Null controllability of the parabolic spherical Grushin equation

Abstract

We investigate the null controllability property of the parabolic equation associated with the Grushin operator defined by the canonical almost-Riemannian structure on the 2-dimensional sphere ${\mathbb{S}}^2$. This is the natural generalization of the Grushin operator $\mathcal{G}={\partial }_x^2+x^2{\partial }_y^2$ on ${\mathbb{R}}^2$ to this curved setting, and presents a degeneracy at the equator of ${\mathbb{S}}^2$. We prove that the null controllability is verified in large time when the control acts as a source term distributed on a subset $\bar{\omega }=\{(x_1,x_2,x_3)\in {\mathbb{S}}^2\mid \alpha <|x_3|<\beta \}$ for some $0\le \alpha <\beta \le 1$. More precisely, we show the existence of a positive time $T^{\ast }>0$ such that the system is null controllable from $\bar{\omega }$ in any time $T\ge T^{\ast }$, and that the minimal time of control from $\bar{\omega }$ satisfies $T_{min}\ge \log (1/\sqrt{1-{\alpha }^2})$. Here, the lower bound corresponds to the Agmon distance of $\bar{\omega }$ from the equator. These results are obtained by proving a suitable Carleman estimate by using unitary transformations and Hardy-Poincaré type inequalities to show the positive null-controllability result. The negative statement is proved by exploiting an appropriate family of spherical harmonics, which concentrates at the equator, to falsify the uniform observability inequality.

References in corpus (2)

• Non null controllability of the Grushin equation in 2D
• From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere