We consider Kolmogorov-type equations on a rectangle domain , that combine diffusion in variable and transport in variable at speed , , with Dirichlet boundary conditions in . We study the null controllability of this equation with a distributed control as source term, localized on a subset of . When the control acts on a horizontal strip with , then the system is null controllable in any time when , and only in large time when (see [K. Beauchard, Math. Control Signals Syst. 26 (2014) 145–176]). In this article, we prove that, when , the system is not null controllable (whatever 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 with ω̅1⊂��, we investigate the null controllability on a toy model, where is replaced by , and is an open subset of . As the original system, this toy model satisfies the controllability properties listed above. We prove that, for and for appropriate domains , then null controllability does not hold (whatever is), when the control acts on a vertical strip 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.
Keywords: Null controllability, degenerate parabolic equation, hypoelliptic operator, geometric control condition
@article{COCV_2015__21_2_487_0, 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}, volume = {21}, number = {2}, year = {2015}, doi = {10.1051/cocv/2014035}, zbl = {1311.93042}, mrnumber = {3348409}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2014035/} }
TY - JOUR AU - Beauchard, Karine AU - Helffer, Bernard AU - Henry, Raphael AU - Robbiano, Luc TI - Degenerate parabolic operators of Kolmogorov type with a geometric control condition JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 487 EP - 512 VL - 21 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014035/ DO - 10.1051/cocv/2014035 LA - en ID - COCV_2015__21_2_487_0 ER -
%0 Journal Article %A Beauchard, Karine %A Helffer, Bernard %A Henry, Raphael %A Robbiano, Luc %T Degenerate parabolic operators of Kolmogorov type with a geometric control condition %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 487-512 %V 21 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014035/ %R 10.1051/cocv/2014035 %G en %F COCV_2015__21_2_487_0
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://www.numdam.org/articles/10.1051/cocv/2014035/
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