Saturated boundary feedback stabilization for LWR traffic flow model

In contrast to the fruitful results in saturated control of ordinary differential equation (ODE) systems, there are few related studies for hyperbolic partial differential equation (PDE) systems. This paper focuses on the saturated boundary feedback stabilization problem for the Lighthill–Whitham–Richards (LWR) traffic flow model, in which a variable speed limit (VSL) device is applied at the downstream boundary in the presence of actuator saturation and a saturated boundary feedback controller is proposed to drive the traffic density to the steady state. By employing the Lyapunov function method along with a modified local sector condition, sufficient conditions for ensuring the local exponential stability of the LWR traffic flow system are developed in C1-norm. Remarkably, the proposed sufficient conditions establish a relationship between the control gain and the region of exponential stability (RES), and thus the maximal RES can be determined by introducing an optimization criterion. Lastly, numerical simulations are performed to verify the effectiveness of the theoretical results.