Null-controllability properties of the generalized two-dimensional Baouendi–Grushin equation with non-rectangular control sets
Annales Henri Lebesgue, Volume 6 (2023), pp. 1479-1522.


Keywords null-controllability, observability, degenerate parabolic equations, resolvant estimates


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.


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