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 at most $2^d$ modes one can achieve controllability of the equation in any finite-dimensional projection of the evolution space $H^{s}(\mathbb{T}^d), \ s>d/2$, as well as approximate controllability in $H^{s}(\mathbb{T}^d)$. We also present negative result regarding exact controllability of cubic Schroedinger equation via a finite-dimensional source term.