Stabilization and Discretization of the Coupled Heat and Wave Equations

In this paper, we consider the stabilization of the coupled heat and wave equations under the static feedback or the dynamic feedback. Moreover, we make the coupled systems discretized by using the finite-volume approach, and then we consider the stabilized properties of the discrete systems. First, for the coupled system under the static feedback, it is shown that the system is exponentially stable by using the Lyapunov method, and then the corresponding discrete system can be shown to be exponentially stable by constucting the discretized Lyapunov function. Second, for the coupled system under the dynamic feedback, we also show that both of the system and its discrete scheme are exponentially stable. Third, numerical simulations are given to show the effectiveness of the stable controllers.