{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T15:05:56Z","timestamp":1772204756927,"version":"3.50.1"},"reference-count":49,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["NHM"],"published-print":{"date-parts":[[2023]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;abstract&gt;&lt;p&gt;In the present work we model multi-lane traffic flow in presence of two population of vehicles: cars and trucks. We first develop a finite-dimensional hybrid system which rely on continuous Bando-Follow-the-Leader dynamics coupled with discrete events motivated by the lane-change maneuvers. Then we rigorously prove that the mean-field limit is given by a system of Vlasov-type PDE with source terms generated by the lane-change maneuvers of the human-driven vehicles.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2023031","type":"journal-article","created":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T04:32:25Z","timestamp":1677472345000},"page":"723-752","source":"Crossref","is-referenced-by-count":5,"title":["Mean-field limit of a hybrid system for multi-lane car-truck traffic"],"prefix":"10.3934","volume":"18","author":[{"given":"Maria Teresa","family":"Chiri","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoqian","family":"Gong","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benedetto","family":"Piccoli","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences and Center for Computational and Integrative Biology, Rutgers University, Camden, NJ, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2023031-1","doi-asserted-by":"publisher","unstructured":"G Albi, N Bellomo, L Fermo, S. Y Ha, J. Kim, L. Pareschi, D. Poyato, J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, <i>Math Models Methods Appl Sci<\/i>, <b>29<\/b> (2019): 1901\u20132005. https:\/\/doi.org\/10.1142\/S0218202519500374","DOI":"10.1142\/S0218202519500374"},{"key":"key-10.3934\/nhm.2023031-2","doi-asserted-by":"crossref","unstructured":"L. Ambrosio, N. Gigli, G. Savar\u00e9, <i>Gradient flows: in metric spaces and in the space of probability measures<\/i>, Berlin: Springer Science &amp; Business Media, 2008.","DOI":"10.1016\/S1874-5717(07)80004-1"},{"key":"key-10.3934\/nhm.2023031-3","doi-asserted-by":"publisher","unstructured":"M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama. Structure stability of congestion in traffic dynamics, <i>Jpn J Ind Appl Math<\/i>, <b>11<\/b> (1994), 203\u2013223. https:\/\/doi.org\/10.1007\/BF03167222","DOI":"10.1007\/BF03167222"},{"key":"key-10.3934\/nhm.2023031-4","doi-asserted-by":"publisher","unstructured":"N. Bellomo, C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, <i>SIAM Rev Soc Ind Appl Math<\/i>, <b>53<\/b> (2011), 409\u2013463. https:\/\/doi.org\/10.1137\/090746677","DOI":"10.1137\/090746677"},{"key":"key-10.3934\/nhm.2023031-5","doi-asserted-by":"crossref","unstructured":"S. Benzoni-Gavage, R. M. Colombo, An $n$-populations model for traffic flow, <i>Eur J Appl Math<\/i>, <b>14<\/b> (2003), 587\u2013612.","DOI":"10.1017\/S0956792503005266"},{"key":"key-10.3934\/nhm.2023031-6","doi-asserted-by":"crossref","unstructured":"V. I. Bogachev, <i>Measure Theory<\/i>, Heidelberg: Springer Berlin, 2007.","DOI":"10.1007\/978-3-540-34514-5"},{"key":"key-10.3934\/nhm.2023031-7","doi-asserted-by":"publisher","unstructured":"R. Borsche, A. Klar, M. Zanella, Kinetic-controlled hydrodynamics for multilane traffic models, <i>Physica A<\/i>, <b>587<\/b> (2022), 126486. https:\/\/doi.org\/10.1016\/j.physa.2021.126486","DOI":"10.1016\/j.physa.2021.126486"},{"key":"key-10.3934\/nhm.2023031-8","doi-asserted-by":"publisher","unstructured":"M. S. Branicky, V. S. Borkar, S. K. Mitter, A unified framework for hybrid control: model and optimal control theory, <i>IEEE Trans. Automat. Contr.