1932

Abstract

This review aims to present recent developments in modeling and control of multiagent systems. A particular focus is set on crowd dynamics characterized by complex interactions among agents, also called social interactions, and large-scale systems. Specifically, in a crowd each individual agent interacts with a field generated by the other agents and the environment. These systems can be modeled at the microscopic scale by ordinary differential equations, while an alternative description at the mesoscopic scale is given by a partial differential equation for the propagation of the probability density of the agents. Control actions can be applied at the individual level as well as at the level of the corresponding fields. This article presents and compares different control types, and the specific application to multilane, multiclass traffic is developed in some detail, showing the main tools at work in a hybrid setting with relevant impacts on autonomous driving.

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2023-05-03
2024-06-24
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Literature Cited

  1. 1.
    Aylaj B, Bellomo N, Gibelli L, Knopoff D. 2020. Crowd Dynamics by Kinetic Theory Modeling: Complexity, Modeling, Simulations, and Safety San Rafael, CA: Morgan & Claypool
    [Google Scholar]
  2. 2.
    Bellomo N, Soler J. 2012. On the mathematical theory of the dynamics of swarms viewed as complex systems. Math. Models Methods Appl. Sci. 22:Suppl. 11140006
    [Google Scholar]
  3. 3.
    Cordier S, Pareschi L, Toscani G. 2005. On a kinetic model for a simple market economy. J. Stat. Phys. 120:253–77
    [Google Scholar]
  4. 4.
    Degond P, Liu JG, Motsch S, Panferov V. 2013. Hydrodynamic models of self-organized dynamics: derivation and existence theory. Methods Appl. Anal. 20:89–114
    [Google Scholar]
  5. 5.
    Herty M, Ringhofer C. 2011. Averaged kinetic models for flows on unstructured networks. Kinet. Relat. Models 4:1081–96
    [Google Scholar]
  6. 6.
    Herty M, Pareschi L. 2010. Fokker-Planck asymptotics for traffic flow models. Kinet. Relat. Models 3:165–79
    [Google Scholar]
  7. 7.
    Gómez-Serrano J, Graham C, Le Boudec JY 2012. The bounded confidence model of opinion dynamics. Math. Models Methods Appl. Sci. 22:1150007
    [Google Scholar]
  8. 8.
    Toscani G. 2006. Kinetic models of opinion formation. Commun. Math. Sci. 4:481–96
    [Google Scholar]
  9. 9.
    Degond P, Motsch S. 2007. Macroscopic limit of self-driven particles with orientation interaction. C. R. Math. Acad. Sci. Paris 345:555–60
    [Google Scholar]
  10. 10.
    Motsch S, Tadmor E. 2014. Heterophilious dynamics enhances consensus. SIAM Rev 56:577–621
    [Google Scholar]
  11. 11.
    Hegselmann R, Krause U. 2002. Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5:32
    [Google Scholar]
  12. 12.
    Ceragioli F, Frasca P, Piccoli B, Rossi F 2021. Generalized solutions to opinion dynamics models with discontinuities. Crowd Dynamics, Vol. 3: Modeling and Social Applications in the Time of COVID-19 N Bellomo, L Gibelli 11–47 Cham, Switz: Birkhaüser
    [Google Scholar]
  13. 13.
    Jabin P, Motsch S. 2014. Clustering and asymptotic behavior in opinion formation. J. Differ. Equ. 257:4165–87
    [Google Scholar]
  14. 14.
    Piccoli B, Rossi F. 2021. Generalized solutions to bounded-confidence models. Math. Models Methods Appl. Sci. 31:1237–76
    [Google Scholar]
  15. 15.
    Helbing D, Molnár P. 1995. Social force model for pedestrian dynamics. Phys. Rev. E 51:4282–86
    [Google Scholar]
  16. 16.
    Lewin K. 1951. Field Theory in Social Science New York: Harper & Brothers
    [Google Scholar]
  17. 17.
    Bellomo N, Clarke D, Gibelli L, Townsend P, Vreugdenhil B. 2016. Human behaviours in evacuation crowd dynamics: from modelling to “big data” toward crisis management. Phys. Life Rev. 18:1–21
    [Google Scholar]
  18. 18.
