Stabilization of Stop-and-Go Waves in Vehicle Traffic Flow

This article investigates the stabilization problem of stop-and-go waves in vehicle traffic flow with bilateral boundary feedback control. The stop-and-go waves induce the discontinuities of vehicle speed and density, which inturn lead to different traffic states on the front and back sides of the shock front. According to the Rankine–Hugoniot condition, a propagation equation of the shock front is proposed depending on the characteristic velocities of the Aw–Rascle–Zhang traffic flow model. Then, the complete dynamics of the stop-and-go waves is formulated as a coupled hyperbolic partial differential equations (PDE– PDE) system with a common moving boundary. The well posedness of the coupled system with the moving boundary is established via the fixed-domain method. To stabilize the discontinuous traffic state and the location of shock front simultaneously, the bilateral boundary feedback control is formulated for the stop-and-go waves of traffic flow. Some sufficient conditions in terms of matrix inequalities are derived for ensuring the local exponential stability of the closed-loop system in the $H^{2}$–norm. Finally, the theoretical results are illustrated with numerical simulations.