Simultaneous Stabilization of Traffic Flow on Two Connected Roads

In this paper we develop a boundary state feedback control law for a cascaded traffic flow network system: one incoming and one outgoing road connected by a junction. The macroscopic traffic dynamics on each road segment are governed by Aw-Rascle-Zhang (ARZ) model, consisting of secondorder nonlinear partial differential equations (PDEs) for traffic density and velocity. Different equilibrium road conditions are considered for the two segments. For stabilization of stop-andgo traffic congestion on the two roads, we consider a ramp metering located at the connecting junction. The traffic flow rate entering from the on-ramp to the mainline junction is actuated. The objective is to simultaneously stabilize the upstream and downstream traffic to given spatially-uniform constant steady states. We design a full state feedback control law for this underactuated network of two systems of two hetero-directional linear first-order hyperbolic PDEs interconnected through the junction boundary. Exponential Convergence to steady states in L² sense is validated by a numerical simulation.