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We consider the 1D nonlinear Schr\"odinger equation with bilinear control. Inthe case of Neumann boundary conditions, local exact controllability of thisequation near the ground state has been proved by Beauchard and Laurent inarXiv:1001.3288. In this paper, we study the case of Dirichlet boundaryconditions. To establish the controllability of the linearised equation, we usea bilinear control acting through four directions: three Fourier modes and onegeneric direction. The Fourier modes are appropriately chosen so that theysatisfy a saturation property. These modes allow to control approximately thelinearised Schr\"odinger equation. We show that the reachable set for thelinearised equation is closed. This is achieved by representing the resolvingoperator as a sum of two linear continuous mappings: one is surjective (herethe control in generic direction is used) and the other is compact. A mappingwith dense and closed image is surjective, so the linearised Schr\"odingerequation is exactly controllable. Then local exact controllability of thenonlinear equation is derived using the inverse mapping theorem.

Local exact controllability of the 1D nonlinear Schr\"odinger equation in the case of Dirichlet boundary conditions