Boundary stabilization of wave equation with velocity recirculation

Nonlocal terms have been the mainstay of the applications of partial differential equation (PDE) backstepping methods to parabolic PDEs. The problem of similar nonlocal terms for wave equations is still open. For wave equations, similar nonlocal terms have not been studied. In this paper, we open the topic of exploration of control of wave PDEs with nonlocal terms. This paper is concerned with the wave equation with in-domain feedback/recirculation of a boundary velocity with a spatially constant coefficient. Due to this nonlocal term, the passivity of the wave equation is destroyed. We first design an explicit state feedback controller to achieve exponential stability for the closed-loop system. Then, we design the output feedback by using infinite-dimensional observer. The backstepping approach is adopted in investigation. It is shown that by using two measurements only, the output feedback makes the closed-loop system exponentially stable.