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Tables of Contents for Optimal Control Theory
Chapter/Section Title
Page #
Page Count
Preface to First Edition
xiii
Preface to Second Edition
xv
What is Optimal Control Theory
1
22
Basic Concepts and Definitions
2
2
Formulation of Simple Control Models
4
3
History of Optimal Control Theory
7
3
Notation and Concepts Used
10
13
The Maximum Principle: Continuous Time
23
34
Statement of the Problem
23
4
The Mathematical Model
24
1
Constraints
24
1
The Objective Function
25
1
The Optimal Control Problem
25
2
Dynamic Programming and the Maximum Principle
27
9
The Hamilton-Jacobi-Bellman Equation
27
4
Derivation of the Adjoint Equation
31
2
The Maximum Principle
33
1
Economic Interpretations of the Maximum Principle
34
2
Elementary Examples
36
8
Sufficiency Conditions
44
4
Solving a TPBVP by Using Spreadsheet Software
48
9
The Maximum Principle: Mixed Inequality Constraints
57
40
A Maximum Principle for Problems with Mixed Inequality Constraints
58
6
Sufficiency Conditions
64
1
Current-Value Formulation
65
4
Terminal Conditions
69
11
Examples Illustrating Terminal Conditions
74
6
Infinite Horizon and Stationarity
80
3
Model Types
83
14
The Maximum Principle: General Inequality Constraints
97
22
Pure State Variable Inequality Constraints: Indirect Method
98
6
Jump Conditions
103
1
A Maximum Principle: Indirect Method
104
7
Current-Value Maximum Principle: Indirect Method
111
2
Sufficiency Conditions
113
6
Applications to Finance
119
34
The Simple Cash Balance Problem
120
9
The Model
120
1
Solution by the Maximum Principle
121
3
An Extension Disallowing Overdraft and Short-Selling
124
5
Optimal Financing Model
129
24
The Model
129
2
Application of the Maximum Principle
131
2
Synthesis of Optimal Control Paths
133
11
Solution for the Infinite Horizon Problem
144
9
Applications to Production and Inventory
153
32
A Production-Inventory System
154
10
The Production-Inventory Model
154
2
Solution by the Maximum Principle
156
3
The Infinite Horizon Solution
159
1
A Complete Analysis of the Constant Positive S Case with Infinite Horizon
160
2
Special Cases of Time Varying Demands
162
2
Continuous Wheat Trading Model
164
9
The Model
165
1
Solution by the Maximum Principle
166
1
Complete Solution of a Special Case
167
3
The Wheat Trading Model with No Short-Selling
170
3
Decision Horizons and Forecast Horizons
173
12
Horizons for the Wheat Trading Model
174
1
Horizons for the Wheat Trading Model with Warehousing Constraint
175
10
Applications to Marketing
185
32
The Nerlove-Arrow Advertising Model
186
8
The Model
186
2
Solution by the Maximum Principle
188
3
A Nonlinear Extension
191
3
The Vidale-Wolfe Advertising Model
194
23
Optimal Control Formulation for the Vidale-Wolfe Model
195
1
Solution Using Green's Theorem when Q is Large
196
9
Solution When Q Is Small
205
1
Solution When T Is Infinite
206
11
The Maximum Principle: Discrete Time
217
24
Nonlinear Programming Problems
217
11
Lagrange Multipliers
218
2
Inequality Constraints
220
7
Theorems from Nonlinear Programming
227
1
A Discrete Maximum Principle
228
6
A Discrete-Time Optimal Control Problem
228
1
A Discrete Maximum Principle
229
2
Examples
231
3
A General Discrete Maximum Principle
234
7
Maintenance and Replacement
241
26
A Simple Maintenance and Replacement Model
242
6
The Model
242
1
Solution by the Maximum Principle
243
2
A Numerical Example
245
2
An Extension
247
1
Maintenance and Replacement for a Machine Subject to Failure
248
6
The Model
249
2
Optimal Policy
251
2
Determination of the Sale Date
253
1
Chain of Machines
254
13
The Model
254
2
Solution by the Discrete Maximum Principle
256
1
Special Case of Bang-Bang Control
257
1
