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Coexisting Automated and Human-Driven Vehicles: Well-Posedness of a Mixed Nonlocal-Local Traffic Model

Rinaldo M. Colombo, Mauro Garavello, Claudia Nocita

TL;DR

The paper addresses a macroscopic traffic model where nonlocal autonomous-vehicle dynamics coexist with local, conventional vehicles, yielding a coupled system of integro-differential PDEs for densities $\rho^i$ and $r$. By decomposing the analysis into a nonlocal subsystem and a local Kruzhkov problem, the authors construct a global semigroup on $\mathcal{BV}^1(\mathbb{R};\mathbb{R}^k) \times \mathcal{BV}(\mathbb{R})$, proving global well-posedness and 1st-order stability with respect to initial data. They establish regularity, positivity, mass conservation, and Lipschitz dependence on initial data, using a recursive approximation scheme that couples nonlocal velocity laws $v_{NL}^i$ and local velocity $v_L$, along with a convolution kernel $\eta$. The results provide a robust analytical foundation for mixed traffic models with coexisting vehicle classes and highlight the nuanced regularity requirements for nonlocal versus local components, enabling potential extensions to nonlocal-to-local limits and higher-population settings. The work advances the mathematical understanding of mixed nonlocal-local traffic dynamics and offers a rigorous framework for future control and optimization studies in heterogeneous traffic scenarios.

Abstract

We present a macroscopic traffic flow model where standard vehicles coexist with vehicles informed on the traffic distribution. The resulting mixed nonlocal-local integro-differential PDEs is proved to generate a locally Lipschitz continuous semigroup whose orbits are uniquely characterized as solutions to the system, according to a natural definition of solution. The norms and function spaces adopted are intrinsic to the different nature of the equations.

Coexisting Automated and Human-Driven Vehicles: Well-Posedness of a Mixed Nonlocal-Local Traffic Model

TL;DR

The paper addresses a macroscopic traffic model where nonlocal autonomous-vehicle dynamics coexist with local, conventional vehicles, yielding a coupled system of integro-differential PDEs for densities and . By decomposing the analysis into a nonlocal subsystem and a local Kruzhkov problem, the authors construct a global semigroup on , proving global well-posedness and 1st-order stability with respect to initial data. They establish regularity, positivity, mass conservation, and Lipschitz dependence on initial data, using a recursive approximation scheme that couples nonlocal velocity laws and local velocity , along with a convolution kernel . The results provide a robust analytical foundation for mixed traffic models with coexisting vehicle classes and highlight the nuanced regularity requirements for nonlocal versus local components, enabling potential extensions to nonlocal-to-local limits and higher-population settings. The work advances the mathematical understanding of mixed nonlocal-local traffic dynamics and offers a rigorous framework for future control and optimization studies in heterogeneous traffic scenarios.

Abstract

We present a macroscopic traffic flow model where standard vehicles coexist with vehicles informed on the traffic distribution. The resulting mixed nonlocal-local integro-differential PDEs is proved to generate a locally Lipschitz continuous semigroup whose orbits are uniquely characterized as solutions to the system, according to a natural definition of solution. The norms and function spaces adopted are intrinsic to the different nature of the equations.

Paper Structure

This paper contains 17 sections, 16 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Analytical Proofs
  4. Stability Property of Solutions
  5. The Nonlocal Problem
  6. Claim 1. The maps $(t,x) \to v(t,x), \partial_x v(t,x) \in \mathbf{C^{0}}\left([0,T] \times {\mathbb{R}}; {\mathbb{R}}^k\right)$. Moreover, for all $t \in [0,T]$: $x \mapsto v(t,x) \in (\mathbf{C^{3}} \cap \mathbf{W^{3,\infty}}) ({\mathbb{R}};{\mathbb{R}}^k)$ and $v \in \mathbf{L^\infty} \left([0,T]; \mathbf{W^{3,\infty}} ({\mathbb{R}};{\mathbb{R}}^k)\right)$.
  7. Claim 2: The following regularities hold: $\forall \, t\in[0,T][(t_o,x_o)\mapstoX (t;t_o,x_o)]\in\mathbf{C^{1}} \left([0,T]\times{\mathbb{R}};{\mathbb{R}}^k\right)\forall \, t\in[0,T][(t_o,x_o)\mapsto\mathcal{E} (t,t_o,x_o)]\in\mathbf{C^{1}} \left([0,T]\times{\mathbb{R}};{\mathbb{R}}^k\right)$ and moreover, for $i=1, \ldots, k$, $(t,t_o,x_o) \in [0,T]\times[0,T]\times{\mathbb{R}}$, ${\left|\mathcal{E}^i(t,t_o,x_o)\right|} \leq e^{QT} \,.$
  8. Claim 3: \ref{['item:32']} holds.
  9. Claim 4: \ref{['item:39']} holds.
  10. Claim 5: \ref{['item:26']} and \ref{['item:47']} hold.
  11. Claim 6: \ref{['item:36']} holds.
  12. Claim 7: \ref{['item:38']} holds.
  13. Bound on the distance between $\rho^i (t)$ and $\widehat{\rho}^i (t)$ in $\mathbf{L^1}({\mathbb{R}}; {\mathbb{R}}^k)$.
  14. Bound on the distance between $\partial_x\rho$ and $\partial_x\widehat{\rho}$ in $\mathbf{C^{0}}\left([0,T]; \mathbf{L^1}({\mathbb{R}}; {\mathbb{R}}^k)\right)$.
  15. The Local Problem
  16. ...and 2 more sections

Key Result

Proposition 2.2

Assume item:17, item:21 and item:hyp-eta. Fix sequences $(\rho_{o,n}, r_{o,n}) \in \mathbf{W^{1,1}} ({\mathbb{R}};{\mathbb{R}}^k) \times \mathbf{L^1} ({\mathbb{R}};{\mathbb{R}})$, $(\varepsilon_n,e_n) \in \mathbf{C^{0}} \left([0,T];\mathbf{W^{1,1}} ({\mathbb{R}}; {\mathbb{R}}^k) \times \mathbf{L^1} admits a solution $(\rho_n,r_n)$ in the sense of Definition def:sol. Moreover, assume that Then, $

Figures (1)

  • Figure 2.1: The initial condition $(\rho_{o,n}, r_{o,n})$ and their sum in \ref{['ex:1']}.

Theorems & Definitions (18)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Example 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Lemma 3.2: Characterization of $\mathcal{BV}^{1} ({\mathbb{R}};{\mathbb{R}}^k)$
  • Lemma 3.4: OperaPrima
  • Lemma 3.5: Fine estimates on $\rho$
  • Lemma 3.6: A priori estimate on $V$
  • ...and 8 more