    Tomlin C, Pappas G, Sastry S. 1998. Conflict resolution for air traffic management: a study in multiagent hybrid systems. IEEE Trans. Autom. Control 43:509–21
    [Google Scholar]
  19. 19.
    Pallottino L, Scordio V, Frazzoli E, Bicchi A. 1998. Decentralized cooperative policy for conflict resolution in multivehicle systems. IEEE Trans. Robot. 23:1170–83
    [Google Scholar]
  20. 20.
    Carrillo JA, Di Francesco M, Figalli A, Laurent T, Slepcev D. 2011. Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156:229–71
    [Google Scholar]
  21. 21.
    Cercignani C, Illner R, Pulvirenti M. 1994. The Mathematical Theory of Dilute Gases New York: Springer
    [Google Scholar]
  22. 22.
    Cristiani E, Piccoli B, Tosin A. 2014. Multiscale Modeling of Pedestrian Dynamics Cham, Switz: Springer
    [Google Scholar]
  23. 23.
    Maury B, Venel J. 2011. A discrete contact model for crowd motion. ESAIM Math. Model. Numer. Anal. 45:145–68
    [Google Scholar]
  24. 24.
    Arechavaleta G, Laumond JP, Hicheur H, Berthoz A. 2008. An optimality principle governing human walking. IEEE Trans. Robot. 24:5–14
    [Google Scholar]
  25. 25.
    Chitour Y, Jean F, Mason P 2012. Optimal control models of goal-oriented human locomotion. SIAM J. Control Optim. 50:147–70
    [Google Scholar]
  26. 26.
    Farina F, Fontanelli D, Garulli A, Giannitrapani A, Prattichizzo D. 2016. When Helbing meets Laumond: the headed social force model. 2016 IEEE 55th Conference on Decision and Control (CDC)3548–53 Piscataway, NJ: IEEE
    [Google Scholar]
  27. 27.
    Cristiani E, Frasca P, Piccoli B. 2011. Effects of anisotropic interactions on the structure of animal groups. J. Math. Biol. 62:569–88
    [Google Scholar]
  28. 28.
    Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O. 1995. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75:1226–29
    [Google Scholar]
  29. 29.
    Cucker F, Smale S. 2007. Emergent behavior in flocks. IEEE Trans. Autom. Control 52:852–62
    [Google Scholar]
  30. 30.
    Ha SY, Ha T, Kim JH. 2010. Emergent behavior of a Cucker–Smale type particle model with nonlinear velocity couplings. IEEE Trans. Autom. Control 55:1679–83
    [Google Scholar]
  31. 31.
    Treiber M, Hennecke A, Helbing D. 2000. Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E 62:1805
    [Google Scholar]
  32. 32.
    Bando M, Hasebe K, Nakayama A, Shibata A, Sugiyama Y. 1995. Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51:1035
    [Google Scholar]
  33. 33.
    Gazis DC, Herman R, Rothery RW. 1961. Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9:545–67
    [Google Scholar]
  34. 34.
    Gong X, Keimer A. 2022. On the well-posedness of the “Bando-follow the leader” car following model and a time-delayed version. ResearchGate RG.2.2.22507.62246. https://doi.org/10.13140/RG.2.2.22507.62246
    [Crossref]
  35. 35.
    Burger M, Capasso V, Morale D. 2007. On an aggregation model with long and short range interactions. Nonlinear Anal. Real World Appl. 8:939–58
    [Google Scholar]
  36. 36.
    Toner J, Tu Y. 1995. Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75:4326–29
    [Google Scholar]
  37. 37.
    Topaz CM, Bertozzi AL. 2004. Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65:152–74
    [Google Scholar]
  38. 38.
    Topaz CM, Bertozzi AL, Lewis MA. 2006. A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68:1601–23
    [Google Scholar]
  39. 39.
    Illner R, Klar A, Materne T. 2003. Vlasov-Fokker-Planck models for multilane traffic flow. Commun. Math. Sci. 1:1–12
    [Google Scholar]
  40. 40.
