Arbitrary decay for boundary stabilization of Schrödinger equation subject to unknown disturbance by Lyapunov approach

This paper deals with the design of boundary control to stabilize one-dimensional Schrödinger equation with general 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 then designed to achieve arbitrary assigned decay rate. The Galerkin approximation scheme is used to show the existence of the solution to the closed-loop system. The exponential stability of the closed-loop system is obtained by the Lyapunov functional method. A numerical example demonstrates the efficiency of the proposed control scheme.