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

Belishev, M. I.; Lasiecka, I.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 143-167.

The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input $\to$ state” map in $L_2$-norms is established. A structure of the reachable sets for arbitrary $T>0$ is studied. In general case, only the first component $u(·,T)$ of the complete state $\{u(·,T),u_t(·,T)\}$ 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 $T_0$ exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input $\to$ output” map) $R^{2T_0}$ uniquely determines $R^T$ for any $T>0$. A procedure recovering $R^{\infty }$ via $R^{2T_0}$ is also described.

Classification : 93C20, 74B05, 35B65, 34K35
Mots clés : isotropic elasticity, dynamical Lame system, regularity of solutions, structure of sets reachable from the boundary in a short time, boundary controllability

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