Lyapunov approach to the boundary stabilization of a beam equation with boundary disturbance

In this paper, we are concerned with the boundary output feedback stabilization of an Euler-Bernoulli beam equation with free boundary at one end and control and disturbance at the other end. A variable structure output feedback stabilizing controller is designed by the Lyapunov function approach. It is shown that the resulting closed-loop system without disturbance 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 finite sum of harmonic disturbance at the control end.