Null controllability of Grushin-type operators in dimension two

We study the null controllability of the parabolic equation associated with the Grushin-type operator $A=\partial_x^2+|x|^{2\gamma}\partial_y^2\,, (\gamma>0),$ in the rectangle $\Omega=(-1,1)\times(0,1)$, under an additive control supported in the strip $\omega=(a,b)\times(0,1)\,, (0<a,b<1)$. We prove that the equation is null controllable in any positive time for $\gamma<1$, and that it fails to be so for $\gamma>1$. In the transition regime $\gamma=1$, we show that both behaviors live together: a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric configuration, null controllability is equivalent to the observability of the Fourier components of the solution of the adjoint system uniformly with respect to the frequency.