Mean-field control of non exchangeable systems

¹ LPSM, Université Paris Cité and Sorbonne University, Paris, France
² Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy
³ LPSM, Sorbonne University and Université Paris Cité, Paris, France
⁴ Ecole Polytechnique, CMAP, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 10 June 2025
Accepted: 19 November 2025

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 L²-set of Wasserstein space-valued functions.