Mean-Field Control for Diffusion Aggregation system with Coulomb Interaction

TL;DR

This paper develops a novel mean-field control framework for a diffusion-aggregation system with Coulomb interaction, proving the existence of optimal controls and establishing strong propagation of chaos without compact support restrictions.

Contribution

It introduces a new approach by controlling the Liouville equation directly and achieves strong propagation of chaos, extending the analysis to unbounded control functions in the aggregation context.

Findings

Existence of optimal control via $\Gamma$-convergence.

Strong propagation of chaos for particle systems.

Global well-posedness in the sub-critical regime.

Abstract

The mean-field control problem for a multi-dimensional diffusion-aggregation system with Coulomb interaction (the so called parabolic elliptic Keller-Segel system) is considered. The existence of optimal control is proved through the $\Gamma$-convergence of the corresponding control problem of the interacting particle system. There are three building blocks in the whole argument. Firstly, for the optimal control problem on the particle level, instead of using classical method for stochastic system, we study directly the control problem of high-dimensional parabolic equation, i.e. the Liouville equation of it. Secondly, we obtain a strong propagation of chaos result for the interacting particle system by combining the convergence in probability and relative entropy method. Due to this strong mean field limit result, we avoid giving compact support requirement for control functions, which has been often used in the literature. Thirdly, because of strong aggregation effect, additional difficulties arise from control function in obtaining the well-posedness theory of the diffusion-aggregation equation, so that the known method cannot be directly applied. Instead, we use a combination of local existence result and bootstrap argument to obtain the global solution in the sub-critical regime.