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Recent Developments in Controlled Crowd Dynamics

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Crowd Dynamics, Volume 2

Abstract

We survey recent results on controlled particle systems. The control aspect introduces new challenges in the discussion of properties and suitable mean field limits. Some of the aspects are highlighted in a detailed discussion of a particular controlled particle dynamics. The applied techniques are shown on this simple problem to illustrate the basic methods. Computational results confirming the theoretical findings are presented and further particle models are discussed.

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References

  1. G. Albi, Kinetic approximation, stability and control of collective behaviour in self-organized systems. PhD thesis, Università degli Studi di Ferrara (2014). http://eprints.unife.it/894/

  2. G. Albi, L. Pareschi, M. Zanella, Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2028), 20140138, 18 (2014)

    Google Scholar 

  3. G. Albi, M. Herty, L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus. Commun. Math. Sci. 13(6), 1407–1429 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Albi, L. Pareschi, M. Zanella, Uncertainty quantification in control problems for flocking models. Math. Probl. Eng. 2015, Art. ID 850124, 14pp. (2015)

    Google Scholar 

  5. G. Albi, M. Bongini, E. Cristiani, D. Kalise, Invisible control of self-organizing agents leaving unknown environments. SIAM J. Appl. Math. 76(4), 1683–1710 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Albi, Y.-P. Choi, M. Fornasier, D. Kalise, Mean field control hierarchy. Appl. Math. Optim. 76(1), 93–135 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Albi, M. Fornasier, D. Kalise, A Boltzmann approach to mean-field sparse feedback control, in 20th IFAC World Congress, vol. 50, ed. by D.P.D. Dochain, D. Henrion (2017), pp. 2898–2903

    Google Scholar 

  8. G. Aletti, G. Naldi, G. Toscani, First-order continuous models of opinion formation. SIAM J. Appl. Math. 67(3), 837–853 (electronic) (2007)

    Google Scholar 

  9. L. Ambrosio, N. Gigli, G. Savare, Gradient Flows in Metric Spaces of Probability Measures. Lectures in mathematics (Birkhäuser, Basel, 2008)

    Google Scholar 

  10. N. Bellomo, G. Ajmone Marsan, A. Tosin, Complex Systems and Society. Modeling and Simulation. SpringerBriefs in mathematics (Springer, Berlin, 2013)

    Google Scholar 

  11. N. Bellomo, P. Degond, E. Tadmor, Active Particles, Volume 1 : Advances in Theory, Models, and Applications. Modeling and simulation in science, engineering and technology (Birkhäuser, Basel, 2017)

    Google Scholar 

  12. E. Ben-Naim, Opinion dynamics, rise and fall of political parties. Europhys. Lett. 69, 671 (2005)

    Article  Google Scholar 

  13. A. Bensoussan, J. Frehse, P. Yam, Mean Field Games and Mean Field Type Control Theory. SpringerBriefs in mathematics (Springer, New York, 2013)

    Google Scholar 

  14. A. Bensoussan, J. Frehse, P. Yam et al., Mean Field Games and Mean Field Type Control Theory, vol. 101 (Springer, Berlin, 2013)

    Book  MATH  Google Scholar 

  15. A. Bensoussan, J. Frehse, S.C.P. Yam, On the interpretation of the master equation. Stoch. Process. Appl. 127(7), 2093–2137 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Bongini, M. Fornasier, D. Kalise, (Un)conditional consensus emergence under perturbed and decentralized feedback controls. Discrete Contin. Dyn. Syst. 35(9), 4071–4094 (2015)

    Google Scholar 

  17. A. Borzì, S. Wongkaew, Modeling and control through leadership of a refined flocking system. Math. Models Methods Appl. Sci. 25(2), 255–282 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: And Well-balanced Schemes for Sources (Springer Science and Business Media, Berlin, 2004)

    Book  MATH  Google Scholar 

  19. F. Bouchut, S. Jin, X. Li, Numerical approximations of pressureless and isothermal gas dynamics. SIAM J. Numer. Anal. 41(1), 135–158 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Boudin, J. Mathiaud, A numerical scheme for the one-dimensional pressureless gases system. Numer. Methods Partial Differ. Equ. 28(6), 1729–1746 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. L. Boudin, F. Salvarani, A kinetic approach to the study of opinion formation. ESAIM: Math. Model. Numer. Anal. 43, 507–522 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. L.M. Briceño Arias, D. Kalise, F.J. Silva, Proximal methods for stationary mean field games with local couplings. SIAM J. Control Optim. 56(2), 801–836 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Burger, L. Caffarelli, P.A. Markowich, M.-T. Wolfram, On a Boltzmann-type price formation model. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469(2157), 20130126, 20 (2013)

