This book is devoted to one of the main questions of the theory of
extremal prob- lems, namely, to necessary and sufficient extremality
conditions. It is intended mostly for mathematicians and also for all
those who are interested in optimiza- tion problems. The book may be
useful for advanced students, post-graduated students, and researchers.
The book consists of four chapters. In Chap. 1 we study the abstract
minimization problem with constraints, which is often called the
mathemati- cal programming problem. Chapter 2 is devoted to one of the
most important classes of extremal problems, the optimal control
problem. In the third chapter we study one of the main objects of the
calculus of variations, the integral quadratic form. In the concluding,
fourth, chapter we study local properties of smooth nonlinear mappings
in a neighborhood of an abnormal point. The problems which are studied
in this book (of course, in addition to their extremal nature) are
united by our main interest being in the study of the so called abnormal
or degenerate problems. This is the main distinction of the present book
from a large number of books devoted to theory of extremal problems,
among which there are many excellent textbooks, and books such as, e.g.,
[13, 38, 59, 78, 82, 86, 101, 112, 119], to mention a few.