Resolution Space, Operators and Systems

278 Pages·1973·11.794 MB·English

by  Dr. R. Saeks (auth.)

Lectu re Notes
in Economics and
Mathematical Systems
Operations Research, Computer Science, Social Science
Edited by M. Beckmann, Providence, G. Goos, Karlsruhe, and
H. P. KUnzi, ZUrich
82
R.Saeks
Resolution Space
Operators and Systems
Springer-Verlag
Berlin· Heidelberg· New York 1973
Advisory Board
H. Albach· A. V. Balakrishnan· F. Ferschl . R. E. Kalman· W. Krelle . G. Seegmiiller
N. Wirth
Dr. R. Saeks
University of Notre Dame
Dept. of Electrical Engineering
Notre Dame, Ind. 46556/USA
AMS Subject Classifications (1970): 28A45, 46E05, 46GlO, 47 A99, 47B99, 93A05, 93A 10,
93B05, 93B35, 93C05, 93C25, 93C45, 93C50, 93D99.
ISBN-13: 978-3-540-06155-7 e-ISBN-13: 978-3-642-80735-0
DOl: 10.1007/978-3-642-80735-0
This work is subject to copyright. All rights· are reserved, whether the whole or part of the material is concerned,
specifically those of translation, reprinting, re-·use of illustrations, broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher,
the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin· Heidelberg 1973. Library of Congress Catalog Card Number 72-97027.
Preface
If one takes the intuitive point of view that a system is a black box
whose inputs and outputs are time functions or time series it is natural
to adopt an operator theoretic approach to the stUdy of such systems. Here
the black box is modeled by an operator which maps an input time function
into an output time function. Such an approach yields a unification of
the continuous (time function) and discrete (time series) theories and
simultaneously allows one to formulate a single theory which is valid for
time-variable distributed and nonlinear systems. Surprisingly, however,
the great potential for such an approach has only recently been realized.
Early attempts to apply classical operator theory typically having failed
when optimal controllers proved to be non-causal, feedback systems unstable
or coupling networks non-lossless. Moreover, attempts to circumvent
these difficulties by adding causality or stability constraints to the
problems failed when it was realized that these time based concepts were
undefined and; in fact, undefinable; in the Hilbert and Banach spaces of
classical operator theory.
Over the past several years these difficulties have been alleviated
with the emergence of a new operator theory wherein time related concepts
such as causality, stability and passivity are well defined and readily
studied. Although the techniques were developed essentially independently
by a number of different authors they are characterized by two consistent
threads. First, the time characteristics of a function (or vector) are
characterized by a family of truncation operators which pick out the part
of the function before and after a given time. Secondly, various operator
valued integrals manifest themselves as the primary tool with which the
theory is implemented.
IV
The most straightforward approach has been that of Willems, Damborg,
Zames and Sandberg who, in the context of their researches into feedback
system stability, adopted the point of view of working with a function
space together with the family of truncation operators defined by
{f(S) ;
t s < t
(E f)(s) =
o ,• s > t
Here Etf is interpreted as the part of f before time t and f - Etf
is interpreted as the part of f after t. As such, by appropriately asso-
ciating the operators on the given function space with the family of
truncation operators, the various stability and related causality concepts
are defined. Closely related to this work is the Fourier analysis of Falb
and Freedman who deal with similar function spaces and families of trunca-
tion operators. Here the Fourier representation and corresponding theorems
are formulated via appropriate weakly convergent operator valued integrals
in a manner somewhat similar to that used in a spectral theoretic context.
Somewhat more abstract than the above is the work of Duttweiler and
Kailath who deal with a Hilbert space of stochastic processes in anestima-
tion theoretic context. Here, however, because of the predictability of
such stochastic processes the truncation operators are replaced by a resolu
tion of the identity wherein Etf is an estimate of f conditioned on measure-
ments made prior to time t, and the integrals of triangular truncation,
originally formulated by the Russian school of spectral theorists, serve
as the primary manipulative tool. An alternative approach, also charac-
terized by truncation operators and operator valued measures is due to
Zemanian. Here Hilbert and/or Banach space valued distributions are
employed for the study of various problems associated with the realizability
v
theory of linear networks.
Finally, we have the resolution space approach of the present mono
graph wherein abstract Hilbert space techniques are combined with classical
spectral theory to obtain a formalism which essentially includes all of
the above, from which many of the ideas and techniques have been drawn.
The approach is due primarily to Porter and the author whose interests in
the subject were derived, respectively, from their research in optimal
control and network synthesis. In this theory a resolution of the identity
on an abstract Hilbert space defines the required truncation operators,
which are manipulated via the Lebesgue and several different types of
Cauchy integral defined over appropriate operator valued measures.
In the first chapter the elements of resolution space and the theory
of causal operators are formulated. These ideas are applied to the
study of the stability, sensitivity and optimization of feedback systems
in the second chapter, and to the controllability, observability, stability
and regulator problems for dynamical systems in the third chapter. The
last chapter deals with the more highly structured concept of a uniform
resolution space wherein time-invariant operators, along with their
Fourier and Laplace transforms, are developed. Finally, there are four
