Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets

Abstract

We consider the null-controllability problem for the generalized Baouendi-Grushin equation $({\partial }_t-{\partial }_x^2-q(x{)}^2{\partial }_y^2)f=1_{\omega }u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q(x)=x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. Our proof relies on several tools: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-{\partial }_x^2+{\nu }^{2}q(x{)}^2$ when $\Re (\nu )>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.