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9783540776468

Optimal Control of Nonlinear Processes

by ; ; ; ;
  • ISBN13:

    9783540776468

  • ISBN10:

    354077646X

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2008-10-04
  • Publisher: Springer Verlag
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List Price: $219.99

Summary

Dynamic optimization is rocket science ' and more. This volume teaches how to harness the modern theory of dynamic optimization to solve practical problems, not only from space flight but also in emerging social applications such as the control of drugs, corruption, and terror. These innovative domains are usefully thought about in terms of populations, incentives, and interventions, concepts which map well into the framework of optimal dynamic control. 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. We celebrate Pontryagin's Maximum Principle ' that crowning intellectual achievement of human understanding ' and push its frontiers by exploring models that display multiple equilibria whose basins of attraction are separated by higher-dimensional DNSS "tipping points". That rich theory is complemented by numerical methods available through a companion web site.

Table of Contents

Prefacep. VII
Acknowledgmentsp. XI
Background
Introductionp. 3
Taking Rocket Science Beyond the Frontiers of Spacep. 3
Why Drugs, Corruption, and Terror?p. 5
Questions Optimal Control Can Answerp. 7
Continuous-Time Dynamical Systemsp. 9
Nonlinear Dynamical Modelingp. 9
One-Dimensional Systemsp. 10
A One-Dimensional Corruption Modelp. 14
Dynamical Systems as ODEsp. 17
Concepts and Definitionsp. 19
Invariant Sets and Stabilityp. 21
Structural Stabilityp. 25
Linearization and the Variational Equationp. 26
Stability Analysis of a One-Dimensional Terror Modelp. 27
ODEs in Higher Dimensionsp. 30
Autonomous Linear ODEsp. 31
Autonomous Nonlinear ODEsp. 42
Stability Behavior in a Descriptive Model of Drug Demandp. 51
Introduction to Bifurcation Theoryp. 55
Terminology and Key Ideas of Bifurcation Theoryp. 56
Normal Forms and the Center Manifold: The Tools of Bifurcation Theoryp. 57
Local Bifurcations in One Dimensionp. 63
Bifurcation Analysis of a One-Dimensional Drug Modelp. 68
The Poincare-Andronov-Hopf Bifurcationp. 71
Higher-Dimensional Bifurcation Analysis of a Drug Modelp. 74
Advanced Topicsp. 78
Stability of Limit Cyclesp. 78
Boundary Value Problemsp. 85
Exercisesp. 89
Notes and Further Readingp. 96
Applied Optimal Control
Tour d'Horizon: Optimal Controlp. 101
Historical Remarksp. 101
A Standard Optimal Control Problemp. 104
The Maximum Principle of Optimal Control Theoryp. 108
Pontryagin's Maximum Principlep. 108
Some General Resultsp. 113
The Maximum Principle for Variable Terminal Timep. 115
Economic Interpretation of the Maximum Principlep. 117
Sufficiency Conditionsp. 119
Existence of an Optimal Solutionp. 122
How to Solve an Optimal Control Problem: A Simple Consumption vs. Investment Modelp. 124
The Principle of Optimalityp. 127
The Hamilton-Jacobi-Bellman Equationp. 127
A Proof of the Maximum Principlep. 130
Singular Optimal Controlp. 131
The Most Rapid Approach Path (MRAP)p. 134
An Example From Drug Control that Excludes Singular Arcsp. 137
An Example From Terror Control with an MRAP Solutionp. 139
The Maximum Principle With Inequality Constraintsp. 142
Mixed Path Constraintsp. 144
General Path Constraintsp. 147
Sufficiency Conditionsp. 154
Infinite Time Horizonp. 155
Definitions of Optimality for Infinite Horizon Problemsp. 155
Maximum Principle for Infinite Time Horizon Problemsp. 156
Sufficiency Conditionsp. 159
Discounted Autonomous Infinite Horizon Modelsp. 159
The Michel Theoremp. 160
The Ramsey Model for an Infinite Time Horizonp. 165
Structural Results on One-State Discounted, Autonomous Systemsp. 167
An Optimal Control Model of a Drug Epidemicp. 168
Model Formulationp. 168
Stability Analysisp. 170
Phase Portrait Analysisp. 176
Exercisesp. 177
Notes and Further Readingp. 183
The Path to Deeper Insight: From Lagrange to Pontryaginp. 189
Introductory Remarks on Optimizationp. 189
Notational Remarksp. 190
Motivation and Insightsp. 190
A Simple Maximization Problemp. 192
Finite-Dimensional Approximation of an Infinite-Dimensional Problemp. 195
Static Maximizationp. 197
Basic Theorems and Definitionsp. 198
Theory and Geometric Interpretation of Lagrange and Karush-Kuhn-Tuckerp. 202
The Envelope Theorem and the Lagrange Multiplierp. 208
The Discrete-Time Maximum Principle as a Static Maximization Problemp. 210
The Calculus of Variationsp. 214
A Simple Variational Examplep. 214
The First Variationp. 216
Deriving the Euler Equation and Weierstrass-Erdmann Conditionsp. 218
Proving the Continuous-Time Maximum Principlep. 223
The Continuous-Time Maximum Principle Revisitedp. 223
Necessary Conditions at Junction Pointsp. 227
Exercisesp. 231
Notes and Further Readingp. 234
Multiple Equilibria, Points of Indifference, and Thresholdsp. 237
Occurrence of Multiple Equilibriap. 238
The Optimal Vector Fieldp. 239
Finite vs. Infinite Time Horizon Modelsp. 239
