Skip to main content
SpringerLink
Log in

Set-Driven Evolution for Multiagent System

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We consider the deterministic evolution in the Euclidean space of a multiagent system with a large number of agents (possibly infinitely many). At each instant of time, besides from time and its current position, the set of velocities available to each agent is influenced by the set described by the current position of all the other agents. The latter is in turn determined by the overall motion of the crowd of all the agents. The interplay to the microscopical point of view of each single agent, and the macroscopical one of the set-evolution yields a non-trivial dynamical system. This two-level multiagent system can be described either by the evolution of a probability measure—describing the instantaneous density of the crowd—or by the evolution of a set—describing the positions where there is at least one agent. In this paper, we precise the links between the two descriptions, providing also some quantitative estimates on the macroscopical admissible evolutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In the sense of Definition 2.1 when (X, d) is replaced by \( ( \textrm{Comp}({\mathbb {R}}^d ), d_{{\mathscr {H}}} )\).

References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008). https://doi.org/10.1007/b137080

    Book  Google Scholar 

  2. Aubin, J.-P., Frankowska, H.: Set-valued Analysis, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2009). https://doi.org/10.1007/978-0-8176-4848-0

    Book  Google Scholar 

  3. Averboukh, Y., Marigonda, A., Quincampoix, M.: Extremal shift rule and viability property for mean field-type control systems. J. Optim. Theory Appl. 189(1), 244–270 (2021). https://doi.org/10.1007/s10957-021-01832-z

    Article  MathSciNet  Google Scholar 

  4. Badreddine, Z., Frankowska, H.: Hamilton–Jacobi inequalities on a metric space. J. Differ. Equ. 271, 1058–1091 (2021). https://doi.org/10.1016/j.jde.2020.09.026

    Article  MathSciNet  Google Scholar 

  5. Bivas, M., Quincampoix, M.: Optimal control for the evolution of deterministic systems. J. Differ. Equ. 269(3), 2228–2263 (2020). https://doi.org/10.1016/j.jde.2020.01.034

    Article  MathSciNet  Google Scholar 

  6. Bivas, M., Quincampoix, M.: Nonsmooth feedback control for multi-agent dynamics. Set-Valued Var Anal. 29, 501–518 (2021). https://doi.org/10.1007/s11228-021-00574-4

    Article  MathSciNet  Google Scholar 

  7. Bonnet, B., Frankowska, H.: Semiconcavity and sensitivity analysis in mean-field optimal control and applications. J. Math. Pures Appl. 9(157), 282–345 (2022). https://doi.org/10.1016/j.matpur.2021.11.001

    Article  MathSciNet  Google Scholar 

  8. Bonnet, B., Frankowska, H.: Necessary optimality conditions for optimal control problems in Wasserstein spaces. Appl. Math. Optim. 84(suppl. 2), S1281–S1330 (2021). https://doi.org/10.1007/s00245-021-09772-w

    Article  MathSciNet  Google Scholar 

  9. Bonnet, B., Frankowska, H.: Differential inclusions in Wasserstein spaces: the Cauchy-Lipschitz framework. J. Differ. Equ. 271, 594–637 (2021). https://doi.org/10.1016/j.jde.2020.08.031

    Article  MathSciNet  Google Scholar 

  10. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580, p. vii+278. Springer, Berlin (1977). https://doi.org/10.1007/BFb0087685

    Book  Google Scholar 

  11. Cavagnari, G., Marigonda, A., Quincampoix, M.: Compatibility of state constraints and dynamics for multiagent control systems. J. Evolut. Equ. 21(4), 4491–4537 (2021). https://doi.org/10.1007/s00028-021-00724-z

    Article  MathSciNet  Google Scholar 

  12. Colombo, R., Lorenz, T., Pogodaev, N.: On the modeling of moving populations through set evolution equations. Discrete Contin. Dyn. Syst. 35, 73–98 (2015). https://doi.org/10.3934/dcds.2015.35.73

    Article  MathSciNet  Google Scholar 

  13. Fornasier, M., Piccoli, B., Rossi, F.: Mean-field sparse optimal control. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 372, 2014 (2028). https://doi.org/10.1098/rsta.2013.0400

    Article  Google Scholar 

  14. Fornasier, M., Solombrino, F.: Mean-field optimal control. ESAIM Control Optim. Calc. Var. 20(4), 1123–1152 (2014). https://doi.org/10.1051/cocv/2014009

    Article  MathSciNet  Google Scholar 

  15. Frankowska, H., Plaskacz, S., Rzezuchowski, T.: Measurable viability theorems and the Hamilton–Jacobi–Bellman equation. J. Differ. Equ. 116(2), 265–305 (1995). https://doi.org/10.1006/jdeq.1995.1036

    Article  MathSciNet  Google Scholar 

  16. Jimenez, C., Marigonda, A., Quincampoix, M.: Optimal control of multiagent systems in the Wasserstein space. Calc. Var. Partial Differ. Equ. 59, 58 (2020). https://doi.org/10.1007/s00526-020-1718-6

    Article  MathSciNet  Google Scholar 

  17. Marigonda, A., Quincampoix, M.: Mayer control problem with probabilistic uncertainty on initial positions. J. Differ. Equ. 264(5), 3212–3252 (2018). https://doi.org/10.1016/j.jde.2017.11.014

    Article  MathSciNet  Google Scholar 

  18. Maury, B., Venel, J.: A mathematical framework for a crowd motion model. C. R. Acad. Sci. Paris Ser. I 346, 1245–1250 (2008). https://doi.org/10.1016/j.crma.2008.10.014

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was partially funded by the Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0010. Rossana Capuani has received funding from the Italian Ministry of University and Research according to D.M. 1062/2021 PON “Ricerca e Innovazione” 2014–2020, Asse IV “Istruzione e ricerca per il recupero” - Azione IV.4 - “Dottorati e contratti di ricerca su tematiche dell’innovazione”, Azione IV.6 “Contratti di ricerca su tematiche Green”. The first two authors were supported by INdAM-GNAMPA Project 2022 “Evoluzione e controllo ottimo in spazi di Wasserstein” (“Evolution and optimal control in Wasserstein spaces”), CUP_E55F22000270001.

Author information

Authors and Affiliations

  1. Department of Economics, Engineering, Society and Business Organization, University of Tuscia, Via del Paradiso, 47, 01100, Viterbo, Italy

    Rossana Capuani

  2. Department of Computer Science, University of Verona, Strada Le Grazie 15, 37134, Verona, Italy

    Antonio Marigonda

  3. Laboratoire de Mathématiques de Bretagne Atlantique, Univ Brest, CNRS UMR 6205, 6, Avenue Victor Le Gorgeu, 29200, Brest, France

    Marc Quincampoix

Authors
  1. Rossana Capuani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonio Marigonda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marc Quincampoix
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rossana Capuani.

Additional information

Communicated by Hélène Frankowska.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Capuani, R., Marigonda, A. & Quincampoix, M. Set-Driven Evolution for Multiagent System. J Optim Theory Appl 200, 280–307 (2024). https://doi.org/10.1007/s10957-023-02344-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02344-8

Keywords

Mathematics Subject Classification

Search

Navigation