A projected Newton-like inertial dynamics for modeling day-to-day traffic evolution with elastic demand

This paper proposes a projected Newton-like inertial dynamics for modeling second-order day-to-day (DTD) traffic evolution with elastic travel demand. The proposed DTD model describes double dynamics of traffic flow and travel cost based on a class of second-order gradient-like dissipative dynamic systems. We use the projection operator to prevent the existence of negative flow, which is regarded as a major pitfall of the existing second-order DTD traffic models. To our knowledge, this would be the first attempt to address the problem of negative flow in the second-order DTD traffic models. Meanwhile, we show that the proposed model inherits the properties of Newton-like inertial dynamics and behaves similarly to the existing second-order DTD models. The proposed model admits a Hessian-driven component, which is closely related to the congestion externality associated with the marginal link travel cost. The proposed model also extends the existing second-order DTD models from the fixed demand case to the elastic demand case. We characterize several theoretical properties of the proposed projected second-order DTD model, such as the equivalence between its fixed points and the user equilibrium with elastic demand, the convergence of the DTD traffic evolution process, and the stability analysis with different stability concepts. We show that the proposed model can be reduced to the well-known network tatonnement model. Finally, we demonstrate the properties of the projected second-order DTD model via numerical examples.