Grass / Caulkins / Feichtinger Optimal Control of Nonlinear Processes

With Applications in Drugs, Corruption, and Terror
1. Auflage 2008
ISBN: 978-3-540-77647-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 552 Seiten

Dynamic optimization is rocket science - and more. This volume teaches researchers and students alike to harness the modern theory of dynamic optimization to solve practical problems. These problems not only cover those in space flight, but also in emerging social applications such as the control of drugs, corruption, and terror. This volume is designed to be a lively introduction to the mathematics and a bridge to these hot topics in the economics of crime for current scholars. The authors celebrate Pontryagin's Maximum Principle - that crowning intellectual achievement of human understanding. The rich theory explored here is complemented by numerical methods available through a companion web site.

Autoren/Hrsg.

Grass, Dieter

Caulkins, Jonathan P.

Feichtinger, Gustav

Tragler, Gernot

Behrens, Doris A.

Inhaltsverzeichnis

1;Preface;7
2;Acknowledgments;11
3;Contents;13
4;Part I Background;21
4.1;1 Introduction;22
4.1.1;1.1 Taking Rocket Science Beyond the Frontiers of Space;22
4.1.2;1.2 Why Drugs, Corruption, and Terror?;24
4.1.3;1.3 Questions Optimal Control Can Answer;26
4.2;2 Continuous-Time Dynamical Systems;28
4.2.1;2.1 Nonlinear Dynamical Modeling;28
4.2.2;2.2 One-Dimensional Systems;29
4.2.3;2.3 A One-Dimensional Corruption Model;33
4.2.4;2.4 Dynamical Systems as ODEs;36
4.2.5;2.5 Stability Analysis of a One-Dimensional Terror Model;46
4.2.6;2.6 ODEs in Higher Dimensions;49
4.2.7;2.7 Stability Behavior in a Descriptive Model of Drug Demand;70
4.2.8;2.8 Introduction to Bifurcation Theory;74
4.2.9;2.9 Bifurcation Analysis of a One-Dimensional Drug Model;87
4.2.10;2.10 The Poincar´ e –Andronov–Hopf Bifurcation;90
4.2.11;2.11 Higher-Dimensional Bifurcation Analysis of a Drug Model;93
4.2.12;2.12 Advanced Topics;97
4.2.13;Exercises;108
4.2.14;Notes and Further Reading;115
5;Part II Applied Optimal Control;118
5.1;3 Tour d’Horizon: Optimal Control;120
5.1.1;3.1 Historical Remarks;120
5.1.2;3.2 A Standard Optimal Control Problem;123
5.1.3;3.3 The Maximum Principle of Optimal Control Theory;127
5.1.4;3.4 The Principle of Optimality;146
5.1.5;3.5 Singular Optimal Control;150
5.1.6;3.6 The Maximum Principle With Inequality Constraints;161
5.1.7;3.7 Infinite Time Horizon;174
5.1.8;3.8 Discounted Autonomous In.nite Horizon Models;178
5.1.9;3.9 An Optimal Control Model of a Drug Epidemic;187
5.1.10;Exercises;196
5.1.11;Notes and Further Reading;202
5.2;4 The Path to Deeper Insight: From Lagrange to Pontryagin;208
5.2.1;4.1 Introductory Remarks on Optimization;208
5.2.2;4.2 Static Maximization;216
5.2.3;4.3 The Calculus of Variations;233
5.2.4;4.4 Proving the Continuous-Time Maximum Principle;242
5.2.5;Exercises;250
5.2.6;Notes and Further Reading;253
5.3;5 Multiple Equilibria, Points of Indi.erence, and Thresholds;256
5.3.1;5.1 Occurrence of Multiple Equilibria;257
5.3.2;5.2 The Optimal Vector Field;258
5.3.3;5.3 A Typical Example;263
5.3.4;5.4 De.ning Indi.erence and DNSS Points;271
5.3.5;5.5 Revisiting the Typical Example;279
5.3.6;5.6 Eradication vs. Accommodation in an Optimal Control Model of a Drug Epidemic;285
5.3.7;Exercises;288
5.3.8;Notes and Further Reading;291
6;Part III Advanced Topics;296
6.1;6 Higher-Dimensional Models;298
6.1.1;6.1 Controlling Drug Consumption;299
6.1.2;6.2 Corruption in Governments Subject to Popularity Constraints;315
6.1.3;6.3 Is It Important to Manage Public Opinion While Fighting Terrorism?;327
6.1.4;Exercises;335
6.1.5;Notes and Further Reading;342
6.2;7 Numerical Methods for Discounted Systems of Infinite Horizon;346
6.2.1;7.1 General Remarks;346
6.2.2;7.2 Numerical Continuation;351
6.2.3;7.3 The Canonical System Without Active Constraints;361
6.2.4;7.4 Calculating Long-Run Optimal Solutions;362
6.2.5;7.5 Continuing the Optimal Solution: Calculating the Stable Manifold;368
6.2.6;7.6 Optimal Control Problems with Active Constraints;378
6.2.7;7.7 Retrieving DNSS Sets;385
6.2.8;7.8 Retrieving Heteroclinic Connections;387
6.2.9;7.9 Numerical Example from Drug Control;389
6.2.10;Exercises;399
6.2.11;Notes and Further Reading;401
6.3;8 Extensions of the Maximum Principle;404
6.3.1;8.1 Multi-Stage Optimal Control Problems;405
6.3.2;8.2 Differential Games;410
6.3.3;8.3 Age-Structured Models;436
6.3.4;8.4 Further Optimal Control Issues;441
6.3.5;Exercises;445
6.3.6;Notes and Further Reading;455
7;Part IV Appendices;460
7.1;A Mathematical Background;462
7.1.1;A.1 General Notation and Functions;462
7.1.2;A.2 Finite-Dimensional Vector Spaces;466
7.1.3;A.3 Topology and Calculus;482
7.2;B Derivations and Proofs of Technical Results;502
7.2.1;B.1 Separation Theorems, Farkas Lemma and Supergradient;502
7.2.2;B.2 Proof of the Michel Theorem;505
7.2.3;B.3 Proof of the Transversality Condition in Proposition 3.74;510
7.2.4;B.4 The In.nite Horizon Transversality Condition Revisited;511
7.2.5;B.5 Monotonicity of the Solution Path;513
7.2.6;B.6 Admissible and Quasi-Admissible Directions;515
7.2.7;B.7 Proof of the Envelope Theorem;517
7.2.8;B.8 The Dimension of the Stable Manifold;518
7.2.9;B.9 Asymptotic Boundary Condition;521
8;References;524
9;Glossary;550
10;Index;554
11;Author Index;564