Stabilization and controllability for the transmission wave equation

We address the problem of control of the transmission wave equation. In particular, we consider the case where, due to total internal reflection of waves at the interface, the system may not be controlled from exterior boundaries. We show that such a system can be controlled by introducing both boundary control along the exterior boundary and distributed control near the transmission boundary and give a physical explanation why the additional control near the transmission boundary might be needed for some domains.