Null controllability of Grushin-type operators in dimension two

We study the null controllability of the parabolic equation associated with the Grushintype operator $A = \partial_x^2 + |x|^{2\gamma}\partial_y^2 (\gamma > 0)$ in the rectangle $\Omega = (-1, 1) \times (0, 1)$, under an additive control supported in an open subset $\omega$ of $\Omega$. We prove that the equation is null controllable in any positive time fo $\gamma<1$ and that there is no time for which it is null controllable for $\gamma>1$. In the transition regime $\gamma=1$ and when $\omega$ is a strip $\omega = (a, b)\times (0, 1) (0 < a, b \leq 1)$, a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric configuration of $\Omega$, 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.