    Google Scholar 

  24. M. Burger, M. Di Francesco, P.A. Markowich, M.-T. Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete Contin. Dyn. Syst. Ser. B 19(5), 1311–1333 (2014)

    MathSciNet  MATH  Google Scholar 

  25. M. Caponigro, M. Fornasier, B. Piccoli, E. Trélat, Sparse stabilization and optimal control of the Cucker-Smale model. Math. Control Relat. Fields 3(4), 447–466 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Caponigro, M. Fornasier, B. Piccoli, E. Trélat, Sparse stabilization and control of alignment models. Math. Models Methods Appl. Sci. 25(3), 521–564 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Cardaliaguet, Notes on mean field games (from P.-L. Lions lectures at the College de France) (2010)

    Google Scholar 

  28. P. Cardaliaguet, The convergence problem in meanfield games with local coupling. Appl. Math. Optim. 76, 177–215 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, A. Poretta, Long time average of mean field games with nonlocal coupling. SIAM J. Control Optim. 51, 3358–3591 (2013)

    Article  MathSciNet  Google Scholar 

  30. R. Carmona, F. Delarue, Probabilistic analysis of mean field games. SIAM J. Control Optim. 51, 2705–2734 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applications I-II (Springer, Berlin, 2018)

    Book  MATH  Google Scholar 

  32. J.A. Carrillo, M. Fornasier, G. Toscani, F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and life sciences. Model. Simul. Sci. Eng. Technol. (Birkhäuser, Boston, 2010), pp. 297–336

    Google Scholar 

  33. C. Cercignani, The Boltzmann Equation and Its Applications (Springer, New York, 1988)

    Book  MATH  Google Scholar 

  34. C. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases. Number 106 in applied mathematical sciences (Springer, New York, 1994)

    Google Scholar 

  35. C. Chalons, D. Kah, M. Massot, Beyond pressureless gas dynamics: quadrature-based velocity moment models. Commun. Math. Sci. 10(4), 1241–1272 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Cristiani, B. Piccoli, A. Tosin, Multiscale Modeling of Pedestrian Dynamics. MS&A. Modeling, simulation and applications, vol. 12 (Springer, Cham, 2014)

    Google Scholar 

  37. F. Cucker, S. Smale, Emergent behavior in flocks. IEEE Trans. Automat. Contr. 52, 852–862 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Degond, J.-G. Liu, C. Ringhofer, Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria. J. Stat. Phys. 154(3), 751–780 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Degond, J.-G. Liu, C. Ringhofer, Evolution of wealth in a nonconservative economy driven by local Nash equilibria. Philos. Trans. R. Soc. A 372 (2014). https://doi.org/10.1098/rsta.2013.0394

  40. P. Degond, J.-G. Liu, C. Ringhofer, Large-scale dynamics of mean-field games driven by local Nash equilibria. J. Nonlinear Sci. 24, 93–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Degond, M. Herty, J.-G. Liu, Meanfield games and model predictive control. Commun. Math. Sci. 5, 1403–1422, 12 (2017)

    Google Scholar 

  42. M. D’Orsogna, Y.-L. Chuang, A. Bertozzi, L. Chayes, Self-propelled particles with soft-core interactions. Patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302 (2006)

    Google Scholar 

  43. B. Dubroca, J.L. Feugeas, Entropic moment closure hierarchy for the radiative transfer equation. C. R. Acad. Sci. Paris Ser. I 329, 915–920 (1999)

    Article  MATH  Google Scholar 

  44. B. Düring, P. Markowich, J.-F. Pietschmann, M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A 465, 3687–3708 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. B. Düring, L. Pareschi, G. Toscani, Kinetic models for optimal control of wealth inequalities. Eur. Phys. J. B 91(10), 265 (2018)

    Google Scholar 

  46. M. Fornasier, F. Solombrino, Mean-field optimal control. ESAIM Control Optim. Calc. Var. 20(4), 1123–1152 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Fornasier, B. Piccoli, F. Rossi, Mean-field sparse optimal control. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2028), 20130400, 21 (2014)

    Google Scholar 

  48. A. Friedman, Differential Games (American Mathematical Society, Providence, 1974). Expository lectures from the CBMS Regional Conference held at the University of Rhode Island, Kingston, R.I., June 4–8, 1973, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 18

    Google Scholar 

  49. F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity (Springer, Berlin, 2016)

    Google Scholar 

  50. D. Gomes, R.M. Velho, M.-T. Wolfram, Socio-economic applications of finite state mean field games. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2028), 20130405, 18 (2014)

    Google Scholar 

  51. O. Guéant, J.-M. Lasry, P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010 (Springer, Berlin, 2011), pp. 205–266

