Local exact controllability of a 1D Bose-Einstein condensate in a time-varying box

We consider a one-dimensional Bose-Einstein condensate in a infinite square-well (box) potential. This is a nonlinear control system in which the state is the wave function of the Bose Einstein condensate and the control is the length of the box. We prove that local exact controllability around the ground state (associated with a fixed length of the box) holds generically with respect to the chemical potential \mu; i.e. up to an at most countable set of \mu-values. The proof relies on the linearization principle and the inverse mapping theorem, as well as ideas from analytic perturbation theory.