Communications in Mathematical Sciences

Volume 18 (2020)

Number 6

A macroscopic traffic flow model with finite buffers on networks: well-posedness by means of Hamilton–Jacobi equations

Pages: 1569 – 1604

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n6.a4

Authors

Nicolas Laurent-Brouty (Centre Inria Sophia Antipolis Médtierranée, Sophia Antipolis, France; and Ecole des Ponts ParisTech Champs-sur-Marne, Marne-la-Vallée, France)

Alexander Keimer (Institute of Transportation Studies, University of California, Berkeley, Cal., U.S.A.)

Paola Goatin (Centre Inria Sophia Antipolis Médtierranée, Sophia Antipolis, France)

Alexandre M. Bayen (Institute of Transportation Studies, University of California, Berkeley, Cal., U.S.A.)

Abstract

We introduce a model dealing with conservation laws on networks and coupled boundary conditions at the junctions. In particular, we introduce buffers of fixed arbitrary size and time-dependent split ratios at the junctions, which represent how traffic is routed through the network, while guaranteeing spill-back phenomena at nodes. Having defined the dynamics at the level of conservation laws, we lift it up to the Hamilton–Jacobi (H‑J) formulation and write boundary datum of incoming and outgoing junctions as functions of the queue sizes and vice-versa. The Hamilton–Jacobi formulation provides the necessary regularity estimates to derive a fixed-point problem in a proper Banach space setting, which is used to prove well-posedness of the model. Finally, we detail how to apply our framework to a non-trivial road network, with several intersections and finite-length links.

Keywords

macroscopic traffic flow models, networks, buffers, fixed-point, Hamilton-Jacobi equations, conservation laws, LWR model

2010 Mathematics Subject Classification

35F21, 35L04, 35L65, 35R02

Full Text (PDF format)

Received 6 May 2019

Accepted 19 March 2020

Published 4 November 2020