Non-convergence in dynamic assignment networks?

In the steady-state assignment model where each link has a non-decreasing cost flow curve we have monotonicity not just at the link level but also at the route level. In our dynamical system we assume that the users swap to cheaper routes. Monotonicity of the route cost function is enough to guarantee that the given function V, detailed later, is in fact a Lyapunov function and hence that the system converges to equilibrium.

In the dynamic assignment model, the route cost function is not a monotone function of route flow, as was shown in [1]. Therefore convergence does not immediately follow, as it does in the steady-state case. This paper essentially shows that the dynamic counterpart of the steady-state Lyapunov function is in fact not a Lyapunov function. This does not at all imply non-convergence of the dynamical system simulating a swap to cheaper routes, but it does raise the question of convergence. Obviously, if another function could be found that satisfies the criteria of being a Lyapunov function this would be sufficient for convergence.