Boundary feedback control for mixed-autonomy traffic flow system with Markov jumping parameters

This paper investigates the boundary feedback control problem for the mixed-autonomy traffic flow system consisting of Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs). Considering different spacing strategies of AVs, the stochastic impact area of AVs is modeled by a continuous Markov chain. The macroscopic traffic dynamics are described by an extended stochastic ARZ traffic model with Markov jumping parameters, consisting of fourth-order Partial Differential Equations (PDEs). Based on the measurements of the speed and density of HVs and AVs at the boundary of the road segment, a boundary feedback control law, which is implemented by regulating the outlet Variable Speed Limits (VSLs) and a ramp metering at the inlet boundary, is designed to drive the traffic states of both HVs and AVs to desired states. Then sufficient conditions for ensuring the exponential mean-square stability of the system are developed in terms of Linear Matrix Inequalities (LMIs) by employing the Lyapunov function method. Finally, the effectiveness of the proposed boundary feedback controller is validated by numerical examples.