# Null controllability of Grushin-type operators in dimension two

### Piermarco Cannarsa

Università di Roma, Italy### Karine Beauchard

École Polytechnique, Palaiseau, France### Roberto Guglielmi

Università di Roma 'Tor Vergata', Italy

## 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 $\omega=(a,b)\times(0,1)\,, (0<a,b\le1)$, 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.

## Cite this article

Piermarco Cannarsa, Karine Beauchard, Roberto Guglielmi, Null controllability of Grushin-type operators in dimension two. J. Eur. Math. Soc. 16 (2014), no. 1, pp. 67–101

DOI 10.4171/JEMS/428