Preface | p. VII |
Acknowledgments | p. XI |
Background | |
Introduction | p. 3 |
Taking Rocket Science Beyond the Frontiers of Space | p. 3 |
Why Drugs, Corruption, and Terror? | p. 5 |
Questions Optimal Control Can Answer | p. 7 |
Continuous-Time Dynamical Systems | p. 9 |
Nonlinear Dynamical Modeling | p. 9 |
One-Dimensional Systems | p. 10 |
A One-Dimensional Corruption Model | p. 14 |
Dynamical Systems as ODEs | p. 17 |
Concepts and Definitions | p. 19 |
Invariant Sets and Stability | p. 21 |
Structural Stability | p. 25 |
Linearization and the Variational Equation | p. 26 |
Stability Analysis of a One-Dimensional Terror Model | p. 27 |
ODEs in Higher Dimensions | p. 30 |
Autonomous Linear ODEs | p. 31 |
Autonomous Nonlinear ODEs | p. 42 |
Stability Behavior in a Descriptive Model of Drug Demand | p. 51 |
Introduction to Bifurcation Theory | p. 55 |
Terminology and Key Ideas of Bifurcation Theory | p. 56 |
Normal Forms and the Center Manifold: The Tools of Bifurcation Theory | p. 57 |
Local Bifurcations in One Dimension | p. 63 |
Bifurcation Analysis of a One-Dimensional Drug Model | p. 68 |
The Poincare-Andronov-Hopf Bifurcation | p. 71 |
Higher-Dimensional Bifurcation Analysis of a Drug Model | p. 74 |
Advanced Topics | p. 78 |
Stability of Limit Cycles | p. 78 |
Boundary Value Problems | p. 85 |
Exercises | p. 89 |
Notes and Further Reading | p. 96 |
Applied Optimal Control | |
Tour d'Horizon: Optimal Control | p. 101 |
Historical Remarks | p. 101 |
A Standard Optimal Control Problem | p. 104 |
The Maximum Principle of Optimal Control Theory | p. 108 |
Pontryagin's Maximum Principle | p. 108 |
Some General Results | p. 113 |
The Maximum Principle for Variable Terminal Time | p. 115 |
Economic Interpretation of the Maximum Principle | p. 117 |
Sufficiency Conditions | p. 119 |
Existence of an Optimal Solution | p. 122 |
How to Solve an Optimal Control Problem: A Simple Consumption vs. Investment Model | p. 124 |
The Principle of Optimality | p. 127 |
The Hamilton-Jacobi-Bellman Equation | p. 127 |
A Proof of the Maximum Principle | p. 130 |
Singular Optimal Control | p. 131 |
The Most Rapid Approach Path (MRAP) | p. 134 |
An Example From Drug Control that Excludes Singular Arcs | p. 137 |
An Example From Terror Control with an MRAP Solution | p. 139 |
The Maximum Principle With Inequality Constraints | p. 142 |
Mixed Path Constraints | p. 144 |
General Path Constraints | p. 147 |
Sufficiency Conditions | p. 154 |
Infinite Time Horizon | p. 155 |
Definitions of Optimality for Infinite Horizon Problems | p. 155 |
Maximum Principle for Infinite Time Horizon Problems | p. 156 |
Sufficiency Conditions | p. 159 |
Discounted Autonomous Infinite Horizon Models | p. 159 |
The Michel Theorem | p. 160 |
The Ramsey Model for an Infinite Time Horizon | p. 165 |
Structural Results on One-State Discounted, Autonomous Systems | p. 167 |
An Optimal Control Model of a Drug Epidemic | p. 168 |
Model Formulation | p. 168 |
Stability Analysis | p. 170 |
Phase Portrait Analysis | p. 176 |
Exercises | p. 177 |
Notes and Further Reading | p. 183 |
The Path to Deeper Insight: From Lagrange to Pontryagin | p. 189 |
Introductory Remarks on Optimization | p. 189 |
Notational Remarks | p. 190 |
Motivation and Insights | p. 190 |
