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Mean-Field Pontryagin Maximum Principle

Author

Listed:
  • Mattia Bongini

    (Technische Universität München)

  • Massimo Fornasier

    (Technische Universität München)

  • Francesco Rossi

    (Aix Marseille Université)

  • Francesco Solombrino

    (Università di Napoli “Federico II”)

Abstract

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.

Suggested Citation

  • Mattia Bongini & Massimo Fornasier & Francesco Rossi & Francesco Solombrino, 2017. "Mean-Field Pontryagin Maximum Principle," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 1-38, October.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:1:d:10.1007_s10957-017-1149-5
    DOI: 10.1007/s10957-017-1149-5
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    References listed on IDEAS

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    1. Rainer Hegselmann & Ulrich Krause, 2002. "Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 5(3), pages 1-2.
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    4. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
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    1. Linjie Zhang & Xizhe Wang & Tao He & Zhongmei Han, 2022. "A Data-Driven Optimized Mechanism for Improving Online Collaborative Learning: Taking Cognitive Load into Account," IJERPH, MDPI, vol. 19(12), pages 1-18, June.

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