Stability of a dynamic model for traffic networks

The dynamic assignment model assumes flow moves towards cheaper routes at each time at a rate proportional to the product of the flow along the more expensive route and the cost difference. Therefore, it is important for the cost function to be monotone so that convergence to equilibrium will occur. Conditions on the bottleneck output function are given for the bottleneck delay function to be monotone, which will imply monotonicity of the route cost function in the single bottleneck per route case. It is shown that for reasonable bottleneck output functions, we have monotonicity of the product of link cost with a decaying exponential. This decay-monotonicity transfers to the route cost in certain given circumstances. This will in turn imply convergence of the dynamical system by applying Lyapunov´s theorem using the appropriate Lyapunov function. It is then important to note that monotonicity of the route cost function implies decay-monotonicity of the route cost function and hence the convergence result is valid for the single bottleneck per route case with monotone link cost functions.