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

We consider the null-controllability problem for the generalized Baouendi-Grushin equation $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_ωu$ on a rectangular domain. Sharp controllability results already exist when the control domain $ω$ 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 $ω$. In some geometries for $ω$, 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 $ω$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-\partial_x^2 + ν^2 q(x)^2$ when $\Re(ν)>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.

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