ALPHONSE, Paul; KOENIG, Armand

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ALPHONSE, Paul
Unité de Mathématiques Pures et Appliquées [UMPA-ENSL]

KOENIG, Armand
Institut de Mathématiques de Bordeaux [IMB]

Langue

Document de travail - Pré-publication

Résumé en anglais

We study the null-controllability properties of heat-like equations posed on the whole Euclidean space $\mathbb R^n$. These evolution equations are associated with Fourier multipliers of the form $\rho(\vert D_x\vert)$, where $\rho\colon[0,+\infty)\rightarrow\mathbb C$ is a measurable function such that $\Re\rho$ is bounded from below. We consider the ``weakly dissipative'' case, a typical example of which is given by the fractional heat equations associated with the multipliers $\rho(\xi) = \xi^s$ in the regime $s\in(0,1)$, for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier $\rho$, in particular assuming that $\rho(\xi) = o(\xi)$, we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier $\rho$.Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.< Réduire

Mots clés en anglais

Null-controllability

Diffusive equations

$\gamma$-thick sets

Cantor-Smith-Volterra sets

URI

https://oskar-bordeaux.fr/handle/20.500.12278/190461

Project ANR

Centre International de Mathématiques et d'Informatique (de Toulouse) - ANR-11-LABX-0040

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251