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Article Dans Une Revue Annales Henri Lebesgue Année : 2023

Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets

Résumé

We consider the null-controllability problem for the generalized Baouendi-Grushin equation (∂t − ∂ 2 x − q(x) 2 ∂ 2 y)f = 1ωu on a rectangular domain. Sharp controllability results already exist when the control domain ω is a vertical strip, or when q(x) = x. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general q and non-rectangular control region ω. In some geometries for ω, the upper bound and the lower-bound are equal, in which case, we know the exact value of the minimal time of null-controllability. Our proof relies on several tools: known results when ω is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the harmonic oscillator −∂ 2 x + ν 2 q(x) 2 when Re(ν) > 0, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.
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Dates et versions

hal-03618415 , version 1 (24-03-2022)
hal-03618415 , version 2 (05-07-2022)
hal-03618415 , version 3 (08-02-2024)

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Jérémi Dardé, Armand Koenig, Julien Royer. Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets. Annales Henri Lebesgue, 2023, 6, pp.1479-1522. ⟨10.5802/ahl.193⟩. ⟨hal-03618415v3⟩
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