<\/i>, <b>43<\/b> (1998), 31\u201345. https:\/\/doi.org\/10.1109\/9.654885","DOI":"10.1109\/9.654885"},{"key":"key-10.3934\/nhm.2023031-9","doi-asserted-by":"publisher","unstructured":"P. Cardaliaguet, N. Forcadel, From heterogeneous microscopic traffic flow models to macroscopic models, <i>SIAM J. Math. Anal.<\/i>, <b>53<\/b> (2021), 309\u2013322. https:\/\/doi.org\/10.1137\/20M1314410","DOI":"10.1137\/20M1314410"},{"key":"key-10.3934\/nhm.2023031-10","unstructured":"M. T. Chiri, X. Q. Gong, B. Piccoli, <i>Hybrid multi-population traffic flow model: Optimal control for a mean-field limit<\/i>, [Preprint], (2021) [cited 2022 Feb 20 ]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/cvgmt.sns.it\/media\/doc\/paper\/5302\/Conference_paper_Car_Truck_with_control.pdf\">https:\/\/cvgmt.sns.it\/media\/doc\/paper\/5302\/Conference_paper_Car_Truck_with_control.pdf<\/ext-link>"},{"key":"key-10.3934\/nhm.2023031-11","doi-asserted-by":"publisher","unstructured":"V. Coscia, M. Delitala, P. Frasca, On the mathematical theory of vehicular traffic flow \u2161: Discrete velocity kinetic models, <i>Int J Non Linear Mech<\/i>, <b>42<\/b> (2007), 411\u2013421. https:\/\/doi.org\/10.1016\/j.ijnonlinmec.2006.02.008","DOI":"10.1016\/j.ijnonlinmec.2006.02.008"},{"key":"key-10.3934\/nhm.2023031-12","doi-asserted-by":"publisher","unstructured":"I. D. Couzin, J. Krause, N. R. Franks, S. A. Levin, Effective leadership and decision-making in animal groups on the move, <i>Nature<\/i>, <b>433<\/b> (2005), 513\u2013516. https:\/\/doi.org\/10.1038\/nature03236","DOI":"10.1038\/nature03236"},{"key":"key-10.3934\/nhm.2023031-13","doi-asserted-by":"publisher","unstructured":"E. Cristiani, B. Piccoli, A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, <i>Multiscale Model Simul<\/i>, <b>9<\/b> (2011), 155\u2013182. https:\/\/doi.org\/10.1137\/100797515","DOI":"10.1137\/100797515"},{"key":"key-10.3934\/nhm.2023031-14","doi-asserted-by":"publisher","unstructured":"F. Cucker, S. Smale, Emergent behavior in flocks, <i>IEEE Trans. Automat. Contr<\/i>., <b>52<\/b> (2007), 852\u2013862. https:\/\/doi.org\/10.1109\/TAC.2007.895842","DOI":"10.1109\/TAC.2007.895842"},{"key":"key-10.3934\/nhm.2023031-15","doi-asserted-by":"publisher","unstructured":"M. Delitala, A. Tosin, Mathematical modeling of vehicular traffic: a discrete kinetic theory approach, <i>Math. Models Methods Appl. Sci.<\/i>, <b>17<\/b> (2007), 901\u2013932. https:\/\/doi.org\/10.1142\/S0218202507002157","DOI":"10.1142\/S0218202507002157"},{"key":"key-10.3934\/nhm.2023031-16","doi-asserted-by":"crossref","unstructured":"M. L. D. Monache, T. Liard, Ana\u00efs Rat, R. Stern, R. Bhadani, B. Seibold, et al., <i>Feedback Control Algorithms for the Dissipation of Traffic Waves with Autonomous Vehicles<\/i>, Cham: Springer International Publishing, 2019,275\u2013299.","DOI":"10.1007\/978-3-030-25446-9_12"},{"key":"key-10.3934\/nhm.2023031-17","doi-asserted-by":"crossref","unstructured":"A. Festa, S. G\u00f6ttlich, A mean field game approach for multi-lane traffic management, <i>IFAC-PapersOnLine<\/i>, <b>51<\/b> (2018), 793\u2013798.","DOI":"10.1016\/j.ifacol.2018.11.448"},{"key":"key-10.3934\/nhm.2023031-18","doi-asserted-by":"crossref","unstructured":"M. Fornasier, B. Piccoli, F. Rossi, Mean-field sparse optimal control, <i>Philos. Trans. Royal Soc.