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

## TL;DR

This paper develops a backstepping-based control strategy for boundary-controlled, anti-stable 1D Schrödinger equations, enabling output regulation without requiring full state measurement, applicable to boundary measurements.

## Contribution

It introduces a novel backstepping approach for boundary control of Schrödinger equations, including both state and output feedback regulators with observers.

## Key findings

- Successfully designed a state feedback regulator for the Schrödinger system.
- Developed a finite-dimensional reference observer and an infinite-dimensional disturbance observer.
- Achieved output regulation with boundary measurements using the proposed control scheme.

## Abstract

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