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

TL;DR

This paper extends the application of an exponential mixing criterion to the complex Ginzburg-Landau equation with noise localized in both physical and Fourier spaces, demonstrating controllability and mixing properties.

Contribution

It shows that the exponential mixing criterion applies to more degenerate noise in the CGL equation, with new controllability results using Agrachev-Sarychev type arguments.

Findings

The linearised CGL equation is almost surely approximately controllable.

The nonlinear CGL equation is approximately controllable with localized control.

The criterion for exponential mixing applies to degenerate noise in both physical and Fourier spaces.

Abstract

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.