Exponential Stabilization of Moving Shockwave in ARZ Traffic Model via Boundary Control: Explicit Gains and Arbitrary Decay Rate

TL;DR

This paper presents boundary feedback controls for stabilizing traffic shockwaves in the ARZ model, achieving arbitrary exponential decay rates through shock-location-based coordinates and Lyapunov functionals.

Contribution

It introduces a novel boundary control method that stabilizes moving shockwaves in the ARZ traffic model with explicit gains and arbitrary decay rates.

Findings

Shock position and system state can be stabilized with any exponential decay rate.

The method transforms a moving-boundary hyperbolic system into a fixed-domain augmented system.

Numerical simulations confirm the effectiveness of the stabilization approach.

Abstract

This paper develops boundary feedback controls to stabilize traffic congestion toward a predefined shock equilibrium in the Aw-Rascle-Zhang (ARZ) traffic flow model. We transform the corresponding moving-boundary $2\times2$ hyperbolic system, covering free and congested flow regimes, respectively, into a shock-free $4\times4$ augmented system on a fixed domain via shock-location-based moving coordinates. By applying the modified Lyapunov functionals concerning shock perturbation, we show that the shock position and the state of the system in $H^2$-norm can be stabilized with an arbitrary exponential decay rate via the given feedback controls. Finally, the stabilization results are demonstrated by numerical simulations.