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
Previous Page Next Page