Null controllability of Kolmogorov-type equations

Karine Beauchard (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

We study the null controllability of Kolmogorov-type equations in a rectangle, under an additive control supported in an open subset of the rectangle. These equations couple a diffusion in variable v with a transport in variable x at speed v^a. For a=1, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support.This improves the previous result [5], in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if a = 1, or if a= 2 and the control support contains the segment {v = 0}, and only in large time if a = 2 and the control support does not contain the segment {v = 0}. Our approach, inspired from [7, 31], is based on 2 key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.

Domaines: Equations aux dérivées partielles [math.AP]

Karine Beauchard. Null controllability of Kolmogorov-type equations. Mathematics of Control, Signals, and Systems, 2014, 26 (1), pp.145-176. ⟨10.1007/s00498-013-0110-x⟩. ⟨hal-00826117⟩