Controllability of the nonlinear Schrodinger equation in the vicinity of the ground state

Local exact controllability of the one-dimensional NLS (subject to zero-boundary conditions) with distributed control is shown to hold in a H1-neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail.