Optimal Urban Networks Via Mass Transportation

by Buttazzo, Giuseppe; Pratelli, Aldo; Solimini, Sergio; Stepanov, Eugene

• ISBN13: 9783540857983
• ISBN10: 3540857982
• Format: Paperback
• Copyright: 2009-01-01
• Publisher: Springer Verlag

Summary

Recently much attention has been devoted to the optimization of transportation networks in a given geographic area. One assumes the distributions of population and of services/workplaces (i.e. the network's sources and sinks) are known, as well as the costs of movement with/without the network, and the cost of constructing/maintaining it. Both the long-term optimization and the short-term, "who goes where," optimization are considered. These models can also be adapted for the optimization of other types of networks, such as telecommunications, pipeline or drainage networks. In the monograph we study the most general problem settings, namely, when neither the shape nor even the topology of the network to be constructed is known a priori.

Table of Contents

Introduction	p. 1
Problem Setting	p. 7
Notation and Preliminaries	p. 7
Properties of Optimal Paths and Relaxed Costs	p. 13
Optimal Connected Networks	p. 25
Optimization Problem	p. 25
Properties of the Optimal Networks	p. 28
Average Distance Problem	p. 33
Relaxed Problem and Existence of Solutions	p. 37
Relaxed Problem Setting	p. 37
Properties of Relaxed Minimizers	p. 41
Non-existence of Classical Solutions	p. 65
Existence of Classical Solutions	p. 71
Topological Properties of Optimal Sets	p. 75
Transiting Mass Function	p. 75
Ordered Transport Path Measures	p. 82
Closedness of Optimal Sets	p. 92
Number of Connected Components of Optimal Sets	p. 95
Optimal Sets and Geodesics in the Two-Dimensional Case	p. 105
Preliminary Constructions	p. 106
Proof of the Main Result	p. 120
Appendix	p. 131
The Mass Transportation Problem	p. 131
Some Tools from Geometric Measure Theory	p. 135
Measures as Duals of the Continuous Functions	p. 135
Push-forward and Tensor Product of Measures	p. 140
Measure Valued Maps and Disintegration Theorem	p. 140
T-convergence	p. 142
References	p. 145
Index	p. 149
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