  Local exact controllability of the 1D NLS (subject to zero boundary
conditions) with distributed control is shown to hold in a $H^1$--neighbourhood
of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to
J.-L. 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.

Optimization and Control;Analysis of PDEs