Optimal Control Theory: Applications to Management Science and Economics
Written by Sethi, Suresh P.; Thompson, Gerald L.
Published by Springer in 2006
Cover
Contents
Preface to First Edition
Preface to Second Edition
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
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 Maximimi Principle
2.3 Elementary Examples
2.4 Sufficiency Conditions
2.5 Solving a TPBVP by Using Spreadsheet Software
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 Terminal Conditions
3.4.1 Examples Illustrating Terminal Conditions
3.5 Infinite Horizon and Stationarity
3.6 Model Types
4 The Maximum Principle: General Inequality C!onstraints
4.1 Pure State Variable Inequality Constraints: Indirect Method
4.1.1 Jimip Conditions
4.2 A Maximimi Principle: Indirect Method
4.3 Current-Value Maximum Principle: Indirect Method
4.4 Sufficiency Conditions
5 Applications to Finance
5.1 The Simple Cash Balance Problem
5.1.1 The Model
5.1.2 Solution by the Maximum Principle
5.1.3 An Extension Disallowing Overdraft and Short-Selling
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
6 Applications to Production and Inventory
6.1 A Production-Inventory System
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 A Complete Analysis of the Constant Positive S Case with Infinite Horizon
6.1.5 Special Cases of Time Varying Demands
6.2 Continuous Wheat Trading Model
6.2.1 The Model
6.2.2 Solution by the Maximum Principle
6.2.3 Complete 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
6.3.2 Horizons for the Wheat Trading Model with Warehousing Constraint
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 A Nonlinear Extension
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
8 The Maximum Principle: Discrete Time
8.1 Nonlinear Programming Problems
8.1.1 Lagrange Multipliers
8.1.2 Inequahty Constraints
8.1.3 Theorems from Nonlinear Programming
8.2 A Discrete Maximimi 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
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 Nimierical 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 Nimierical Example
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
11 Economic Applications
11.1 Models of Optimal Economic Growth
11.1.1 An Optimal Capital Accumulation Model
11.1.2 Solution by the Maximimi Principle
11.1.3 A One-Sector Model with 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 Maxim-um Principle
11.3.3 Phase Diagram Analysis
11.4 Miscellaneous Applications
12 Differential Games, Distributed Systems, and Impulse Control
12.1 Differential Games
12.1.1 Two Person Zero-Sum Differential Games
12.1.2 Nonzero-Simm Differential Games
12.1.3 An Application to the Common-Property Fishery Resources
12.2 Distributed Parameter Systems
12.2.1 The Distributed Parameter Maximum Principle
12.2.2 The Cattle Ranching Problem
12.2.3 Interpretation of the Adjoint Function
12.3 Impulse Control
12.3.1 The Oil Driller's Problem
12.3.2 The Maximimi Principle for Impulse Optimal Control
12.3.3 Solution of the Oil Driller's Problem
12.3.4 Machine Maintenance and Replacement
12.3.5 Application of the Impulse Maximum Principle
13 Stochastic Optimal Control
13.1 The Kahnan Filter
13.2 Stochastic Optimal Control
13.3 A Stochastic Production Planning Model
13.3.1 Solution for the Production Planning Problem
13.4 A Stochastic Advertising Problem
13.5 An Optimal Consumption-Investment Problem
13.6 Concluding Remarks
A: Solutions of Linear Differential Equations
A.1 Linear Differential Equations with Constant Coefficients
A.2 Homogeneous Equations of Order One
A.3 Homogeneous Equations of Order Two
A.4 Homogeneous Equations of Order η
A.5 Particular Solutions of Linear D.E. with Constant Coefficients
A.6 Integrating Factor
A.7 Reduction of Higher-Order Linear Equations to Systems of First-Order Linear Equations
A.8 Solution of Linear Two-Point Boundary Value Problems
A.9 Homogeneous Partial Differential Equations
A.10 Inhomogeneous Partial Differential Equations
A.11 Solutions of Finite Difference Equations
A.11.1Changing Polynomials in Powers of κ into Factorial Powers of κ
A. 11.2 Changing Factorial Powers of k into Ordinary Powers of κ
B: Calculus of Variations and Optimal Control Theory
B.1 The Simplest Variational Problem
B.2 The Euler 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 the 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 Linear-Quadratic Problems
D.1.1 Certainty Equivalence or Separation Principle
D.2 Second-Order Variations
D.3 Singular Control
E: Answers to Selected Exercises
Bibliography
Index
List of Figures
List of Tables
Last Page
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