Degenerate parabolic operators of Kolmogorov type with a geometric control condition.

Karine Beauchard (1) , Bernard Helffer (2) , Raphael Henry (2) , Luc Robbiano (3)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 UVSQ - Université de Versailles Saint-Quentin-en-Yvelines

We consider Kolmogorov-type equations on a rectangle domain, that combine diff usion in variable v and transport in variable x at speed v^m (m is an integer), 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 of the rectangle domain. In dimension one, when the control acts on a horizontal strip, that does no contain {x=0}, then the system is null controllable in any time T > 0 when m= 1, and only in large time T > T_min > 0 when m= 2 (see [10]). In this article, we prove that, when m > 2, the system is not null controllable (whatever T is) in this confi guration. This is due to the diff usion weakening produced by the first order term. When the control acts on a vertical strip, we investigate the null controllability on a toy model, where (d/dx, x in the 1D torus) is replaced by ( \sqrt{-\Delta},x in \Omega_1), and \Omega_1 is an open subset of R^N. As the original system, this toy model satis fies the controllability properties listed above. We prove that, for m=1 and for appropriate domains \Omega_1, then null controllability does not hold (whatever T > 0 is), when the control acts on a vertical strip. 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.

Domaines: Equations aux dérivées partielles [math.AP]

Karine Beauchard, Bernard Helffer, Raphael Henry, Luc Robbiano. Degenerate parabolic operators of Kolmogorov type with a geometric control condition.. 2013. ⟨hal-00863056⟩