Jackellyny Dassy do Nascimento Carvalho

arXiv2026

Control of the Schrödinger equation in $\mathbb{R}^3$: The critical case

This article deals with the $H^{1}$--level local null controllability for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$. Firstly, we demonstrate that the problem under consideration is well-posed using Strichartz estimates. Moreover, through the Hilbert uniqueness method, we prove the linear Schrödinger equation to be controllable. Finally, we use a perturbation argument and show local controllability for the critical nonlinear Schrödinger equation.

math.AP↗

arXiv2024

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

This article addresses the stabilizability of a perturbed quintic defocusing Schrödinger 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 Schrödinger 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.

math.AP↗