Null-Controllability of a System of Linear Thermoelasticity

Abstract

We consider a linear system of thermoelasticity in a compact, C infin, n-dimensional connected Riemannian manifold. This system consists of a wave equation coupled to a heat equation. When the boundary of the manifold is non‐empty, Dirichlet boundary conditions are considered. We study the controllability properties of this system when the control acts in the hyperbolic equation (and not in the parabolic one) and has its support restricted to an open subset of the manifold. We show that, if the control time and the support of the control satisfy the geometric control condition for the wave equation, this system of thermoelasticity is null-controllable. More precisely, any finite‐energy solution can be driven to zero at the control time. An analogous result is proved when the control acts on the parabolic equation. Finally, when the manifold has no boundary, the null‐controllability of the linear system of three‐dimensional thermoelastic ity is proved.