Contemporary Mathematics

Volume 209, 1997

BOUNDARY OBSERVABILITY AND CONTROLLABILITY

OF LINEAR ELASTODYNAMIC SYSTEMS

FATIHA ALA8AU AND VILMOS KOMORNIK

ABSTRACT. In [10) Lions obtained several boundary observability and exact

controllability results for homogeneous and isotropic linear elastodynamic sys-

tems. We generalize these results to the non-isotropic case. This requires some

new identities. We also study the optimality of our estimates.

1.

08SERVABILITY

Let

(aijkt)

be a tensor such that

aijkl = a;ikl = aklij

(all indices run over the integers 1, 2, 3), satisfying for some a 0 the ellipticity

condition

(1)

for every symmetric tensor

€ij.

(Here and in the sequel we shall use the summation

convention for repeated indices.)

Let {} be a non-empty bounded open set in

JR3

having a boundary

r

of class C2

•

Given a function

e

=

(e17 e2

,

6) : 0

-t

IR3

,

we shall use the notations

€ij

=

~(ei,j

+

e;,i),

O'ij

=

aijkl€kl,

where

ei,i = aeif8x;

and

e;,i = ae;/8xi.

Hit is necessary to be more precise, we

shall write

ei;(e)

and

ai;(e)

instead of

ei;,

aii·

Consider the problem

{ e~'- O'ij,j =

0 in {}

X

IR,

ei

=

0 on

r

X

IR,

~i(o) =

e?

and

eHo)

=

el

in

n,

~

=

1,2,3.

(2)

We recall (see e.g. [2] or [4]) that this problem is well-posed in the following

sense:

(i) Given

(eo,el)

E

HJ(0)3 x L2 (0)

3

arbitrarily, the problem (2) has a unique

(so-called weak) solution

e

E

C(IR; HJ(0)3

)

n

C

1

(1R; L

2

(0)

3

);

1991 Mathematics Subject Classification. 35L55, 35Q72, 93805, 93807, 93C20.

Key words and phrases. observability, controllability, partial differential equation, elasticity.

The second author is grateful to the organisers of the conference for their invitation and to the

INRIA Lorraine (Projet Numath) for supporting his travel expenses.

©

1997 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/209/02754