case of thermoviscoelasticity. However, the case ;¹ = and

a

(

x

) 0 is worth discussing in more detail. Indeed, from previous articles published in the literature, one may think that system (2.8) generates a semigroup in the following space with zero average:

H

0 =ⁿ(

uv

)²(

H

¹())ⁿ(

L

²())ⁿ

L

²() :

Z

u

(

x

)

dx

=

Z

v

(

x

)

dx

= 0^o

:

Actually this is a mistake. To see this, we dene the function

f

(

t

) by

f

(

t

) =

Z

u

(

t

)

dx:

Take (

u

⁰

u

¹

⁰)^2H0 such that

Z

;

m u

⁰

d

;⁶= 0

(2

:

9) and let

u

be the solution of (2.8) corresponding to this initial data. We

claim that ^Z

;

m u

⁰(

t

)

d

;⁶0

:

(2

:

10) If this is not true, then ^R;

m u

(

t

)

d

; is constant in time. In addition, we can show that the energy ^k(

u

(

t

)

u

⁰(

t

)

(

t

))^k2H decays to zero exponentially

as

t

^{! 1}. Consequently, ^;

m u

(

t

)

d

; also decays to zero exponentially since there is a positive constant

C

such that

j Z

;

m u

(

t

)

d

;^j

C

^k(

u

(

t

)

u

⁰(

t

)

(

t

))^kH

:

It therefore follows that

Z

;

m u

(

t

)

d

;0

which contradicts (2.9). By (2.10), we have

f

⁰⁰(

t

) =

Z

u

⁰⁰(

t

)

dx

=

Z

u

+ (

+

)^rdiv

u

^;

^r

]

dx

=

Z

;

@u @

^{+ (}

+

)div(

u

)

^;

]

d

;

=^;

Z

;

m u

⁰(

t

)

d

;

60

:

Hence,^R

u

(

t

)

dx

and ^R

u

⁰(

t

)

dx

are not always equal to zero along the so- lution trajectories of (2.8). Thus, ^H0 is not invariant under the ow given by (2.8). Consequently, system (2.8) does not generate a semigroup in^H0.

On the other hand,^H is invariant under the ow given by (2.8). In fact, the function

g

(

t

) =

Z

;

m u

(

t

)

d

; +

Z

u

⁰(

t

)

dx

is constant along the solution trajectories of (2.8). This is because

g

⁰(

t

) =

Z

;

m u

⁰(

t

)

d

; +

Z

u

⁰⁰(

t

)

dx

=

Z

;

m u

⁰(

t

)

d

; +

Z

u

+ (

+

)^rdiv

u

^;

^r

]

dx

=

Z

;

m u

⁰(

t

)

d

; +

Z

;

@u @

^{+ (}

+

)div(

u

)

^;

]

d

;

= 0

:

This is why the chosen space for solving (2.8) is^H.

Generally speaking, one can say that a semigroup preserves a quantity if and only if its generator does it. More precisely, let us consider

u

⁰ =

Au t >

0

u

(0) =

u

⁰

⁽²

:

11)

on a Banach space

X

, where

A

is a unbounded linear operator on

X

with the domain

D

(

A

) and

u

^{0 2}

X

. Assume that

A

generates a semigroup. let

L

:

X

^{! R}be a linear and bounded functional. Then, the following two conditions are equivalent:

(i)

Lu

(

t

) remains constant in time for every solution

u

of (2.11) (ii)

L

(

Au

⁰) = 0 for all

u

^{0 2}

D

(

A

).

To see that (ii) implies (i), we rst assume that

u

^{0 2}

D

(

A

), and then

u

(

t

)²

D

(

A

) for every

t

0. Applying

L

to equation (2.11), we obtain

dL

(

u

(

t

))

dt

⁼

L

(

u

⁰) =

L

(

A

(

u

(

t

)) = 0

:

Therefore,

L

(

u

(

t

)) remains constant in time. If

u

^{0 2}

X

, by density, we can show that (i) still holds.

For the other implication, note that

Au

⁰ is the limit of (

u

(

t

)^;

u

(0))

=t

in

X

as

t

^! 0,

u

being the solution of the equation with initial data

u

⁰. Now, in view of (i), we have

L

((

u

(

t

)^;

u

(0))

=t

) = 0 for all

t

. On the other hand, this quantity should converge to

L

(

Au

⁰) as

t

^! 0. This shows that (ii) holds.

