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 𝕊2. This is the natural generalization of the Grushin operator 𝒢 = ∂x2 + x2∂y2 on ℝ2 to this curved setting and presents a degeneracy at the equator of 𝕊2. We prove that the null controllability is verified in large time when the control acts as a source term distributed on a subset ω̅ = {(x1, x2, x3) ∈ 𝕊2 | α < | x3 | < β} for some 0 ≤ α < β ≤ 1. More precisely, we show the existence of a positive time T* > 0 such that the system is null controllable from ω̅ in any time T ≥ T*, and that the minimal time of control from ω̅ satisfies Tmin ≥ log(1/√(1 - α2)) . Here, the lower bound corresponds to the Agmon distance of ω̅ from the equator. These results are obtained by proving a suitable Carleman estimate 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, concentrating at the equator, to falsify the uniform observability inequality.