Optimal urban networks via mass transportation / Giuseppe Buttazzo ... [and others].

Format:

Book

Contributor:

Buttazzo, Giuseppe.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1961.

Lecture notes in mathematics ; 1961

Language:

English

Subjects (All):

Transportation--Mathematical models.

Transportation.

Mathematical optimization.

Physical Description:

x, 150 pages : illustrations, maps ; 24 cm.

Place of Publication:

Berlin : Springer, [2009]

Summary:

Lecture Notes in Mathematics

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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.

Contents:

1 Introduction 1

2 Problem Setting 7

2.1 Notation and Preliminaries 7

2.2 Properties of Optimal Paths and Relaxed Costs 13

3 Optimal Connected Networks 25

3.1 Optimization Problem 25

3.2 Properties of the Optimal Networks 28

3.3 Average Distance Problem 33

4 Relaxed Problem and Existence of Solutions 37

4.1 Relaxed Problem Setting 37

4.2 Properties of Relaxed Minimizers 41

4.3 Non-existence of Classical Solutions 65

4.4 Existence of Classical Solutions 71

5 Topological Properties of Optimal Sets 75

5.1 Transiting Mass Function 75

5.2 Ordered Transport Path Measures 82

5.3 Closedness of Optimal Sets 92

5.4 Number of Connected Components of Optimal Sets 95

6 Optimal Sets and Geodesics in the Two-Dimensional Case 105

6.1 Preliminary Constructions 106

6.2 Proof of the Main Result 120

Appendix 131

A The Mass Transportation Problem 131

B Some Tools from Geometric Measure Theory 135

B.1 Measures as Duals of the Continuous Functions 135

B.2 Push-forward and Tensor Product of Measures 140

B.3 Measure Valued Maps and Disintegration Theorem 140

B.4 T-convergence 142.

Notes:

Includes bibliographical references and index.

ISBN:

9783540857983

3540857982

3540857990

9783540857990

OCLC:

258102457