Exponential boundary stabilization of a one-dimensional anti-damped wave equation via integral transformation

Abstract

This work investigates the boundary stabilization of a {one-dimensional} wave equation subject to anti-damping effects. The objective is to derive an explicit boundary control that guarantees exponential dissipation of the system energy within a suitable functional framework. The approach is based on a Volterra integral transformation, which converts the original unstable dynamics into a target system exhibiting favorable stability properties. The associated kernels are characterized as solutions of a coupled hyperbolic system posed on a triangular domain. The well-posedness of this target model is established using semigroup theory. A Lyapunov functional is then constructed to design a stabilizing boundary feedback for the transformed system. By combining this control law with the inverse transformation, an explicit feedback controller for the original system is obtained. In addition, a similarity relationship between the semigroups of the original and target systems is derived, allowing the exponential stability of the latter to be transferred to the closed-loop dynamics of the former. Finally, numerical experiments are provided to illustrate the performance and effectiveness of the proposed control methodology.