Mean-Field Pontryagin Maximum Principle
Abstract
We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as -limits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the discrete optimal control problems, under a suitable scaling of the adjoint variables.
Cite
@article{arxiv.1504.02236,
title = {Mean-Field Pontryagin Maximum Principle},
author = {Mattia Bongini and Massimo Fornasier and Francesco Rossi and Francesco Solombrino},
journal= {arXiv preprint arXiv:1504.02236},
year = {2015}
}