  In the paper arXiv:1802.03250, a criterion for exponential mixing is
established for a class of random dynamical systems. In that paper, the
criterion is applied to PDEs perturbed by a noise localised in the Fourier
space. In the present paper, we show that, in the case of the complex
Ginzburg-Landau (CGL) equation, that criterion can be used to consider even
more degenerate noise that is localised both in physical and Fourier spaces.
This is achieved by checking that the linearised equation is almost surely
approximately controllable. We also study the problem of controllability of the
nonlinear CGL equation. Using Agrachev-Sarychev type arguments, we prove an
approximate controllability property in the case of a control force which is
again localised in physical and Fourier spaces.
