Boundary state feedback exponential stabilization for a one-dimensional wave equation with velocity recirculation

In this paper, we consider boundary state feedback stabilization of a one-dimensional wave equation with in-domain feedback/recirculation of an intermediate point velocity. We firstly construct an auxiliary control system which has a nonlocal term of the displacement at the same intermediate point. Then by choosing a well-known exponentially stable wave equation as its target system, we find one backstepping transformation from which a state feedback law for this auxiliary system is proposed. Finally, taking the resulting closed-loop of the auxiliary system as a new target system, we obtain another backstepping transformation from which a boundary state feedback controller for the original system is designed. By the equivalence of three systems, the closed-loop of original system is proved to be well-posed and exponentially stable. Some numerical simulations are presented to validate the theoretical results. (C) 2019 Elsevier Ltd. All rights reserved.