Incorporation into the Wagner-Whitin Framework for a Complete Solution
258
1
A Numerical Example
259
8
Applications to Natural Resources
267
22
The Sole Owner Fishery Resource Model
268
5
The Dynamics of Fishery Models
268
1
The Sole Owner Model
269
1
Solution by Green's Theorem
270
3
An Optimal Forest Thinning Model
273
6
The Forestry Model
273
1
Determination of Optimal Thinning
274
2
A Chain of Forests Model
276
3
An Exhaustible Resource Model
279
10
Formulation of the Model
279
3
Solution by the Maximum Principle
282
7
Economic Applications
289
18
Models of Optimal Economic Growth
289
6
An Optimal Capital Accumulation Model
290
1
Solution by the Maximum Principle
290
1
A One-Sector Model with a Growing Labor Force
291
1
Solution by the Maximum Principle
292
3
A Model of Optimal Epidemic Control
295
4
Formulation of the Model
295
1
Solution by Green's Theorem
296
3
A Pollution Control Model
299
4
Model Formulation
299
1
Solution by the Maximum Principle
300
1
Phase Diagram Analysis
301
2
Miscellaneous Applications
303
4
Differential Games, Distributed Systems, and Impulse Control
307
32
Differential Games
308
7
Two Person Zero-Sum Differential Games
308
2
Nonzero-Sum Differential Games
310
2
An Application to the Common-Property Fishery Resources
312
3
Distributed Parameter Systems
315
7
The Distributed Parameter Maximum Principle
317
1
The Cattle Ranching Problem
318
4
Interpretation of the Adjoint Function
322
1
Impulse Control
322
17
The Oil Driller's Problem
324
1
The Maximum Principle for Impulse Optimal Control
325
2
Solution of the Oil Driller's Problem
327
4
Machine Maintenance and Replacement
331
1
Application of the Impulse Maximum Principle
332
7
Stochastic Optimal Control
339
24
The Kalman Filter
340
5
Stochastic Optimal Control
345
2
A Stochastic Production Planning Model
347
5
Solution for the Production Planning Problem
350
2
A Stochastic Advertising Problem
352
3
An Optimal Consumption-Investment Problem
355
5
Concluding Remarks
360
3
A Solutions of Linear Differential Equations
363
16
A.1 Linear Differential Equations with Constant Coefficients
363
1
A.2 Homogeneous Equations of Order One
364
1
A.3 Homogeneous Equations of Order Two
364
1
A.4 Homogeneous Equations of Order n
365
1
A.5 Particular Solutions of Linear D.E. with Constant Coefficients
366
2
A.6 Integrating Factor
368
1
A.7 Reduction of Higher-Order Linear Equations to Systems of First-Order Linear Equations
369
3
A.8 Solution of Linear Two-Point Boundary Value Problems
372
1
A.9 Homogeneous Partial Differential Equations
372
2
A.10 Inhomogeneous Partial Differential Equations
374
1
A.11 Solutions of Finite Difference Equations
375
4
A.11.1 Changing Polynomials in Powers of k into Factorial Powers of k
376
1
A.11.2 Changing Factorial Powers of k into Ordinary Powers of k
377
2
B Calculus of Variations and Optimal Control Theory
379
14
B.1 The Simplest Variational Problem
379
1
B.2 The Euler Equation
380
3
B.3 The Shortest Distance Between Two Points on the Plane
383
1
B.4 The Brachistochrone Problem
384
2
B.5 The Weierstrass-Erdmann Corner Conditions
386
2
B.6 Legendre's Conditions: The Second Variation
388
1
B.7 Necessary Condition for a Strong Maximum
389
1
B.8 Relation to the Optimal Control Theory
390
3
C An Alternative Derivation of the Maximum Principle
393
8
C.1 Needle-Shaped Variation
394
2
C.2 Derivation of the Adjoint Equation and the Maximum Principle
396
5
D Special Topics in Optimal Control
401
8
D.1 Linear-Quadratic Problems
401
4
D.1.1 Certainty Equivalence or Separation Principle
403
2
D.2 Second Order Variations
405
2
D.3 Singular Control
407
2
E Answers to Selected Exercises
409
8
Bibliography
417
66
Index
483
18
List of Figures
501
4
List of Tables
505