Controllability of the cubic Schroedinger equation via a low-dimensional source term

We study controllability of d-dimensional defocusing cubic Schroedinger equation under periodic boundary conditions. The control is applied additively, via a source term, which is a linear combination of few complex exponentials (modes) with time-variant coefficients - controls. We manage to prove that controlling 2(d) modes one can achieve controllability of the equation in each finite-dimensional projection of the evolution space H-s (T-d), s > d/2, as well as approximate controllability in H-s(T-d). We also present a negative result regarding exact controllability of cubic Schroedinger equation via a finite-dimensional source term.