On a Mean Field Optimal Control Problem

TL;DR

This paper studies a mean field optimal control problem involving agent interactions, noise, and nonlocal effects, establishing the existence of solutions for the coupled PDE system that models the dynamics and control of the population.

Contribution

It provides the first existence result for a mean field control system with nonlocal interactions and an aggregation-diffusion constraint, extending mean field game theory.

Findings

Existence of solutions for the coupled Hamilton-Jacobi and Fokker-Planck system.

Analysis of nonlocal interaction effects on the PDE system.

Connection with and extension of mean-field game models.

Abstract

In this paper we consider a mean field optimal control problem with an aggregation-diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton-Jacobi and a Fokker-Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker-Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.