Minimal time for null controllability of the parabolic spherical Baouendi-Grushin equation

Abstract

We study null controllability for the parabolic equation on $\mathbb{S}^{2}$ endowed with its canonical almost-Riemannian structure. For a spherical crown $\omega=\{\alpha<x_3<\beta\}$, where $0\le \alpha<\beta\le1$, we prove the sharp minimal time formula $T_{\min}(\omega)=\ln(1/\sqrt{1-\alpha^{2}})$ for null controllability in $\omega$. We also prove that, whenever the control region contains the equator, null controllability holds in every positive time. The proof combines two complementary tools. First, after Fourier decomposition with respect to the periodic variable, we establish observability estimates for a family of one-dimensional singular parabolic equations, with constants uniform with respect to the Fourier mode; the singularities at the poles are handled via a Hardy-Poincar\'e inequality. Second, for crowns away from the equator, we use the moment method to construct controls on the pole-touching crown $\alpha<x_3< 1$ from sharp weighted lower bounds on associated Legendre functions, and then pass to a general crown $\alpha<x_3<\beta$ by a cut-off argument on the full domain combined with the arbitrary-time controllability of crowns containing the equator. The result closes the large-time gap left in earlier work and gives the exact null-controllability threshold for the canonical almost-Riemannian heat equation on $\mathbb S^2$.