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 one-dimensional 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 makes the corresponding time delay system unstable.We then 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.