<\/i>, <b>372<\/b> (2014), 20130400.","DOI":"10.1098\/rsta.2013.0400"},{"key":"key-10.3934\/nhm.2023031-19","doi-asserted-by":"publisher","unstructured":"M. Garavello, B. Piccoli, Hybrid necessary principle, <i>SIAM J Control Optim<\/i>, <b>43<\/b> (2005), 1867\u20131887. https:\/\/doi.org\/10.1137\/S0363012903416219","DOI":"10.1137\/S0363012903416219"},{"key":"key-10.3934\/nhm.2023031-20","doi-asserted-by":"publisher","unstructured":"D. C. Gazis, R. Herman, R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, <i>Oper. Res.<\/i>, <b>9<\/b> (1961), 545\u2013567. https:\/\/doi.org\/10.1287\/opre.9.4.545","DOI":"10.1287\/opre.9.4.545"},{"key":"key-10.3934\/nhm.2023031-21","doi-asserted-by":"publisher","unstructured":"R. Goebel, R. G. Sanfelice, A. R. Teel, Hybrid dynamical systems, <i>IEEE Control Syst. Mag.<\/i>, <b>29<\/b> (2009), 28\u201393. https:\/\/doi.org\/10.1109\/MCS.2008.931718","DOI":"10.1109\/MCS.2008.931718"},{"key":"key-10.3934\/nhm.2023031-22","unstructured":"X. Q. Gong, A. Keimer, <i>On the well-posedness of the \"bando-follow the leader\" car following model and a \"time-delayed version\"<\/i>, [Preprint], (2022) [cited 2022 Feb 24 ]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"http:\/\/dx.doi.org\/10.13140\/RG.2.2.22507.62246\">10.13140\/RG.2.2.22507.62246<\/ext-link>"},{"key":"key-10.3934\/nhm.2023031-23","unstructured":"X. Q. Gong, B. Piccoli, G. Visconti, <i>Mean-field limit of a hybrid system for multi-lane multi-class traffic<\/i>, arXiv: 2007.14655, [Preprint], (2020) [cited 2022 Feb 24 ]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.48550\/arXiv.2007.14655\">https:\/\/doi.org\/10.48550\/arXiv.2007.14655<\/ext-link>"},{"key":"key-10.3934\/nhm.2023031-24","unstructured":"Jack K. Hale, <i>Ordinary differential equations<\/i>, Roberte E. New York: Krieger Publishing company, 1980."},{"key":"key-10.3934\/nhm.2023031-25","doi-asserted-by":"publisher","unstructured":"M Herty, R Illner, A Klar, V Panferov, Qualitative properties of solutions to systems of fokker-planck equations for multilane traffic flow, <i>Transport Theor Stat Phys<\/i>, <b>35<\/b> (2006), 31\u201354. https:\/\/doi.org\/10.1080\/00411450600878573","DOI":"10.1080\/00411450600878573"},{"key":"key-10.3934\/nhm.2023031-26","doi-asserted-by":"publisher","unstructured":"M. Herty, S. Moutari, G. Visconti, Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, <i>SIAM J. Appl. Math.<\/i>, <b>78<\/b> (2018), 2252\u20132278. https:\/\/doi.org\/10.1137\/17M1151821","DOI":"10.1137\/17M1151821"},{"key":"key-10.3934\/nhm.2023031-27","doi-asserted-by":"crossref","unstructured":"M. Herty, L. Pareschi, Fokker-Planck asymptotics for traffic flow models, <i>Kinet. Relat. Models<\/i>, <b>3<\/b> (2010), 165\u2013179.","DOI":"10.3934\/krm.2010.3.165"},{"key":"key-10.3934\/nhm.2023031-28","unstructured":"M. Herty, G. Puppo, G. Visconti, <i>Model of vehicle interactions with autonomous cars and its properties<\/i>, arXiv: 2107.14081, [Preprint], (2021) [cited 2022 Feb 20 ]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.48550\/arXiv.2107.14081\">https:\/\/doi.org\/10.48550\/arXiv.2107.14081<\/ext-link>"},{"key":"key-10.3934\/nhm.2023031-29","doi-asserted-by":"crossref","unstructured":"M. Herty, G. Visconti, Analysis of risk levels for traffic on a multi-lane highway, <i>IFAC-PapersOnLine<\/i>, <b>51<\/b> (2018), 43\u201348.","DOI":"10.1016\/j.ifacol.2018.07.008"},{"key":"key-10.3934\/nhm.2023031-30","doi-asserted-by":"publisher","unstructured":"H. Holden, N. H. Risebro, Models for dense multilane vehicular traffic, <i>SIAM J. Math. Anal.