Adaptive stabilization for a class of wave equations with uncertain nonlinear boundary actuator dynamics

This paper investigates the stabilization of a wave equation with uncertain boundary actuator dynamics. The remarkable characters of the system under investigation are revealed by the essential nonlinearities involved in the boundary actuator and the serious parametric unknowns coming from anti-damping coefficient at uncontrolled boundary and those in the actuator, which result into the incapability of the traditional control methods on this topic. For this, a novel adaptive control strategy is proposed in this paper. Specifically, a Riemann variables based state transformation is first introduced to change the original system into a new one whose stabilization implies that of the original system. Then, a state-feedback controller joint with some adaptive laws which are smartly chosen for the compensation of unknown parameters is explicitly designed by a finite-dimensional backstepping procedure. Owing to the compensation of boundary actuator dynamics, the performance analysis of the closed-loop system is more complicated than that of the related literature. For this, a rigorous recursive step is given to show the stability of the resulting closed-loop system. Finally, a numerical example is provided to verify the effectiveness of the proposed theoretical results.