Stabilization of one-dimensional Schrödinger equation with variable coefficient under delayed boundary output feedback

In this article, we are concerned with the stabilization of one-dimensional Schrödinger equation with variable coefficient and boundary observation. The boundary observation is suffered from an arbitrary given time delay. We use the observer and predictor to solve the stabilization problem with time delay: The state is estimated in the time span where the observation is available; and the state is predicted in the time interval where the observation is not available. It is shown that the estimated state feedback stabilizes exponentially the system. The numerical simulation is presented to illustrate the effect of the stabilizing controller.