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Extremal Shift Rule and Viability Property for Mean Field-Type Control Systems

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  • Yurii Averboukh

    (Krasovskii Institute of Mathematics and Mechanics
    Ural Federal University
    Higher School of Economics)

  • Antonio Marigonda

    (University of Verona)

  • Marc Quincampoix

    (Univ Brest)

Abstract

We investigate when a mean field-type control system can fulfill a given constraint. Namely, given a closed set of probability measures on the torus, starting from any initial probability measure belonging to this set, does there exist a solution to the mean field control system remaining in it for any time? This property—the so-called viability property—is equivalently characterized through a property involving normals to the given set of probability measures. We prove that, if the Hamiltonian is nonpositive at any normal distribution to the given set, then the feedback strategy realizing the extremal shift rule provides the approximate viability. This implies the usual viability property. Conversely, the Hamiltonian is nonpositive at any normal distribution if the given set is viable. Our approach enables us to derive generalized feedback laws which ensure the trajectory to fulfill the constraint. This generalized feedback called here extremely shift rule is inspired by constructive motions developed by Krasovskii and Subbotin for differential games.

Suggested Citation

  • Yurii Averboukh & Antonio Marigonda & Marc Quincampoix, 2021. "Extremal Shift Rule and Viability Property for Mean Field-Type Control Systems," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 244-270, April.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:1:d:10.1007_s10957-021-01832-z
    DOI: 10.1007/s10957-021-01832-z
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    References listed on IDEAS

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    1. P. Cardaliaguet & M. Quincampoix, 2008. "Deterministic Differential Games Under Probability Knowledge Of Initial Condition," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 10(01), pages 1-16.
    2. Yurii Averboukh, 2019. "Krasovskii–Subbotin Approach to Mean Field Type Differential Games," Dynamic Games and Applications, Springer, vol. 9(3), pages 573-593, September.
    3. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, July.
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    Cited by:

    1. Rossana Capuani & Antonio Marigonda & Marc Quincampoix, 2024. "Set-Driven Evolution for Multiagent System," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 280-307, January.

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