Carleman Estimates and Exact Controllability Results for a Transmission System of two Nonconservative wave Equations

We consider a transmission system of two nonconservative wave equations with nonconstant coefficients in the one-dimensional setting. We investigate the exact (boundary and internal) controllability of this system using a single control. To solve these exact controllability problems, we rely on the duality approach which consists in proving inverse or observability inequality for the corresponding adjoint system. To prove these observability inequalities, we establish new Carleman estimates for the corresponding uncontrolled transmission system. These Carleman estimates require the introduction of new weight functions adapted to the coefficients of the principal operators. In particular, our Carleman estimates with internal observation are new in the framework of transmission systems.