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A Fokker–Planck approach to control collective motion

Author

Listed:
  • Souvik Roy

    (Universität Würzburg)

  • Mario Annunziato

    (Università degli Studi di Salerno)

  • Alfio Borzì

    (Universität Würzburg)

  • Christian Klingenberg

    (Universität Würzburg)

Abstract

A Fokker–Planck control strategy for collective motion is investigated. This strategy is formulated as the minimisation of an expectation objective with a bilinear optimal control problem governed by the Fokker–Planck equation modelling the evolution of the probability density function of the stochastic motion. Theoretical results on existence and regularity of optimal controls are provided. The resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, $$L^1$$ L 1 stability, and second-order accuracy of the forward solution. A projected non-linear conjugate gradient scheme is used to solve the optimality system. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the efficiency of the proposed control framework.

Suggested Citation

  • Souvik Roy & Mario Annunziato & Alfio Borzì & Christian Klingenberg, 2018. "A Fokker–Planck approach to control collective motion," Computational Optimization and Applications, Springer, vol. 69(2), pages 423-459, March.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:2:d:10.1007_s10589-017-9944-3
    DOI: 10.1007/s10589-017-9944-3
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    References listed on IDEAS

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    1. Néstor Sepúlveda & Laurence Petitjean & Olivier Cochet & Erwan Grasland-Mongrain & Pascal Silberzan & Vincent Hakim, 2013. "Collective Cell Motion in an Epithelial Sheet Can Be Quantitatively Described by a Stochastic Interacting Particle Model," PLOS Computational Biology, Public Library of Science, vol. 9(3), pages 1-12, March.
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    Cited by:

    1. Tim Breitenbach & Alfio Borzì, 2020. "The Pontryagin maximum principle for solving Fokker–Planck optimal control problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 499-533, June.
    2. Butt, Muhammad Munir, 2021. "Two-level difference scheme for the two-dimensional Fokker–Planck equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 276-288.
    3. Simone Göttlich & Ralf Korn & Kerstin Lux, 2019. "Optimal control of electricity input given an uncertain demand," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(3), pages 301-328, December.
    4. Suvra Pal & Souvik Roy, 2021. "On the estimation of destructive cure rate model: A new study with exponentially weighted Poisson competing risks," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 75(3), pages 324-342, August.

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