Optima andequilibria for traffic flow on networkswith backward propagating queues

Alberto Bressan; Khai T. Nguyen

doi:10.3934/nhm.2015.10.717

Article Contents

2015, Volume 10, Issue 4: 717-748. Doi: 10.3934/nhm.2015.10.717

This issue Previous Article Next Article Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks

Optima and equilibria for traffic flow on networks with backward propagating queues

Alberto Bressan^1, and
Khai T. Nguyen^2,

1.
Department of Mathematics, Penn State University, University Park, Pa.16802
2.
Department of Mathematics, Penn State University, University Park, PA 16802

Received: January 2015

Revised: April 2015

Published: December 2015

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Abstract

This paper studies an optimal decision problem for several groups of drivers on a network of roads. Drivers have different origins and destinations, and different costs, related to their departure and arrival time. On each road the flow is governed by a conservation law, while intersections are modeled using buffers of limited capacity, so that queues can spill backward along roads leading to a crowded intersection. Two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination.

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Mathematics Subject Classification: Primary: 49K35, 35L65; Secondary: 90B20.

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References

[1]	N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modelling, Math. Models Methods Appl. Sci., 12 (2002), 1801-1843.doi: 10.1142/S0218202502002343.
[2]	A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM J. Math. Anal., 43 (2011), 2384-2417.doi: 10.1137/110825145.
[3]	A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers, ESAIM; Control, Optim. Calc. Var., 18 (2012), 969-986.doi: 10.1051/cocv/2011198.
[4]	A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks, Networks & Heter. Media, 8 (2013), 627-648.doi: 10.3934/nhm.2013.8.627.
[5]	A. Bressan, C. J. Liu, W. Shen and F. Yu, Variational analysis of Nash equilibria for a model of traffic flow, Quarterly Appl. Math., 70 (2012), 495-515.doi: 10.1090/S0033-569X-2012-01304-9.
[6]	A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads, Networks & Heter. Media, 10 (2015), 255-293.doi: 10.3934/nhm.2015.10.255.
[7]	A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction, Discr. Cont. Dyn. Syst., 35 (2015), 4149-4171.doi: 10.3934/dcds.2015.35.4149.
[8]	A. Cascone, C. D'Apice, B. Piccoli and L. Rarità, Optimization of traffic on road networks, Math. Models Methods Appl. Sci., 17 (2007), 1587-1617.doi: 10.1142/S021820250700239X.
[9]	Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete Contin. Dyn. Syst. B, 5 (2005), 599-630.doi: 10.3934/dcdsb.2005.5.599.
[10]	G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.doi: 10.1137/S0036141004402683.
[11]	R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh, A 133 (2003), 759-772.doi: 10.1017/S0308210500002663.
[12]	C. D'Apice, P. I. Kogut and R. Manzo, Efficient controls for traffic flow on networks, J. Dyn. Control Syst., 16 (2010), 407-437.doi: 10.1007/s10883-010-9099-3.
[13]	L. C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, Providence, RI, 2010.doi: 10.1090/gsm/019.
[14]	T. Friesz, Dynamic Optimization and Differential Games, Springer, New York, 2010.doi: 10.1007/978-0-387-72778-3.
[15]	T. Friesz and K. Han, Dynamic Network User Equilibrium, Springer, 2013.
[16]	T. Friesz, K. Han, P. A. Neto, A. Meimand and T. Yao, Dynamic user equilibrium based on a hydrodynamic model, Transp. Res., B 47 (2013), 102-126.doi: 10.1016/j.trb.2012.10.001.
[17]	T. Friesz, T. Kim, C. Kwon and M. A. Rigdon, Approximate network loading and dual-time-scale dynamic user equilibrium, Transp. Res., B 45 (2011), 176-207.doi: 10.1016/j.trb.2010.05.003.
[18]	M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. 32 (2012), 1915-1938.doi: 10.3934/dcds.2012.32.1915.
[19]	M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Springfield, Mo., 2006.
[20]	M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincar\'e, 26 (2009), 1925-1951.doi: 10.1016/j.anihpc.2009.04.001.
[21]	M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in Advances in Dynamic Network Modeling in Complex Transportation Systems, Complex Networks and Dynamic Systems, S V. Ukkusuri and K. Ozbay eds., Springer, New York, 2 (2013), 143-161.doi: 10.1007/978-1-4614-6243-9_6.
[22]	M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optim. Theory Appl., 126 (2005), 589-616.doi: 10.1007/s10957-005-5499-z.
[23]	K. Han,T. Friesz and T. Yao, Existence of simultaneous route and departure choice dynamic user equilibrium, Transp. Res., B 53 (2013), 17-30.doi: 10.1016/j.trb.2013.01.009.
[24]	M. Herty, C. Kirchner and A. Klar, Instantaneous control for traffic flow, Math. Methods Appl. Sci., 30 (2007), 153-169.doi: 10.1002/mma.779.
[25]	M. Herty, J. P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826.doi: 10.3934/nhm.2009.4.813.
[26]	M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Netw. Heter. Media, 1 (2006), 275-294.doi: 10.3934/nhm.2006.1.275.
[27]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Reprint of the 1980 original. SIAM, Philadelphia, PA, 2000.doi: 10.1137/1.9780898719451.
[28]	M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345.doi: 10.1098/rspa.1955.0089.
[29]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.doi: 10.1287/opre.4.1.42.