Mean-Field Pontryagin Maximum Principle

  • Mattia Bongini
  • Massimo Fornasier
  • Francesco RossiEmail author
  • Francesco Solombrino


We derive a maximum principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. 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 optimal control problem for the discretized density, under a suitable scaling of the adjoint variables.


Sparse optimal control Mean-field limit \(\varGamma \)-limit Optimal control with ODE–PDE constraints Subdifferential calculus Hamiltonian flows 

Mathematics Subject Classification




The authors acknowledge the support of the PHC-PROCOPE Project “Sparse Control of Multiscale Models of Collective Motion.” Mattia Bongini and Massimo Fornasier additionally acknowledge the support of the ERC-Starting Grant Project “High-Dimensional Sparse Optimal Control.” Francesco Rossi additionally acknowledges the support of the ANR project CroCo ANR-16-CE33-0008.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Fakultät MathematikTechnische Universität MünchenGarching bei MünchenGermany
  2. 2.CNRS, ENSAM, Université de Toulon, LSIS UMR 7296Aix Marseille UniversitéMarseilleFrance
  3. 3.Dipartimento di Matematica e ApplicazioniUniversità di Napoli “Federico II”NaplesItaly

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