  We develop backstepping state feedback control to stabilize a moving
shockwave in a freeway segment under bilateral boundary actuations of traffic
flow. A moving shockwave, consisting of light traffic upstream of the shockwave
and heavy traffic downstream, is usually caused by changes of local road
situations. The density discontinuity travels upstream and drivers caught in
the shockwave experience transitions from free to congested traffic. Boundary
control design in this paper brings the moving shockwave front to a static
setpoint 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 differential
equations (PDEs). Each represents the traffic density of a spatial domain
segregated by the moving interface. By Rankine-Hugoniot condition, the
interface position is driven by flux discontinuity and thus governed by a PDE
state dependent ordinary differential equation (ODE). For the PDE-ODE coupled
system. the control objective is to stabilize both the PDE states of traffic
density and the ODE state of moving shock position to setpoint values. Using
delay representation and backstepping method, we design predictor feedback
controllers to cooperatively compensate state-dependent input delays to the
ODE. From Lyapunov stability analysis, we show local stability of the
closed-loop system in $H^1$ norm. The performance of controllers is
demonstrated by numerical simulation.
