Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

Beauchard, Karine

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 34, 24 p.

The goal of this note is to present the results of the references [5] and [4]. We study the null controllability of the parabolic equations associated with the Grushin-type operator ${\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_y^2$ ($\gamma >0$) in the rectangle $(x,y)\in (-1,1)\times (0,1)$ or with the Kolmogorov-type operator $v^{\gamma }{\partial }_{x}f+{\partial }_v^{2}f$ ($\gamma \in \{1,2\}$) in the rectangle $(x,v)\in 𝕋\times (-1,1)$, under an additive control supported in an open subset $\omega$ of the space domain.

We prove that the Grushin-type equation is null controllable in any positive time for $\gamma <1$ and that there is no time for which it is null controllable for $\gamma >1$. In the transition regime $\gamma =1$ and when $\omega$ is a strip $\omega =(a,b)\times (0,1),(0<a,b\le 1)$, a positive minimal time is required for null controllability.

For the Kolmogorov-type equation with $\gamma =1$ and periodic-type boundary conditions (in $v$), we prove that null controllability holds in any positive time, with any control support $\omega$. This improves the previous result [6], in which the control support was a strip $\omega =𝕋\times (a,b)$.

For the Kolmogorov-type equation with Dirichlet boundary conditions and a strip $\omega =𝕋\times (a,b)$ ($0<a<b<1$) as control support, we prove that null controllability holds in any positive time if $\gamma =1$, and only in large time if $\gamma =2$.

Our approach, inspired from [8, 33], is based on 2 key ingredients: the observability of the Fourier components of the solution of the adjoint system (a heat equation with potential), uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.

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