Stability for the Timoshenko Beam System with Local Kelvin-Voigt Damping

In this paper, we consider a vibrating beam with one segment made of viscoelastic material of a Kelvin.Voigt (shorted as K.V) type and other parts made of elastic material by means of the Timoshenko model. We have deduced mathematical equations modelling its vibration and studied the stability of the semigroup associated with the equation system. We obtain the exponential stability under certain hypotheses of the smoothness and structural condition of the coefficients of the system, and obtain the strong asymptotic stability under weaker hypotheses of the coefficients.