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Controlling a Nonlinear Fokker–Planck Equation via Inputs with Nonlocal Action

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Abstract

This paper concerns an optimal control problem (P) associated to a nonlinear Fokker–Planck equation via inputs with nonlocal action. The Fokker–Planck equation describes the dynamics of the probability density of a population under a control that produces a repellent vector field which displaces the population. Actually, problem (P) asks to optimally displace a population via the repellent action produced by the control. The problem is deeply related to a stochastic optimal control problem \((P_S)\) for a McKean–Vlasov equation. The existence of an optimal control is obtained for the deterministic problem (P). The existence of an optimal control is established and necessary optimality conditions are derived for a penalized optimal control problem \((P_h)\) related to a backward Euler approximation of the nonlinear Fokker–Planck equation (with a constant discretization step h). Using a passing-to-the-limit-like argument (as \(h\rightarrow 0\)) one derives the necessary optimality conditions for problem (P). Some possible extensions are discussed as well.

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Acknowledgements

The author would like to thank the reviewers for their comments and suggestions.

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  1. “Octav Mayer” Institute of Mathematics of the Romanian Academy, Bd. Carol I no 8, 700505, Iaşi, Romania

    Ştefana-Lucia Aniţa

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Aniţa, ŞL. Controlling a Nonlinear Fokker–Planck Equation via Inputs with Nonlocal Action. Appl Math Optim 89, 68 (2024). https://doi.org/10.1007/s00245-024-10135-4

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