Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

Karine Beauchard; Michela Egidi; Karel Pravda-Starov

doi:10.5802/crmath.79

Équations aux dérivées partielles

Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

Karine Beauchard ¹ ; Michela Egidi ² ; Karel Pravda-Starov ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
² Ruhr Universität Bochum, Fakultät für Mathematik, Gebaude IB 3/55, Universitätsstr. 150, 44780 Bochum, Germany

Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 651-700.

Résumé

We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein–Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein–Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability.

Reçu le : 2019-08-27
Révisé le : 2020-05-25
Accepté le : 2020-05-26
Publié le : 2020-10-02

DOI : 10.5802/crmath.79

Classification : 93B05, 35H10

Affiliations des auteurs :

Karine Beauchard ¹ ; Michela Egidi ² ; Karel Pravda-Starov ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
² Ruhr Universität Bochum, Fakultät für Mathematik, Gebaude IB 3/55, Universitätsstr. 150, 44780 Bochum, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {651--700},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2020},
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Karine Beauchard; Michela Egidi; Karel Pravda-Starov. Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 651-700. doi : 10.5802/crmath.79. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.79/

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