Graduate Studies in Mathematics 2002; 226 pp; hardcover Volume: 41 ISBN10: 0821829777 ISBN13: 9780821829776 List Price: US$40 Member Price: US$32 Order Code: GSM/41
 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. Readership Graduate students and research mathematicians interested in ordinary differential equations, calculus of variations, optimal control, and optimization. Reviews "The material of the book may very well be used for an introductory lecture on differential inclusions."  Jahresbericht der DMV "The book is well written and contains a number of excellent problems."  Zentralblatt MATH Table of Contents Foundations  Convex analysis
 Setvalued analysis
 Nonsmooth analysis
Differential inclusions  Existence theorems
 Viability and invariance
 Controllability
 Optimality
 Stability
 Stabilization
 Comments
 Bibliography
 Index
