Null controllability of Kolmogorov-type equations

We study the null controllability of Kolmogorov-type equations ∂t f + v ∂x f − ∂2 v f = u(t, x, v)1ω(x, v) in a rectangle , under an additive control supported in an open subset ω of . For γ = 1, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support ω. This improves the previous result by Beauchard and Zuazua (Ann Ins H Poincaré Anal Non Linéaire 26:1793–1815, 2009), 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 γ = 1 or if γ = 2 and ω contains the segment {v = 0}, and only in large time if γ = 2 and ω does not contain the segment {v = 0}. Our approach, inspired from Benabdallah et al. (C R Math Acad Sci Paris 344(6):357–362, 2007), Lebeau and Robbiano (Commun Partial Differ Equ 20:335–356, 1995), is based on two 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.