Lyapunov Stabilization for Nonlocal Traffic Flow Models

Using a nonlocal second-order traffic flow model we present an approach to control the dynamics toward a steady state. The system is controlled by the leading vehicle driving at a prescribed velocity and also determines the steady state. Thereby, we consider both the microscopic (trajectory based) and macroscopic (density based) scales. We show that the fixed point of the microscopic traffic flow model is (locally) asymptotically stable for any kernel function. To obtain global stabilization, we present Lyapunov functions for both the microscopic and the macroscopic scale and compute the explicit rates at which the vehicles influenced by the nonlocality tend toward the stationary solution. We obtain stabilization results for a constant kernel function and arbitrary initial data or concave kernels and monotone initial data. In particular, the stabilization is exponential in time. Numerical examples demonstrate the theoretical results.