This paper develops boundary control law for autonomous vehicles to stabilize the stop-and-go traffic on freeway. The macroscopic traffic dynamics is described by the Aw-Rascle-Zhang (ARZ) model in a time and state dependent domain. The leading autonomous vehicle aims to regulate the traffic behind it to uniform equilibrium and the domain length of the traffic to a setpoint. The traffic density and speed is governed by second-order, nonlinear hyperbolic partial differential equations (PDEs), coupled with a state-dependent ODE for the leading autonomous vehicle. The actuation is the speed of autonomous vehicle at the moving front boundary of the domain. We linearize the system around a uniform velocity and density reference and certain physical properties are discussed for the model validity. The linearized model describes the dynamics of deviations of density and velocity from the reference. By transforming the linearized system in a moving coordinate, we obtain a domain with a fixed boundary at one end and a state-dependent moving boundary at the other end. The well-posedness of the system is proved and the linear instability of open-loop system is shown. We further map the system to Riemann variables and based on it, propose the boundary feedback control law actuated by the leading autonomous vehicle. The exponential stability of state variables in L2 norm and convergence to the setpoint domain length is achieved for the closed-loop system.

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