Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied the in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behaviour of the NLS equation with random potential. More precisely, we assume that the amplitude of the potential is a random process whose law is 1-periodic in time and nondegenerate. Combining the controllability with a stopping time argument and the Markov property, we show that the trajectories of the random equation are almost surely unbounded in regular Sobolev spaces.