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

Abstract

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.

Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates

“…In [5], the authors also proved the local controllability of a nonlinear Schrödinger equation with Neumann boundary conditions. The case of Dirichlet boundary conditions has been treated later by Duca and Nersesyan in [26].…”

Small-time bilinear control of Schrödinger equations with application to rotating linear molecules

“…Saturation techniques have been introduced by Agrachev and Sarychev [1,2] to study the approximate controllability of 2D Navier-Stokes and Euler systems with additive controls, and extended to the 3D case in [23,24]. Other recent developments of these techniques are given, e.g., in [13] to study small-time controllability properties of semiclassical Schr ödinger equations, and in [15] to study local exact controllability of 1D Schr ödinger equations with Dirichlet boundary conditions.…”