<\/i>, <b>51<\/b> (2019), 3694\u20133713. https:\/\/doi.org\/10.1137\/19M124318X","DOI":"10.1137\/19M124318X"},{"key":"key-10.3934\/nhm.2023031-31","doi-asserted-by":"crossref","unstructured":"R. Illner, A. Klar, T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, <i>Commun. Math. Sci.<\/i>, <b>1<\/b> (2003), 1\u201312.","DOI":"10.4310\/CMS.2003.v1.n1.a1"},{"key":"key-10.3934\/nhm.2023031-32","unstructured":"N. Kardous, A. Hayat, S. McQuade, X. Q. Gong, S. Truong, P. Arnold, et al., <i>A rigorous multi-population multi-lane hybrid traffic model and its mean-field limit for dissipation of waves via autonomous vehicles<\/i>, arXiv: 2205.06913, [Preprint], (2021) [cited 2022 Feb 20 ]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.48550\/arXiv.2205.06913\">https:\/\/doi.org\/10.48550\/arXiv.2205.06913<\/ext-link>"},{"key":"key-10.3934\/nhm.2023031-33","doi-asserted-by":"publisher","unstructured":"A. Kesting, M. Treiber, D. Helbing, General lane-changing model mobil for car-following models, <i>Trans Res Rec<\/i>, <b>1999<\/b> (2007), 86\u201394. https:\/\/doi.org\/10.3141\/1999-10","DOI":"10.3141\/1999-10"},{"key":"key-10.3934\/nhm.2023031-34","doi-asserted-by":"publisher","unstructured":"A. Klar, R. Wegener, Enskog-like kinetic models for vehicular traffic, <i>J. Stat. Phys.<\/i>, <b>87<\/b> (1997), 91\u2013114. https:\/\/doi.org\/10.1007\/BF02181481","DOI":"10.1007\/BF02181481"},{"key":"key-10.3934\/nhm.2023031-35","doi-asserted-by":"publisher","unstructured":"S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis, <i>Trans. Res.<\/i>, <b>9<\/b> (1975), 225\u2013235. https:\/\/doi.org\/10.1016\/0041-1647(75)90063-5","DOI":"10.1016\/0041-1647(75)90063-5"},{"key":"key-10.3934\/nhm.2023031-36","doi-asserted-by":"publisher","unstructured":"B. Piccoli, A. Tosin, M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory, <i>Z Angew Math Phys<\/i>, <b>71<\/b> (2020), 1\u201325. https:\/\/doi.org\/10.1007\/s00033-019-1224-x","DOI":"10.1007\/s00033-019-1224-x"},{"key":"key-10.3934\/nhm.2023031-37","doi-asserted-by":"crossref","unstructured":"B. Piccoli, Hybrid systems and optimal control, <i>Proceedings of the 37th IEEE Conference on Decision and Control<\/i>, <b>1<\/b> (1998), 13\u201318.","DOI":"10.1109\/CDC.1998.760582"},{"key":"key-10.3934\/nhm.2023031-38","doi-asserted-by":"publisher","unstructured":"B. Piccoli, F. Rossi, Generalized wasserstein distance and its application to transport equations with source, <i>Arch Ration Mech Anal<\/i>, <b>211<\/b> (2014), 335\u2013358. https:\/\/doi.org\/10.1007\/s00205-013-0669-x","DOI":"10.1007\/s00205-013-0669-x"},{"key":"key-10.3934\/nhm.2023031-39","doi-asserted-by":"crossref","unstructured":"B. Piccoli, A. Tosin, Vehicular traffic: A review of continuum mathematical models, <i>Encycl. Complex. Syst. Sci.<\/i>, <b>22<\/b> (2009), 9727\u20139749.","DOI":"10.1007\/978-0-387-30440-3_576"},{"key":"key-10.3934\/nhm.2023031-40","unstructured":"I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, In: R. Herman, editor, <i>Theory of traffic flow<\/i>, Amsterdam: Elsevier, 1961,158\u2013164."