Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions

Alessandro Duca (1) , Vahagn Nersesyan (2, 1)

1 LMV - Laboratoire de Mathématiques de Versailles
2 NYU–ECNU Institute of Mathematical Sciences at NYU Shanghai

We consider the 1D nonlinear Schrödinger equation with bilinear control. In the case of Neumann boundary conditions, local exact controllability of this equation near the ground state has been proved by Beauchard and Laurent [BL10]. In this paper, we study the case of Dirichlet boundary conditions. To establish the controllability of the linearised equation, we use a bilinear control acting through four directions: three Fourier modes and one generic direction. The Fourier modes are appropriately chosen so that they satisfy a saturation property. These modes allow to control approximately the linearised Schrödinger equation. We show that the reachable set for the linearised equation is closed. This is achieved by representing the resolving operator as a sum of two linear continuous mappings: one is surjective (here the control in generic direction is used) and the other is compact. A mapping with dense and closed image is surjective, so the linearised Schrödinger equation is exactly controllable. Then local exact controllability of the nonlinear equation is derived using the inverse mapping theorem.

Mots clés: AMS subject classifications: 35Q55; 81Q93; 93B05 Nonlinear Schrödinger equation; local exact controllability; linearisation; approximate controllability; saturation property

Domaines: Mathématiques [math]

Alessandro Duca, Vahagn Nersesyan. Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions. 2022. ⟨hal-03579006⟩