Polynomial stability of the Timoshenko system by one boundary damping

In this paper, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. This system, which models the dynamics of a beam, is a hyperbolic system with two wave speeds. Assuming that the wave speeds are equal, we prove non-uniform stability and an optimal polynomial energy decay rate is obtained. Otherwise, if the ratio of the wave speeds is a rational number, we show that the decay rate is of polynomial type.