Note that, in our situation, the functional

L

is given by

L

(

uv

) =

Z

;

m ud

; +

Z

vdx:

Since system (2.8) generates a strongly continuous semigroup

S

(

t

) in ^H, then for every initial data

(

u

⁰

u

¹

⁰)^2H system (2.8) has a unique solution (

uu

⁰

) with

(

uu

⁰

)²

C

(0

¹)^H)

:

Moreover, if

(

u

⁰

u

¹

⁰)²

D

^;1

then (

uu

⁰

)²

C

(0

¹)

D

^;1)

:

In order to ensure that the solution

u

has su!cient regularity to perform the integrations by parts we will do, in this paper, we suppose that

;^1\;² =

:

(2

:

12)

Under this assumption, by the standard elliptic regularity properties, we

have

u

²

C

(0

¹)(

H

²())ⁿ)

:

(2

:

13)

This regularity property is needed for the proof of the following theorems. If (2.12) does not hold, then (2.13) fails in general even in the case of the wave equation (see 18]). However, even though (2.12) fails, Komornik and Zuazua 16] still proved the uniform boundary stabilization of the wave equation in the case where

n

3. Their proof was based on an inequality established

by Grisvard 11] for the solution of the wave equation with such boundary singularity. Whether or not Komornik and Zuazua's result still holds for the system of thermoelasticity with such boundary singularity is an open problem as the similar inequality of Grisvard has not been proved yet in the literature.

In order to state our main results, we introduce some constants as follows.

Set

R

(

x

⁰) = max

x2

j

m

(

x

)^j= max

x2

j

n

X

k =1

(

x

^k;

x

^0k)^{2 j1=2}

(2

:

14)

K

(

a

) = 2

A

²

R

(

x

⁰)²

^{+ (2;}

n

)

A

(2

:

15) where

m

(

x

) =

x

^;

x

⁰ and

A

= max

x2;

2

a

(

x

). Let

be the smallest positive constant such that

Z

;

2

j

u

^j2

d

;

^2k

u

^k2H1()

⁸

u

²

H

¹()

:

(2

:

16) Let

⁰ be the smallest positive constant such that

k(

uv

)^kH1()L2()

^0k(

uv

)^kW

⁸(

uv

)²

W:

(2

:

17) The thermoelastic energy of (2.8) is dened by

E

(

ut

) =^k(

u

(

t

)

u

⁰(

t

)

(

t

))^k2H

:

(2

:

18) We now state our main results of this paper.

Theorem 2.1. Let ;¹ and ;² be given by (1.4) and (1.5), respectively, satisfying (2.12). If the function

a

(

x

) satises

K

(

a

)

R

(

x

⁰)

²

²⁰

<

2

for

n

2

(2

:

19) or

a

(

x

) (

n

^;2)

2

R

²(

x

⁰) on ;²

for

n

3

(2

:

20) then there exists a positive constant

!

, independent of (

u

⁰

u

¹

⁰), such that

E

(

ut

)

E

(

u

0)

e

^{1;! t}

⁸

t

0

(2

:

21) for all solutions of (2.8) with (

u

⁰

u

¹

⁰)^2H. Further, the constant

!

can be given by

!

=

k

^;1

(2

:

22) where

k

=

8

<

:

k1+k2+k3

2;K(a)R(x 0

) 2

2

0

;"(R(x 0

) 2

2

0 +

2

+ 2

0

)

n

2,

k

1 +k

2 +k

3

2;"(R(x 0

) 2

2

0 +

2

+ 2

0

)

n

3, (2

:

23)

k

¹ = 2

R

²(

x

⁰)

^{+ (}

n

^;1)²

4

"

^{+ 1}

(2

:

24)

k

² = 4max 2

²

R

²(

x

⁰) (

n

^;1)²

²⁰

4

(2

:

25)

k

³ =

R

²(

x

⁰)

"

⁺

(

n

^;1)²

4

"

⁺

²⁰

(2

:

26) 0

< " <

8

<

:

2;K(a)R(x 0

) 2

2

0

R(x 0

) 2

2

0 +

2

+ 2

0

n

2,

2

R(x 0

) 2

2

0 +

2

+ 2

0

n

3. (2

:

27) In the case of the Lame system (1.8), Theorem 2.1 still holds even though

a

is large. Namely, we have

Theorem 2.2. Let ;¹ and ;² be given by (1.4) and (1.5), respectively, satisfying (2.12). Let

a

(

x

) be any nonnegative function on ;² satisfying (1.7). Then there are positive constants

M !

, independent of (

u

⁰

u

¹), such that

E

(

ut

)

ME

(

u

0)

e

^{;! t}

⁸

t

0

(2

:

28) for all solutions of (1.8) with(

u

⁰

u

¹)²

W

.

In this case, the energy

E

(

ut

) is given by

E

(

ut

_{) = 12}^Z

h

j

u

⁰(

xt

)^j2+

^jr

u

(

xt

)^j2+ (

+

)^jdiv

u

(

xt

)^j2i

dx

+ 12

Z

;

2

am

^j

u

(

t

)^j2

d

;

:

⁽²

:

29) If the function

a

(

x

) satises (2.19) or (2.20), then the constants

M

and

!

in Theorem 2.2, of course, are the same as in Theorem 2.1. However, if

a

(

x

) does not satisfy (2.19) and (2.20), then

M

and

!

can not be explicitly given since in this case the proof of Theorem 2.2 is based on a control-theoretic method (see 39] and 46]), which is not constructive.

