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El. knyga: Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror

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  • Formatas: PDF+DRM
  • Išleidimo metai: 24-Jul-2008
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540776475
Kitos knygos pagal šią temą:
Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and TerrorDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Išleidimo metai: 24-Jul-2008
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540776475
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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.

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. 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. The authors celebrate Pontryagin's Maximum Principle - that crowning intellectual achievement of human understanding. Yet they go further, pushing 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.

Recenzijos

From the reviews:









"This textbook on optimal control of nonlinear processes introduces several innovations compared to standard textbooks on this topic. The software used is available through a companion website. The book can be highly recommended to students, teachers, and researchers interested in optimal control." (Walter Alt, Zentralblatt MATH, Vol. 1149, 2008)

Preface vii
Acknowledgments xi
Part I Background
Introduction
3(6)
Taking Rocket Science Beyond the Frontiers of Space
3(2)
Why Drugs, Corruption, and Terror?
5(2)
Questions Optimal Control Can Answer
7(2)
Continuous-Time Dynamical Systems
9(92)
Nonlinear Dynamical Modeling
9(1)
One-Dimensional Systems
10(4)
A One-Dimensional Corruption Model
14(3)
Dynamical Systems as ODEs
17(10)
Concepts and Definitions
19(2)
Invariant Sets and Stability
21(4)
Structural Stability
25(1)
Linearization and the Variational Equation
26(1)
Stability Analysis of a One-Dimensional Terror Model
27(3)
ODEs in Higher Dimensions
30(21)
Autonomous Linear ODEs
31(11)
Autonomous Nonlinear ODEs
42(9)
Stability Behavior in a Descriptive Model of Drug Demand
51(4)
Introduction to Bifurcation Theory
55(13)
Terminology and Key Ideas of Bifurcation Theory
56(1)
Normal Forms and the Center Manifold: The Tools of Bifurcation Theory
57(6)
Local Bifurcations in One Dimension
63(5)
Bifurcation Analysis of a One-Dimensional Drug Model
68(3)
The Poincare-Andronov-Hopf Bifurcation
71(3)
Higher-Dimensional Bifurcation Analysis of a Drug Model
74(4)
Advanced Topics
78(23)
Stability of Limit Cycles
78(7)
Boundary Value Problems
85(4)
Exercises
89(7)
Notes and Further Reading
96(5)
Part II Applied Optimal Control
Tour d'Horizon: Optimal Control
101(88)
Historical Remarks
101(3)
A Standard Optimal Control Problem
104(4)
The Maximum Principle of Optimal Control Theory
108(19)
Pontryagin's Maximum Principle
108(5)
Some General Results
113(2)
The Maximum Principle for Variable Terminal Time
115(2)
Economic Interpretation of the Maximum Principle
117(2)
Sufficiency Conditions
119(3)
Existence of an Optimal Solution
122(2)
How to Solve an Optimal Control Problem: A Simple Consumption vs. Investment Model
124(3)
The Principle of Optimality
127(4)
The Hamilton-Jacobi-Bellman Equation
127(3)
A Proof of the Maximum Principle
130(1)
Singular Optimal Control
131(11)
The Most Rapid Approach Path (MRAP)
134(3)
An Example From Drug Control that Excludes Singular Arcs
137(2)
An Example From Terror Control with an MRAP Solution
139(3)
The Maximum Principle With Inequality Constraints
142(13)
Mixed Path Constraints
144(3)
