On the global stability of departure time user equilibrium: A Lyapunov approach

In (Jin, 2018), a new day-to-day dynamical system was proposed for drivers' departure time choice at a single bottleneck. Based on three behavioral principles, the nonlocal departure and arrival times choice problems were converted to the local scheduling payoff choice problem, whose day-to-day dynamics are described by the Lighthill-Whitham-Richards (LWR) model on an imaginary road of increasing scheduling payoff. Thus the departure time user equilibrium (DTUE), the arrival time user equilibrium (ATUE), and the scheduling payoff user equilibrium (SPUE) are uniquely determined by the stationary state of the LWR model, which was shown to be locally, asymptotically stable with analysis of the discrete approximation of the LWR model and through a numerical example. In this study attempt to analytically prove the global stability of the SPUE, ATUE, and DTUE. We first generalize the conceptual models for arrival time and scheduling payoff choices developed in (Jin, 2018) for a single bottleneck with a generalized scheduling cost function, which includes the cost of the free-flow travel time. Then we present the LWR model for the day-to-day dynamics for the scheduling payoff choice as well as the SPUE. We further formulate a new optimization problem for the SPUE and demonstrate its equivalent to the optimization problem for the ATUE in (Iryo and Yoshii, 2007). Finally we show that the objective functions in the two optimization formulations are equal and can be used as the potential function for the LWR model and prove that the stationary state of the LWR model, and therefore, the SPUE, DTUE, and ATUE, are globally, asymptotically stable, by using Lyapunov's second method. Such a globally stable behavioral model can provide more efficient departure time and route choice guidance for human drivers and connected and autonomous vehicles in more complicated networks.Comment: 17 pages, 3 figure

Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph

We consider a class of non-linear dynamics on a graph that contains and generalizes various models from network systems and control and study convergence to uniform agreement states using gradient methods. In particular, under the assumption of detailed balance, we provide a method to formulate the governing ODE system in gradient descent form of sum-separable energy functions, which thus represent a class of Lyapunov functions; this class coincides with Csisz\'{a}r's information divergences. Our approach bases on a transformation of the original problem to a mass-preserving transport problem and it reflects a little-noticed general structure result for passive network synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed gradient formulation extends known gradient results in dynamical systems obtained recently by M. Erbar and J. Maas in the context of porous medium equations. Furthermore, we exhibit a novel relationship between inhomogeneous Markov chains and passive non-linear circuits through gradient systems, and show that passivity of resistor elements is equivalent to strict convexity of sum-separable stored energy. Eventually, we discuss our results at the intersection of Markov chains and network systems under sinusoidal coupling