Polynomial stability of the Rao-Nakra beam with a single internal viscous damping

In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations. For a few special cases, optimal orders are confirmed. We also study the synchronization of the model with viscous damping on the transversal displacement. Our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations.