Null-controllability of hypoelliptic quadratic differential equations

Beauchard, Karine; Pravda-Starov, Karel

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 1-43.

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.

We study the null-controllability of parabolic equations associated with a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated with these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated with hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated with any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space.

Classification : 93B05, 35H10
Keywords: Null-controllability, observability, quadratic differential operators, Ornstein-Uhlenbeck operators, Fokker-Planck operators, hypoellipticity
Mot clés : Contrôlabilité à zéro, observabilité, opérateurs différentiels quadratiques, opérateurs de Ornstein-Uhlenbeck, opérateurs de Fokker-Planck, hypoellipticité

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