no. 4
Bilinear local controllability to the trajectories of the Fokker–Planck equation with a localized control
[Contrôlabilité bilinéaire locale aux trajectoires de l’équation de Fokker–Planck avec un contrôle localisé]
Duprez, Michel1 ; Lissy, Pierre2
1 Inria, Université de Strasbourg, ICUBE, équipe MIMESIS Strasbourg (France)
2 CEREMADE, Université Paris-Dauphine & CNRS UMR 7534, Université PSL 75016 Paris (France)
Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1621-1659.

Ce travail est consacré au contrôle de l’équation de Fokker–Planck, posée sur un domaine borné régulier de ℝ d , avec un terme de dérive localisé. Nous démontrons que cette équation est localement contrôlable aux trajectoires régulières non nulles. De plus, sous certaines conditions, nous expliquons comment réduire le nombre de contrôles autour du contrôle de référence. Les résultats sont obtenus à l’aide d’une méthode de linéarisation standard et la méthode de contrôle fictif. Les principales nouveautés sont les suivantes. Premièrement, la résolubilité algébrique est effectuée et utilisée directement sur le problème adjoint. Deuxièmement, nous démontrons une nouvelle inégalité de Carleman pour l’équation de la chaleur avec terme du premier ordre dépendant du temps et de l’espace : le membre de droite est le gradient de la solution localisée sur un sous-ouvert. Pour finir, nous donnons un exemple de trajectoire régulière autour de laquelle l’équation de Fokker–Planck n’est pas contrôlable ave un nombre réduit de contrôles, pour souligner que nos conditions sont pertinentes.

This work is devoted to the control of the Fokker–Planck equation, posed on a smooth bounded domain of ℝ d , with a localized drift force. We prove that this equation is locally controllable to regular nonzero trajectories. Moreover, under some conditions, we explain how to reduce the number of controls around the reference control. The results are obtained thanks to a standard linearization method and the fictitious control method. The main novelties are twofold. First, the algebraic solvability is performed and used directly on the adjoint problem. We then prove a new Carleman inequality for the heat equation with a space-time varying first-order term: the right-hand side is the gradient of the solution localized on an open subset. We finally give an example of regular trajectory around which the Fokker–Planck equation is not controllable with a reduced number of controls, to highlight that our conditions are relevant.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3501
Classification : 93B05, 93B07, 93B25, 93C10, 35K40
Keywords: Controllability, Parabolic equations, Carleman estimates, Fictitious control method, Algebraic solvability
Mot clés : contrôlabilité, équations paraboliques, estimations de Carleman, méthode de contrôle fictif, résolubilité algébrique

Duprez, Michel 1 ; Lissy, Pierre 2

1 Inria, Université de Strasbourg, ICUBE, équipe MIMESIS Strasbourg (France)
2 CEREMADE, Université Paris-Dauphine & CNRS UMR 7534, Université PSL 75016 Paris (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Bilinear local controllability to the trajectories of the {Fokker{\textendash}Planck} equation with a localized control},
     journal = {Annales de l'Institut Fourier},
     pages = {1621--1659},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Duprez, Michel; Lissy, Pierre. Bilinear local controllability to the trajectories of the Fokker–Planck equation with a localized control. Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1621-1659. doi : 10.5802/aif.3501. https://aif.centre-mersenne.org/articles/10.5802/aif.3501/

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