  In this paper, we consider a parabolic PDE on a torus of arbitrary dimension.
The nonlinear term is a smooth function of polynomial growth of any degree. In
this general setting, the corresponding Cauchy problem is not necessarily well
posed. We show that the equation in question is approximately controllable by
only a finite number of Fourier modes. This result is proved by using some
ideas from the geometric control theory introduced by Agrachev and Sarychev.
