Optimal Control Theory: Applications to Management Science and Economics

Preface to Fourth Edition
Preface to Third Edition
Preface to Second Edition
Preface to First Edition
Biography
Contents
List of Figures
List of Tables
1 What Is Optimal Control Theory?
1.1 Basic Concepts and Definitions
1.2 Formulation of Simple Control Models
1.3 History of Optimal Control Theory
1.4 Notation and Concepts Used
1.4.1 Differentiating Vectors and Matrices with Respect to Scalars
1.4.2 Differentiating Scalars with Respect to Vectors
1.4.3 Differentiating Vectors with Respect to Vectors
1.4.4 Product Rule for Differentiation
1.4.5 Miscellany
1.4.6 Convex Set and Convex Hull
1.4.7 Concave and Convex Functions
1.4.8 Affine Function and Homogeneous Function of Degree k
1.4.9 Saddle Point
1.4.10 Linear Independence and Rank of a Matrix
1.5 Plan of the Book
Exercises for Chap.1
References
2 The Maximum Principle: Continuous Time
2.1 Statement of the Problem
2.1.1 The Mathematical Model
2.1.2 Constraints
2.1.3 The Objective Function
2.1.4 The Optimal Control Problem
2.2 Dynamic Programming and the Maximum Principle
2.2.1 The Hamilton-Jacobi-Bellman Equation
2.2.2 Derivation of the Adjoint Equation
2.2.3 The Maximum Principle
2.2.4 Economic Interpretations of the Maximum Principle
2.3 Simple Examples
2.4 Sufficiency Conditions
2.5 Solving a TPBVP by Using Excel
Exercises for Chap.2
References
3 The Maximum Principle: Mixed Inequality Constraints
3.1 A Maximum Principle for Problems with Mixed Inequality Constraints
3.2 Sufficiency Conditions
3.3 Current-Value Formulation
3.4 Transversality Conditions: Special Cases
3.5 Free Terminal Time Problems
3.6 Infinite Horizon and Stationarity
3.7 Model Types
Exercises for Chap.3
References
4 The Maximum Principle: Pure State and Mixed Inequality Constraints
4.1 Jumps in Marginal Valuations
4.2 The Optimal Control Problem with Pure and Mixed Constraints
4.3 The Maximum Principle: Direct Method
4.4 Sufficiency Conditions: Direct Method
4.5 The Maximum Principle: Indirect Method
4.6 Current-Value Maximum Principle: Indirect Method
Exercises for Chap.4
References
5 Applications to Finance
5.1 The Simple Cash Balance Problem
5.1.1 The Model
5.1.2 Solution by the Maximum Principle
5.2 Optimal Financing Model
5.2.1 The Model
5.2.2 Application of the Maximum Principle
5.2.3 Synthesis of Optimal Control Paths
5.2.4 Solution for the Infinite Horizon Problem
Exercises for Chap.5
References
6 Applications to Production and Inventory
6.1 Production-Inventory Systems
6.1.1 The Production-Inventory Model
6.1.2 Solution by the Maximum Principle
6.1.3 The Infinite Horizon Solution
6.1.4 Special Cases of Time Varying Demands
6.1.5 Optimality of a Linear Decision Rule
6.1.6 Analysis with a Nonnegative Production Constraint
6.2 The Wheat Trading Model
6.2.1 The Model
6.2.2 Solution by the Maximum Principle
6.2.3 Solution of a Special Case
6.2.4 The Wheat Trading Model with No Short-Selling
6.3 Decision Horizons and Forecast Horizons
6.3.1 Horizons for the Wheat Trading Model with No Short-Selling
6.3.2 Horizons for the Wheat Trading Model with No Short-Selling and a Warehousing Constraint
Exercises for Chap.6
References
7 Applications to Marketing
7.1 The Nerlove-Arrow Advertising Model
7.1.1 The Model
7.1.2 Solution by the Maximum Principle
7.1.3 Convex Advertising Cost and Relaxed Controls
7.2 The Vidale-Wolfe Advertising Model
7.2.1 Optimal Control Formulation for the Vidale-Wolfe Model
7.2.2 Solution Using Green's Theorem When Q Is Large
7.2.3 Solution When Q Is Small
7.2.4 Solution When T Is Infinite
Exercises for This Chap.7
References
8 The Maximum Principle: Discrete Time
8.1 Nonlinear Programming Problems
8.1.1 Lagrange Multipliers
8.1.2 Equality and Inequality Constraints
8.1.3 Constraint Qualification
8.1.4 Theorems from Nonlinear Programming
8.2 A Discrete Maximum Principle
8.2.1 A Discrete-Time Optimal Control Problem
