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We investigate the regulator problem (tracking and disturbance rejection) fora system (plant) described by a boundary controlled anti-stable linearone-dimensional Schrodinger equation, using the backstepping approach. Theoutput to be controlled is not required to be measurable and its observationoperator is assumed to be admissible for a certain operator semigroup that isrelated to the operator semigroup of the original plant. We consider both thestate feedback and the output feedback regulator problem. In the latter case,the measurement from the Schrodinger equation is taken at the boundary. Firstwe show that the open-loop system is well-posed. We design a state feedbackcontrol law that solves the regulator problem by the backstepping method. Then,a finite-dimensional reference observer and an infinite-dimensional disturbanceobserver are designed. Putting these together, we obtain an output feedbackcontroller with internal loop that achieves output regulation.

Solving the regulator problem for the one-dimensional Schrodinger equation via backstepping