On uniform controllability of 1D transport equations in the vanishing viscosity limit

Camille Laurent; Matthieu Léautaud

doi:10.5802/crmath.405

Théorie du contrôle

On uniform controllability of 1D transport equations in the vanishing viscosity limit

Camille Laurent ¹ ; Matthieu Léautaud ²

¹ CNRS UMR 7598 and Sorbonne Universités UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
² Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Saclay, CNRS, Bâtiment 307, 91405 Orsay Cedex France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 265-312.

Résumé

We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.

We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schrödinger operator. They improve, in this particular situation, the results obtained in the companion paper [38].

The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [40], together with estimates on the spectral gap [33, 1]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.

Reçu le : 2022-03-04
Révisé le : 2022-07-06
Accepté le : 2022-07-08
Publié le : 2023-01-12

DOI : 10.5802/crmath.405

Classification : 93B07, 93B05, 35B25, 35F05, 35K05, 93C73

Affiliations des auteurs :

Camille Laurent ¹ ; Matthieu Léautaud ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On uniform controllability of {1D} transport equations in the vanishing viscosity limit},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {265--312},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.405},
     language = {en},
}

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Camille Laurent; Matthieu Léautaud. On uniform controllability of 1D transport equations in the vanishing viscosity limit. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 265-312. doi : 10.5802/crmath.405. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.405/

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