W. O. Assis and B. E. Milani, Generation of optimal schedules for metro lines using model predictive control, Automatica, vol.40, issue.8, pp.1397-1404, 2004.

V. Cacchiani, An overview of recovery models and algorithms for real-time railway rescheduling, Transp. Res. B, Methodol, vol.63, pp.15-37, 2014.

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc, vol.78, issue.3, pp.385-390, 1980.

J. Cury, F. Gomide, and M. Mendes, A methodology for generation of optimal schedules for an underground railway system, IEEE Trans. Autom. Control, vol.25, issue.2, pp.217-222, 1980.

O. Engelhardt-funke and M. Kolonko, Analysing stability and investments in railway networks using advanced evolutionary algorithms, Int. Trans. Oper. Res, vol.11, issue.4, pp.381-394, 2004.

N. Farhi, M. Goursat, and J. Quadrat, Derivation of the fundamental traffic diagram for two circular roads and a crossing using minplus algebra and Petri net modeling, Proc. IEEE CDC-ECC, pp.2119-2124, 2005.

N. Farhi, M. Goursat, and J. P. Quadrat, Fundamental traffic diagrams of elementary road networks, Proc. ECC, pp.4944-4951, 2007.

N. Farhi, Modélisation minplus et commande du trafic de villes Régulières, Ph.D. dissertation, Univ. Paris 1 Panthéon-Sorbonne, 2008.

N. Farhi, Modeling and control of elementary 2D-traffic systems using Petri nets and minplus algebra, Proc. IEEE CDC/CCC, pp.2292-2297, 2009.

N. Farhi, M. Goursat, and J. Quadrat, The traffic phases of road networks, Transp. Res. Part C, Emerg. Technol, vol.19, issue.1, pp.85-102, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00860175

N. Farhi, Piecewise linear car-following modeling, Transp. Res. C, Emerg. Technol, vol.25, pp.100-112, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00880035

N. Farhi, C. Nguyen-van-phu, H. Haj-salem, and J. Lebacque, Traffic modeling and real-time control for metro lines, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01611051

N. Farhi, C. Nguyen-van-phu, H. Haj-salem, and J. Lebacque, Traffic modeling and real-time control for metro lines. Part I-A max-plus algebra model explaining the traffic phases of the train dynamics, Proc. ACC, pp.3834-3839, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01611051

N. Farhi, C. Nguyen-van-phu, H. Haj-salem, and J. Lebacque, Traffic modeling and real-time control for metro lines. Part II-The effect of passenger demand on the traffic phases, Proc. ACC, pp.3828-3833, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01611051

S. Gaubert and J. Gunawardena, A non-linear hierarchy for discrete event dynamical systems, Proc. 4th IEE Workshop Discrete Event Syst. (WODES), 1998.

S. Gaubert and J. Gunawardena, Existence of eigenvectors for monotone homogeneous functions, HP Labs, 1999.

, This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. FARHI: PHYSICAL MODELS AND CONTROL OF THE TRAIN DYNAMICS IN A METRO LINE WITHOUT JUNCTION 9

C. J. Goodman and S. Murata, Metro traffic regulation from the passenger perspective, Proc. Inst. Mech. Eng. Part F, J. Rail Rapid Transit, vol.215, issue.2, pp.137-147, 2001.

J. Gunawardena and M. Keane, On the existence of cycle times for some non-expansive maps, HP Labs, 1995.

, IEEE Standard for Communications-Based Train Control (CBTC) Performance and Functional Requirements, IEEE Standard, 1474.

W. Lin and J. Sheu, Metro traffic regulation by adaptive optimal control, IEEE Trans. Intell. Transp. Syst, vol.12, issue.4, pp.1064-1073, 2011.

F. Schanzenbacher, R. Chevrier, and N. Farhi, Fluidification du trafic transilien: Approche prédictive et optimisation quadratique, Proc. ROADEF, 2016.

F. Schanzenbächer, N. Farhi, Z. Christoforou, F. Leurent, and G. Gabriel, A discrete event traffic model explaining the traffic phases of the train dynamics in a metro line system with a junction, Proc. IEEE CDC, pp.6283-6288, 2017.

F. Schanzenbächer, N. Farhi, F. Leurent, and G. Gabriel, A discrete event traffic model explaining the traffic phases of the train dynamics on a linear metro line with demand-dependent control, Proc. Amer. Control Conf, 2018.

X. Sun, S. Zhang, H. Dong, and H. Zhu, Optimal train schedule with headway and passenger flow dynamic models, Proc. IEEE Int. Conf. Intell. Rail Transp. (ICIRT), pp.307-312, 2013.

V. V. Breusegem, G. Campion, and G. Bastin, Traffic modeling and state feedback control for metro lines, Proc. IEEE Trans. Autom. Control, vol.36, issue.7, pp.770-784, 1991.