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Karine Beauchard; Karel Pravda-Starov
Null-controllability of hypoelliptic quadratic differential equations
(Contrôlabilité à zéro d’équations aux dérivées partielles quadratiques hypoelliptiques)
Journal de l'École polytechnique — Mathématiques, 5 (2018), p. 1-43, doi: 10.5802/jep.62
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Class. Math.: 93B05, 35H10
Mots clés: Contrôlabilité à zéro, observabilité, opérateurs différentiels quadratiques, opérateurs de Ornstein-Uhlenbeck, opérateurs de Fokker-Planck, hypoellipticité

Résumé - Abstract

Nous étudions la contrôlabilité à zéro d’équations paraboliques associées à une classe générale d’opérateurs différentiels quadratiques hypoelliptiques. Les opérateurs différentiels quadratiques sont les opérateurs définis, en quantification de Weyl, par un symbole quadratique à valeurs complexes. Dans ce travail, nous considérons la classe des opérateurs quadratiques accrétifs avec espace singulier réduit au singleton zéro. Ces opérateurs différentiels, possiblement dégénérés et non auto-adjoints, sont hypoelliptiques et génèrent des semi-groupes de contractions, régularisant dans des espaces de Gelfand-Shilov particuliers, en tout temps strictement positif. Grâce à cet effet régularisant, nous démontrons, en adaptant la méthode de Lebeau-Robbiano, que les équations paraboliques associées sont contrôlables à zéro en tout temps strictement positif, lorsque les contrôles sont localisés sur un sous domaine, assurant classiquement la contrôlabilité à zéro de l’équation de la chaleur. Nous déduisons de ce résultat la contrôlabilité à zéro d’équations paraboliques associées à des opérateurs hypoelliptiques de Ornstein-Uhlenbeck agissant sur des espaces $L^2$ à poids, dont le poids est la mesure invariante. La même stratégie fournit la contrôlabilité à zéro, en tout temps strictement positif, avec le même support de contrôle, pour les équations paraboliques associées aux opérateurs de Ornstein-Uhlenbeck hypoelliptiques agissant sur l’espace $L^2$ plat, étendant ainsi le résultat connu pour l’équation de la chaleur et l’équation de Kolmogorov posées sur tout l’espace.

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