Null controllability of the parabolic spherical Grushin equation

TL;DR

This paper establishes the conditions and minimal time for null controllability of a degenerate parabolic equation on the sphere, extending known results from flat to curved geometries using Carleman estimates and spherical harmonics.

Contribution

It generalizes null controllability results of the Grushin operator to the spherical setting, identifying the minimal control time and conditions on the control region.

Findings

Null controllability holds for large enough time when control acts on specific regions.

The minimal control time is bounded below by the logarithm of the inverse of the control region’s distance from the equator.

A Carleman estimate and spherical harmonics are used to prove positive and negative controllability results.

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 $T^{*}>0$ such that the system is null controllable from $\bar{\omega}$ in any time $T\ge T^{*}$, 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-Poincar\'e 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.