Pedestrian Dynamics: Mathematical Theory and Evacuation Control

Homeland security, transportation, and city planning depend upon well-designed evacuation routes. You can’t wait until the day of to realize your plan won’t work. Designing successful evacuation plans requires an in-depth understanding of models and control designs for the problems of traffic flow, construction and road closures, and the intangible human factors. Pedestrian Dynamics: Mathematical Theory and Evacuation Control clearly delineates the derivation of mathematical models for pedestrian dynamics and how to use them to design feedback controls for evacuations.

The book includes:


Mathematical models derived from basic principles
Mathematical analysis of the model
Details of past work
MATLAB® code
65 figures and 400 equations

Unlike most works on traffic flow, this book examines the development of optimal methods to effectively control and improve pedestrian traffic flow. The work of a leading expert, it examines the differential equations applied to conservation laws encountered in the study of pedestrian dynamics and evacuation control problem. The author presents new pedestrian traffic models for multi-directional flow in two dimensions. He considers a range of control models in various simulations, including relaxed models and those concerned with direction and magnitude velocity commands. He also addresses questions of time, cost, and scalability. The book clearly demonstrates what the future challenges are and provides the tools to meet them.

Introduction
Motivation
Literature Survey
Outline

Derivation of Conservation Laws
Mass Conservation
Momentum Conservation
Energy Conservation
Combined Equations
General Conservation

Traffic Models: One Dimensional Case
Lighthill-Whitham-Richards Model
Payne-Whitham Model
Aw-Rascle Model
Zhang Model
Pedestrian and Control Models in One Dimension

Traffic Models: Two-Dimensional Case
Two-Dimensional LWR Model
Two-Dimensional Payne-Whitham Model
Two-Dimensional Aw-Rascle Model
Two-Dimensional Zhang Model

Conservation Law Solutions
Method of Characteristics
Classical or Strong Solutions
Weak Solutions
Scalar Riemann Problem
Admissibility Conditions
Kruzkov’s Entropy Function
Well-posedness
Oleinik Entropy Condition
Scalar Initial-Boundary Problem

Traffic Control
Scalar Conservation Law Solution
Dynamical Systems and C0-Semigroups
Optimal Control
Optimal Flux Control for Scalar Conservation Law
Feedback Control for Scalar Law
Advective Feedback Control for Relaxation Systems
Wellposedness for Bounded Advection Control

Simulations for Advective Control
Godunov’s Method
Simulation Results for Advective Control

Conclusions
Summary
Contributions
Future Work