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

Fei Cao, Xiaoqian Gong, Alexander Keimer

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.

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

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 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 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 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.
Paper Structure (10 sections, 11 theorems, 74 equations, 8 figures)

This paper contains 10 sections, 11 theorems, 74 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Bando-FtL model
  3. The well-posedness and equilibrium of the Bando-FtL model on an infinite time horizon
  4. The well-posedness of the Bando-FtL model on an infinite time horizon for two vehicles
  5. The existence and uniqueness of equilibria of the Bando-FtL model
  6. Long-term behavior of the Bando-FtL model
  7. Asymptotic convergence in the two-vehicle Bando-FtL Model
  8. Alternative approach to the long-term convergence
  9. Long-term behavior of the Bando-FtL model with N + 1 vehicles
  10. Conclusion and Future Work

Key Result

Theorem 2.6

Let $T \in \mathbb R_{>0}$ be a finite time horizon. Let the assumptions in defi:solution hold. Then there exists a unique solution $({x},{v})\in W^{2,\infty}((0,T); \mathbb R^{N+1}) \times W^{1, \infty}((0,T); \mathbb R_{\geq 0}^{N+1})$ for system system_dyn in the sense of defi:solution over the f Furthermore, the solution $({x},{v})\in W^{2,\infty}((0,T); \mathbb R^{N+1}) \times W^{1, \infty}((

Figures (8)

  • Figure 1: One choice for the optimal velocity function $V$ in \ref{['eq:optimal_velocity_function_example']} with parameters $c=1$, $l=4.5$, $d_{\textnormal{s}}=2.5$, and $v_{\max}=30$.
  • Figure 1: Left: The follower's space headway, with the uniform minimum distance shown as a dotted line. Right: The two vehicles' velocities.
  • Figure 1: Left: Follower's space headway. Right: Leader and follower velocities with follower optimal velocity.
  • Figure 2: Left: The follower's space headway with the uniform maximum distance shown as a dotted line. Right: The two vehicles' velocities.
  • Figure 2: Six subregions of the $(V,v)$ plane.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 2.1: Bando-FtL acceleration
  • Remark 2.2: some comments on the Bando-FtL acceleration
  • Remark 2.3: an example for the optimal velocity function $V$
  • Definition 2.4
  • Definition 2.5: a solution to the Bando-FtL system \ref{['system_dyn']}
  • Theorem 2.6
  • Definition 3.1
  • Theorem 3.2: existence of a uniform minimum distance over a small time interval $T$
  • Proof 1
  • Remark 3.3: further discussion on $h_{\min}$
  • ...and 28 more