    Carrillo JA, D'Orsogna MR, Panferov V 2009. Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2:363–78
    [Google Scholar]
  41. 41.
    Chuang Y, D'Orsogna M, Marthaler D, Bertozzi A, Chayes L 2007. State transition and the continuum limit for the 2D interacting, self-propelled particle system. Phys. D 232:33–47
    [Google Scholar]
  42. 42.
    Degond P, Motsch S. 2008. Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18:Suppl. 11193–215
    [Google Scholar]
  43. 43.
    Ha SY, Tadmor E. 2008. From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 1:415–35
    [Google Scholar]
  44. 44.
    Boudin L, Salvarani F. 2009. A kinetic approach to the study of opinion formation. ESAIM Math. Model. Numer. Anal. 43:507–22
    [Google Scholar]
  45. 45.
    Degond P, Herty M, Liu JG. 2014. Flow on sweeping networks. Multiscale Model. Simul. 12:538–65
    [Google Scholar]
  46. 46.
    Fornasier M, Haskovec J, Toscani G. 2011. Fluid dynamic description of flocking via the Povzner-Boltzmann equation. Phys. D 240:21–31
    [Google Scholar]
  47. 47.
    Carrillo JA, Choi YP, Hauray M 2014. The derivation of swarming models: mean-field limit and Wasserstein distances. Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation A Muntean, F Toschi 1–46 Vienna: Springer
    [Google Scholar]
  48. 48.
    Herty M, Kalise D. 2018. Suboptimal nonlinear feedback control laws for collective dynamics. 2018 IEEE 14th International Conference on Control and Automation (ICCA)556–61 Piscataway, NJ: IEEE
    [Google Scholar]
  49. 49.
    Ha SY, Liu JG. 2009. A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7:297–325
    [Google Scholar]
  50. 50.
    Carrillo JA, Fornasier M, Rosado J, Toscani G. 2010. Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42:218–36
    [Google Scholar]
  51. 51.
    Canizo JA, Carrillo JA, Rosado J. 2010. A well-posedness theory in measures for some kinetic models of collective motion. Math. Models Methods Appl. Sci. 21:515–39
    [Google Scholar]
  52. 52.
    Jabin P. 2014. A review of the mean fields limits for vlasov equations. Kinet. Relat. Models 7:661–711
    [Google Scholar]
  53. 53.
    Choi YP, Kalise D, Peszek J, Peters AA. 2019. A collisionless singular Cucker-Smale model with decentralized formation control. SIAM J. Appl. Dyn. Syst. 18:1954–81
    [Google Scholar]
  54. 54.
    Oh KK, Park MC, Ahn HS. 2015. A survey of multi-agent formation control. Automatica 53:424–40
    [Google Scholar]
  55. 55.
    Peters AA, Middleton RH, Mason O. 2014. Leader tracking in homogeneous vehicle platoons with broadcast delays. Automatica 50:64–74
    [Google Scholar]
  56. 56.
    Freudenthaler G, Meurer T. 2020. PDE-based multi-agent formation control using flatness and backstepping: analysis, design and robot experiments. Automatica 115:108897
    [Google Scholar]
  57. 57.
    Dyer JRG, Johansson A, Helbing D, Couzin ID, Krause J. 2009. Leadership, consensus decision making and collective behaviour in humans. Philos. Trans. R. Soc. B 364:781–89
    [Google Scholar]
  58. 58.
    Burger M, Pinnau R, Totzeck C, Tse O, Roth A 2020. Instantaneous control of interacting particle systems in the mean-field limit. J. Comput. Phys. 405:109181
    [Google Scholar]
  59. 59.
    Herty M, Ringhofer C. 2011. Feedback controls for continuous priority models in supply chain management. Comput. Methods Appl. Math. 11:206–13
    [Google Scholar]
  60. 60.
    Degond P, Göttlich S, Herty M, Klar A. 2007. A network model for supply chains with multiple policies. Multiscale Model. Simul. 6:820–37
    [Google Scholar]
  61. 61.
    Tosin A, Zanella M. 2019. Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles. Multiscale Model. Simul. 17:716–49
    [Google Scholar]
  62. 62.
