Degenerate parabolic operators of Kolmogorov type with a geometric control condition

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 0w henγ =1 , and only in large time T> T min > 0w henγ = 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 ω1 ⊂ T, we investigate the null controllability on a toy model, where (∂x ,x ∈ T )i s replaced by (i(−Δ) 1/2 ,x ∈ Ω1), and Ω1 is an open subset of R N.A s the original system, this toy model satisfies the controllability properties listed above. We prove that, for γ =1 , 2 and for appropriate domains (Ω1 ,ω 1), then null controllability does not hold (whatever T> 0 is), when the control acts on a vertical strip ω = ω1 × (−1, 1) with ω1 ⊂ Ω1. 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.