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In the paper arXiv:1802.03250, a criterion for exponential mixing isestablished for a class of random dynamical systems. In that paper, thecriterion is applied to PDEs perturbed by a noise localised in the Fourierspace. In the present paper, we show that, in the case of the complexGinzburg-Landau (CGL) equation, that criterion can be used to consider evenmore degenerate noise that is localised both in physical and Fourier spaces.This is achieved by checking that the linearised equation is almost surelyapproximately controllable. We also study the problem of controllability of thenonlinear CGL equation. Using Agrachev-Sarychev type arguments, we prove anapproximate controllability property in the case of a control force which isagain localised in physical and Fourier spaces.

The complex Ginzburg-Landau equation perturbed by a force localised both in physical and Fourier spaces