Prescribed-time estimation and output regulation of the linearized Schrodinger equation by backstepping

We study state estimation of the linearized Schrodinger equation within a prescribed terminal time. We make use of a time-varying, complex-valued observer gain and boundary measurements to construct the observer, where the gain is designed such that the estimate error converges to zero within the terminal time. The observer gain proposed herein is developed via the backstepping method by selecting a target error equation that stabilizes to zero within the terminal time. Our time-varying observer gain diverges as time approaches the terminal time. Nevertheless, we can guarantee prescribed-time stabilization of the estimator error equation by characterizing the growth-in-time of the observer gain and comparing it to the stability of the target error equation. We develop the full-state feedback dual result, and we combine the boundary estimation and control results to develop prescribed-time output regulation. (C) 2020 European Control Association. Published by Elsevier Ltd. All rights reserved.