General Path Constraints
147(7)
Sufficiency Conditions
154(1)
Infinite Time Horizon
155(4)
Definitions of Optimality for Infinite Horizon Problems
155(1)
Maximum Principle for Infinite Time Horizon Problems
156(3)
Sufficiency Conditions
159(1)
Discounted Autonomous Infinite Horizon Models
159(9)
The Michel Theorem
160(5)
The Ramsey Model for an Infinite Time Horizon
165(2)
Structural Results on One-State Discounted, Autonomous Systems
167(1)
An Optimal Control Model of a Drug Epidemic
168(21)
Model Formulation
168(2)
Stability Analysis
170(6)
Phase Portrait Analysis
176(1)
Exercises
177(6)
Notes and Further Reading
183(6)
The Path to Deeper Insight: From Lagrange to Pontryagin
189(48)
Introductory Remarks on Optimization
189(8)
Notational Remarks
190(1)
Motivation and Insights
190(2)
A Simple Maximization Problem
192(3)
Finite-Dimensional Approximation of an Infinite-Dimensional Problem
195(2)
Static Maximization
197(17)
Basic Theorems and Definitions
198(4)
Theory and Geometric Interpretation of Lagrange and Karush-Kuhn-Tucker
202(6)
The Envelope Theorem and the Lagrange Multiplier
208(2)
The Discrete-Time Maximum Principle as a Static Maximization Problem
210(4)
The Calculus of Variations
214(9)
A Simple Variational Example
214(2)
The First Variation
216(2)
Deriving the Euler Equation and Weierstrass-Erdmann Conditions
218(5)
Proving the Continuous-Time Maximum Principle
223(14)
The Continuous-Time Maximum Principle Revisited
223(4)
Necessary Conditions at Junction Points
227(4)
Exercises
231(3)
Notes and Further Reading
234(3)
Multiple Equilibria, Points of Indifference, and Thresholds
237(42)
Occurrence of Multiple Equilibria
238(1)
The Optimal Vector Field
239(5)
Finite vs. Infinite Time Horizon Models
239(4)
Discounted Autonomous Models for an Infinite Time Horizon
243(1)
A Typical Example
244(8)
Existence and Stability of the Equilibria
245(2)
Determining the Optimal Vector Field and the Optimal Costate Rule
247(5)
Defining Indifference and DNSS Points
252(8)
Multiplicity and Separability
253(1)
Definitions
254(2)
Conclusions from the Definitions
256(4)
Revisiting the Typical Example
260(6)
Eradication vs. Accommodation in an Optimal Control Model of a Drug Epidemic
266(13)
Exercises
269(3)
Notes and Further Reading
272(7)
Part III Advanced Topics
Higher-Dimensional Models
279(48)
Controlling Drug Consumption
280(16)
Model of Controlled Drug Demand
280(3)
Deriving the Canonical System
283(3)
The Endemic Level of Drug Demand
286(1)
Optimal Dynamic Policy away from the Endemic State
287(5)
Optimal Policies for Different Phases of a Drug Epidemic
292(4)
Corruption in Governments Subject to Popularity Constraints
296(12)
The Modeled Incentive for Being Corrupt
297(2)
Optimality Conditions
299(1)
Insights About the Incentive to Be Corrupt
300(2)
Is Periodic Behavior Caused by Rational Optimization?
302(6)
Is It Important to Manage Public Opinion While Fighting Terrorism?
308(19)
What One Should Know when Fighting Terrorism
309(1)
Derivation of the Canonical System
310(1)
Numerical Calculations
311(3)
Optimal Strategy for a Small Terror Organization
314(2)
Exercises
316(7)
Notes and Further Reading
323(4)
Numerical Methods for Discounted Systems of Infinite Horizon
327(58)
General Remarks
327(5)
Problem Formulation and Assumptions
328(1)
Notation
329(1)
Numerical Methods for Solving Optimal Control Problems
330(1)
Boundary Value Problems from Optimal Control
330(2)
Numerical Continuation
332(10)
Continuation Algorithms
333(5)
Continuing the Solution of a BVP
338(4)
