Narrow Results

In this paper we present a second-order model based on the Aw, Rascle, Zhang model (ARZ) for vehicular traffics subject to point constraints on the flow, its motivation being, for instance, the modeling of traffic lights along a road. We first introduce a definition of entropy solution by choosing a family of entropy pairs analogous to the Kruzhkov entropy pairs for scalar conservation laws; then we apply the wave-front tracking method to prove existence and a priori bounds for the entropy solutions of constrained Cauchy problem for ARZ with initial data of bounded variation and piecewise constant constraints. The case of solutions attaining values at the vacuum is considered. We construct an explicit example to describe some qualitative features of the solutions.

Macroscopic models for both vehicular and pedestrian traffic are based on conservation laws. The mathematical description of toll gates along roads or of the escape dynamics for crowds needs the introduction of unilateral constraints on the observable flow. This note presents a rigorous approach to these constraints, and numerical integrations of the resulting models are included to show their practical usability.