Asymptotic solution and effective Hamiltonian of a Hamilton–Jacobi equation in the modeling of traffic flow on a homogeneous signalized road

Abstract

In this paper, we study the large time behavior of solutions to a 1-d time-dependent Hamilton–Jacobi equation which models the traffic flow on a ring road with a traffic signal. The existence and precise properties of the long time average of the flux (effective Hamiltonian) have been obtained. We also prove that the car density and flux eventually exhibit a periodic pattern in time. Its minimum period is the same as the cycle length when the cycle length is large, and might be an integer multiple (≥2) of the cycle length when the cycle length is small.