A Simple Maximization Problem | p. 192 |
Finite-Dimensional Approximation of an Infinite-Dimensional Problem | p. 195 |
Static Maximization | p. 197 |
Basic Theorems and Definitions | p. 198 |
Theory and Geometric Interpretation of Lagrange and Karush-Kuhn-Tucker | p. 202 |
The Envelope Theorem and the Lagrange Multiplier | p. 208 |
The Discrete-Time Maximum Principle as a Static Maximization Problem | p. 210 |
The Calculus of Variations | p. 214 |
A Simple Variational Example | p. 214 |
The First Variation | p. 216 |
Deriving the Euler Equation and Weierstrass-Erdmann Conditions | p. 218 |
Proving the Continuous-Time Maximum Principle | p. 223 |
The Continuous-Time Maximum Principle Revisited | p. 223 |
Necessary Conditions at Junction Points | p. 227 |
Exercises | p. 231 |
Notes and Further Reading | p. 234 |
Multiple Equilibria, Points of Indifference, and Thresholds | p. 237 |
Occurrence of Multiple Equilibria | p. 238 |
The Optimal Vector Field | p. 239 |
Finite vs. Infinite Time Horizon Models | p. 239 |
Discounted Autonomous Models for an Infinite Time Horizon | p. 243 |
A Typical Example | p. 244 |
Existence and Stability of the Equilibria | p. 245 |
Determining the Optimal Vector Field and the Optimal Costate Rule | p. 247 |
Defining Indifference and DNSS Points | p. 252 |
Multiplicity and Separability | p. 253 |
Definitions | p. 254 |
Conclusions from the Definitions | p. 256 |
Revisiting the Typical Example | p. 260 |
Eradication vs. Accommodation in an Optimal Control Model of a Drug Epidemic | p. 266 |
Exercises | p. 269 |
Notes and Further Reading | p. 272 |
Advanced Topics | |
Higher-Dimensional Models | p. 279 |
Controlling Drug Consumption | p. 280 |
Model of Controlled Drug Demand | p. 280 |
Deriving the Canonical System | p. 283 |
The Endemic Level of Drug Demand | p. 286 |
Optimal Dynamic Policy away from the Endemic State | p. 287 |
Optimal Policies for Different Phases of a Drug Epidemic | p. 292 |
Corruption in Governments Subject to Popularity Constraints | p. 296 |
The Modeled Incentive for Being Corrupt | p. 297 |
Optimality Conditions | p. 299 |
Insights About the Incentive to Be Corrupt | p. 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 Terrorism | p. 309 |
Derivation of the Canonical System | p. 310 |
Numerical Calculations | p. 311 |
Optimal Strategy for a Small Terror Organization | p. 314 |
Exercises | p. 316 |
Notes and Further Reading | p. 323 |
Numerical Methods for Discounted Systems of Infinite Horizon | p. 327 |
General Remarks | p. 327 |
Problem Formulation and Assumptions | p. 328 |
Notation | p. 329 |
Numerical Methods for Solving Optimal Control Problems | p. 330 |
Boundary Value Problems from Optimal Control | p. 330 |
Numerical Continuation | p. 332 |
Continuation Algorithms | p. 333 |
Continuing the Solution of a BVP | p. 338 |
The Canonical System Without Active Constraints | p. 342 |
Calculating Long-Run Optimal Solutions | p. 343 |
Equilibria | p. 344 |
Limit Cycles | p. 346 |
Continuing the Optimal Solution: Calculating the Stable Manifold | p. 349 |
Stable Manifold of an Equilibrium | p. 350 |
Stable Manifold of Limit Cycles | p. 354 |
Optimal Control Problems with Active Constraints | p. 359 |
The Form of the Canonical System for Mixed Path Constraints | p. 360 |
