Rapid stabilization and finite time stabilization of the bilinear Schr\"odinger equation

TL;DR

This paper introduces a novel method for rapid and finite time stabilization of the bilinear Schrödinger equation and its linearized form, utilizing Grammian operators and new analytical tools inspired by optimal control theory.

Contribution

It develops a new approach for stabilization of the bilinear Schrödinger system, combining techniques from control theory and new estimates for control costs in small time.

Findings

Established rapid stabilization of the bilinear Schrödinger system.

Achieved finite time stabilization of the linearized system.

Introduced new control cost estimates in small time.

Abstract

We propose a method to establish the rapid stabilization of the bilinear Schr\"odinger control system and its linearized system, and the finite time stabilization of the linearized system using the Grammian operators. The analysis of the rapid stabilization involves a new quantity (variable) which is inspired by the adjoint state in the optimal control theory and is proposed in our recent work on control systems associated with strongly continuous group. The analysis of the finite time stabilization follows the strategy introduced by Coron and Nguyen in the study of the finite time stabilization of the heat equation and incorporate a new ingredient involving the estimate of the cost of controls of the linearized system in small time derived in this paper.