Null controllability of Grushin-type operators in dimension two

Abstract

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 an open subset $\omega$ of $\Omega$. We prove that the equation is null controllable in any positive time for $\gamma <1$ 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 $\omega =(a,b)\times (0,1),(0<a,b\le 1)$, 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.