Delay compensated control of a wave equation with velocity recirculation

This paper considers the control design and stability analysis of a wave equation with velocity recirculation and an input delay. When both an in-domain nonlocal term involving the velocity at an intermediate point and an input delay are present, the instability of the wave system is enhanced, posing significant challenges to the control design and stability analysis. We first transform the input delay into a first-order transport equation, resulting in a wave-transport coupled system. Then, we design three novel transformations to convert the wave-transport coupled system into two intermediate auxiliary systems and a final stable target system. Simultaneously, a delay-compensated control is proposed for the wave system with velocity recirculation using the backstepping method. Finally, we prove that the closed-loop system is well-posed and exponentially stable. Furthermore, we provide some numerical simulations to demonstrate the validity of the control.