Continuity of the Path Delay Operator for LWR-Based Network Loading with Spillback Ke Han a Benedetto Piccoli b Terry L. Friesz c a Department of Civil and Environmental Engineering Imperial College London, London SW7 2BU, UK Corresponding author. Email: k.han@imperial.ac.uk b Department of Mathematical Sciences and CCIB Rutgers University, Camden, NJ 08102, USA Email: piccoli@camden.rutgers.edu c Department of Industrial and Manufacturing Engineering Pennsylvania State University, University Park, PA 16802, USA Email: tfriesz@psu.edu Abstract This paper establishes the continuity of the path delay operators for dynamic net- work loading (DNL) problems based on the Lighthill-Whitham-Richards model, which explicitly captures vehicle spillback. This DNL aims at describing and predicting the spatial-temporal evolution of traffic flow and congestion on a network, which is consis- tent with established route and departure time choices of travelers. We formulate this LWR-based DNL model as a system of partial differential algebraic equations (PDAEs). Continuous dependence of a merge and a diverge junction model with respect to their initial/boundary conditions is investigated in detail, which leads to the continuity of the delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide a novel method for estimating the minimum network supply without resort to numerical computations. As a result, it is shown that gridlock can never occur in finite time horizon. Key words: path delay operator; continuity; dynamic network loading; LWR model; spillback; gridlock 1 Introduction Dynamic traffic assignment (DTA) is usually viewed as the descriptive modeling of time vary- ing flows on vehicular networks consistent with established traffic flow. DTA models describe and predict departure rates, departure times and route choices of travelers over a given plan- ning horizon. It seeks to describe the dynamic evolution of traffic in networks in a fashion consistent with the fundamental notions of traffic flow and travel demand; see Peeta and Zil- iaskopoulos (2001) for some review on DTA models and recent developments. Dynamic user equilibrium (DUE) of the open-loop type, which is one type of DTA, remains a major modern 1 arXiv:1501.04241v1 [math.AP] 17 Jan 2015