Null controllability of Grushin-type operators in dimension two

TL;DR

This paper investigates the null controllability of a parabolic equation with a Grushin-type operator in two dimensions, establishing conditions under which controllability is possible or fails, depending on the parameter or.

Contribution

It provides a precise characterization of null controllability for Grushin-type operators in 2D, identifying the critical parameter or and the minimal time needed.

Findings

Null controllability holds for or<1 in any positive time.

Null controllability fails for or>1.

At or=1, controllability depends on minimal time.

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 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.