Traffic congestion control for ARZ model with an arbitrarily large input delay

This paper addresses the stabilisation problem for Aw-Rascle-Zhang (ARZ) traffic model in the presence of an arbitrarily large input delay. The linearised ARZ model is a 2×2 hyperbolic partial differential equation (PDE) system with proximal reflection, which introduces significant analytical challenges when combined with input delays. To tackle this problem, we propose a backstepping-based boundary controller capable of stabilising the linearised ARZ model under these conditions. The input delay is modelled as a transport PDE, which reformulates the entire system into a 3×3 hyperbolic PDE system. A backstepping transformation is designed to map the original system into a stable target system, enabling the design of a delay-compensated controller. A key technical contribution of this work is that for hyperbolic PDEs with delays, we develop a characteristic-region-wise construction for kernel functions subject to two boundary constraints and close the proof via successive approximation. Another contribution is that we utilise the small-gain theorem for input-to-state stability (ISS) of hyperbolic PDEs. Two simulations are provided to illustrate the effectiveness of the proposed delay-compensated controller: one compares it with a controller without compensation, and the other employs real-world vehicle trajectory data to validate its effectiveness.