Mean field sparse optimal control of systems with additive white noise

TL;DR

This paper develops a mean-field optimal control framework for multi-agent systems with additive white noise, focusing on sparse interventions on leaders and deriving a limiting PDE-ODE system.

Contribution

It introduces a rigorous limit process from finite-dimensional controlled SDEs to an infinite-dimensional PDE-ODE mean-field control problem.

Findings

Derivation of a PDE-ODE system representing the mean-field limit.

Establishment of a Γ-limit linking finite and infinite-dimensional control problems.

Framework for controlling noisy multi-agent systems with sparse interventions.

Abstract

We analyze the problem of controlling a multi-agent system with additive white noise through parsimonious interventions on a selected subset of the agents (leaders). For such a controlled system with a SDE constraint, we introduce a rigorous limit process towards an infinite dimensional optimal control problem constrained by the coupling of a system of ODE for the leaders with a McKean-Vlasov-type SDE, governing the dynamics of the prototypical follower. The latter is, under some assumptions on the distribution of the initial data, equivalent with a (nonlinear parabolic) PDE-ODE system. The derivation of the limit mean-field optimal control problem is achieved by linking the mean-field limit of the governing equations together with the $\Gamma$-limit of the cost functionals for the finite dimensional problems.