On a link-based day-to-day traffic assignment model Lanshan Han a,⇑ , Lili Du b a School of Civil Engineering, Purdue University, 550 Stadium Mall Dr., West Lafayette, IN 47907, USA b NEXTRANS Center, Purdue University, 3000 Kent Ave., West Lafayette, IN 47906, USA article info Article history: Received 8 April 2011 Received in revised form 11 September 2011 Accepted 11 September 2011 Keywords: Day-to-day dynamics User equilibrium Lyapunov stability Finite-dimensional variational inequality abstract In this paper, we perform a rigorous analysis on a link-based day-to-day traffic assignment model recently proposed in He et al. (2010). Several properties, including the invariance set and the constrained stability, of this dynamical process are established. An extension of the model to the asymmetric case is investigated and the stability result is also established under slightly more restrictive assumptions. Numerical experiments are conducted to demonstrate the findings. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction and motivation Day-to-day dynamics is often applied to study the learning and equilibration process in a congested traffic network. In particular, these models describe how traffic flows, starting from a disequilibrium state, evolve over time. Compared to the classical static traffic assignment models, day-to-day (DTD) dynamics capture the traffic fluctuations and the learning behav- iors of the drivers rather than the final static equilibrium states. A fundamental research question regarding the DTD dynamics is whether the dynamical process leads to a user equilibrium state, or mathematically speaking, whether the dynamics is sta- ble or convergent. If an affirmative answer can be obtained, then the dynamics is mathematically consistent to the axiom of user equilibrium, namely, after a long time of evolution, the traffic network will eventually reach a user equilibrium state. Due to this reason, almost all the DTD dynamics studied in the literature are constructed in such a way that the set of steady states of the dynamics coincides with the set of user equilibria. However, these carefully selected models do not automatically guar- antee that the dynamics will eventually evolve to a user equilibrium state since the dynamics itself may not be stable. There- fore, it is necessary to carefully examine the dynamics and show the stability mathematically. The proposed study in this paper aims to provide a rigorous stability analysis for the DTD dynamics recently proposed in He et al. (2010). In recent years, several different deterministic DTD dynamics models have appeared in the literature. These models in- clude the proportional flow adjusting model Smith (1984), the gap function based model Friesz et al. (1994), the projected dynamical system model Zhang and Nagurney (1996), the first-in-first-out dynamical system model Jin (2007), and a link flow dynamics Zhang et al., 2001. See also a recent paper Yang and Zhang (2009). In He et al. (2010), a new link-based DTD dynamics model was proposed and has been applied to the study of traveler’s behavior following the collapse of I-35W Mississippi River Bridge in Minneapolis, Minnesota, see He and Liu (in press). It has also been shown in He et al. (2010) that by properly choosing a function to measure the difference between two link flow vectors, the proposed model is robust to adding dummy nodes into the network. Moreover, it was established that the equilibrium states of the dynamics 0191-2615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2011.09.005 ⇑ Corresponding author. E-mail addresses: han144@purdue.edu (L. Han), ldu@purdue.edu (L. Du). Transportation Research Part B 46 (2012) 72–84 Contents lists available at SciVerse ScienceDirect Transportation Research Part B journal homepage: www.elsevier.com/locate/trb