Null controllability of Grushin-type operators in dimension two

We study the null controllability of the parabolic equation associated with the Grushin-type operator A = ∂ x+|x|∂ y , (γ > 0), in the rectangle Ω = (−1, 1) × (0, 1), under an additive control supported in an open subset ω of Ω. We prove that the equation is null controllable in any positive time for γ < 1 and that there is no time for which it is null controllable for γ > 1. In the transition regime γ = 1 and when ω is a strip ω = (a, b)× (0, 1) , (0 < a, b ≤ 1), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric con guration of Ω, null controllability is closely linked to the one-dimensional observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the Fourier frequency.