Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type

Clotilde Fermanian Kammerer; Cyril Letrouit

doi:10.5802/jep.176

Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type
[Observabilité et contrôlabilité de l’équation de Schrödinger sur des quotients de groupes de type Heisenberg]

Clotilde Fermanian Kammerer ^1,² ; Cyril Letrouit ^3,⁴

¹ Univ Gustave Eiffel, LAMA F-77447 Marne-la-Vallée, France
² Univ Paris Est Creteil, CNRS, LAMA F-94010 Creteil, France
³ DMA, École normale supérieure, CNRS, PSL Research University 75005 Paris, France
⁴ Sorbonne Université, Université Paris-Diderot, CNRS, Inria, LJLL 75005 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1459-1513.

Résumé
Abstract

Dans cet article, nous donnons des conditions nécessaires et des conditions suffisantes pour la contrôlabilité d’une équation de Schrödinger impliquant un opérateur sous-elliptique sur une variété compacte. Cet opérateur est le sous-laplacien d’une variété obtenue en quotientant un groupe de type Heisenberg par l’un de ses sous-groupes discrets. Cette classe de groupes nilpotents est un exemple important de groupes de Lie de pas 2. Le sous-laplacien est alors un opérateur sous-elliptique et nous montrons qu’à la différence de ce qui se passe pour le cas elliptique sur le tore ou sur des surfaces à courbures négatives, il existe un temps minimal de contrôlabilité pour l’équation de Schrödinger associée à ce sous-laplacien. Les principaux outils que nous utilisons sont des mesures semi-classiques à valeurs opérateurs construites via la théorie des représentations et une notion de paquets d’ondes semi-classiques que nous introduisons ici dans le contexte des groupes de type Heisenberg.

We give necessary and sufficient conditions for the controllability of a Schrödinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schrödinger equations is subelliptic, and, contrary to what happens for the usual elliptic Schrödinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

Reçu le : 2020-10-08
Accepté le : 2021-08-16
Publié le : 2021-09-03

DOI : 10.5802/jep.176

Classification : 35R03, 35H20, 35J10, 93B07, 35Q93
Keywords: Sub-elliptic operator, control theory, observability, nilmanifold, H-type group
Mot clés : Opérateur sous-elliptique, théorie du contrôle, nilvariété, groupe de type Heisenberg

Affiliations des auteurs :

Clotilde Fermanian Kammerer ^{1, 2} ; Cyril Letrouit ^{3, 4}

¹ Univ Gustave Eiffel, LAMA F-77447 Marne-la-Vallée, France
² Univ Paris Est Creteil, CNRS, LAMA F-94010 Creteil, France
³ DMA, École normale supérieure, CNRS, PSL Research University 75005 Paris, France
⁴ Sorbonne Université, Université Paris-Diderot, CNRS, Inria, LJLL 75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Observability and controllability for {the~Schr\"odinger} equation on quotients of groups of {Heisenberg} type},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Clotilde Fermanian Kammerer; Cyril Letrouit. Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1459-1513. doi : 10.5802/jep.176. https://jep.centre-mersenne.org/articles/10.5802/jep.176/

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