受物理空间和傅里叶空间中局部力扰动的复Ginzburg-Landau方程
The complex Ginzburg-Landau equation perturbed by a force localised both in physical and Fourier spaces
摘要：
在arXiv：1802.03250中，对一类随机动力系统建立了指数混合的判别准则.在该文件中，该标准适用于扰动的噪声局部化在傅立叶空间的偏微分方程。在本文中，我们表明，在复杂的金斯堡-朗道（CGL）方程的情况下，该标准可以用来考虑更退化的噪声，是本地化的物理和傅立叶空间。这是通过检查线性化方程几乎肯定近似可控。我们还研究了非线性CGL方程的能控性问题。使用Agrachev-Sarychev型参数，我们证明了一个近似的可控性性质的情况下，控制力再次定位在物理和傅立叶空间。
英文摘要：
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.
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