## Degenerate parabolic operators of Kolmogorov type with a
geometric control condition^{∗}

^{1} Centre de Mathématiques Laurent
Schwartz, Ecole Polytechnique, 91128
Palaiseau cedex,
France.

Karine.Beauchard@math.polytechnique.fr
^{2} Département de Mathématiques,
Batiment 425, Université Paris Sud, 91405
Orsay cedex,
France.

Bernard.Helffer@math.u-psud.fr;
Raphael.Henry@math.u-psud.fr
^{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.

luc.robbiano@uvsq.fr

Received:
13
September
2013

Revised:
4
June
2014

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}, *γ* ∈ 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 *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 ω̅_{1}⊂𝕋, we investigate the null controllability on a toy
model, where (*∂*_{x}*,x*
∈T) is replaced by (*i*( − *Δ*)^{1 / 2}*,x* ∈
*Ω*_{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 ω̅_{1}⊂𝕋. 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.

Mathematics Subject Classification: 93C20 / 93B05 / 93B07

Key words: Null controllability / degenerate parabolic equation / hypoelliptic operator / geometric control condition

*© EDP Sciences, SMAI, 2015*