A second-order model for vehicular traffics with local point constraints on the flow

Abstract

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

Existence of BV solutions for a non-conservative constrained Aw–Rascle–Zhang model for vehicular traffic

“…Remark 1. Note that the fully conservative RS q 1 -solutions studied in [2] satisfy Definition 2.1, indeed for such solutions the second term on the left-hand side is zero for a.e. t > 0.…”

“…This implies that the functional is not controlled by the TV(U 0 ). For this reason in Theorem 2.1 we have to assume that the initial datum is such that Υ(U 0 ) is bounded, as it was the case in [2].…”

“…The existence of RS q 1 -solutions to general constrained Cauchy problems for the ARZ system has been proved in [2] for the case of initial data with bounded variation and piece-wise constant constraint. The existence of RS q 2 -solutions has been addressed by [14] in the case of constant constraint functions and under assumptions on the initial data ensuring that the waves of the first family have negative (propagation) speeds.…”

Entropy solutions for a traffic model with phase transitions

“…Second order macroscopic models close the system (1), (2) by adding a further conservation law. The most celebrated second order macroscopic model is the Aw, Rascle [8] and Zhang [54] model (ARZ).…”

An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic

“…As in [2,5,11], the proof of the above theorem is based on the wave-front tracking algorithm, see [8,17] and the references therein. The details of the proof are deferred to Section 4.…”