Stabilization of a wave equation with a distributed control

In this paper, a distributed control isproposed to consider the stabilization of a wave equation. First, we transfer the distributed control problem into a boundary control problem via introducing a simple transformation including a new control function. Second, combining with the stabilization controller of the intermediate boundary control system and hidden regularity of the wave equation, we obtain the distributed feedback control of the original wave system. Third, we prove the exponential stability of the closed-loop wave system. Then, we point out that the presented procedure for the wave system is also valid for the stabilization of the heat equation with a distributed control considered by Tsubakino et al. (2012). Finally, a numerical result is given, which shows the validity of the distributed feedback controller for the wave system designed by backstepping method.