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Computational Optimization and Applications

, Volume 69, Issue 2, pp 423–459 | Cite as

A Fokker–Planck approach to control collective motion

  • Souvik RoyEmail author
  • Mario Annunziato
  • Alfio Borzì
  • Christian Klingenberg
Article

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, \(L^1\) 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.

Keywords

Fokker–Planck equation Alternate direction method Chang–Cooper scheme Projected gradient method Control constrained PDE optimization 

Mathematics Subject Classification

35Q84 35Q91 35Q93 49K20 49J20 65C20 

Notes

Acknowledgements

The authors would like to gratefully acknowledge the comments by the referees which helped to improve this paper. S. Roy would like to thank A. S. Vasudeva Murthy and Praveen Chandrashekar for several fruitful discussions during the initial phases of this work. This work was supported in part by the European Union under Grant Agreement No. 304617 Marie Curie Research Training Network “Multi-ITN STRIKE—Novel Methods in Computational Finance” and the BMBF project “ROENOBIO”. S. Roy was also supported by the DAAD Passage to India Program and the AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems established in TIFR/ICTS, Bangalore.

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Authors and Affiliations

  1. 1.Institut für MathematikUniversität WürzburgWürzburgGermany
  2. 2.Dipartimento di MatematicaUniversità degli Studi di SalernoFiscianoItaly
  3. 3.Institut für MathematikUniversität WürzburgWürzburgGermany

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