Parameter estimation and stabilization for one-dimensional Schrödinger equation with boundary output constant disturbance and non-collocated control

We consider parameter estimation and stabilization for a one-dimensional Schrödinger equation with an unknown constant disturbance suffered from the boundary observation at one end and the non-collocated control at other end. An adaptive observer is designed in terms of measured position with unknown constant by the Lyapunov functional approach. By a backstepping transformation for infinite-dimensional systems, it is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. The numerical experiments are carried out to illustrate the proposed approach.