Regulator Design for a Congested Continuum Traffic Model with App-Routing Instability
DOI: 10.23919/acc45564.2020.9147386
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Summary
This paper addresses the instability in highway traffic caused by real-time navigation applications (e.g., Waze, Google Maps), which introduce destabilizing feedback by greedily routing vehicles based on current conditions. The authors propose a control design methodology for a linearized continuum traffic model in the congested regime to attenuate this instability. The traffic flow is modeled using the Aw-Rascle-Zhang (ARZ) model, a second-order hyperbolic partial differential equation (PDE) system that captures both density and velocity dynamics, unlike first-order models that fail to represent oscillatory stop-and-go traffic. The model is augmented with a non-local boundary condition representing the influx of cars driven by app-routing decisions, which acts as an adversarial feedback loop. The study employs the method of infinite-dimensional backstepping to design a multi-tiered boundary controller. The authors first linearize the ARZ model around an equilibrium congestion profile and diagonalize the convection operator. To handle the non-local boundary condition arising from app-routing, they introduce an extended backstepping transformation involving Volterra integral operators with specific kernels. This transformation shifts the destabilizing interior coupling and non-local boundary feedback into the boundary, where it can be neutralized by a designed ramp metering control input. The existence of the companion kernels for this transformation is characterized through a system of hyperbolic PDEs. The analysis focuses on $H^1$ solutions, ensuring pointwise boundedness of traffic states to prevent physical violations such as negative density or velocity exceeding free-flow limits. The main result demonstrates that the designed feedback controller exponentially stabilizes the equilibrium congestion solution in the $H^1$ norm. Specifically, for sufficiently small initial data, the closed-loop system guarantees the existence of solutions on the infinite time interval and ensures that deviations from the equilibrium decay exponentially. The controller effectively counteracts the destabilizing effects of the app-routing feedback, maintaining traffic within admissible bounds. This work extends prior literature on traffic stabilization by addressing the specific challenge of non-local feedback introduced by digital navigation tools, providing a rigorous mathematical framework for stabilizing continuum traffic models under such conditions.
Key finding
The proposed infinite-dimensional backstepping controller exponentially stabilizes the linearized traffic model with app-routing feedback, ensuring global existence of solutions for small initial perturbations.
Methodology
theoretical
Provenance
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|---|---|---|---|---|---|---|
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| verify | success | — | — | — | 2 | 2026-06-10 |
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