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.