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