Degenerate parabolic operators of Kolmogorov type with a geometric control condition
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 487-512.

We consider Kolmogorov-type equations on a rectangle domain (x,v)Ω=T×(-1,1), that combine diffusion in variable v and transport in variable x at speed v γ , γN * , 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 ω of Ω. When the control acts on a horizontal strip ω=T×(a,b) with 0<a<b<1, then the system is null controllable in any time T>0 when γ=1, and only in large time T>T min >0 when γ=2 (see [K. Beauchard, Math. Control Signals Syst. 26 (2014) 145–176]). In this article, we prove that, when γ>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 ω=ω 1 ×(-1,1) with ω̅1⊂��, we investigate the null controllability on a toy model, where ( x ,xT) is replaced by (i(-Δ) 1/2 ,xΩ 1 ), and Ω 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 γ=1,2 and for appropriate domains (Ω 1 ,ω 1 ), then null controllability does not hold (whatever T>0 is), when the control acts on a vertical strip ω=ω 1 ×(-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.

DOI: 10.1051/cocv/2014035
Classification: 93C20, 93B05, 93B07
Keywords: Null controllability, degenerate parabolic equation, hypoelliptic operator, geometric control condition
Beauchard, Karine 1; Helffer, Bernard 2; Henry, Raphael 2; Robbiano, Luc 3

1 Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau cedex, France.
2 Département de Mathématiques, Batiment 425, Université Paris Sud, 91405 Orsay cedex, France.
3 Laboratoire de Mathématiques de Versailles (LM-Versailles), Université de Versailles Saint-Quentin-en-Yvelines, CNRS UMR 8100, 45 Avenue des Etats-Unis, 78035 Versailles, France.
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     author = {Beauchard, Karine and Helffer, Bernard and Henry, Raphael and Robbiano, Luc},
     title = {Degenerate parabolic operators of {Kolmogorov} type with a geometric control condition},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {487--512},
     publisher = {EDP-Sciences},
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Beauchard, Karine; Helffer, Bernard; Henry, Raphael; Robbiano, Luc. Degenerate parabolic operators of Kolmogorov type with a geometric control condition. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 487-512. doi : 10.1051/cocv/2014035. http://archive.numdam.org/articles/10.1051/cocv/2014035/

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