Skip to main content
main-content

Über dieses Buch

In preparing the second edition, I have taken advantage of the opportunity to correct errors as well as revise the presentation in many places. New material has been included, in addition, reflecting relevant recent work. The help of many colleagues (and especially Professor J. Stoer) in ferreting out errors is gratefully acknowledged. I also owe special thanks to Professor v. Sazonov for many discussions on the white noise theory in Chapter 6. February, 1981 A. V. BALAKRISHNAN v Preface to the First Edition The title "Applied Functional Analysis" is intended to be short for "Functional analysis in a Hilbert space and certain of its applications," the applications being drawn mostly from areas variously referred to as system optimization or control systems or systems analysis. One of the signs of the times is a discernible tilt toward application in mathematics and conversely a greater level of mathematical sophistication in the application areas such as economics or system science, both spurred undoubtedly by the heightening pace of digital computer usage. This book is an entry into this twilight zone. The aspects of functional analysis treated here are rapidly becoming essential in the training at the advance graduate level of system scientists and/or mathematical economists. There are of course now available many excellent treatises on functional analysis.

Inhaltsverzeichnis

Frontmatter

1. Basic Properties of Hilbert Spaces

Abstract
This is an introductory chapter in which we study the basic properties of Hilbert spaces, indispensable for an understanding of the sequel. Although it is fairly complete in itself, this chapter is necessarily brief in many areas and the reader would find it helpful to have had an elementary introduction to linear spaces, and Hilbert spaces in particular, such as one finds in the standard texts on real analysis.
A. V. Balakrishnan

2. Convex Sets and Convex Programming

Abstract
In this chapter we concentrate on properties of convex sets in a Hilbert space and some of the related problems of importance in application to convex programming: variational problems for convex functions over convex sets, central to which are the Kuhn-Tucker theorem and the minimax theorem of von Neumann, which in turn are based on the “separation” theorems for convex sets. A related result is the Farkas lemma in finite dimensions which finds application in network flow problems.
A. V. Balakrishnan

3. Functions, Transformations, Operators

Abstract
This chapter presents the core of operator theory essential to our purposes. Thus much standard material has had to be omitted and more specialized topics included, as for example the theory of Hilbert-Schmidt and Nuclear and Volterra operators, whereas the spectral representation theory of self adjoint operators has been limited to compact operators. Examples illustrating the theory are included as often as possible.
A. V. Balakrishnan

4. Semigroups of Linear Operators

Abstract
In this chapter we present an introductory treatment of the theory of semigroups of linear operators over a Hilbert space, emphasizing those aspects which are of importance in applications. As a rule we shall not strive for generality and instead shall dwell on special classes of semigroups such as compact semigroups and Hilbert-Schmidt semigroups. Semigroup theory is generally accepted as an integral part of functional analysis and is included in most standard treatises on functional analysis which should be consulted for details if necessary. We have taken some pains to illustrate the application to partial differential equations; the abstract parts of the theory are in many ways easier than the specialization to partial differential equations. Nevertheless the abstract formulation has the advantage that it provides a direct generalization of finite dimensional models and makes the transition more transparent, especially in the application to control problems.
A. V. Balakrishnan

5. Optimal Control Theory

Abstract
The theory of optimal control is one of the major areas of application of mathematics today. From its early inception to meet the demands of automatic control system design in engineering, it has grown steadily in scope and now has spread to many other far removed areas such as economics. Until recently the theory has been limited to “lumped parameter systems”— systems governed by ordinary differential equations. In fact, it is most developed for linear ordinary differential equations—particularly feedback control for quadratic performance index—where the results are most complete and closest to use in practical design. The extension to partial differential equations (and delay differential equations) is currently an active area of research and holds much promise. It is natural that this extension deal with linear systems not only for mathematical reasons but also for reasons of practicality. The theory of semigroups of linear operators developed in the last chapter lends a convenient setting for this purpose and offers many advantages. It provides a useful degree of generality and serves, for instance, to distinguish between those aspects peculiar to the particular partial differential equation involved and those which are more general. Not the least advantage is the structural similarity to the familiar finite-dimensional model. Of course, semigroup theory per se applies only to time invariant systems; but this is not a serious limitation.
A. V. Balakrishnan

6. Stochastic Optimization Theory

Abstract
This final chapter deals with a class of stochastic optimization problems. For this purpose we introduce a measure theoretic structure on top of the topological structure, and the resulting interplay brings a new set of questions interesting on their own as well. The measure theory is nonclassical in that the measures are only finitely additive on the field of cylinder sets, the canonical example being the Gauss measure. The notion of a weak random variable suffices for the stochastic extension of the control problems of the previous chapter, a crucial notion being that of “white noise,” leading to a treatment that is novel with this book, of filtering and control problems embracing in particular linear stochastic partial differential equations. Important tools in the development are the Krein factorization theorem and the Riccati equation. For nonlinear operations we develop a “nonlinear” white noise theory in which the notion of a physical random variable plays a crucial role, as in the calculation of the Radon-Nikodym derivative of finitely additive Gaussian measures. Within the scope of the present work we can but touch upon the general theory of nonlinear stochastic differential equations.
A. V. Balakrishnan

Backmatter

Weitere Informationen