Finite-Time Boundary Stabilization for LWR Traffic Flow Model

Finite-time stability means that the trajectory of dynamical system converges to a Lyapunov stable equilibrium state in finite time. The finite time stabilization of traffic flow model can improve the transportation efficiency and reduce the occurrence of traffic congestion. In this letter, the finite-time boundary stabilization problem for the Lighthill-Whitham-Richards (LWR) traffic flow model is considered, in which a variable speed limit (VSL) device is applied at the downstream boundary and a finite-time boundary controller is designed to drive the traffic density to the steady state in finite time. By utilizing the method of characteristics to illustrate the property of finite-time convergence and the Lyapunov function approach to prove stability, the local finite-time stability of LWR traffic flow system is ensured in $H^{2}$ -norm. Moreover, the settling-time can be given depended on the initial value and the parameters of finite-time boundary controller. Lastly, a numerical simulation example is presented to verify the effectiveness of the theoretical results.