Controllability of a coupled wave system with a single control and different speeds

Abstract

We consider an exact controllability problem in a smooth bounded domain, for a coupled wave system, with different speeds and a single control acting on a subdomain satisfying the Geometric Control Condition and on one speed only. Actions for the wave equations with the second speed are obtained through a coupling term. Firstly, we construct appropriate state spaces with compatibility conditions associated with the coupling structure. Secondly, in these well-prepared spaces, we prove that the coupled wave system is exactly controllable if and only if the coupling structure satisfies an operator Kalman rank condition.