    Han Y, Hegyi A, Yuan Y, Hoogendoorn S, Papageorgiou M, Roncoli C. 2017. Resolving freeway jam waves by discrete first-order model-based predictive control of variable speed limits. Transp. Res. C 77:405–20
    [Google Scholar]
  63. 63.
    Stern RE, Cui S, Delle Monache ML, Bhadani R, Bunting M et al. 2018. Dissipation of stop-and-go waves via control of autonomous vehicles: field experiments. Transp. Res. C 89:205–21
    [Google Scholar]
  64. 64.
    Estrada-Rodriguez G, Gimperlein H. 2020. Interacting particles with Lévy strategies: limits of transport equations for swarm robotic systems. SIAM J. Appl. Math. 80:476–98
    [Google Scholar]
  65. 65.
    Albi G, Pareschi L, Zanella M. 2014. Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. A 372:20140138
    [Google Scholar]
  66. 66.
    Garnier J, Papanicolaou G, Yang TW. 2017. Consensus convergence with stochastic effects. Vietnam J. Math. 45:51–75
    [Google Scholar]
  67. 67.
    Fornasier M, Solombrino F. 2014. Mean-field optimal control. ESAIM Control Optim. Calc. Var. 20:1123–52
    [Google Scholar]
  68. 68.
    Fornasier M, Lisini S, Orrieri C, Savaré G. 2019. Mean-field optimal control as gamma-limit of finite agent controls. Eur. J. Appl. Math. 30:1153–86
    [Google Scholar]
  69. 69.
    Fornasier M, Piccoli B, Rossi F. 2014. Mean-field sparse optimal control. Philos. Trans. R. Soc. A 372:20130400
    [Google Scholar]
  70. 70.
    Briceño Arias LM, Kalise D, Silva FJ 2018. Proximal methods for stationary mean field games with local couplings. SIAM J. Control Optim. 56:801–36
    [Google Scholar]
  71. 71.
    Albi G, Choi YP, Fornasier M, Kalise D. 2017. Mean field control hierarchy. Appl. Math. Optim. 76:93–135
    [Google Scholar]
  72. 72.
    Aduamoah M, Goddard BD, Pearson JW, Roden JC. 2020. PDE-constrained optimization models and pseudospectral methods for multiscale particle dynamics. arXiv:2009.09850 [math.NA]
  73. 73.
    Liu S, Jacobs M, Li W, Nurbekyan L, Osher SJ. 2020. Computational methods for nonlocal mean field games with applications. arXiv:2004.12210 [math.OC]
  74. 74.
    Caponigro M, Fornasier M, Piccoli B, Trélat E. 2015. Sparse stabilization and control of alignment models. Math. Models Methods Appl. Sci. 25:521–64
    [Google Scholar]
  75. 75.
    Bailo R, Bongini M, Carrillo JA, Kalise D. 2018. Optimal consensus control of the Cucker-Smale model. IFAC-PapersOnLine 51:131–6
    [Google Scholar]
  76. 76.
    Albi G, Kalise D. 2018. (Sub)optimal feedback control of mean field multi-population dynamics. IFAC-PapersOnLine 51:386–91
    [Google Scholar]
  77. 77.
    Caponigro M, Fornasier M, Piccoli B, Trélat E. 2013. Sparse stabilization and optimal control of the Cucker-Smale model. Math. Control Relat. Fields 3:447–66
    [Google Scholar]
  78. 78.
    Borzì A, Wongkaew S. 2015. Modeling and control through leadership of a refined flocking system. Math. Models Methods Appl. Sci. 25:255–82
    [Google Scholar]
  79. 79.
    Herty M, Pareschi L, Steffensen S. 2015. Mean–field control and Riccati equations. Netw. Heterog. Media 10:699
    [Google Scholar]
  80. 80.
    Garrard WL. 1972. Suboptimal feedback control of linear gyroscopic systems. J. Optim. Theory Appl. 10:404–14
    [Google Scholar]
  81. 81.
    Thevenet L, Buchot JM, Raymond JP. 2010. Nonlinear feedback stabilization of a two-dimensional Burgers equation. ESAIM Control Optim. Calc. Var. 16:929–55
    [Google Scholar]
  82. 82.
