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.