2022/02/17 by Alessandro Duca, Duca, Alessandro, Vahagn Nersesyan +1
Mathematics · Physics and Astronomy · #35Q55 #81Q93 #93B05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2202.08723
openalex publication_date 2022/02/17 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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 in arXiv:1001.3288. 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.