Calculus of Variations and Geometric Measure Theory

M. Bongini - M. Fornasier - F. Rossi - F. Solombrino

Mean-field Pontryagin Maximum Principle

created by solombrin on 08 Apr 2015
modified on 08 Oct 2017


Published Paper

Inserted: 8 apr 2015
Last Updated: 8 oct 2017

Journal: Journal of Optimization Theory and Applications
Volume: 175
Number: 1
Pages: 1-38
Year: 2017
Doi: 10.1007/s10957-017-1149-5


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 $\Gamma$-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.