The Form of the Canonical System for Pure State Constraints | p. 360 |
Solutions Exhibiting Junction Points | p. 362 |
Retrieving DNSS Sets | p. 366 |
Locating a DNSS Point | p. 366 |
Continuing a DNSS Point | p. 368 |
Retrieving Heteroclinic Connections | p. 368 |
Locating a Heteroclinic Connection | p. 368 |
Continuing a Heteroclinic Connection in Parameter Space | p. 369 |
Numerical Example from Drug Control | p. 370 |
Stating the Necessary Conditions | p. 370 |
Equilibria of the Canonical System | p. 372 |
Numerical Analysis | p. 372 |
Optimal Vector Field for v = 4,000 | p. 373 |
Optimal Vector Field for v = 12,000 | p. 377 |
Exercises | p. 380 |
Notes and Further Reading | p. 382 |
Extensions of the Maximum Principle | p. 385 |
Multi-Stage Optimal Control Problems | p. 386 |
Necessary Optimality Conditions for Two-Stage Control Problems | p. 386 |
Two-Stage Models of Drug Control | p. 387 |
Counter-Terror Measures in a Multi-Stage Scenario | p. 388 |
Differential Games | p. 391 |
Terminology | p. 392 |
Nash Equilibria | p. 394 |
Tractable Game Structures | p. 397 |
A Corrupt Politician vs. the Tabloid Press | p. 397 |
Leader-Follower Games | p. 404 |
A Post September 11th Game on Terrorism | p. 407 |
Age-Structured Models | p. 417 |
A Maximum Principle for Distributed Parameter Systems | p. 419 |
Age-Structured Drug Initiation | p. 420 |
Further Optimal Control Issues | p. 422 |
Delayed Systems | p. 422 |
Stochastic Optimal Control | p. 424 |
Impulse Control and Jumps in the State Variables | p. 425 |
Nonsmooth Systems | p. 426 |
Exercises | p. 426 |
Notes and Further Reading | p. 436 |
Appendices | |
Mathematical Background | p. 443 |
General Notation and Functions | p. 443 |
Finite-Dimensional Vector Spaces | p. 447 |
Vector Spaces, Linear Dependence, and Basis | p. 447 |
Linear Transformations and Matrices | p. 450 |
Inverse Matrices and Linear Equations | p. 453 |
Determinants | p. 455 |
Linear Form and Dual Space | p. 457 |
Eigenvalues and Eigenvectors | p. 459 |
Euclidean Vector Space R[superscript n] | p. 461 |
Topology and Calculus | p. 463 |
Open Set, Neighborhood, and Convergence | p. 463 |
Continuity and Differentiability | p. 464 |
Maximization of Real-Valued Functions in R[superscript n] | p. 471 |
Convex Analysis | p. 473 |
Taylor Theorem and Implicit Function Theorem | p. 475 |
Integration Theory | p. 477 |
Distributions | p. 481 |
Derivations and Proofs of Technical Results | p. 483 |
Separation Theorems, Farkas Lemma and Supergradient | p. 483 |
Proof of the Michel Theorem | p. 486 |
Augmented and Truncated Problem | p. 487 |
Optimal Solution of Problem (B.8) | p. 487 |
Necessary Conditions for Problem (B.8) | p. 488 |
Limit of Solutions for Increasing Time Sequence | p. 489 |
Proof of the Transversality Condition in Proposition 3.74 | p. 491 |
The Infinite Horizon Transversality Condition Revisited | p. 492 |
Monotonicity of the Solution Path | p. 494 |
Admissible and Quasi-Admissible Directions | p. 496 |
Proof of the Envelope Theorem | p. 498 |
The Dimension of the Stable Manifold | p. 499 |
Asymptotic Boundary Condition | p. 502 |
Equilibrium | p. 502 |
Limit Cycle | p. 503 |
References | p. 505 |
Glossary | p. 531 |
Index | p. 535 |
Author Index | p. 545 |
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