Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller - Laboratoire Jacques-Louis Lions Accéder directement au contenu
Article Dans Une Revue Analysis & PDE Année : 2021

Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller

Résumé

We are concerned with the short-time observability constant of the heat equation from a subdomain ω of a bounded domain M. The constant is of the form eK∕T, where K depends only on the geometry of M and ω. Luc Miller (J. Differential Equations 204:1 (2004), 202–226) conjectured that K is (universally) proportional to the square of the maximal distance from ω to a point of M. We show in particular geometries that K may blow up like ∣∣log(r)∣∣2 when ω is a ball of radius r, hence disproving the conjecture. We then prove in the general case the associated upper bound on this blowup. We also show that the conjecture is true for positive solutions of the heat equation. The proofs rely on the study of the maximal vanishing rate of (sums of) eigenfunctions. They also yield lower and upper bounds for other geometric constants appearing as tunneling constants or approximate control costs. As an intermediate step in the proofs, we provide a uniform Carleman estimate for Lipschitz metrics. The latter also implies uniform spectral inequalities and observability estimates for the heat equation in a bounded class of Lipschitz metrics, which are of independent interest.
Fichier principal
Vignette du fichier
Laurent et Léautaud - 2021 - Observability of the heat equation, geometric cons.pdf (850.83 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03206393 , version 1 (23-04-2021)

Identifiants

Citer

Camille Laurent, Matthieu Léautaud. Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller. Analysis & PDE, 2021, 14 (2), pp.355-423. ⟨10.2140/apde.2021.14.355⟩. ⟨hal-03206393⟩
50 Consultations
43 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More