  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.
