Optimal Control Theory with Applications in Economics

✍ Scribed by Thomas A. Weber

• Publisher: The MIT Press
• Year: 2011
• Tongue: English
• Leaves: 373
• Category: Library

✦ Synopsis

This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. The theory of ordinary differential equations (ODEs) is the backbone of the theory developed in the book, and chapter 2 offers a detailed review of basic concepts in the theory of ODEs, including the solution of systems of linear ODEs, state-space analysis, potential functions, and stability analysis. Following this, the book covers the main results of optimal control theory, in particular necessary and sufficient optimality conditions; game theory, with an emphasis on differential games; and the application of control-theoretic concepts to the design of economic mechanisms. Appendixes provide a mathematical review and full solutions to all end-of-chapter problems. The material is presented at three levels: single-person decision making; games, in which a group of decision makers interact strategically; and mechanism design, which is concerned with a designer's creation of an environment in which players interact to maximize the designer's objective. The book focuses on applications; the problems are an integral part of the text. It is intended for use as a textbook or reference for graduate students, teachers, and researchers interested in applications of control theory beyond its classical use in economic growth. The book will also appeal to readers interested in a modeling approach to certain practical problems involving dynamic continuous-time models.

✦ Table of Contents

Cover......Page 1
Contents......Page 8
Foreword......Page 10
Acknowledgments......Page 12
1- Introduction......Page 14
1.1- Outline......Page 16
1.3- A Brief History of Optimal Control......Page 18
1.4- Notes......Page 28
2.1- Overview......Page 30
2.2.2- Some Explicit Solutions When n = 1......Page 33
2.2.3- Well-Posed Problems......Page 41
2.2.4- State-Space Analysis......Page 52
2.2.5- Exact ODEs and Potential Function......Page 53
2.2.6- Autonomous Systems......Page 57
2.2.7- Stability Analysis......Page 58
2.2.8- Limit Cycles and Invariance......Page 74
2.3.1- Reducing Higher-Order ODEs to First-Order ODEs......Page 80
2.3.2- Solution Techniques......Page 81
2.4- Notes......Page 88
2.5- Exercises......Page 89
3.1- Overview......Page 94
3.2- Control Systems......Page 96
3.2.1- Linear Controllability and Observability......Page 98
3.3.1- A Simple Optimal Control Problem......Page 101
3.3.2- Sufficient Optimality Conditions......Page 102
3.3.3- Necessary Optimality Conditions......Page 108
3.4- Finite-Horizon Optimal Control......Page 116
3.5- Infinite-Horizon Optimal Control......Page 126
3.6- Supplement 1: A Proof of the Pontryagin Maximum Principle......Page 132
3.6.1- Problems without State-Control Constraints......Page 133
3.6.2- Problems with State-Control Constraints......Page 138
3.7- Supplement 2: The Filippov Existence Theorem......Page 148
3.8- Notes......Page 153
3.9- Exercises......Page 154
4.1- Overview......Page 162
4.2.1- Static Games of Complete Information......Page 168
4.2.2- Static Games of Incomplete Information......Page 174
4.2.3- Dynamic Games of Complete Information......Page 179
4.2.4- Dynamic Games of Incomplete Information......Page 193
4.3- Differential Games......Page 201
4.3.1- Markovian Equilibria......Page 202
4.3.2- Non-Markovian Equilibria......Page 209
4.4- Notes......Page 215
4.5- Exercises......Page 216
5.1- Motivation......Page 220
5.2- A Model with Two Types......Page 221
5.3- The Screening Problem......Page 228
5.4- Nonlinear Pricing......Page 233
5.5- Notes......Page 239
5.6- Exercises......Page 240
A.1- Algebra......Page 244
A.2- Normed Vector Spaces......Page 246
A.3- Analysis......Page 253
A.4- Optimization......Page 259
A.5- Notes......Page 264
B.1- Numerical Methods......Page 266
B.2- Ordinary Differential Equations......Page 271
B.3- Optimal Control Theory......Page 284
B.4- Game Theory......Page 315
B.5- Mechanism Design......Page 337
Appendix C: Intellectual Heritage......Page 346
References......Page 348
Index......Page 362