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We develop backstepping state feedback control to stabilize a movingshockwave in a freeway segment under bilateral boundary actuations of trafficflow. A moving shockwave, consisting of light traffic upstream of the shockwaveand heavy traffic downstream, is usually caused by changes of local roadsituations. The density discontinuity travels upstream and drivers caught inthe shockwave experience transitions from free to congested traffic. Boundarycontrol design in this paper brings the moving shockwave front to a staticsetpoint position, hindering the upstream propagation of traffic congestion.The traffic dynamics are described with Lighthill-Whitham-Richard (LWR) model,leading to a system of two first-order hyperbolic partial differentialequations (PDEs). Each represents the traffic density of a spatial domainsegregated by the moving interface. By Rankine-Hugoniot condition, theinterface position is driven by flux discontinuity and thus governed by a PDEstate dependent ordinary differential equation (ODE). For the PDE-ODE coupledsystem. the control objective is to stabilize both the PDE states of trafficdensity and the ODE state of moving shock position to setpoint values. Usingdelay representation and backstepping method, we design predictor feedbackcontrollers to cooperatively compensate state-dependent input delays to theODE. From Lyapunov stability analysis, we show local stability of theclosed-loop system in $H^1$ norm. The performance of controllers isdemonstrated by numerical simulation.

Bilateral Boundary Control of Moving Shockwave in LWR Model of Congested Traffic