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# Model of vehicle interactions with autonomous cars and its properties

• *Corresponding author: Giuseppe Visconti
• We study a hierarchy of models based on kinetic equations for the descriptions of traffic flow in presence of autonomous and human–driven vehicles. The autonomous cars considered in this paper are thought of as vehicles endowed with some degree of autonomous driving which decreases the stochasticity of the drivers' behavior. Compared to the existing literature, we do not model autonomous cars as externally controlled vehicles. We investigate whether this feature is enough to provide a stabilization of traffic instabilities such as stop and go waves. We propose two indicators to quantify traffic instability and we find, with analytical and numerical tools, that traffic instabilities are damped as the penetration rate of the autonomous vehicles increases.

Mathematics Subject Classification: Primary: 76A30, 35Q20, 35Q70.

 Citation: • • Figure 1.  Left: flux–density diagram (4). Middle: variance of microscopic speeds at equilibrium (9) as function of the density $\rho$. Right: diffusion coefficient (13) in the diffusive limit of the BGK model (11) as function of the density $\rho$. The plots are obtained using the Maxwellian of the spatially homogeneous model (3)–(6) with $\Delta v = \frac{ v_{\max}}{3}$. Here, $\rho_{\max} = 1$ and $v_{\max} = 1$

Figure 2.  Schematic representation of the Ansatz (29) for the kinetic distribution with $\rho = 0.8 \rho_{\max}$ and $\delta_1 = 1$ (dotted line), $\delta_2 = 4$ (dashed line), $\delta_3 = 7$ (solid line). The values $\overline{f}_i$, $i = 1,2,3$, are found in order to satisfy mass conservation

Figure 3.  Fundamental diagrams of the single–distribution model (21) for autonomous and human–driven vehicles for three choices of the parameter $\bar{\rho}$. The different diagrams refer to different penetration rates, $p\leq 0.4$. Here $v_{\max} = 1 = \rho_{\max} = 1$, $\Delta v = \frac13$ and $\hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v$

Figure 4.  Variance of microscopic speeds at equilibrium of the single–distribution model (21) for autonomous and human–driven vehicles. Three choices of the parameter $\bar{\rho}$ and different penetration rates $p$ are considered. Here $v_{\max} = 1 = \rho_{\max} = 1$, $\Delta v = \frac13$ and $\hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v$

Figure 5.  Top row: sign of the diffusion coefficient $\mu(\rho)$ (13) of the single–distribution model (21) for autonomous and human–driven vehicles. Three choices of the parameter $\bar{\rho}$ and different penetration rates $p$ are considered. Here $v_{\max} = 1 = \rho_{\max} = 1$, $\Delta v = \frac13$ and $\hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v$. Bottom row: left boundary $\alpha$ and right boundary $\beta$ of the interval of instability, and its amplitude $\Gamma$, as function of the penetration rate $p$

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