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.
Keywords: Null controllability, Carleman estimates, singular/degenerate parabolic equations, Hardy-Poincaré type inequalities, Grushin operator, unitary transformation, spherical harmonics, almost-Riemannian geometry
@article{COCV_2022__28_1_A70_0,
author = {Tamekue, Cyprien},
title = {Null {Controllability} of the {Parabolic} {Spherical} {Grushin} {Equation}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022055},
mrnumber = {4512971},
zbl = {1511.93022},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022055/}
}
TY - JOUR AU - Tamekue, Cyprien TI - Null Controllability of the Parabolic Spherical Grushin Equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022055/ DO - 10.1051/cocv/2022055 LA - en ID - COCV_2022__28_1_A70_0 ER -
%0 Journal Article %A Tamekue, Cyprien %T Null Controllability of the Parabolic Spherical Grushin Equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022055/ %R 10.1051/cocv/2022055 %G en %F COCV_2022__28_1_A70_0
Tamekue, Cyprien. Null Controllability of the Parabolic Spherical Grushin Equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 70. doi: 10.1051/cocv/2022055
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This work was supported by a grant from the “Fondation CFM pour la Recherche”. It was also partially supported by the iCODE Institute, research project of the IDEXParis-Saclay.
