Skip to main content
SpringerLink
Log in
Book cover

Crowd Dynamics, Volume 1 pp 137–165Cite as

Birkhäuser

Measure-Theoretic Models for Crowd Dynamics

  • Chapter
  • First Online:

Abstract

This chapter revises some modeling, analysis, and simulation contributions for crowd dynamics using time-evolving measures. Two key features are strictly related to the use of measures: on one side, this setting permits to generalize both microscopic and macroscopic crowd models. On the other side, it allows an easy description of multi-scale crowd models, e.g., with leaders and followers. The main analytical tool for studying measure evolution is to endow the space of measures with the Wasserstein distance.

This chapter also describes our recent contributions about crowd modeling with time-varying total mass. This requires to use a more flexible metric tool in the space of measures, that we called generalized Wasserstein distance.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. G. Albi, M. Bongini, F. Rossi, and F. Solombrino. Leader formation with mean-field birth and death models. in preparation.

    Google Scholar 

  2. G. Arechavaleta, J.-P. Laumond, H. Hicheur, and A. Berthoz. An optimality principle governing human walking. IEEE Transactions on Robotics, 24:5–14, 2008.

    Article  Google Scholar 

  3. N. Bellomo and A. Bellouquid. On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms. Networks and Heterogeneous Media, 6:383–399, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  4. N. Bellomo, D. Clarke, L. Gibelli, P. Townsend, and B. Vreugdenhil. Human behaviours in evacuation crowd dynamics: From modelling to big data toward crisis management. Physics of Life Reviews, 18:1–21, 2016.

    Article  Google Scholar 

  5. N. Bellomo, B. Piccoli, and A. Tosin. Modeling crowd dynamics from a complex system viewpoint. Mathematical models and methods in applied sciences, 22(supp02):1230004, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Bellomo and C. Dogbe. On the modeling of traffic and crowds: a survey of models, speculations, and perspectives. SIAM Rev., 53(3):409–463, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Bellomo and L. Gibelli. Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds. Mathematical Models and Methods in Applied Sciences, 25(13):2417–2437, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bongini, M. Fornasier, F. Rossi, and F. Solombrino. Mean-Field Pontryagin Maximum Principle. Journal of Optimization Theory and Applications, 175(1):1–38, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Camazine, J. Deneubourg, N. Franks, J. Sneyd, G. Theraulaz, and E. Bonabeau. Self organization in biological systems. Princeton University Press, 2003.

    Google Scholar 

  10. M. Caponigro, B. Piccoli, F. Rossi, and E. Trélat. Mean-field sparse Jurdjevic-Quinn control. M3AS: Math. Models Meth. Appl. Sc., 27(7):1223–1253, 2017.

    MathSciNet  MATH  Google Scholar 

  11. J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepčev. Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J., 156(2):229–271, 02 2011.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil. Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, chapter Particle, kinetic, and hydrodynamic models of swarming, pages 297–336. Birkhäuser Boston, Boston, 2010.

    Chapter  MATH  Google Scholar 

  13. C. Cercignani, R. Illner, and M. Pulvirenti. The mathematical theory of dilute gases, volume 106 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994.

    MATH  Google Scholar 

  14. Y. Chitour, F. Jean, and P. Mason. Optimal control models of goal-oriented human locomotion. SIAM Journal on Control and Optimization, 50(1):147–170, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Chizat, G. Peyré, B. Schmitzer, and F.-X. Vialard. An interpolating distance between optimal transport and Fisher–Rao metrics. Foundations of Computational Mathematics, 18(1):1–44, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Cristiani, P. Frasca, and B. Piccoli. Effects of anisotropic interactions on the structure of animal groups. Journal of mathematical biology, 62(4):569–588, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Cristiani, B. Piccoli, and A. Tosin. Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul., 9(1):155–182, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Cucker and S. Smale. Emergent behavior in flocks. IEEE Trans. Automat. Control, 52(5):852–862, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Di Marzo Serugendo, M.-P. Gleizes, and A. Karageorgos. Self-organization in multi-agent systems. The Knowledge Engineering Review, 20(2):165–189, 2005.

    Article  Google Scholar 

  20. B. Düring, P. Markowich, J.-F. Pietschmann, and M.-T. Wolfram. Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, page rspa20090239. The Royal Society, 2009.

    Google Scholar 

  21. L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. CRC press, 1991.

    Google Scholar 

  22. F. Farina, D. Fontanelli, A. Garulli, A. Giannitrapani, and D. Prattichizzo. When Helbing meets Laumond: The Headed Social Force Model. In 2016 IEEE 55th Conference on Decision and Control (CDC), pages 3548–3553, Dec 2016.