    Beeler SC, Tran HT, Banks HT. 2000. Feedback control methodologies for nonlinear systems. J. Optim. Theory Appl. 107:1–33
    [Google Scholar]
  83. 83.
    Kalise D, Kunisch K. 2018. Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs. SIAM J. Sci. Comput. 40:A629–52
    [Google Scholar]
  84. 84.
    Banks HT, Lewis BM, Tran HT. 2007. Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach. Comput. Optim. Appl. 37:177–218
    [Google Scholar]
  85. 85.
    Cloutier JR, D'Souza CN, Mracek CP 1997. Nonlinear regulation and nonlinear H control via the state-dependent Riccati equation technique: part 2, examples. Proceedings of the First International Conference on Nonlinear Problems in Aviation and Aerospace: May 9–11, 1996, Daytona Beach, Florida, USA Daytona Beach, FL: Embry-Riddle Aeronaut. Univ. Press
    [Google Scholar]
  86. 86.
    Albi G, Herty M, Kalise D, Segala C. 2022. Moment-driven predictive control of mean-field collective dynamics. SIAM J. Control Optim. 60:814–41
    [Google Scholar]
  87. 87.
    Dolgov S, Kalise D, Kunisch K. 2019. Tensor decompositions for high-dimensional Hamilton-Jacobi-Bellman equations. arXiv:1908.01533 [math.OC]
  88. 88.
    Azmi B, Kalise D, Kunisch K. 2020. Optimal feedback law recovery by gradient-augmented sparse polynomial regression. arXiv:2007.09753 [math.OC]
  89. 89.
    Mayne DQ, Rawlings JB, Rao CV, Scokaert PO. 2000. Constrained model predictive control: stability and optimality. Automatica 36:789–814
    [Google Scholar]
  90. 90.
    Camacho EF, Alba CB. 2013. Model Predictive Control London: Springer
    [Google Scholar]
  91. 91.
    Grüne L, Pannek J. 2017. Nonlinear model predictive control. Nonlinear Model Predictive Control: Theory and Algorithms43–66 London: Springer
    [Google Scholar]
  92. 92.
    Azmi B, Kunisch K. 2019. A hybrid finite-dimensional RHC for stabilization of time-varying parabolic equations. SIAM J. Control Optim. 57:3496–526
    [Google Scholar]
  93. 93.
    Albi G, Herty M, Pareschi L. 2015. Kinetic description of optimal control problems and applications to opinion consensus. Commun. Math. Sci. 13:1407–29
    [Google Scholar]
  94. 94.
    Caponigro M, Piccoli B, Rossi F, Trélat E. 2017. Mean-field sparse Jurdjevic-Quinn control. Math. Models Methods Appl. Sci. 27:1223–53
    [Google Scholar]
  95. 95.
    Piccoli B, Rossi F, Trélat E. 2015. Control to flocking of the kinetic Cucker-Smale model. SIAM J. Math. Anal. 47:4685–719
    [Google Scholar]
  96. 96.
    Colombo RM, Pogodaev N. 2012. Confinement strategies in a model for the interaction between individuals and a continuum. SIAM J. Appl. Dyn. Syst. 11:741–70
    [Google Scholar]
  97. 97.
    Colombo RM, Pogodaev N. 2013. On the control of moving sets: positive and negative confinement results. SIAM J. Control Optim. 51:380–401
    [Google Scholar]
  98. 98.
    Düring B, Markowich P, Pietschmann JF, Wolfram MT. 2009. Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A 465:3687–708
    [Google Scholar]
  99. 99.
    Fornasier M, Solombrino F. 2014. Mean-field optimal control. ESAIM Control Optim. Calc. Var. 20:1123–52
    [Google Scholar]
  100. 100.
    Bensoussan A, Frehse J, Yam P. 2013. Mean Field Games and Mean Field Type Control Theory New York: Springer
    [Google Scholar]
  101. 101.
    Lasry JM, Lions PL. 2007. Mean field games. Jpn. J. Math. 2:229–60
    [Google Scholar]
  102. 102.
