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We consider the Klein-Gordon system posed in an inhomogeneous medium withsmooth boundary subject to a local viscoelastic damping distributed around aneighborhoodof the boundary according to the Geometric Control Condition. Weshow that the energy of the system goes uniformly and exponentially to zero forall initial data of finite energy taken in bounded sets of finite energyphase-space. For this purpose, refined microlocal analysis arguments areconsidered by exploiting ideas due to Burq and Gerard . By using sharp Carlemanestimates we prove a unique continuation property for coupled systems.

Uniform stabilization for the Klein-Gordon system in a inhomogeneous medium with locally distributed damping