Stabilization for a Cascaded ODE-Shrödinger System via a New Transformation

In this work, we are concerned with the exponential stabilization for a cascaded ODE-Schrodinger system with two interconnections flowed from PDE to ODE at both ends. In absence of disturbance, a new transformation is introduced such that the PDE variable is unaltered and the ODE variable is changed. This is quite different from the transformation corresponding to the PDE backstepping where the ODE variable is unaltered and the PDE variable is changed. An exponential stability controller is constructed. When internal uncertainty and external disturbance are considered, we design an infinite- dimensional disturbance estimator to estimate the total disturbance. Based on such estimator, a Luenberger state observer as well as controller is designed. It is proved that the original system is exponentially stable and the whole closed-loop system is bounded. Some simulations are presented.