Controllability of semilinear Schroedinger equation via low-dimensional source term

We study controllability of 2D defocusing cubic Schroedinger equation under periodic boundary conditions and control applied via source term (additively). The source term is a linear combination of few complex exponentials (modes) with time-variant coefficients - controls. We manage to prove that controlling just 4 modes one can achieve controllability of this equation in any finite-dimensional projection of its evolution space H^(1+\sigma), as well as approximate controllability in H^(1+\sigma) (\sigma>0). We also present negative result regarding exact controllability of cubic Schroedinger equation via a finite-dimensional source term.