On the Stabilization for Timoshenko System with Past History and Frictional Damping Controls

Abstract. In this paper, we consider vibrating system of Timoshenko type in one-dimensional bounded domain with complementary past history and frictional damping controls acting only in the equation for the rotation angle. We show that the dissipation given by this complementary controls is strong enough to guarantee the stability of the system in case of the same wave propagation speeds as well as in the opposite case. We establish in each case a general decay estimate of the solution. In the particular case when the wave propagation speeds are different and the frictional damping is linear, we give a relationship between the smoothness of the initial data and the decay rate of the energy. At the end of the paper, we discuss the well-posedness and smoothness of solution using the semigroup theory.