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Softcover ISBN:  9781470468545 
Product Code:  GSM/41.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470420925 
Product Code:  GSM/41.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470468545 
eBook ISBN:  9781470420925 
Product Code:  GSM/41.S.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsGraduate Studies in MathematicsVolume: 41; 2002; 226 ppMSC: Primary 34; 49; Secondary 93;
A differential inclusion is a relation of the form \(\dot x \in F(x)\), where \(F\) is a setvalued 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 righthand 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 setvalued 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.ReadershipGraduate students and research mathematicians interested in ordinary differential equations, calculus of variations, optimal control, and optimization.

Table of Contents

Part 1. Foundations

Chapter 1. Convex analysis

Chapter 2. Setvalued analysis

Chapter 3. Nonsmooth analysis

Part 2. Differential inclusions

Chapter 4. Existence theorems

Chapter 5. Viability and invariance

Chapter 6. Controllability

Chapter 7. Optimality

Chapter 8. Stability

Chapter 9. Stabilization


Additional Material

Reviews

The material of the book may very well be used for an introductory lecture on differential inclusions.
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A differential inclusion is a relation of the form \(\dot x \in F(x)\), where \(F\) is a setvalued 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 righthand 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 setvalued 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.
Graduate students and research mathematicians interested in ordinary differential equations, calculus of variations, optimal control, and optimization.

Part 1. Foundations

Chapter 1. Convex analysis

Chapter 2. Setvalued analysis

Chapter 3. Nonsmooth analysis

Part 2. Differential inclusions

Chapter 4. Existence theorems

Chapter 5. Viability and invariance

Chapter 6. Controllability

Chapter 7. Optimality

Chapter 8. Stability

Chapter 9. Stabilization

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