Output Feedback Stabilization of a One-Dimensional Schrödinger Equation by Boundary Observation With Time Delay

In this article, we are concerned with the stabilization of a 1-D Schrodinger equation with boundary control and observation. The boundary observation is suffered from an arbitrary given time delay. We first show that the proportional output feedback control that stabilizes exponentially the time delay free system can not stabilize the corresponding time delay system anymore. We then use the observer and predictor to solve the stabilization problem of time delay system: the state is estimated in the time span where the observation is available; and the state is predicted in the time interval when the observation is not available. It is shown that the estimated state feedback stabilizes exponentially the time delay system. The numerical simulation is presented to illustrate the effect of the stabilizing controller.