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
