Degenerate parabolic operators of Kolmogorov type with a geometric control condition

Beauchard, Karine; Helffer, Bernard; Henry, Raphael; Robbiano, Luc

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 487-512

We consider Kolmogorov-type equations on a rectangle domain $(x,v)\in \Omega =T\times (-1,1)$, that combine diffusion in variable $v$ and transport in variable $x$ at speed $v^{\gamma }$, $\gamma \in N^{*}$, with Dirichlet boundary conditions in $v$. We study the null controllability of this equation with a distributed control as source term, localized on a subset $\omega$ of $\Omega$. When the control acts on a horizontal strip $\omega =T\times (a,b)$ with $0<a<b<1$, then the system is null controllable in any time $T>0$ when $\gamma =1$, and only in large time $T>T_{min}>0$ when $\gamma =2$ (see [K. Beauchard, Math. Control Signals Syst. 26 (2014) 145–176]). In this article, we prove that, when $\gamma >3$, the system is not null controllable (whatever $T$ is) in this configuration. This is due to the diffusion weakening produced by the first order term. When the control acts on a vertical strip $\omega =\omega {}_1\times (-1,1)$ with ω̅₁⊂��, we investigate the null controllability on a toy model, where $(\partial {}_x,x\in T)$ is replaced by $(i{(-\Delta )}^{1/2},x\in \Omega {}_1)$, and $\Omega {}_1$ is an open subset of $R^N$. As the original system, this toy model satisfies the controllability properties listed above. We prove that, for $\gamma =1,2$ and for appropriate domains $(\Omega {}_1,\omega {}_1)$, then null controllability does not hold (whatever $T>0$ is), when the control acts on a vertical strip $\omega =\omega {}_1\times (-1,1)$ with ω̅₁⊂��. Thus, a geometric control condition is required for the null controllability of this toy model. This indicates that a geometric control condition may be necessary for the original model too.

Classification : 93C20, 93B05, 93B07
Keywords: Null controllability, degenerate parabolic equation, hypoelliptic operator, geometric control condition

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