A decentralized algorithm for a mean field control problem of piecewise deterministic Markov processes

¹ Adrien Séguret is with PSL Research University, Universite Paris-Dauphine, CEREMADE, Place de Lattre de Tassigny, 75016 Paris, France; Finance for Energy Market Research Centre (FIME), Paris, France; Osiris, EDF R&D, 91120 Palaiseau, France
² Thomas Le Corre is with DI ENS, CNRS, PSL University, INRIA Paris, France
³ Nadia Oudjane is with Osiris, EDF R&D, 91120 Palaiseau, France and with Finance for Energy Market Research Centre (FIME), Paris, France

^* Corresponding author: seguret.adrien@outlook.fr

Received: 27 July 2023
Accepted: 1 December 2023

Abstract

This paper provides a decentralized approach for the control of a population of N agents to minimize an aggregate cost. Each agent evolves independently according to a Piecewise Deterministic Markov dynamics controlled via unbounded jumps intensities. The N-agent high dimensional stochastic control problem is approximated by the limiting mean field control problem. A Lagrangian approach is proposed. Although the mean field control problem is not convex, it is proved to achieve zero duality gap. A stochastic version of the Uzawa algorithm is shown to converge to the primal solution. At each dual iteration of the algorithm, each agent solves its own small dimensional sub problem by means of the Dynamic Programming Principal, while the dual multiplier is updated according to the aggregate response of the agents. Finally, this algorithm is used in a numerical simulation to coordinate the charging of a large fleet of electric vehicles in order to track a target consumption profile.