Stabilisation of Timoshenko beam system with a tip payload under the unknown boundary external disturbances

This paper is concerned with the boundary stabilisation of a Timoshenko beam with a tip payload under the boundary external disturbances. Nonlinear feedback control laws are designed to reduce the effects of the external disturbances. Since the controlled system is nonlinear, the well posedness of the nonlinear closed-loop system is investigated using the theory of nonlinear maximal monotone operators and the variational principle. In addition, we show the exponential stability of the nonlinear controlled system using the Lyapunov direct method. In the end, the numerical simulations are displayed to illustrate the effectiveness of the proposed boundary controls.