Boundary controllers and observers for Schrödinger equation

We consider a problem of stabilization of linearized Schrodinger equation using boundary actuation and measurements. We propose two different control designs. First, a simple proportional collocated boundary controller is shown to exponentially stabilize the system. However, the decay rate of the closed loop system cannot be prescribed. The second, full-state feedback boundary control design achieves an arbitrary decay rate. We formally view the Schrodinger equation as a heat equation in complex variables and apply the backstepping method, recently developed for boundary control of reaction-advection-diffusion equations. The resulting controller is then supplied with the backstepping observer to obtain an output- feedback compensator. The designs are illustrated with simulations.