Differential Games: A Concise Introduction

A Concise Introduction

Jiongmin Yong

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814596244
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This book uses a small volume to present the most basic results for deterministic two-person differential games. The presentation begins with optimization of a single function, followed by a basic theory for two-person games. For dynamic situations, the author first recalls control theory which is treated as single-person differential games. Then a systematic theory of two-person differential games is concisely presented, including evasion and pursuit problems, zero-sum problems and LQ differential games.

The book is intended to be self-contained, assuming that the readers have basic knowledge of calculus, linear algebra, and elementary ordinary differential equations. The readership of the book could be junior/senior undergraduate and graduate students with majors related to applied mathematics, who are interested in differential games. Researchers in some other related areas, such as engineering, social science, etc. will also find the book useful.

Contents:

• Introduction
• Control Theory — Single-Player Differential Games
• Two-Person Zero Sum Differential Games — A General Consideration
• Differential Games with Unbounded Controls
• Differential Games of Pursuit and Evasion
• Linear-Quadratic Differential Games
• Differential Games with Switching Strategies

Readership: Researchers in optimization and control theory and also game theory and economics. The book will be a supplementary volume to relevant textbooks and/or a good research reference book in certain areas.

Chapter 1

Introduction

1.1 Optimization Problems

Let S be a nonempty set and ℝ be the set of all real numbers. Let f : S → ℝ be a map. We pose the following problem.

Problem (O). Find an

∈ S such that

Such a problem is called a minimization problem. A point

∈ S satisfying (1.1) is called a solution of Problem (O), or equivalently, a minimum of f(·) over S. If such an

∈ S (uniquely) exists, we say that Problem (O) is (uniquely) solvable. Note that we may also pose the following problem.

Problem (O)′. Find an

∈ S such that

We call the above a maximization problem. It is clear that by considering −f(·) instead of f(·), one reduces Problem (O)′ to Problem (O). Therefore, it suffices to just consider Problem (O). One usually refers to a minimization or maximization problem as an optimization problem.

Let us now look at some results on Problem (O). First, let

be a finite set, which is the case, in particular, if S itself is finite. In this case, Problem (O) is solvable, by directly comparing the values in f(S). Theoretically, this is trivial. However, although it will not be pursued further in this book, we point out that, practically, finding a minimum in f(S) might be quite a nontrivial job if the number of elements in set f(S) is very large.

Next, we let f(S) be infinite. Then the existence of a minimum is not gu...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1. Introduction
• 2. Control Theory — Single-Player Differential Games
• 3. Two-Person Zero Sum Differential Games — A General Consideration
• 4. Differential Games with Unbounded Controls
• 5. Differential Games of Pursuit and Evasion
• 6. Linear-Quadratic Differential Games
• 7. Differential Games with Switching Strategies
• Bibliography
• Index