8.2.2 A Discrete Maximum Principle
8.2.3 Examples
8.3 A General Discrete Maximum Principle
Exercises for Chap.8
References
9 Maintenance and Replacement
9.1 A Simple Maintenance and Replacement Model
9.1.1 The Model
9.1.2 Solution by the Maximum Principle
9.1.3 A Numerical Example
9.1.4 An Extension
9.2 Maintenance and Replacement for a Machine Subject to Failure
9.2.1 The Model
9.2.2 Optimal Policy
9.2.3 Determination of the Sale Date
9.3 Chain of Machines
9.3.1 The Model
9.3.2 Solution by the Discrete Maximum Principle
9.3.3 Special Case of Bang-Bang Control
9.3.4 Incorporation into the Wagner-Whitin Framework for a Complete Solution
9.3.5 A Numerical Example
Exercises for Chap.9
References
10 Applications to Natural Resources
10.1 The Sole-Owner Fishery Resource Model
10.1.1 The Dynamics of Fishery Models
10.1.2 The Sole Owner Model
10.1.3 Solution by Green's Theorem
10.2 An Optimal Forest Thinning Model
10.2.1 The Forestry Model
10.2.2 Determination of Optimal Thinning
10.2.3 A Chain of Forests Model
10.3 An Exhaustible Resource Model
10.3.1 Formulation of the Model
10.3.2 Solution by the Maximum Principle
Exercises for Chap.10
References
11 Applications to Economics
11.1 Models of Optimal Economic Growth
11.1.1 An Optimal Capital Accumulation Model
11.1.2 Solution by the Maximum Principle
11.1.3 Introduction of a Growing Labor Force
11.1.4 Solution by the Maximum Principle
11.2 A Model of Optimal Epidemic Control
11.2.1 Formulation of the Model
11.2.2 Solution by Green's Theorem
11.3 A Pollution Control Model
11.3.1 Model Formulation
11.3.2 Solution by the Maximum Principle
11.3.3 Phase Diagram Analysis
11.4 An Adverse Selection Model
11.4.1 Model Formulation
11.4.2 The Implementation Problem
11.4.3 The Optimization Problem
11.5 Miscellaneous Applications
Exercises for Chap.11
References
12 Stochastic Optimal Control
12.1 Stochastic Optimal Control
12.2 A Stochastic Production Inventory Model
12.2.1 Solution for the Production Planning Problem
12.3 The Sethi Advertising Model
12.4 An Optimal Consumption-Investment Problem
12.5 Concluding Remarks
Exercises for Chap.12
References
13 Differential Games
13.1 Two-Person Zero-Sum Differential Games
13.2 Nash Differential Games
13.2.1 Open-Loop Nash Solution
13.2.2 Feedback Nash Solution
13.2.3 An Application to Common-Property Fishery Resources
13.3 A Feedback Nash Stochastic Differential Game in Advertising
13.4 A Feedback Stackelberg Stochastic Differential Game of Cooperative Advertising
Exercises for Chap.13
References
A Solutions of Linear Differential Equations
A.1 First-Order Linear Equations
A.2 Second-Order Linear Equations with Constant Coefficients
A.3 System of First-Order Linear Equations
A.4 Solution of Linear Two-Point Boundary Value Problems
A.5 Solutions of Finite Difference Equations
A.5.1 Changing Polynomials in Powers of k into Factorial Powers of k
A.5.2 Changing Factorial Powers of k into Ordinary Powers of k
B Calculus of Variations and Optimal Control Theory
B.1 The Simplest Variational Problem
B.2 The Euler-Lagrange Equation
B.3 The Shortest Distance Between Two Points on the Plane
B.4 The Brachistochrone Problem
B.5 The Weierstrass-Erdmann Corner Conditions
B.6 Legendre's Conditions: The Second Variation
B.7 Necessary Condition for a Strong Maximum
B.8 Relation to Optimal Control Theory
C An Alternative Derivation of the Maximum Principle
C.1 Needle-Shaped Variation
C.2 Derivation of the Adjoint Equation and the Maximum Principle
D Special Topics in Optimal Control
D.1 The Kalman Filter
D.2 Wiener Process and Stochastic Calculus
D.3 The Kalman-Bucy Filter
D.4 Linear-Quadratic Problems
D.4.1 Certainty Equivalence or Separation Principle
D.5 Second-Order Variations
D.6 Singular Control
D.7 Global Saddle Point Theorem
D.8 The Sethi-Skiba Points
D.9 Distributed Parameter Systems
E Answers to Selected Exercises
Bibliography
Index