Control of the Schr\"{o}dinger equation in $\mathbb{R}^3$: The critical case

Abstract

This article deals with the $H^{1}$--level local null controllability for the energy-critical nonlinear Schr\"{o}dinger 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 Schr\"{o}dinger equation to be controllable. Finally, we use a perturbation argument and show local controllability for the critical nonlinear Schr\"{o}dinger equation.