Unicité et contrôle pour le système de Lamé (Uniqueness and controllability for the Lamé system). (French) Zbl 1007.35006

Bellassoued, Mourad

ESAIM, Control Optim. Calc. Var. 6, 561-592 (2001).

The Lamé system of the form \[ Pu\triangleq \biggl[\bigl( \partial^2_t- \mu(t,x)\Delta \bigr)id_n- \nu(t,x)\nabla (\text{div} \cdot)+R (t, x; \partial_t,\partial_x) \biggr]u=0 \] is considered and it is shown to have local Cauchy uniqueness across a noncharacteristic surface. The proof requires a general Fourier transform to be applied to the symbol of \(P\) and certain micro-local techniques to obtain a generalized Carleman inequality from which uniqueness follows by standard arguments.
The result is applied to the boundary control of the Lamé system for both Dirichlet and Neumann boundary conditions using the Hilbert uniqueness approach of Lions.

Reviewer: Stephan Paul Banks (Sheffield)

MSC:

35B37	PDE in connection with control problems (MSC2000)
93B05	Controllability
74B05	Classical linear elasticity

Keywords:

elastic wave equation; local Cauchy uniqueness; non-characteristic surface; Fourier transform; Carleman inequality; boundary control; Hilbert uniqueness approach of Lions

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