Exponential Convergence for Wave Equations with Dynamic Boundary Condition

This paper is devoted to the output-feedback control of wave equations with dynamic boundary condition. Essentially different from the related literature, less information is available for feedback and more general boundary condition is imposed at the origin. To solve the problem, an output-feedback control scheme is proposed by flexibly using the backstepping method, operator semigroup theory and Lyapunov function method. First, an appropriate observer is constructed to estimate the unmeasured states of the original system. Then, by the idea of backstepping method, an output-feedback controller is successfully constructed in two steps. It can be proven that, by choosing the design parameters appropriately and using the operator semigroup theory, the output-feedback controller guarantees that the resulting closed-loop system is well-posed and exponentially stable in the sense of certain norm.