Optimal control problems driven by nonlinear degenerate Fokker-Planck equations

TL;DR

This paper analyzes the well-posedness of optimal control problems involving nonlinear degenerate Fokker-Planck equations coupled with ODEs, relevant for mean-field models of multi-agent dynamics with non-Lipschitz nonlinearities.

Contribution

It introduces a framework combining stability estimates and dissipativity assumptions to establish well-posedness for complex coupled PDE-ODE control problems.

Findings

Established well-posedness under non-Lipschitz nonlinearities

Developed stability estimates for measure solutions

Ensured compactness and lower semicontinuity of the cost functional

Abstract

The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean H\"{o}lder and sublinearity conditions.