Stabilization of a perturbed quintic defocusing Schr\"odinger equation in $\mathbb{R}^{3}$

TL;DR

This paper proves local exponential stabilization of a perturbed quintic defocusing Schr"odinger equation in three-dimensional space using microlocal analysis and profile decomposition, demonstrating effective control of solutions at the energy level.

Contribution

It introduces a novel stabilization method for the perturbed Schr"odinger equation in D, combining profile decomposition and microlocal analysis techniques.

Findings

Established profile decomposition for linear and nonlinear systems.

Proved local exponential stabilization of solutions.

Derived an observability inequality for the system.

Abstract

This article addresses the stabilizability of a perturbed quintic defocusing Schr\"odinger equation in $\mathbb{R}^{3}$ at the $H^1$--energy level, considering the influence of a damping mechanism. More specifically, we establish a profile decomposition for both linear and nonlinear systems and use them to show that, under certain conditions, the sequence of nonlinear solutions can be effectively linearized. Lastly, through microlocal analysis techniques, we prove the local exponential stabilization of the solution to the perturbed Schr\"odinger equation in $\mathbb{R}^{3}$ showing an observability inequality for the solution of the system under consideration, which is the key result of this work.