Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation

Fatiha Alabau-Boussouira; Piermarco Cannarsa; Cristina Urbani

doi:10.5802/crmath.567

Article de recherche - Théorie du contrôle

Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation
[Contrôle bilinéaire d’équations d’évolution avec opérateur de contrôle non borné. Application à l’équation de Fokker–Planck]

Fatiha Alabau-Boussouira ¹ ; Piermarco Cannarsa ² ; Cristina Urbani ³

¹ Laboratoire Jacques-Louis Lions, Sorbonne Université, 75005, Paris, Université de Lorraine, France
² Dipartimento di Matematica, Università di Roma Tor Vergata, 00133, Roma, Italy
³ Dipartimento di Scienze Tecnologiche e dell’Innovazione, Universitas Mercatorum, Piazza Mattei 10, 00186, Roma, Italy

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 511-545.

Résumé
Abstract

Nous étudions la contrôlabilité exacte de l’équation d’évolution

u^{'} (t) + A u (t) + p (t) B u (t) = 0

où A est un opérateur auto-adjoint non négatif dans l’espace de Hilbert X et B est un opérateur linéaire non borné de X, dominé par la racine carrée de l’opérateur A. L’action du contrôle est bilinéaire et sous forme d’entrée scalaire, ce qui signifie que le contrôle est la fonction scalaire p, supposée ne dépendre que du temps et être de carré intégrable. Notre résultat principal est la contrôlabilité locale exacte de l’équation d’évolution au voisinage de la solution de l’état fondamental, c’est-à-dire au voisinage de la solution de l’équation d’évolution de donnée initiale égale à la première fonction propre de A.

Nous avons traité un problème similaire sous une forme plus générale dans notre article précédent [Contrôlabilité exacte aux solutions propres pour les équations d’évolution de type parabolique via contrôle bilinéaire, Alabau-Boussouira F., Cannarsa P. et Urbani C., Nonlinear Diff. Eq. Appl. (2022)] dans les cas où l’opérateur B est borné. L’extension actuelle aux opérateurs non bornés permet de nombreuses autres applications, comme celle de l’équation de Fokker–Planck en dimension un d’espace. Elle permet également de considérer une classe plus large d’opérateurs de contrôle.

We study the exact controllability of the evolution equation

u^{'} (t) + A u (t) + p (t) B u (t) = 0

where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$ , which is dominated by the square root of $A$ . The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$ , which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of $A$ , as initial data.

The analogous problem (in a more general form) was addressed in our previous paper [Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, Alabau-Boussouira F., Cannarsa P. and Urbani C., Nonlinear Diff. Eq. Appl. (2022)] for a bounded operator $B$ . The current extension to unbounded operators allows for many more applications, including the Fokker–Planck equation in one space dimension, and a larger class of control actions.

Reçu le : 2023-03-12
Révisé le : 2023-07-06
Accepté le : 2023-09-05
Publié le : 2024-05-31

DOI : 10.5802/crmath.567

Classification : 35Q93, 93C25, 93C10, 93B05, 35K90

Affiliations des auteurs :

Fatiha Alabau-Boussouira ¹ ; Piermarco Cannarsa ² ; Cristina Urbani ³

¹ Laboratoire Jacques-Louis Lions, Sorbonne Université, 75005, Paris, Université de Lorraine, France
² Dipartimento di Matematica, Università di Roma Tor Vergata, 00133, Roma, Italy
³ Dipartimento di Scienze Tecnologiche e dell’Innovazione, Universitas Mercatorum, Piazza Mattei 10, 00186, Roma, Italy

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Fatiha Alabau-Boussouira and Piermarco Cannarsa and Cristina Urbani},
     title = {Bilinear control of evolution equations with unbounded lower order terms. {Application} to the {Fokker{\textendash}Planck} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {511--545},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.567},
     language = {en},
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Fatiha Alabau-Boussouira; Piermarco Cannarsa; Cristina Urbani. Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 511-545. doi : 10.5802/crmath.567. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.567/

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