Stability result for the timoshenko system with a time-varying delay term in the internal feedbacks

We study the exponential stability of the Timoshenko beam system by interior time-dependent delay term feedbacks. The beam is clamped at the two hand points subject to two internal feedbacks: one with a time-varying delay and the other without delay. Using the variable norm technique of Kato, it is proved that the system is well-posed whenever an hypothesis between the weight of the delay term in the feedback, the weight of the term without delay and the wave speeds. By introducing an appropriate Lyapunov functional the exponential stability of the system is proved. Under the imposed constrain on the weights of the feedbacks and the wave speeds, the exponential decay of the energy is established via a suitable Lyapunov functional.