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

We consider the null-controllability problem for the generalized Baouendi-Grushin equation \((\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_\omega u\) on a rectangular domain. Sharp controllability results already exist when the control domain \(\omega\) is a vertical strip, or when \(q(x) =...

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Bibliographic Details
Published inarXiv.org
Main Authors Dardé, Jérémi, Koenig, Armand, Royer, Julien
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 07.07.2022
Subjects
Controllability
Domains
Lower bounds
Operators (mathematics)
Polynomials
Spectrum analysis
Strip
Upper bounds
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Summary:We consider the null-controllability problem for the generalized Baouendi-Grushin equation \((\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_\omega u\) on a rectangular domain. Sharp controllability results already exist when the control domain \(\omega\) 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 \(\omega\). In some geometries for \(\omega\), 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 \(\omega\) is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schr\"odinger operator \(-\partial_x^2 + \nu^2 q(x)^2\) when \(\Re(\nu)>0\), pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.
ISSN:2331-8422