The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance

In this paper, we are concerned with the boundary stabilization of a one-dimensional anti-stable wave equation with the boundary external disturbance. The backstepping method is first applied to transform the anti-stability from the free end to the control end. A variable structure feedback stabilizing controller is designed by the Lyapunov function approach. It is shown that the resulting closed-loop system is associated with a nonlinear semigroup and is asymptotically stable. In addition, we show that this controller is robust to the external disturbance in the sense that the vibrating energy of the closed-loop system is also convergent to zero as time goes to infinity in the presence of bounded deterministic disturbance at the control end. The existence and uniqueness of the solution are also developed by the Galerkin approximation scheme.