Stabilization and control for the biharmonic Schr\"odinger equation

TL;DR

This paper demonstrates that a nonlinear fourth order Schrödinger system on a periodic domain can be globally stabilized and exactly controlled using internal controls, leveraging properties of Bourgain spaces and propagation of compactness.

Contribution

It establishes the global stabilization and exact controllability of a fourth order nonlinear Schrödinger system on a periodic domain, a novel result in control theory for such equations.

Findings

The linear system is globally exponentially stabilizable.

The nonlinear system is globally exactly controllable.

Control is achieved via internal controls on sub-domains.

Abstract

The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schr\"odinger system: $$i\partial_tu +\partial_x^2u-\partial_x^4u=\lambda |u|^2u,$$ on a periodic domain $\mathbb{T}$ with internal control supported on an arbitrary sub-domain of $\mathbb{T}$. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exact controllability ensures that fourth order nonlinear Schr\"odinger is globally exactly controllable.