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

IF 1.6 2区数学 Q2 MATHEMATICS, APPLIED

Applied Mathematics and Optimization Pub Date : 2024-04-26 DOI:10.1007/s00245-024-10135-4

Ştefana-Lucia Aniţa

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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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通过非局部作用输入控制非线性福克-普朗克方程

本文涉及通过非局部作用输入与非线性福克-普朗克方程相关的最优控制问题 (P)。福克-普朗克方程描述了在产生排斥矢量场的控制下种群概率密度的动态变化，而排斥矢量场会使种群发生位移。实际上，问题（P）要求通过控制产生的排斥作用，以最佳方式迁移一个种群。这个问题与麦金-弗拉索夫方程的随机最优控制问题（(P_S)\）有很深的联系。在确定性问题（P）中得到了最优控制的存在性。对于与非线性福克-普朗克方程的后向欧拉近似（离散化步长为 h）相关的受惩罚最优控制问题（(P_h)\），确定了最优控制的存在性并导出了必要的最优性条件。通过使用类似于极限的论证（如 \(h\rightarrow 0\) ），可以推导出问题（P）的必要最优条件。同时还讨论了一些可能的扩展。

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来源期刊

Applied Mathematics and Optimization 数学-应用数学

CiteScore

3.30

自引率

5.60%

发文量

103

审稿时长

>12 weeks

期刊介绍： The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.