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

Abstract

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 = ω₁ × ( − 1,1) with ω̅₁⊂𝕋, we investigate the null controllability on a toy model, where (∂_x,x ∈T) is replaced by (i( − Δ)^{1 / 2},x ∈ Ω₁), and Ω₁ 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 (Ω₁,ω₁), then null controllability does not hold (whatever T> 0 is), when the control acts on a vertical strip ω = ω₁ × ( − 1,1) with ω̅₁⊂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.