},{"key":"key-10.3934\/nhm.2023031-41","unstructured":"I. Prigogine, R. Herman, <i>Kinetic theory of vehicular traffic<\/i>, New York: American Elsevier Publishing, 1971."},{"key":"key-10.3934\/nhm.2023031-42","doi-asserted-by":"publisher","unstructured":"G. Puppo, M. Semplice, A. Tosin, G. Visconti, Analysis of a multi-population kinetic model for traffic flow, <i>Commun Math Sci<\/i>, <b>15<\/b> (2017), 379\u2013412. https:\/\/doi.org\/10.4310\/CMS.2017.v15.n2.a5","DOI":"10.4310\/CMS.2017.v15.n2.a5"},{"key":"key-10.3934\/nhm.2023031-43","unstructured":"A Reuschel, Vehicle movements in a platoon with uniform acceleration or deceleration of the lead vehicle, <i>Zeitschrift des Oesterreichischen Ingenieur-und Architekten-Vereines<\/i>, <b>95<\/b> (1950), 50\u201362."},{"key":"key-10.3934\/nhm.2023031-44","doi-asserted-by":"publisher","unstructured":"J. Song, S. Karni, A second order traffic flow model with lane changing, <i>J. Sci. Comput.<\/i>, <b>81<\/b> (2019), 1429\u20131445. https:\/\/doi.org\/10.1007\/s10915-019-01023-z","DOI":"10.1007\/s10915-019-01023-z"},{"key":"key-10.3934\/nhm.2023031-45","doi-asserted-by":"publisher","unstructured":"Raphael E. Stern, S. Cui, Maria Laura Delle Monache, R. Bhadani, M. Bunting, M. Churchill, et al., Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, <i>Transp Res Part C Emerg Technol<\/i>, <b>89<\/b> (2018), 205\u2013221. https:\/\/doi.org\/10.1016\/j.trc.2018.02.005","DOI":"10.1016\/j.trc.2018.02.005"},{"key":"key-10.3934\/nhm.2023031-46","doi-asserted-by":"publisher","unstructured":"A. B. Sukhinova, M. A. Trapeznikova, B. N. Chetverushkin, N. G. Churbanova, Two-dimensional macroscopic model of traffic flows, <i>Math. Models Comput. Simul.<\/i>, <b>1<\/b> (2009), 669\u2013676. https:\/\/doi.org\/10.1134\/S2070048209060027","DOI":"10.1134\/S2070048209060027"},{"key":"key-10.3934\/nhm.2023031-47","doi-asserted-by":"publisher","unstructured":"C. Tomlin, G. J. Pappas, S. Sastry, Conflict resolution for air traffic management: a study in multiagent hybrid systems, <i>IEEE Trans. Automat. Contr.<\/i>, <b>43<\/b> (1998), 509\u2013521. https:\/\/doi.org\/10.1109\/9.664154","DOI":"10.1109\/9.664154"},{"key":"key-10.3934\/nhm.2023031-48","doi-asserted-by":"publisher","unstructured":"T. Trimborn, L. Pareschi, M. Frank, Portfolio optimization and model predictive control: A kinetic approach, <i>Discrete Cont. Dyn.-B<\/i>, <b>24<\/b> (2019), 6209\u20136238. https:\/\/doi.org\/10.3934\/dcdsb.2019136","DOI":"10.3934\/dcdsb.2019136"},{"key":"key-10.3934\/nhm.2023031-49","doi-asserted-by":"publisher","unstructured":"P Zhang, R. X Liu, S. C. Wong, S. Q Dai, Hyperbolicity and kinematic waves of a class of multi-population partial differential equations, <i>Eur J Appl Math<\/i>, <b>17<\/b> (2006), 171\u2013200. https:\/\/doi.org\/10.1017\/S095679250500642X","DOI":"10.1017\/S095679250500642X"}],"container-title":["Networks and Heterogeneous Media"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2023031?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,15]],"date-time":"2024-10-15T05:50:54Z","timestamp":1728971454000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2023031"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023]]},"references-count":49,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023]]}},"URL":"https:\/\/doi.org\/10.3934\/nhm.2023031","relation":{},"ISSN":["1556-1801"],"issn-type":[{"value":"1556-1801","type":"print"}],"subject":[],"published":{"date-parts":[[2023]]}}}