Bilinear local controllability to the trajectories of the Fokker–Planck equation with a localized control

Abstract

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.Résumé. -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.

Control of reaction-diffusion models in biology and social sciences

“…In this section we have shown that, even when L > L * , we can build a path of admissible steady-states outside Γ connecting u L and 0 (not satisfying (25)). But we have also seen that this path cannot be used to control the dynamics while preserving the constraints.…”

“…Therefore at the final time, t = T , only the final target v(•, T ) ≡ 0 will be reached, regardless of the value of the control. We refer to [25,15,58,59] and to [74,81,79,86,50,82,80,83,24] for the analysis of bilinear control on population dynamics systems.…”

“…constant steady-states as targets, one can use the same methods to control to any element of the constructed paths of steady-states even if they are not constant. In particular, if the system can be controlled to the constant steady-state θ in large time, one can also control it to any target (possibly requiring a larger time) that is path-connected in an admissible way to θ (as long as the path is away of the prescribed bounds, see (25) in Theorem 4.2 and Subsection 7.2).…”

Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science

“…Finally, we mention some recent papers about the approximate multiplicative controllability for reaction-diffusion equations between sign-changing states: in [16] by the author with Cannarsa and Khapalov regarding a semilinear uniformly parabolic system, and [37] by the author with Nitsch and Trombetti, concerning degenerate parabolic equations. Furthermore, some interesting contributions about exact controllability issues for evolution equations via bilinear controls have recently appeared, in particular we mention [2] by Alabau-Boussouira, Cannarsa and Urbani, and [29] by Duprez and Lissy.…”