Theorem 2.2 is an answer to the open question raised by Alabau and Komornik 2], who proved the exponential stability of the Lame system for the case where

a

(

x

)

m

a

⁰is a small enough constant and is a ball, and then conjectured that this result probably remains true even for

a

⁰ large.

We now consider the controllability problem:

8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

u

^00;

u

^;(

+

)^rdiv

u

+

^r

= 0 in (0

¹)

^0;

+

div

u

⁰= 0 in (0

¹)

= 0 on ;(0

¹)

u

= 0 on ;¹(0

¹)

@u @

^{+ (}

+

)div(

u

)

+

am u

=

on ;²(0

¹)

u

(0) = 0

u

⁰(0) = 0 (0) = 0 in

(2

:

30)

where

= (

^{1 n}) is a control acting on the boundary ;².

Given

T >

0, let

Q

= (0

T

)

# = ;(0

T

)

#¹ = ;¹(0

T

) and

#² = ;²(0

T

).

The partial exact controllability problem can be stated as follows: Given

T >

0

for every state(

u

⁰

u

¹)²

W

, we want to nd a control

in a suitable function space such that the solution of (2.30) satises

u

(

xT

) =

u

⁰(

x

)

u

⁰(

xT

) =

u

¹(

x

) in

(2

:

31)

disregarding the values of the temperature.

As stated in 30, p. 34-35], this is equivalent to steering every initial state (

u

⁰

u

¹) of the displacement in the function space to the state (

u

(

T

)

u

⁰(

T

)) = (0

0), disregarding the values of the temperature.

Let

be the smallest positive constant such that

kdiv

u

^k;1

^k

u

^k0

⁸

u

²

L

²()

:

(2

:

32) Theorem 2.3. Let ;¹ and ;² be given by (1.4) and (1.5), respectively, satisfying (2.12). Assume that the function

a

(

x

) satises (2.19) or (2.20).

Let

T

⁰ be large enough such that if

T T

⁰ then

2(1 +

²

²

²

T

²)

e

^{1;! T}

<

1

(2

:

33) where

!

is the constant in Theorem 2.1. Then for any (

u

⁰

u

¹)²

W

, there exists a boundary control function

(

xt

) with

p

m

²⁽

L

²(#²))ⁿ

such that the solution of (2.30) satises (2.31). Moreover, there exists pos- itive constant

c

, independent of (

u

⁰

u

¹), such that

k

p

m

^k(L2(2))n

c

^k(

u

⁰

u

¹)^kW

:

(2

:

34) In comparison with the existing literature, the main contribution of The- orem 2.3 is that there is no smallness restriction on the coupling parameters

and

.

In the case of the Lame system

8

>

>

>

<

>

>

>

:

u

^00;

u

^;(

+

)^rdiv

u

= 0 in (0

¹)

u

= 0 on ;¹(0

¹)

@u @

^{+ (}

+

)div(

u

)

+

am u

=

on ;²(0

¹)

u

(0) = 0

u

⁰(0) = 0 in

(2

:

35) Theorem 2.3 still holds even though

a

is large. Namely, we have

Theorem 2.4. Let ;¹ and ;² be given by (1.4) and (1.5), respectively, satisfying (2.12). Let

a

(

x

) be any nonnegative function on ;² satisfying (1.7). Let

T

⁰ be large enough such that if

T T

⁰ then

Me

^{;! T}

<

1

(2

:

36) where

M

and

!

are the constants in Theorem 2.2. Then for any (

u

⁰

u

¹) ²

W

, there exists a boundary control function

(

xt

) with

p

m

²⁽

L

²(#²))ⁿ

such that the solution of (2.35) satises (2.31). Moreover, there exist positive constants

c

¹

c

², independent of (

u

⁰

u

¹), such that

c

^1k(

u

⁰

u

¹)^{kW k}

p

m

^k(L2(2))n

c

^2k(

u

⁰

u

¹)^kW

:

(2

:

37)

3. Proof of main results

Let us rst describe our steps of proof of Theorems 2.1-2.4. We rst prove Theorem 2.1, and then prove Theorem 2.3. Since the Lame system is a special case of the thermoelastic system, Theorems 2.2 and 2.4 are simulta- neously proved in the case that

a

(

x

) satises (2.19) or (2.20). Finally, we prove Theorems 2.2 and 2.4 in the case that

a

(

x

) is large (i.e.,

a

(

x

) does not necessarily satisfy (2.19) or (2.20)).

