Control of a thermo-viscoelastic translational Timoshenko beam

In this paper, we study a beam fixed to a base in a translational motion at one end and to a tip mass at its free end. The beam is subject to some vibrations and it is modelled as a Timoshenko structure coupled with a heat equation of hyperbolic type. The only damping we employ is the thermal effect. We come up with a control force capable of driving the system to the equilibrium state in an exponential manner. It is also explained how to get any desirable rate of decay. Our method is based on the multiplier technique.