Intro
Preface
Contents
1 Introduction
1.1 Introduction
1.2 Some DTA and DUE Literature
1.3 Vocabulary of DUE Modeling
1.4 Alternative Formulations of DUE
1.5 The Structure of DUE Models
1.6 Dynamic Network Loading Models
1.6.1 Vickrey's Model of a Traffic Bottleneck and Its Extension
1.6.2 Network Loading Based on Link Dynamics
1.6.3 Network Loading as a LWR-Based PDAE System
1.6.3.1 The Perakis-Kachani DNL Models
1.6.3.2 The Han-Friesz Within-Link Dynamicsfor DNL
1.6.4 Network Loading Based on the CTM

1.6.5 Network Loading Based on the Variational Approach
1.6.6 LWR Network Loading Based on Closed-Form Operators
1.6.7 Dynamic User Equilibrium in Continuous Time
1.6.8 Dynamic User Equilibrium in Discrete Time
1.7 Other Considerations in Classifying DUE Models
1.8 Unresolved/Partially Resolved Fundamental Challenges
References and Suggested Reading
2 Mathematical Preliminaries
2.1 Selected Topics in Functional Analysis
2.1.1 Hilbert Spaces
2.1.2 Topological Vector Spaces
2.1.3 Compactness
2.1.4 The Contraction Mapping Theorem
2.2 Nonlinear Programming

2.2.1 Nonlinear Program Defined
2.2.2 The Fritz John Conditions
2.2.3 The Kuhn-Tucker Conditions
2.2.4 Kuhn-Tucker Conditions Sufficient
2.2.5 Kuhn-Tucker Conditions for Variational Inequalities
2.3 Calculus of Variations
2.3.1 The Space C1[t0,tf]
2.3.2 The Concept of a Variation
2.4 Optimal Control
2.4.1 The State Operator
2.4.2 Necessary Conditions for Continuous-TimeOptimal Control
2.4.3 Sufficiency in Optimal Control
2.4.3.1 The Mangasarian Theorem
2.4.3.2 The Arrow Theorem
2.5 Differential Variational Inequalities
2.5.1 Problem Definition

2.5.2 Regularity Conditions for DIV
2.5.3 Necessary Conditions
2.6 Nash Games and Differential Nash Games
2.6.1 Nash Equilibria and Normal Form Games
2.6.2 Differential Nash Games and Differential Nash Equilibria
2.6.3 Generalized Differential Nash Equilibria
2.7 The Scalar Conservation Law
2.7.1 Definition and Examples of the Scalar Conservation Law
2.7.2 Characteristics, Shock Waves, and Weak Solutions
2.7.3 Non-uniqueness of Integral Solutions, EntropyConditions
2.8 The Hamilton-Jacobi Equations and the Variational Principle
2.8.1 The Hamilton-Jacobi Equation

2.8.2 The Variational Theory
2.8.2.1 The Classical Lax-Hopf Formula
2.8.2.2 The Generalized Lax-Hopf Formula
References and Suggested Reading
3 The Variational Inequality Formulation of Dynamic User Equilibria
3.1 Notation and Essential Background
3.2 The VI Formulation of DUE with Fixed Demand
3.2.1 Definition of DUE with Fixed Demand
3.2.2 Variational Inequality Problem Equivalent to DUE with Fixed Demand
3.3 The VI Formulation of DUE with Elastic Demand
3.3.1 Definition of DUE with Elastic Demand