    Google Scholar 

  23. I. Giardina. Collective behavior in animal groups: theoretical models and empirical studies. Human Frontier Science Program Journal, (205–219), 2008.

    Article  Google Scholar 

  24. P. Glensdorf and I. Prigozhin. Thermodynamic Theory of Structure, Stability and Fluctuations. Wiley Interscience, 1971.

    Google Scholar 

  25. S.-Y. Ha and E. Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic & Related Models, 1(3):415–435, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Haskovec. Flocking dynamics and mean-field limit in the Cucker–Smale-type model with topological interactions. Physica D: Nonlinear Phenomena, 261:42–51, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Helbing and P. Molnár. Social force model for pedestrian dynamics. Physical Review E, 51(5):4282–4286, 1995.

    Article  Google Scholar 

  28. S. Kondratyev, L. Monsaingeon, and D. Vorotnikov. A new optimal transport distance on the space of finite Radon measures. Adv. Differential Equations, 21(11-12):1117–1164, 2016.

    MathSciNet  MATH  Google Scholar 

  29. P. Krugman. The Self Organizing Economy. Blackwell Publisher, 1995.

    Google Scholar 

  30. J. Lehn. Supramolecular chemistry—scope and perspectives molecules, supermolecules, and molecular devices (nobel lecture). Angewandte Chemie International Edition in English, 27(1):89–112, 1988.

    Article  Google Scholar 

  31. K. Lewin. Field Theory in Social Science. Harper & Brothers, 1951.

    Google Scholar 

  32. M. Liero, A. Mielke, and G. Savaré. Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves. SIAM J. Math. Anal., 48(4):2869–2911, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Maury and J. Venel. A discrete contact model for crowd motion. ESAIM: Mathematical Modelling and Numerical Analysis, 45(1):145–168, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Muntean, J. Rademacher, and A. Zagaris. Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity. Springer, 2016.

    Google Scholar 

  35. H. Neunzert. An introduction to the nonlinear Boltzmann-Vlasov equation. In C. Cercignani, editor, Kinetic Theories and the Boltzmann Equation, pages 60–110, Berlin, Heidelberg, 1984. Springer Berlin Heidelberg.

    Chapter  Google Scholar 

  36. L. Pallottino, V. Scordio, E. Frazzoli, and A. Bicchi. Decentralized cooperative policy for conflict resolution in multivehicle systems. IEEE Transactions on Robotics, 23(6):1170–1183, 1998.

    Article  Google Scholar 

  37. L. Perea, P. Elosegui, and G. Gómez. Extension of the Cucker-Smale Control Law to Space Flight Formations. Journal of Guidance, Control, and Dynamics, 32:527–537, 2009.

    Article  Google Scholar 

  38. B. Piccoli and F. Rossi. Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes. Acta Appl. Math., 124:73–105, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  39. B. Piccoli and F. Rossi. Generalized Wasserstein distance and its application to transport equations with source. Arch. Ration. Mech. Anal., 211(1):335–358, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  40. B. Piccoli and F. Rossi. On properties of the generalized Wasserstein distance. Arch. Ration. Mech. Anal., 222(3):1339–1365, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  41. B. Piccoli, F. Rossi, and M. Tournus. A norm for signed measures. Application to non local transport equation with source term. preprint, hal-01665244, Dec 2017.

    Google Scholar 

  42. B. Piccoli and A. Tosin. Time-evolving measures and macroscopic modeling of pedestrian flow. Archive for Rational Mechanics and Analysis, 199(3):707–738, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  43. S. T. Rachev. Probability metrics and the stability of stochastic models, volume 269. John Wiley & Son Ltd, 1991.

    Google Scholar 

  44. C. Tomlin, G. Pappas, and S. Sastry. Conflict resolution for air traffic management: A study in multiagent hybrid systems. IEEE Transactions on automatic control, 43(4):509–521, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  45. C. Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  46. N. Wiener. The Mathematics of Self-Organising Systems. Recent Developments in Information and Decision Processes. Macmillan, 1962.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Rutgers University – Camden, Camden, NJ, USA

    Benedetto Piccoli

  2. Department of Mathematics “Tullio Levi–Civita”, University of Padova, Padua, Italy

    Francesco Rossi

Authors
  1. Benedetto Piccoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Rossi .

Editor information

Editors and Affiliations

  1. School of Engineering, University of Edinburgh, Edinburgh, UK

    Livio Gibelli

  2. Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Nicola Bellomo

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Piccoli, B., Rossi, F. (2018). Measure-Theoretic Models for Crowd Dynamics. In: Gibelli, L., Bellomo, N. (eds) Crowd Dynamics, Volume 1. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-05129-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation