Dynamic feedback stabilization of an unstable wave equation

In this paper, we consider the stabilization of an unstable wave equation through a dynamic boundary compensator. The measurement, which is one internal point of the state of the wave equation, is fluxed into the compensator while the output of the compensator is forced into the boundary of the wave equation. The well-posedness of the closed-loop system is proved by using the semigroup method. By choosing appropriate controller parameters and adopting Nyquist criterion for distributed parameter systems, the eigenvalues of the closed-loop systems are shown to be inside the left-half complex plane. Then the closed-loop system is proved to be exponentially stable by using the Riesz basis approach. Numerical simulations are presented to verify the effectiveness of the proposed compensator.