Lyapunov stabilization for nonlocal traffic flow models

TL;DR

This paper introduces a Lyapunov-based control approach for nonlocal second-order traffic flow models, ensuring stability towards a steady state across microscopic and macroscopic scales, with explicit convergence rates and numerical validation.

Contribution

It develops Lyapunov functions for nonlocal traffic models and proves asymptotic stability for various kernel functions and initial conditions, advancing control methods in traffic flow modeling.

Findings

Microscopic fixed point is asymptotically stable for any kernel.

Explicit Lyapunov functions and convergence rates are derived.

Numerical examples confirm theoretical stability results.

Abstract

Using a nonlocal second-order traffic flow model we present an approach to control the dynamics towards a steady state. The system is controlled by the leading vehicle driving at a prescribed velocity and also determines the steady state. Thereby, we consider both, the microscopic and macroscopic scales. We show that the fixed point of the microscopic traffic flow model is asymptotically stable for any kernel function. Then, we present Lyapunov functions for both, the microscopic and macroscopic scale, and compute the explicit rates at which the vehicles influenced by the nonlocal term tend towards the stationary solution. We obtain the stabilization effect for a constant kernel function and arbitrary initial data or concave kernels and monotone initial data. Numerical examples demonstrate the theoretical results.