Discounted Autonomous Models for an Infinite Time Horizonp. 243
A Typical Examplep. 244
Existence and Stability of the Equilibriap. 245
Determining the Optimal Vector Field and the Optimal Costate Rulep. 247
Defining Indifference and DNSS Pointsp. 252
Multiplicity and Separabilityp. 253
Definitionsp. 254
Conclusions from the Definitionsp. 256
Revisiting the Typical Examplep. 260
Eradication vs. Accommodation in an Optimal Control Model of a Drug Epidemicp. 266
Exercisesp. 269
Notes and Further Readingp. 272
Advanced Topics
Higher-Dimensional Modelsp. 279
Controlling Drug Consumptionp. 280
Model of Controlled Drug Demandp. 280
Deriving the Canonical Systemp. 283
The Endemic Level of Drug Demandp. 286
Optimal Dynamic Policy away from the Endemic Statep. 287
Optimal Policies for Different Phases of a Drug Epidemicp. 292
Corruption in Governments Subject to Popularity Constraintsp. 296
The Modeled Incentive for Being Corruptp. 297
Optimality Conditionsp. 299
Insights About the Incentive to Be Corruptp. 300
Is Periodic Behavior Caused by Rational Optimization?p. 302
Is It Important to Manage Public Opinion While Fighting Terrorism?p. 308
What One Should Know when Fighting Terrorismp. 309
Derivation of the Canonical Systemp. 310
Numerical Calculationsp. 311
Optimal Strategy for a Small Terror Organizationp. 314
Exercisesp. 316
Notes and Further Readingp. 323
Numerical Methods for Discounted Systems of Infinite Horizonp. 327
General Remarksp. 327
Problem Formulation and Assumptionsp. 328
Notationp. 329
Numerical Methods for Solving Optimal Control Problemsp. 330
Boundary Value Problems from Optimal Controlp. 330
Numerical Continuationp. 332
Continuation Algorithmsp. 333
Continuing the Solution of a BVPp. 338
The Canonical System Without Active Constraintsp. 342
Calculating Long-Run Optimal Solutionsp. 343
Equilibriap. 344
Limit Cyclesp. 346
Continuing the Optimal Solution: Calculating the Stable Manifoldp. 349
Stable Manifold of an Equilibriump. 350
Stable Manifold of Limit Cyclesp. 354
Optimal Control Problems with Active Constraintsp. 359
The Form of the Canonical System for Mixed Path Constraintsp. 360
The Form of the Canonical System for Pure State Constraintsp. 360
Solutions Exhibiting Junction Pointsp. 362
Retrieving DNSS Setsp. 366
Locating a DNSS Pointp. 366
Continuing a DNSS Pointp. 368
Retrieving Heteroclinic Connectionsp. 368
Locating a Heteroclinic Connectionp. 368
Continuing a Heteroclinic Connection in Parameter Spacep. 369
Numerical Example from Drug Controlp. 370
Stating the Necessary Conditionsp. 370
Equilibria of the Canonical Systemp. 372
Numerical Analysisp. 372
Optimal Vector Field for v = 4,000p. 373
Optimal Vector Field for v = 12,000p. 377
Exercisesp. 380
Notes and Further Readingp. 382
Extensions of the Maximum Principlep. 385
Multi-Stage Optimal Control Problemsp. 386
Necessary Optimality Conditions for Two-Stage Control Problemsp. 386
Two-Stage Models of Drug Controlp. 387
Counter-Terror Measures in a Multi-Stage Scenariop. 388
Differential Gamesp. 391
Terminologyp. 392
Nash Equilibriap. 394
Tractable Game Structuresp. 397
A Corrupt Politician vs. the Tabloid Pressp. 397
Leader-Follower Gamesp. 404
A Post September 11th Game on Terrorismp. 407
Age-Structured Modelsp. 417
A Maximum Principle for Distributed Parameter Systemsp. 419
Age-Structured Drug Initiationp. 420
Further Optimal Control Issuesp. 422
Delayed Systemsp. 422
Stochastic Optimal Controlp. 424
Impulse Control and Jumps in the State Variablesp. 425
Nonsmooth Systemsp. 426
Exercisesp. 426
Notes and Further Readingp. 436
Appendices
Mathematical Backgroundp. 443
General Notation and Functionsp. 443
Finite-Dimensional Vector Spacesp. 447
Vector Spaces, Linear Dependence, and Basisp. 447
Linear Transformations and Matricesp. 450
Inverse Matrices and Linear Equationsp. 453
Determinantsp. 455
Linear Form and Dual Spacep. 457
Eigenvalues and Eigenvectorsp. 459
Euclidean Vector Space R[superscript n]p. 461
Topology and Calculusp. 463
Open Set, Neighborhood, and Convergencep. 463
Continuity and Differentiabilityp. 464
Maximization of Real-Valued Functions in R[superscript n]p. 471
Convex Analysisp. 473
Taylor Theorem and Implicit Function Theoremp. 475
Integration Theoryp. 477
Distributionsp. 481
Derivations and Proofs of Technical Resultsp. 483
Separation Theorems, Farkas Lemma and Supergradientp. 483
Proof of the Michel Theoremp. 486
Augmented and Truncated Problemp. 487
Optimal Solution of Problem (B.8)p. 487
Necessary Conditions for Problem (B.8)p. 488
Limit of Solutions for Increasing Time Sequencep. 489
Proof of the Transversality Condition in Proposition 3.74p. 491
The Infinite Horizon Transversality Condition Revisitedp. 492
Monotonicity of the Solution Pathp. 494
Admissible and Quasi-Admissible Directionsp. 496
Proof of the Envelope Theoremp. 498
The Dimension of the Stable Manifoldp. 499
Asymptotic Boundary Conditionp. 502
Equilibriump. 502
Limit Cyclep. 503
Referencesp. 505
Glossaryp. 531
Indexp. 535
Author Indexp. 545
Table of Contents provided by Ingram. All Rights Reserved.

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