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The dynamical Lame system: regularity of solutions, boundary controllability and boundary data continuation

Published online by Cambridge University Press:  15 August 2002

M. I. Belishev  and
I. Lasiecka
M. I. Belishev
Affiliation:
Saint-Petersburg Department of the Steklov Mathematical Institute (POMI), Fontanka 27, St. Petersburg 191011, Russia; belishev@pdmi.ras.ru.
I. Lasiecka
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22901, USA; il2v@weyl.math.virginia.edu.
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Abstract

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 $u(\cdot ,T)$ of the complete state $\{ u(\cdot ,T),u_t(\cdot ,T)\}$ 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) $R^{2T_0}$ uniquely determines RT for any T>0. A procedure recovering R∞via$R^{2T_0}$ is also described.

Keywords

Isotropic elasticitydynamical Lame systemregularity of solutionsstructure of sets reachable from the boundary in a short timeboundary controllability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 143 - 167
Copyright
© EDP Sciences, SMAI, 2002

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