Abstract
This paper recasts the Friesz et al. (1993) measure theoretic model of dynamic network user equibrium as a controlled variational inequality problem involving Riemann integrals. This restatement is done to make the model and its foundations accessible to a wider audience by removing the need to have a background in functional analysis. Our exposition is dependent on previously unavailable necessary conditions for optimal control problems with state-dependent time lags. These necessary conditions, derived in an Appendix, are employed to show that a particular variational inequality control problem has solutions that are dynamic network user equilibria. Our analysis also shows that use of proper flow propagation constraints obviates the need to explicitly employ the arc exit time functions that have complicated numerical implementations of the Friesz et al. (1993) model heretofore. We close by describing the computational implications of numerically determining dynamic user equilibria from formulations based on state-dependent time lags.
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Friesz, T.L., Bernstein, D., Suo, Z. et al. Dynamic Network User Equilibrium with State-Dependent Time Lags. Networks and Spatial Economics 1, 319–347 (2001). https://doi.org/10.1023/A:1012896228490
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DOI: https://doi.org/10.1023/A:1012896228490