Abstract
The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads.   If the queue sizes within the buffer are known, then the initial-boundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula.   Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection.   The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general $L^\infty $ data, and continuity w.r.t. weak convergence of the initial densities.
Highlights
Optimal traffic assignment and dynamic user equilibria on networks have been widely discussed in the engineering literature [10, 11]
Aim of the present paper is to develop a new class of models describing traffic flow at intersections, with more realistic features, including the backward propagation of queues
On the other hand, we formulate the boundary conditions by assigning an upper bound on the flux through the boundary, at each time t
Summary
Optimal traffic assignment and dynamic user equilibria on networks have been widely discussed in the engineering literature [10, 11]. (iii) Our model of traffic flow at intersections achieves well-posedness for general L∞ data, and continuity w.r.t. weak convergence Because of these properties, it is ideally suited to study optimization and Nash equilibrium problems, as shown in the forthcoming paper [6]. On the other hand, we formulate the boundary conditions by assigning an upper bound on the flux through the boundary, at each time t This bound depends on the solution itself, through the measurable coefficients θij. As remarked earlier, this property is essential toward the analysis of optimization problems. On the uniqueness of solutions to ODEs with measurable right hand side, are collected in the Appendix
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