A multi-class non-local macroscopic model with time delay for mixed autonomous / human-driven traffic

TL;DR

This work develops a global-in-time, non-local, time-delayed macroscopic model for mixed autonomous and human-driven traffic across $M$ vehicle classes, with class-specific reaction times and look-ahead ranges. It proves global well-posedness by constructing Hilliges-Weidlich finite-volume approximations that satisfy uniform $L^\infty$ and BV bounds, establishing $\mathbf{L}^1$-stability and uniqueness via entropy methods, and showing convergence to the non-delayed model as delays vanish. The analysis hinges on a saturation mechanism $f_i$ to enforce a maximum density and yields BV controls that handle the non-local interactions and delays; the authors also provide numerical simulations. Numerics illustrate that autonomous vehicles improve traffic flow and dampen stop-and-go waves, even without external control, by leveraging larger look-ahead and shorter reaction times. Overall, the paper advances the mathematical theory of non-local, delayed multi-class traffic models and demonstrates practical implications for AV-enabled traffic stabilization.

Abstract

In this paper, we present a class of systems of non-local conservation laws in one space-dimension incorporating time delay, which can be used to investigate the interaction between autonomous and human-driven vehicles, each characterized by a different reaction time and interaction range. We construct approximate solutions using a Hilliges-Weidlich scheme and we provide uniform L $\infty$ and BV estimates which ensure the convergence of the scheme, thus obtaining existence of entropy weak solutions of bounded variation. Uniqueness follows from an L 1 stability result derived from the entropy condition. Additionally, we provide numerical simulations to illustrate applications to mixed autonomous / human-driven traffic flow modeling. In particular, we show that the presence of autonomous vehicles improves overall traffic flow and stability.

Figures (9)

• Figure 1: Comparison between the solution of the model \ref{['biclasse']} with no saturation ($f_i\equiv 1$ for $i=1,2$) and the solution corresponding to the saturation functions \ref{['expf']}. The initial data is given by \ref{['data_trasl']} and the final time is $T=30$. Left column: density $\rho_1(T,\cdot)$ of fast cars ($V_1=0.04$); Right column: density $\rho_2(T,\cdot)$ of slow cars ($V_2=0.015$).
• Figure 2: Comparison between the model \ref{['biclasse']} considered in this work, and the modified model \ref{['mod_biclasse']}, where the saturation functions depend on the total density $r=\rho_1+\rho_2$. The initial data is \ref{['data_trasl']}, the kernels are constant and the saturation functions are both exponential functions. Top row: Total density. Bottom row: Singular densities taken individually.
• Figure 3: Convergence of the delayed model \ref{['biclasse']} with initial data $\rho_1^0(x)=\rho_2^0(x)=\frac{1}{2}r^0(x)$ and initial total density given by \ref{['eq:tests']} to the model with no delay \ref{['nodelay_bis']} with the same initial data, as ${\left\|\boldsymbol\tau\right\|}_1\rightarrow 0$. Top row: total density $\rho_1+\rho_2$; Bottom row: densities $\rho_1$ and $\rho_2$ plotted individually.
• Figure 4: Comparison between the total density of the solution of \ref{['biclasse']} with initial data \ref{['eq:tests']}-\ref{['initial1']}-\ref{['initial2']} and respectively constant (AVs) and linear decreasing (HVs) kernels, corresponding to different values of the penetration rate $p\in[0,1]$.
• Figure 5: Density profiles of each class taken individually, corresponding to the total densities shown in Figure \ref{['confrontop']}. Two top rows: HVs. Two bottom rows: AVs.

Theorems & Definitions (17)

• Definition 1: Weak solution
• Definition 2: Entropy weak solution
• Lemma 1: Positivity
• Lemma 2: $\mathbf{L^1}$-bound
• Lemma 3: $\mathbf{L^\infty}$-bound / weak maximum principle
• Remark 1
• Lemma 4: $\mathbf{L^\infty}$-bound / weak maximum principle
• Proposition 1: Spatial $\mathbf{BV}$-bound
• Remark 2: Dependence on the parameters
• Remark 3: General case of $M$ classes