## Car-following models

Car-following models are microscopic traffic flow modelling approaches developed since the 1950s. The models are continuous in time and space. They are first or second-order differential equation systems describing the speed or the acceleration of a vehicle according to the distance spacing and speed of the predecessor. Car-following models are notably used for the longitudinal motion planning of automated vehicles (adaptive cruise control systems).

#### ATG model

The time gap is a fundamental variable for the modelling of car-following situations. Classical regulation policies recommend keeping a constant time gap during a pursuit (ranging between 0.8 and 2.2 s, see the ISO 15622 Standard for ACC systems). In the adaptive time gap (ATG) model, the time gap is relaxed to a safety time gap parameter.  The ATG model and its extensions can describe stable and collision-free dynamics.

#### CFOV model

The optimal velocity (OV) model is a famous car-following model introduced in 1995 by Bando et al. Despite its simplicity, the model describes stop-and-go dynamics for fine-tuning the parameters. Unfortunately, unrealistic collisions and backward motion are not to exclude. The collision-free optimal velocity (CFOV) model is a minimal first-order version of the optimal velocity model. The CFOV model describes stop-and-go dynamics for large reaction time as the OV model does, but without the drawback of collisions or backward movement (see the simulations #1, #2, #3). Derivation of the microscopic model results in a parabolic convection-diffusion partial differential macroscopic equation.

## Stop-and-go waves

Vehicular flows tend to describe stop-and-go waves for congested density levels (see the experiments by Sugiyama et al. or more recently by Stern et al.). Stop-and-go waves negatively impact road user safety and comfort and the environment. Indeed stop-and-go waves generate more energy consumption and pollutant emission than homogeneous streaming.

#### Delay-induced stop-and-go waves

Stop-and-go waves are typical self-organized collective phenomena emerging from driver interactions. They are observed in pedestrian and bicycle flow as well. In traffic engineering, stop-and-go dynamics are explained by delay and low reactivity of the drivers using non-linear car-following models, describing a linear instability and phase transition. The phase transition generally induces capacity drop and hysteresis phenomena.

#### Noise-induced stop-and-go waves

Pioneer works with cellular automata models by Nagel, Schreckenberg, and Schadschneider have shown that stop-and-go dynamics can result from noise effects. In continuous-time, it turns out that colored noises such as Brownian noises given by the Ornstein-Uhlenbeck process allow describing stop-and-go phenomena as well. However, technically, the approach is different from the linear instability of delayed models. Stop-and-go dynamics emerge even in linearly stable framework, due to stochastic effects and nonlinear instability mechanisms.

## Stability analysis

Stability properties of car-following models are expected features of adaptive cruise control systems in traffic engineering. Indeed, several recent experiments (see e.g. the experiments by Gunter et al. or the OpenACC experiments) have shown that current ACC systems describe unstable collective behaviors.

#### Local stability

In the local stability analysis (also platoon stability), some vehicles follow a leader with assigned speed. The local stability is called over-damped when the convergence is oscillation-free. Such a feature allows ensuring (at least partially) the absence of oscillations in the dynamics.

#### Global stability

The global stability analysis (also string stability) tackles vehicles on an infinite line or with periodic boundary conditions. Global stability conditions contain inductive and convective perturbations that may locally vanish. They are, in general, more restrictive than local conditions.

Local stability analysis generally relies on Laplace transform, while circulant matrix properties or exponential ansatz allow tackling the global stability for finite periodic and infinite systems, respectively. The stability conditions are determined by solving the characteristic equation in the complex plane. Stable solutions only exist in the left half-plane (Hurwitz polynomial), with overdamped solutions living on the real semi-axis. The characteristic equations are polynomials for ODE systems while they are exponential-polynomials for delayed systems. Only few delayed models in the literature have explicit stability conditions.

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