Regulator Design for a Congested Continuum Traffic Model with App-Routing Instability

In this paper, we propose a control design methodology for a linearized continuum traffic model in the congested regime. The continuum traffic flow on a highway is modeled using a linearized version of the quasilinear system of hyperbolic partial differential equations known as the Aw-Rascle-Zhang (ARZ) model. The linear traffic model is augmented with a novel non-local boundary condition representing car influx due to the use of routing apps such as Google Maps and Waze. The routing apps act as real-time previews for highway traffic, introducing potentially destabilizing feedback in the app-based navigation decision process, necessitating the development of a feedback controller. We first study small-time H¹ solutions of the linearized model with the addition of the app-routing for sufficiently small initial data. We introduce an extended, multi-tiered boundary control design based upon the method of infinite-dimensional backstepping. Using an intermediate decoupling transformation, we account for the non-local boundary condition arising from routing app feedback. It is shown that for sufficiently small H¹ data, the equilibrium congestion solution is exponentially stable and guarantees the existence of closed-loop solutions on the infinite time interval. The existence of the extended backstepping method is studied by characterizing the existence of the companion kernels.