**Graduate Studies in Mathematics**

Volume: 41;
2002;
226 pp;
Hardcover

MSC: Primary 34; 49;
Secondary 93

Print ISBN: 978-0-8218-2977-6

Product Code: GSM/41

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

**Electronic ISBN: 978-1-4704-2092-5
Product Code: GSM/41.E**

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

# Introduction to the Theory of Differential Inclusions

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*Georgi V. Smirnov*

A differential inclusion is a relation of the form \(\dot x \in
F(x)\), where \(F\) is a set-valued map associating any point \(x
\in R^n\) with a set \(F(x) \subset R^n\). As such, the notion of
a differential inclusion generalizes the notion of an ordinary differential
equation of the form \(\dot x = f(x)\). Therefore, all problems usually
studied in the theory of ordinary differential equations (existence and
continuation of solutions, dependence on initial conditions and parameters,
etc.) can be studied for differential inclusions as well. Since a differential
inclusion usually has many solutions starting at a given point, new types of
problems arise, such as investigation of topological properties of the set of
solutions, selection of solutions with given properties, and many others.

Differential inclusions play an important role as a tool in the study of
various dynamical processes described by equations with a discontinuous or
multivalued right-hand side, occurring, in particular, in the study of dynamics
of economical, social, and biological macrosystems. They also are very useful in
proving existence theorems in control theory.

This text provides an introductory treatment to the theory of differential
inclusions. The reader is only required to know ordinary differential
equations, theory of functions, and functional analysis on the elementary
level.

Chapter 1 contains a brief introduction to convex analysis. Chapter 2
considers set-valued maps. Chapter 3 is devoted to the Mordukhovich version of
nonsmooth analysis. Chapter 4 contains the main existence theorems and gives
an idea of the approximation techniques used throughout the text. Chapter 5 is
devoted to the viability problem, i.e., the problem of selection of a solution
to a differential inclusion that is contained in a given set. Chapter 6 considers
the controllability problem. Chapter 7 discusses extremal problems for
differential inclusions. Chapter 8 presents stability theory, and Chapter 9
deals with the stabilization problem.

#### Readership

Graduate students and research mathematicians interested in ordinary differential equations, calculus of variations, optimal control, and optimization.

#### Reviews & Endorsements

The material of the book may very well be used for an introductory lecture on differential inclusions.

-- Jahresbericht der DMV

#### Table of Contents

# Table of Contents

## Introduction to the Theory of Differential Inclusions

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xi12 free
- Introduction xiii14 free
- Part 1. Foundations 118 free
- Part 2. Differential Inclusions 85102
- Chapter 4. Existence Theorems 87104
- §4.1. Background notes 88105
- §4.2. Lipschitzian differential inclusions 90107
- §4.3. Upper semi-continuous differential inclusions 96113
- §4.4. Discontinuous differential equations 103120
- §4.5. Existence of optimal solutions 106123
- §4.6. Dependence on initial conditions 109126
- §4.7. Discrete approximations 113130
- §4.8. Problems 116133

- Chapter 5. Viability and Invariance 119136
- Chapter 6. Controllability 139156
- Chapter 7. Optimality 157174
- Chapter 8. Stability 171188
- Chapter 9. Stabilization 199216

- Comments 213230
- Bibliography 219236
- Index 225242
- Back Cover Back Cover1245