Homogenization of a 1D pursuit law with delay and a counter-example

TL;DR

This paper derives a macroscopic Hamilton-Jacobi model from a microscopic 1D pursuit law with delay, identifying conditions for homogenization and providing counter-examples where the macroscopic limit fails.

Contribution

It introduces a homogenization process for delayed pursuit laws, establishing a threshold for reaction times and demonstrating cases where the macroscopic model does not apply.

Findings

Homogenization holds for reaction times below a specific threshold.

A strict comparison principle is established for the infinite system.

Counter-examples show failure of homogenization at higher reaction times.

Abstract

In this paper, we consider a one dimensional pursuit law with delay which is derived from traffic flow modelling. It takes the form of an infinite system of first order coupled delayed equations. Each equation describes the motion of a driver who interacts with the preceding one, taking into account his reaction time. We derive a macroscopic model, namely a Hamilton-Jacobi equation, by a homogenization process for reaction times that are below an explicit threshold. The key idea is to show, that below this threshold, a strict comparison principle holds for the infinite system. In a second time, for well-chosen dynamics and higher reaction times, we show that there exist some microscopic pursuit laws that do not lead to the previous macroscopic model.