This book is devoted to macroscopic models for traffic on a network, with possible applications to car traffic, telecommunications and supply-chains.
The rapidly increasing number of circulating cars in modern cities renders the problem of traffic control of paramount importance, affecting productivity, pollution, life-style etc. The solution of the such problems has thus great socio-economical impact.
Starting from classical and recent fluid-dynamic approaches to describe car traffic on a single road, the book develops an original theory to deal with arbitrarily complex networks. Moreover, efficient numerical schemes are obtained, real urban networks are well described and tests with real data are convenient and easy to implement.
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Contents Author Bio.
This book has no competitors in this field and will certainly be welcome by the applied math community.
The choice of topics covered in the book and its overall presentation are very good.
This is a quite nice manuscript, mathematically oriented, but strongly motivated by very applied questions. It is very clearly written.
To View /Download Contents and Chapter1 (Introduction)
Reviews
Professor Michel Rascle
This is a very nice book, on an important subject: the description of traffic flow on networks, implicitly the description via Partial Differential Equations. The book is rather mathematically oriented, which could be a drawback in view of the wide domain of applications of this field and therefore of the huge cultural gaps between potential readers. However, on one hand modeling the numerous varieties of junctions can be so tricky and so obscure that each scientific community needs a clear exposition and clear statements, and on the other hand, precisely the book is very well written and uses the minimal amount of sophisticated mathematics compatible with the subject. I definitely recommend it in particular to any graduate student willing to enter this field.
1 Introduction . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 1
1.1 Book Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Entropy Admissible Solutions . . . . . . . . . . . . . . . . . . . 16
2.3.1 Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Functions with Bounded Variation . . . . . . . . . . . . . . . . . 26
2.6 Wave-Front Tracking and Existence of Solutions . . . . . . . . . . 32
2.7 Uniqueness and Continuous Dependence . . . . . . . . . . . . . . . 38
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.9 Bibliographical Note . . . . . . . . . . . . . . . . . . . . . . . 44
3 Macroscopic Traffic Models . . . . . . . . . . . . . . . . . . . . . 47
3.1 Lighthill-Whitham-Richards Model . . . . . . . . . . . . . . . . . 47
3.2 Payne-Whitham Model . . . . . . . . . . . . . . . . . . . . . . . .55
3.3 Drawbacks of Second Order Models . . . . . . . . . . . . . . . . . 56
3.4 Aw-Rascle Model . . . . . . . . . . . . . . . . . . . . . . . . . .57
3.5 Third Order Models . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Hyperbolic Phase Transition Model . . . . . . . . . . . . . . . . .61
3.7 Multilane Model . . . . . . . . . . . . . . . . . . . . . . . . . .64
3.8 Multipopulation Model . . . . . . . . . . . . . . . . . . . . . . .66
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
3.10 Bibliographical Note . . . . . . . . . . . . . . . . . . . . . . .70
4 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 Basic Definitions and Assumptions . . . . . . . . . . . . . . . . 71
4.2 Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . .72
4.3 Wave-Front Tracking . . . . . . . . . . . . . . . . . . . . . . . .74
4.4 A Case Study for Riemann Solvers . . . . . . . . . . . . . . . . . 81
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
4.6 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .94
5 Lighthill-Whitham-Richards Model on Networks . . . . . . . . . . . . 95
5.1 Basic Definitions and Assumptions. . . . . . . . . . . . . . . . . 95
5.2 The Riemann Problem at Junctions . . . . . . . . . . . . . . . . . 99
5.3 Interaction Estimates . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Lipschitz Continuous Dependence . . . . . . . . . . . . . . . . . 111
5.5 Time Dependent Traffic . . . . . . . . . . . . . . . . . . . . . .115
5.6 Total Variation of the Fluxes. . . . . . . . . . . . . . . . . . .117
5.7 Total Variation of the Densities . . . . . . . . . . . . . . . . .118
5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.9 Bibliographical Note . . . . . . . . . . . . . . . . . . . . . . .121
6 Aw-Rascle Model on Networks . . . . . . . . . . . . . . . . . . . . 133
6.1 Basic Definitions and Assumptions . . . . . . . . . . . . . . . . 133
6.2 Riemann Problems at Junctions . . . . . . . . . . . . . . . . . . 134
6.3 Stability of Solutions to Riemann Problems at Junctions . . . . . 147
6.4 Existence of Solutions to a Cauchy Problem . . . . . . . . . . . .155
6.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.6 Bibliographical Note . . . . . . . . . . . . . . . . . . . . . . .158
7 Source Destination Model . . . . . . . . . . . . . . . . . . . . . .161
7.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . 166
7.3 Wave-Front Tracking Algorithm . . . . . . . . . . . . . . . . . . 173
7.4 Basic Estimates of Interactions . . . . . . . . . . . . . . . . . 173
7.5 Perturbations of an equilibrium . . . . . . . . . . . . . . . . . 183
7.6 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.7 Bibliographical Note . . . . . . . . . . . . . . . . . . . . . . .185
8 An Example of Traffic Regulation: Circles vs Lights . . . . . . . . 187
8.1 Flux Control for Traffic Lights . . . . . . . . . . . . . . . . . 188
8.2 Single Lane Traffic Circle with low Traffic . . . . . . . . . . . 191
8.3 Single Lane Traffic Circle with heavy Traffic . . . . . . . . . . 193
8.4 Multi Lane Traffic Circle with no Interaction . . . . . . . . . . 197
8.5 Traffic Light vs Traffic Circle . . . . . . . . . . . . . . . . . 198
9 Telecommunication Network . . . . . . . . . . . . . . . . . . . . . 201
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.2 Packets Loss and Velocity Functions on Transmission Lines . . . . 203
9.3 Riemann Solver at Nodes . . . . . . . . . . . . . . . . . . . . . 206
9.4 Estimates on Density Variation . . . . . . . . . . . . . . . . . .210
9.5 Uniqueness and Lipschitz Continuous Dependence . . . . . . . . . .215
10 Numerics on Networks . . . . . . . . . . . . . . . . . . . . . . . 217
10.1 Numerical Approximation . . . . . . . . . . . . . . . . . . . . .217
10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.3 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .225
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Benedetto Piccoli received the Laurea degree (cum laude) from the University of Padova, Italy, and the Ph.D. degree from the International School for Advanced Studies (SISSA/ISAS), Trieste, Italy, in 1991 and 1994, respectively. Currently, he is Research Director at the Istituto per le Applicazioni del Calcolo "Mauro Picone" of the C.N.R.
He held visiting positions at Rutgers university and Universite de Paris Sud Orsay. He has published more than 80 research papers in journals, books, and refereed conferences.
He is editor in chief of Networks and Heterogeneous Media and he is in the editorial board of SIAM Journal on Control and Optimization, Journal of Dynamical and Control Systems and ESAIM Control Optimization and Calculus of Variation.
His main research interests are in the fields of control theory (optimal control, hybrid systems, quantized systems), mathematical finance, conservation laws and traffic flow.
Mauro Garavello received the Laurea degree (cum laude) from the University of Padova, Italy, and the Ph.D. degree from the International School for Advanced Studies (SISSA/ISAS), Trieste, Italy under the supervision of B. Piccoli. He is currently with the University of Milano - Bicocca. His research interests include: viscosity solutions to HJB equations, hybrid optimal control, control of pdes and traffic flow.