    Degond P, Herty M, Liu JG. 2017. Meanfield games and model predictive control. Commun. Math. Sci. 15:1403–22
    [Google Scholar]
  103. 103.
    Barker M. 2019. From mean field games to the best reply strategy in a stochastic framework. J. Dyn. Games 6:291–314
    [Google Scholar]
  104. 104.
    Herty M, Ringhofer C. 2019. Consistent mean field optimality conditions for interacting agent systems. Commun. Math. Sci. 17:1095–108
    [Google Scholar]
  105. 105.
    Herty M, Zanella M. 2017. Performance bounds for the mean-field limit of constrained dynamics. Discrete Contin. Dyn. Syst. 37:2023–43
    [Google Scholar]
  106. 106.
    Albi G, Pareschi L, Zanella M. 2019. Boltzmann games in heterogeneous consensus dynamics. J. Stat. Phys. 175:97–125
    [Google Scholar]
  107. 107.
    Albi G, Pareschi L, Toscani G, Zanella M 2017. Recent advances in opinion modeling: control and social influence. Active Particles, Vol. 1: Advances in Theory, Models, and Applications N Bellomo, P Degond, E Tadmor 49–98 Cham, Switz: Birkhäuser
    [Google Scholar]
  108. 108.
    Trimborn T, Pareschi L, Frank M. 2019. Portfolio optimization and model predictive control: a kinetic approach. Discrete Contin. Dyn. Syst. B 24:6209–38
    [Google Scholar]
  109. 109.
    Laval JA, Daganzo CF. 2006. Lane-changing in traffic streams. Transp. Res. B 40:251–64
    [Google Scholar]
  110. 110.
    Jin WL. 2013. A multi-commodity Lighthill-Whitham-Richards model of lane-changing traffic flow. Proc. Soc. Behav. Sci. 80:658–77
    [Google Scholar]
  111. 111.
    Zheng Z. 2014. Recent developments and research needs in modeling lane changing. Transp. Res. B 60:16–32
    [Google Scholar]
  112. 112.
    Herty M, Visconti G. 2018. Analysis of risk levels for traffic on a multi-lane highway. IFAC-PapersOnLine 51:943–48
    [Google Scholar]
  113. 113.
    Li X, Li X, Xiao Y, Jia B 2016. Modeling mechanical restriction differences between car and heavy truck in two-lane cellular automata traffic flow model. Phys. A 451:49–62
    [Google Scholar]
  114. 114.
    Chiri MT, Gong X, Piccoli B. 2023. Mean-field limit of a hybrid system for multi-lane car-truck traffic. Netw. Heterog. Media 18:723–52
    [Google Scholar]
  115. 115.
    Kardous N, Hayat A, McQuade ST, Gong X, Truong S et al. 2022. A rigorous multi-population multi-lane hybrid traffic model and its mean-field limit for dissipation of waves via autonomous vehicles. arXiv:2205.06913 [eess.SY]
  116. 116.
    Herty M, Puppo G, Visconti G. 2023. Model of vehicle interactions with autonomous cars and its properties. Discrete Cont. Dyn. B 28:833–53
    [Google Scholar]
  117. 117.
    Gong X, Piccoli B, Visconti G. 2022. Mean-field limit of a hybrid system for multi-lane multi-class traffic. arXiv:2007.14655 [math.AP]
  118. 118.
    Gong X, Piccoli B, Visconti G. 2021. Mean-field of optimal control problems for hybrid model of multilane traffic. IEEE Control Syst. Lett. 5:1964–69
    [Google Scholar]
  119. 119.
    Chiri MT, Gong X, Piccoli B. 2021. Hybrid multi-population traffic flow model: optimal control for a mean-field limit. ResearchGate RG.2.2.22377.21606. https://doi.org/10.13140/RG.2.2.22377.21606
    [Crossref]
  120. 120.
    Bonnet B, Cipriani C, Fornasier M, Huang H. 2021. A measure theoretical approach to the mean-field maximum principle for training NeurODEs. HAL hal-03289521. https://hal.archives-ouvertes.fr/hal-03289521
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