Null controllability of Kolmogorov-type equations

Abstract

We study the null controllability of Kolmogorov-type equations \(\partial _t f + v^\gamma \partial _x f - \partial _v^2 f = u(t,x,v) 1_{\omega }(x,v)\) in a rectangle \(\Omega \), under an additive control supported in an open subset \(\omega \) of \(\Omega \). For \(\gamma =1\), with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support \(\omega \). This improves the previous result by Beauchard and Zuazua (Ann Ins H Poincare Anal Non Lineaire 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 \(\gamma =1\) or if \(\gamma =2\) and \(\omega \) contains the segment \(\{v=0\}\), and only in large time if \(\gamma =2\) and \(\omega \) 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.