Control and Stabilization for the Nonlinear Schrödinger equation

• Ano de defesa: 2024
• Autor(a) principal: CARVALHO, Jackellyny Dassy do Nascimento
• Orientador(a): SILVA, Pablo Gustavo Albuquerque Braz e
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de Pernambuco
• Programa de Pós-Graduação: Programa de Pos Graduacao em Matematica
• País: Brasil
• Palavras-chave em Português: Estabilização; Controle; Estimativas de Strichartz; Decomposição em perfis; Medida microlocal; Expoente crítico
• Link de acesso: https://repositorio.ufpe.br/handle/123456789/58025

This thesis brings together some results related to the nonlinear H1-critical Schrödinger equation in R3, in particular, a null controllability result where, using Strichartz estimates, the controllability of the linear system (HUM method) and a perturbation argument, the con- trollability for the nonlinear system is achieved. Furthermore, for the aforementioned equation with a perturbation term, we prove exponential decay for some solutions that are bounded in energy space but small at a lower norm. This result is a consequence of a profile decom- position obtained for linear and nonlinear solutions combined with a propagation result that involves arguments from microlocal analysis, namely the defect measure theory. After showing that a sequence of nonlinear solutions can be linearized under some conditions, we prove an observability estimate that implies the stabilization result.