| Issue |
ESAIM: COCV
Volume 8, 2002
A tribute to JL Lions
|
|
|---|---|---|
| Page(s) | 143 - 167 | |
| DOI | https://doi.org/10.1051/cocv:2002058 | |
| Published online | 15 August 2002 | |
The dynamical Lame system: regularity of solutions, boundary controllability and boundary data continuation
1
Saint-Petersburg Department of
the Steklov Mathematical Institute (POMI), Fontanka 27,
St. Petersburg 191011, Russia; belishev@pdmi.ras.ru.
2
Department of Mathematics, University of Virginia,
Charlottesville, VA 22901, USA; il2v@weyl.math.virginia.edu.
Received:
31
December
2001
The boundary control problem for the dynamical Lame system
(isotropic elasticity model) is considered. The continuity of
the “input → state" map in L2-norms is established. A structure of the
reachable sets for arbitrary T>0 is studied.
In general case, only the first component 
of the
complete state
may be controlled, an approximate controllability occurring in
the subdomain filled with the shear (slow) waves.
The controllability results are applied to the problem of the boundary
data continuation. If T0 exceeds the time needed
for shear waves to fill the entire domain, then the response
operator (“input → output" map) 
uniquely determines
RT for any T>0. A procedure recovering R∞
via
is also described.
Mathematics Subject Classification: 93C20 / 74B05 / 35B65 / 34K35
Key words: Isotropic elasticity / dynamical Lame system / regularity of solutions / structure of sets reachable from the boundary in a short time / boundary controllability.
© EDP Sciences, SMAI, 2002
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