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

In this paper, we are concerned with the boundary output feedback stabilisation of an Euler-Bernoulli beam equation with one free boundary end and control/disturbance on the other end. A variable structure output feedback stabilising control law is designed by the Lyapunov functional approach. It is shown that the resulting closed-loop system without disturbance is associated with a nonlinear semigroup and asymptotically stable except the zero dynamics. In addition, we show that this control law is robust to the external disturbance in the sense that the vibrating energy of the closed-loop system outside of the zero dynamics converges to zero as time goes to infinity in spite of the presence of finite sum of harmonic disturbance on the control end. The existence of the Filippov solution with disturbance is developed by the Galerkin approximation scheme.