    Book  MATH  Google Scholar 

  52. R. Hegselmann, U. Krause, Opinion dynamics and bounded confidence, models, analysis and simulation. J. Artif. Soc. Soc. Simul. 5(3), 2 (2002)

    Google Scholar 

  53. M. Herty, C. Ringhofer, Consistent mean field optimality conditions for interacting agent systems. Commun. Math. Sci. 17, 1095–1108 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Herty, M. Zanella, Performance bounds for the mean-field limit of constrained dynamics. Discrete Contin. Dyn. Syst. 37(4), 2023–2043 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  55. M. Herty, S. Steffensen, L. Pareschi, Mean-field control and Riccati equations. Netw. Heterog. Media 10(3), 699–715 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  56. M. Herty, S. Steffensen, L. Pareschi, Control strategies for the dynamics of large particle systems, in Active Particles, ed. by P.D.N. Bellomo, E. Tadmor. Advances in theory, models, and applications, vol. 2. Model. Simul. Sci. Eng. Technol. (Birkhäuser, Basel, 2019)

    Google Scholar 

  57. M. Huang, R.P. Malhame, P.E. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6, 221–252 (2006)

    MathSciNet  MATH  Google Scholar 

  58. M. Huang, P.E. Caines, R.P. Malhame, An invariance principle in large population stochastic dynamic games. J. Syst. Sci. Complex. 20, 162–172 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  59. M. Huang, P.E. Caines, R.P. Malhame, The NCE mean field principle with locality dependent cost interactions. IEEE Trans. Automat. Contr. 55, 2799–2805 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. P. Jabin, A review of the mean fields limits for Vlasov equations. Kinet. Relat. Models 7, 661–711 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. S. Jin, Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48(3), 235–276 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  62. D. Kalise, K. Kunisch, Z. Rao, Infinite horizon sparse optimal control. J. Optim. Theory Appl. 172(2), 481–517 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  63. J.-M. Lasry, P.-L. Lions, Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  64. C. D. Levermore, Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  65. S. Motsch, E. Tadmor, Heterophilious dynamics enhances consensus. SIAM Rev. 4, 577–621 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  66. G. Naldi, L. Pareschi, G. Toscani, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Modeling and simulation in science, engineering and technology (Birkhauser, Boston, 2010)

    Google Scholar 

  67. L. Pareschi, G. Toscani, Interacting Multi-Agent Systems. Kinetic Equations & Monte Carlo Methods (Oxford University Press, Oxford, 2013)

    Google Scholar 

  68. B. Piccoli, N.P. Duteil, E. Trélat, Sparse control of Hegselmann–Krause models: black hole and declustering. SIAM J. Control Optim. 57(4), 2628–2659 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  69. E.D. Sontag, Mathematical Control Theory: Deterministic Finite-Dimensional Systems,Texts in Applied Mathematics, vol. 6, 2nd ed. (Springer, New York, 1998)

    Google Scholar 

  70. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991)

    Book  MATH  Google Scholar 

  71. T. Trimborn, L. Pareschi, M. Frank, A relaxation scheme for the approximation of the pressureless Euler equations. Numer. Methods Partial Differ. Equ. Int. J. 22(2), 484–505 (2006)

    Article  MathSciNet  Google Scholar 

  72. T. Trimborn, L. Pareschi, M. Frank, Portfolio optimization and model predictive control: a kinetic approach. Discrete Contin. Dyn. Syst. B 24(11), 6209–6238 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  73. T. Vicsek, A. Zafeiris, Collective motion. Phys. Rep. 517, 71–140 (2012)

    Article  Google Scholar 

  74. M.-T. Wolfram, Opinion formation in a heterogeneous society, in Econophysics of Order-Driven Markets. New Econ. Windows (Springer, Milan, 2011), pp. 277–288

    Google Scholar 

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Acknowledgements

We acknowledge the support by the National Research Foundation of South Africa (Grant number: 93099 and 93476), DFG HE5386/14,15, 18, BMBF ENets 05M18PAA, Cluster of Excellence Internet of Production (ID 390621612) and NSF RNMS grant No. 1107291 (KI-Net).

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, South Africa

    M. K. Banda

  2. IGPM, RWTH Aachen, Aachen, Germany

    M. Herty & T. Trimborn

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  1. M. K. Banda
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  2. M. Herty
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  1. Institute for Multiscale Thermofluids, University of Edinburgh, Edinburgh, UK

    Livio Gibelli

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Banda, M.K., Herty, M., Trimborn, T. (2020). Recent Developments in Controlled Crowd Dynamics. In: Gibelli, L. (eds) Crowd Dynamics, Volume 2. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-50450-2_7

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