Optimal Control of Nonlinear Processes (eBook)

With Applications in Drugs, Corruption, and Terror

Dieter Grass, Jonathan P. Caulkins, Gustav Feichtinger, Gernot Tragler, Doris A. Behrens (Autoren)

eBook Download: PDF

2008
XX, 552 Seiten
Springer Berlin (Verlag)
978-3-540-77647-5 (ISBN)

Infos

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 24.7.2008
• Zusatzinfo: XX, 552 p. 91 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Medizin / Pharmazie
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Corruption • Counter-terror • Drug policy • Nonlinear Processes • optimal control theory • Optimization
• ISBN-10: 3-540-77647-8 / 3540776478
• ISBN-13: 978-3-540-77647-5 / 9783540776475