Rapid Stabilization of a Linearized Bilinear $1-D$ Schr\"odinger Equation

TL;DR

This paper demonstrates a method for rapidly stabilizing a linearized 1-D Schrödinger equation with bilinear control using a transformation-based feedback law, ensuring exponential stability.

Contribution

It introduces a novel transformation approach to achieve rapid stabilization of the linearized Schrödinger equation with bilinear control.

Findings

Successful design of a feedback law for rapid stabilization

Transformation ensures exponential stability of the system

Conditions guarantee the uniqueness and invertibility of the transformation

Abstract

We consider the one dimensional Schr\"odinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.