Control and Stabilization for the Nonlinear Schrödinger equation

Abstract

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.