Existence of optima and equilibria for traffic flow on networks

Alberto Bressan; Ke Han; Alberto Bressan; Ke Han

doi:10.3934/nhm.2013.8.627

Networks and Heterogeneous Media

2013, Volume 8, Issue 3: 627-648. doi: 10.3934/nhm.2013.8.627

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Existence of optima and equilibria for traffic flow on networks

Alberto Bressan ^{1
,},
Ke Han ^{2
,}

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

Received: 01 August 2012 Revised: 01 February 2013
Primary: 35L65, 49K20; Secondary: 90B20, 91A10.

This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, 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. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.
- Traffic network,
- scalar conservation law,
- global optima,
- user equilibria
Citation: Alberto Bressan, Ke Han. Existence of optima and equilibria for traffic flow on networks[J]. Networks and Heterogeneous Media, 2013, 8(3): 627-648. doi: 10.3934/nhm.2013.8.627

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Abstract

This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, 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. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.

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