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

TL;DR

This paper investigates the null-controllability of a generalized Baouendi-Grushin equation on a rectangular domain with non-rectangular control sets, providing bounds and exact values for the minimal control time in various geometries.

Contribution

It extends existing controllability results to more general control regions and functions q(x), offering bounds and exact minimal times for null-controllability.

Findings

Derived upper and lower bounds for minimal control time.

Identified geometries where exact control time is known.

Extended controllability results to 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 Schr\"odinger 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.