Long-time behaviour of a translational thermoelastic Timoshenko system with second sound

We consider a Timoshenko system coupled with heat equations modelled by Cattaneo's law. The coupling is through the transverse displacement. Both ends of the beam are dynamic. One end of the beam is fixed to a base in a translational motion and a tip mass is attached to the other end. We design a feedback control acting at the base. It is shown that this feedback control is a reasonable one and is capable of stabilizing the system. We prove an exponential and a polynomial stability result using the multiplier technique. To this end, we introduce new functionals to form a suitable Lyapunov functional.