REALIZABILITY THEORY FOR CONTINUOUS LINEAR SYSTEMS
A.H. Zemanian
PREFACE

``Realizability theory'' is part of mathematical systems theory and
is concerned with the following ideas. Any physical system
defines a relation between the stimuli imposed on the system
and the corresponding responses. Moreover, any such system
is always causal and may possess other properties such as time-invariance
and passivity. Two questions: How are the physical properties
of the system reflected in various mathematical descriptions of the
relation? Conversely, given a mathematical description of a relation,
does there exist a corresponding physical system possessing
certain specified properties? If the latter is true,
the mathematical description is said to be realizable.
Considerations of this sort arise in a number of physical sciences.
Examples of this are the works of McMillan, Newcomb, and Wohlers
on electrical networks, those of Toll and Wu on scattering phenomena,
and Gross, Love, and Meixner on viscoelasticity.

This book is an exposition of realizability theory as applied to
the operators generated by physical systems as mappings of stimuli
into responses. This constitutes the so-called ``black box'' approach
since we do not concern ourselves with the internal structure of the
system at hand. Physical characteristics such as linearity,
causality, time-invariance, and passivity are defined
as mathematical restrictions on a given operator. Then, the two
questions are answered by obtaining a description of the operator
in the form of a kernel or convolution representation and establishing a variety of
necessary and sufficient conditions for that representation to possess the
indicated properties. Thus, the present work is an abstraction
of classical realizability theory in the following way. A given representation
is realized not by a physical system but rather by an operator possessing
mathematically defined properties, such as causality and passivity,
which have physical significance. We may state this in another
way. Our primary concern is the study of physical properties
and their mathematical characterizations and not the design of particular
systems.

Two properties we shall always impose on any operator under
consideration are linearity and continuity. They are quite
commonly (but by no means always) possessed by physical systems.
Of course, continuity only has a meaning with respect to the
topologies of the domain and range spaces of the operator. We can in general
take into account a wider class of continuous linear operators
by choosing a smaller domain space with a stronger topology
and a larger range space with a weaker topology. With this as
our motivation, we choose the basic testing-function space of
distribution theory as the domain for our operators and the space
of distributions as the range space. The imposition of other
properties upon the operator will in general allow us to extend
the operator onto wider domains in a continuous fashion. For example,
time-invariance implies that the operator has a convolution
representation and can therefore be extended onto the space of all
distributions with compact supports. This distributional setting
also provides the following facility. It allows us to obtain certain
results, such as Schwartz's kernel theorem, which simply do not hold under any
formulation that permits the use of only ordinary function. Thus,
distribution theory provides a natural language for the realizability
theory of continuous linear systems.

Still another facet of this book should be mentioned. Almost all
the realizability theories for electrical systems deal with signals that take their
instantaneous values in n-dimensional Euclidean space. However,
there are many systems whose signals have instantaneous values in a
Hilbert space or Banach space. Section 4.2 gives an example of this.
For this reason, we assume that the domain and range spaces for the
operator at hand consist of Banach-space-valued distributions. Many of
the results of earlier realizability theories readily carry over to
this more general setting, other results go over but with difficulty,
and some do not generalize at all. Moreover, the theory of
Banach-space-valued distributions is somewhat more complicated than
that of scalar distributions; Chapter 3 presents an exposition of it.
Still other analytical tools we shall need as a consequence of our use
of Banach-space-valued distributions are the elementary calculus of
functions taking their values in locally convex spaces, which is given
in Chapter 1, and Hackenbroch's theory for the integration of
Banach-space-valued functions with respect to operator-valued measures,
a subject we discuss in Chapter 2.

The systems theory in this book occurs in Chapters 4, 5, 7, and 8.
Chapter 4 is a development of Schwartz's kernel theorem in the present
context and ends with a kernel representation for our continuous linear
operators. Causality appears as a support condition on the kernel.
How time-invariance converts a kernel operator into a convolution
operator is indicated in Chapter 5. We digress in Chapter 6 to develop
those properties of the Laplace transformation that will be needed
in our subsequent frequency-domain discussions. Passivity is a very strong
assumption; it is from this that we get the richest realizability
theory. Chapter 7 imposes a passivity condition that is appropriate
for scattering phenomena, whereas a passivity condition that is suitable
for an admittance formulism is exploited in Chapter 8.

It is assumed that the reader is familiar with the material found in the
customary undergraduate courses on advanced calculus, Lebesgue integration,
and functions of a complex variable. Furthermore, a variety of standard
results concerning topological linear spaces and the Bochner integral
will be used. In order to make this book accessible to readers who
may be unfamiliar with either of these topics, a survey of them is given
in the appendices. Although no proofs are presented, enough definitions
and discussions are given to make what is presented there understandable,
it is hoped, to someone with no knowledge of either subject. Almost
every result concerning the aforementioned two topics that is used in this
book can be found in the appendices, and a reference to the particular
appendix where it occurs is usually given. For the few remaining results
of this nature that are employed, we provide references to the literature.

The problems usually ask the reader either to supply the proofs of certain
assertions that were made but not proved in the text or to extend the
theory in various ways. On occasion, we employ a result that was stated
only in a previous problem. For this reason, it is advisable for the
reader to pay some attention to the problems.

All theorems, corollaries, lemmas, examples, and figures are triple-numbered;
the first two numbers coincide with the corresponding section numbers.
On the other hand, equations are single-numbered starting with (1) in
each section.

A.H. Zemanian
Stony Brook, New York