Mean-field control of non exchangeable systems

TL;DR

This paper develops a theoretical framework for optimal control of mean-field systems with heterogeneous interactions, establishing well-posedness, dynamic programming principles, and viscosity solutions in Wasserstein spaces.

Contribution

It introduces a novel approach to control non-exchangeable mean-field systems, including well-posedness, law invariance, and a dynamic programming principle in Wasserstein space.

Findings

Proved well-posedness of controlled mean-field systems with heterogeneous interactions.

Established a law invariance property for the value function.

Derived a viscosity solution characterization in Wasserstein space.

Abstract

We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal probability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a $L^2$-set of Wasserstein space-valued functions.