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 ( ∂ t - ∂ x 2 - q ( x ) 2 ∂ y 2 ) f = 1 ω u 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 - ∂ 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.