Exponential stabilization of a neutrally delayed viscoelastic Timoshenko beam

A Timoshenko type beam with a viscoelastic damping in the rotational displacement is considered. Taking into account a neutral type delay, we intend to establish sufficient conditions ensuring a fast stability result, despite the previously observed destabilizing effect due to delays in the system. This requires determining suitable classes of kernels in the neutral delay and suitable relaxation functions in the memory term. We shall proceed with the multiplier technique and try to find appropriate Lyapunov type functional with which we will modify our classical energy functional. This modified energy functional must satisfy a differential inequality or an integro-differential inequality leading to a decay with some specific rate of convergence.

A Timoshenko type beam with a viscoelastic damping in the rotational displacement is considered. Taking into account a neutral type delay, we intend to establish sufficient conditions ensuring a fast stability result, despite the previously observed destabilizing effect due to delays in the system. This requires determining suitable classes of kernels in the neutral delay and suitable relaxation functions in the memory term. We shall proceed with the multiplier technique and try to find appropriate Lyapunov type functional with which we will modify our classical energy functional. This modified energy functional must satisfy a differential inequality or an integro-differential inequality leading to a decay with some specific rate of convergence.