Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control

We consider systems of Timoshenko type in a one-dimensional bounded domain. The physical system is damped by a single feedback force, only in the equation for the rotation angle, no direct damping is applied on the equation for the transverse displacement of the beam. Moreover the damping is assumed to be nonlinear with no growth assumption at the origin, which allows very weak damping. We establish a general semi-explicit formula for the decay rate of the energy at infinity in the case of the same speed of propagation in the two equations of the system. We prove polynomial decay in the case of different speed of propagation for both linear and nonlinear globally Lipschitz feedbacks.