Stabilization of an ODE-Schrodinger Cascade System with Time Delay in Observation

In this paper, we are concerned with the stabilization of a cascaded ODE-Schrodinger system subject to boundary observation with time delay. Both observer and predictor systems are designed to estimate the state variable on the time interval $[0,t-\tau]$ when the observation is available, and to predict the state variable on the time interval $[t-\tau,\ t]$ when the observation is not available respectively. Based on the estimated state variable, the output feedback stabilizing controller by backstepping method is presented and the closed-loop system is shown to be exponentially stable.