Selected mathematical problems of traffic flow theory

Abstract

In spite of the long history of traffic flow theory development, a lot of its aspects are still unsettled, initiate debates and criticism and require further research. The main reasons for such a situation are the high level of complexity of the study object and difficulties in formalizing its behaviour as a complex social technical system. This article considers mathematical statements of problems and some analytical results from the follow-the-leader models to large-scale network models. For the follow-the-leader model, the corresponding class of nonlinear systems of ordinary differential equations is presented. Existence conditions for bounded motion under certain restrictions are obtained. The physical concepts behind models of mathematical physics for traffic flow are discussed and mathematical results obtained by Soviet mathematicians are presented. Selected open problems are introduced related to new mixed traffic models combining deterministic and stochastic approaches, that is, classical dynamics and probability of random walk. An approach to traffic flow modelling on networks is also discussed.

Keywords:

• traffic flow
• traffic jam
• shock waves
• follow-the-leader models
• conservation laws
• macroscopic and microscopic models

2000 AMS Subject Classifications :

• 34D05
• 34D05
• 35B05
• 35B40
• 35Q58

Acknowledgements

This work has been supported by the Russian Foundation for Basic Research, Grant No. 08-01-00959-a. Section 7 of this paper has been supported by RFBR No. 08-07-00158-a, 10-07-00620-a; RHSF No. 08-02-00347; PFI OMN RAN No. 3; PFI Prezidium RAN P-2; FCP ‘Scientific Innovation Russia 2009–2013’ P949, P1490.

Notes

^†Section 7 belongs to Ass. Prof. A.V. Gasnikov of this article, other sections belong to A.P. Buslaev, M.V. Yashina.