Analysis of the long-term behavior of the "Bando--follow-the-leader'' car-following model

TL;DR

This paper analyzes the long-term stability and behavior of the Bando--follow-the-leader car-following model, proving collision avoidance and exponential convergence of following vehicles under certain conditions, supported by theoretical and numerical evidence.

Contribution

It establishes collision-free behavior and exponential convergence of vehicles in the Bando--follow-the-leader model using Lyapunov functions and perturbation methods, extending understanding of its long-term dynamics.

Findings

Vehicles maintain safe headways over time.

Following vehicles converge to a common velocity and headway.

Numerical simulations confirm theoretical convergence results.

Abstract

In this article, we investigate the long-term behavior of the ``Bando--follow-the-leader'' car-following model, whose well-posedness and stability with respect to delay were analyzed in a recent work \cite{gong2023well}. We first establish the collision-free property of the model with \(N+1\in\N_{\geq2}\) vehicles over an infinite time horizon, assuming that the trajectory of the first vehicle is prescribed, by demonstrating the existence of a uniform strictly positive lower bound on the space headway between adjacent vehicles. Furthermore, assuming that the first vehicle travels at a constant velocity and \(N\in\N_{\geq1}\) vehicles follow it according to the Bando--follow-the-leader model on a single lane, our main results state that, with certain reasonable constraints imposed on the modeling parameters, all \(N\) following vehicles will eventually (i.e., when time goes to infinity) converge to the same headway and velocity with a globally exponential convergence rate. The analytical methods are based on Lyapunov functions and a perturbation argument. Numerical simulations are also provided to illustrate the obtained theoretical convergence guarantees.