In the sequel, the summation convention is assumed. We recall the con- stants

R

(

x

⁰)

K

(

a

) and

⁰ are given by (2.14)-(2.17), respectively.

Proof of Theorem 2.1. The idea of the proof of Theorem 2.1 is simple.

According to Theorem 8.1 of 15, p. 103], it su!ces to show that

Z

1

s

E

(

ut

)

dt

kE

(

us

)

⁸

s

0

where

k

is given by (2.23). However, the proof of this inequality is generally not easy. We use the multiplier techniques to attack it.

Given 0

s < T

, let

Q

^s = (

sT

)

#^s = ;(

sT

)

#^1s= ;¹(

sT

) and #^2s= ;² (

sT

).

We may as well assume that (

u

⁰

u

¹

⁰) ²

D

^;1 since the general case (

u

⁰

u

¹

⁰)^2Hcan be handled by density. Then (

u

) is a classical solution of (2.8). Multiplying the rst equation of (2.8) by

m

^k

@u

ⁱ

@x

^k and integrating on

Q

^s, we have

Z

Q

s

m

^k

@u

ⁱ

@x

^k

u

⁰⁰ⁱ

dxdt

=

u

⁰ⁱ(

t

)

m

^k

@u

ⁱ(

t

)

@x

^k

T

s

;

12

Z

s

m

^k

^{k j}

u

^0ij2

d

# +

n

2

Z

Qs

j

u

^0ij2

dxdt

⁽³

:

1)

Z

Q

s

m

^k

@u

ⁱ

@x

^k

u

ⁱ

dxdt

=

Z

s

@u

ⁱ

@ m

^k

@u

ⁱ

@x

^k

d

#^;

Z

Qs

@u

ⁱ

@x

^j

@

@x

^j

m

^k

@u

ⁱ

@x

^k

dxdt

=

Z

s

@u

ⁱ

@ m

^k

@u

ⁱ

@x

^k

d

#^;

Z

Qs

jr

u

^ij2

dxdt

;

12

Z

Q

s

m

^k

@

@x

^{k jr}

u

^ij2

dxdt

=^Z

s

@u

ⁱ

@ m

^k

@u

ⁱ

@x

^k

d

#^;1 2

Z

s

m

^k

^kjr

u

^ij2

d

# +

n

^;2

2

Z

Q

s

jr

u

^ij2

(3

:

2)

Qs

m

^k

@u

ⁱ

@x

^k

@

@x

^i(div

u

)

dxdt

=

Z

s

div(

u

)

m

^k

@u

ⁱ

@x

^k

ⁱ

d

#^;

Z

Q

s

div(

u

)

@

@x

ⁱ

m

^k

@u

ⁱ

@x

^k

dxdt

=

Z

s

div(

u

)

m

^k

@u

ⁱ

@x

^k

ⁱ

d

#^;

Z

Q

s

jdiv(

u

)^j2

dxdt

;

12

Z

Q

s

m

^k

@

@x

^k

jdiv(

u

)^j2

dxdt

=

Z

s

div(

u

)

m

^k

@u

ⁱ

@x

^k

ⁱ

d

#^; 1 2

Z

s

m

^k

^kjdiv(

u

)^j2

d

# +

n

^;2

2

Z

Qs

jdiv(

u

)^j2

dxdt:

(3

:

3)

Multiplying the rst equation of (2.8) by

u

ⁱ and integrating over

Q

^s, we obtain

Z

Q

s

j

u

^{i0j2 ;}

^jr

u

^{ij2 ;}(

+

)^jdiv

u

^j2

dxdt

= (

u

⁰ⁱ(

t

)

u

ⁱ)^T

s

+

^Z

Q

s

u

ⁱ

@

@x

ⁱ

dxdt

;

^Z

s

@u

ⁱ

@ u

ⁱ

d

#^;(

+

)

Z

s

u

ⁱ

ⁱdiv(

u

)

d

#

:

(3

:

4)

It therefore follows from (2.8) and (3.1)-(3.4) that

2

Z

T

s

E

(

ut

)

dt

=

Z

s

m

^k

^kj

u

^{0ij2 ;}

^jr

u

^ij2;(

+

)^jdiv(

u

)^j2]

d

# + 2

Z

s

@u @

^{i + (}

+

)div(

u

)

ⁱ]

m

^k

@u

ⁱ

@x

^k

d

# + (

n

^;1)

Z

s

@u @

^{i+ (}

+

)div(

u

)

ⁱ]

u

ⁱ

d

# +

Z

2s

am

^j

u

^ij2

d

#

;2

u

⁰ⁱ(

t

)

m

^k

@u

ⁱ(

t

)

@x

^{k +}

n

^;1 2

u

^iT

s

;2

^Z

Q

s

r

(

m

^k

@u

@x

^{k +}

n

^;1

2

u

)

dxdt

+

Z

Q

s

j

^j2

dxdt