Stability of a damped hyperbolic timoshenko system coupled with a heat equation

We consider a one-dimensional damped hyperbolic Timoshenko beam that is coupled with a heat equation. When its wave speeds are different, it is known that the Timoshenko beam that is coupled with a heat equation, under Cattaneo's law, does not have exponential stability. With two internal dampings being introduced, in this paper, we show that the system under Cattaneo's law is exponentially stable. We also show the exponential stability when the system is considered under Fourier's law.