The Canonical System Without Active Constraints
342(1)
Calculating Long-Run Optimal Solutions
343(6)
Equilibria
344(2)
Limit Cycles
346(3)
Continuing the Optimal Solution: Calculating the Stable Manifold
349(10)
Stable Manifold of an Equilibrium
350(4)
Stable Manifold of Limit Cycles
354(5)
Optimal Control Problems with Active Constraints
359(7)
The Form of the Canonical System for Mixed Path Constraints
360(1)
The Form of the Canonical System for Pure State Constraints
360(2)
Solutions Exhibiting Junction Points
362(4)
Retrieving DNSS Sets
366(2)
Locating a DNSS Point
366(2)
Continuing a DNSS Point
368(1)
Retrieving Heteroclinic Connections
368(2)
Locating a Heteroclinic Connection
368(1)
Continuing a Heteroclinic Connection in Parameter Space
369(1)
Numerical Example from Drug Control
370(15)
Stating the Necessary Conditions
370(2)
Equilibria of the Canonical System
372(1)
Numerical Analysis
372(1)
Optimal Vector Field for ν = 4,000
373(4)
Optimal Vector Field for ν = 12,000
377(3)
Exercises
380(2)
Notes and Further Reading
382(3)
Extensions of the Maximum Principle
385(120)
Multi-Stage Optimal Control Problems
386(5)
Necessary Optimality Conditions for Two-Stage Control Problems
386(1)
Two-Stage Models of Drug Control
387(1)
Counter-Terror Measures in a Multi-Stage Scenario
388(3)
Differential Games
391(26)
Terminology
392(2)
Nash Equilibria
394(3)
Tractable Game Structures
397(1)
A Corrupt Politician vs. the Tabloid Press
397(7)
Leader-Follower Games
404(3)
A Post September 11th Game on Terrorism
407(10)
Age-Structured Models
417(5)
A Maximum Principle for Distributed Parameter Systems
419(1)
Age-Structured Drug Initiation
420(2)
Further Optimal Control Issues
422(21)
Delayed Systems
422(2)
Stochastic Optimal Control
424(1)
Impulse Control and Jumps in the State Variables
425(1)
Nonsmooth Systems
426(1)
Exercises
426(10)
Notes and Further Reading
436(7)
Part IV Appendices
A Mathematical Background
443(40)
A.1 General Notation and Functions
443(4)
A.2 Finite-Dimensional Vector Spaces
447(1)
A.2.1 Vector Spaces, Linear Dependence, and Basis
447(3)
A.2.2 Linear Transformations and Matrices
450(3)
A.2.3 Inverse Matrices and Linear Equations
453(2)
A.2.4 Determinants
455(2)
A.2.5 Linear Form and Dual Space
457(2)
A.2.6 Eigenvalues and Eigenvectors
459(2)
A.2.7 Euclidean Vector Space Rn
461(2)
A.3 Topology and Calculus
463(1)
A.3.1 Open Set, Neighborhood, and Convergence
463(1)
A.3.2 Continuity and Differentiability
464(7)
A.3.3 Maximization of Real-Valued Functions in Rn
471(2)
A.3.4 Convex Analysis
473(2)
A.3.5 Taylor Theorem and Implicit Function Theorem
475(2)
A.3.6 Integration Theory
477(4)
A.3.7 Distributions
481(2)
B Derivations and Proofs of Technical Results
483(22)
B.1 Separation Theorems, Farkas Lemma and Supergradient
483(3)
B.2 Proof of the Michel Theorem
486(1)
B.2.1 Augmented and Truncated Problem
487(1)
B.2.2 Optimal Solution of Problem (B.8)
487(1)
B.2.3 Necessary Conditions for Problem (B.8)
488(1)
B.2.4 Limit of Solutions for Increasing Time Sequence
489(2)
B.3 Proof of the Transversality Condition in Proposition 3.74
491(1)
B.4 The Infinite Horizon Transversality Condition Revisited
492(2)
B.5 Monotonicity of the Solution Path
494(2)
B.6 Admissible and Quasi-Admissible Directions
496(2)
B.7 Proof of the Envelope Theorem
498(1)
B.8 The Dimension of the Stable Manifold
499(3)
B.9 Asymptotic Boundary Condition
502(1)
B.9.1 Equilibrium
502(1)
B.9.2 Limit Cycle
503(2)
References 505(26)
Glossary 531(4)
Index 535(10)
Author Index 545
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