Delayed displacement feedback stabilization of two coupled wave equations with joint anti-damping

In this paper, we consider the stabilization problem of two coupled wave equations with joint anti-damping. By designing a novel transformation, the wave system is transformed into a system with damping term at the joint point, and the feedback controller with delayed displacement is designed. The invertibility of the transformation is proven through the mathematical induction. Furthermore, by using the Riesz basis method and the PDE approach, the well-posedness and exponential stability of the non-dissipative closed-loop system are established. Simulation results are presented to verify the effectiveness of the feedback control law.