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 (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.