appendices. The first three of these are devoted to a review (without
proof) of results from topological group theory, operator valued integra
tion, and spectral theory, respectively, which are needed in the main body
of the text, while the representation theory for resolution and uniform
resolution spaces is formulated in the fourth appendix.
Although the placement of the applied chapters in the middle of the
text may seem to be somewhat unusual, this ordering was chosen so that
the three elementary chapters (equals a first graduate course in real
VI
analysis) would appear first with the more advanced material (equals
harmonic analysis and non-self-adjoint spectral theory) in the last
chapter and appendices. As such, the first three chapters may be used for
a first or second year graduate course in either engineering or mathematics.
This being aided by rather long discussion and problem sections at the
end of each chapter. These include problems which range from the simple
through difficult to the open. In fact, many generalizations of the
theory and related topics are introduced in the problems.
References to results in the monograph are made by a three digit
characterization, for instance (2.D.3) denotes the third paragraph in the
fourth section of the second chapter. In general theorems, lemmas and
equations are unnumbered with references made to the chapter section and
paragraph only. Since the paragraphs contain at most one "significant"
result, this causes no ambiguity. Similarly, references to books and
technical papers are made via a three digit characterization such as
(rG-3) which denotes the third paper by author FG. With regard to such
references, except for classical theorems, we have adopted the policy of
not attributing theorems to specific people though we often give a list
of papers wherein results related to a specific theorem appear. This
policy having been necessitated by the numerous hands through which most
of the results presented have passed in the process of reaching the degree
of generality in which they are presented.
Finally, acknowledgements to those who have contributed to this work
via their comments and suggestions are in order. To list but a few:
R.A. Goldstein, R.J. Leake, N. Levan, S.R. Liberty, R.-W. Liu, P. l1asani,
VII
W.A. Porter, M.K. Sain and A.H. Zemanian. Moreover, I would like to
express my gratitude to R.M. DeSantis for his careful reading and detailed
comments on the manuscript, to P.R. Halmos for recommending inclusion of the
manuscript in the Springer-Verlag Lecture Notes series, and to Mrs. Louise
Mahoney for her careful typing of the manuscript.
Contents
1. Causality .....................................•.......•..•.•••. 1
A. Resolution Space........................................... 3
B. Causal Operators........................................... 8
C. Closure Theorems .....•.................••..............•.•. 14
D. The Integrals of Triangular Truncation ............•...••••. 22
E. Strictly Causal Operators.................................. 29
F. Operator Decomposition..................................... 41
G. Problems and Discussion ..................•............•••.. 50
2. Feedback Systems .•.......................................•.•.•. 61
A. Well-Posedness .......................................•••••• 63
B. Stability. .................................•...........•... 72
C. Sensitivity. .•...............•......... '" ..............••. 82
D. Opt imal Controllers........................................ 89
E. Problems and Discussion.................................... 100
3. Dynamical Systems.............................................. 114
A. State Decomposition. ........•............•................. 116
B. Controllability, Observability and Stability. .............. 132
C. The Regulator Problem...................................... 140
D. Problems and Discussion. •..•.............................•. 148
4. Time-Invariance. ......•...........•............................ 155
A. Uniform Resolution Space. ...•.............................. 156
B. Spaces of Time-Invariant Operators. ...........•............ 160
C. The Fourier Transform. .................... " ., ...........•. 168
D. The Laplace Transform........................ .. . . . . . . . . • . .. 178
E. Problems and Discussion ...............................•••• 191
x
Appendices
A. Topological Groups......................................... 204
A. Elementary Group Concepts ..•.•...•.•.•...•....•..•.•.•• 205
B. Character Groups .••.••.••.•.•..•...•.••••...••.••.•••.• 206
C. Ordered Groups •.•.•.•.••.••••..•.•....•.•.•.•..•.•••..• 209
D. Integration on (LCA) Groups .•.•..•..••....••.....•.•••• 211
E. Differentiation on (LCA) Groups .•.•....•.•.......••.•.. 214
B. Operator Valued Integration .•.•.•.•...••..•.•..••••••••.•.• 217
A. Operator Valued Measures.. . . • • . . • • . • • • • • . • • • • . • . • . . • • •• 219
B. The Lebesgue Integral.................................. 222
C. The Cauchy Integrals .•••••••...•.•.•••..•••••.••..••••• 225
D. Integration over Spectral Measures •.•..••••.•••••••.•.• 229
C. Spectral Theory ............................................. 232
A. Spectral Theory for Unitary Groups .•••..•.•..••.•.•...• 233
B. Spectral Multiplicity Theory .••.•.•••••..••..••...•.••• 237
C. Spectral Theory for Contractive Semigroups ••••..•••.•.• 240
D. Representat ion Theory. • • • • • • . • . • . . . • • • • . • • . . • • . • • • • . • • • • • •• 244
A. Resolution Space Representation Theory ••••••..•.•••.••• 245
B. Uniform Resolution Space Representation Theory ••••••••. 248
-_____ ...••.•..••..•••..•.•••.. _ ... _. 258
1. CAUSALITY
In this chapter the basic concepts of resolution space and causal
operators which prove to be of fundamental importance throughout the work
are introduced. The results presented are due primarily to W. A. Porter
(PR-l, PR-2, PR-3, PR-4), R. M. DeSantis (DE-l, DE-2, DE-3) and the author
(SA-4) in their present formulation. Similar results have, however, been
obtained in the context of other formalisms (WL-3. DA-l. DA-2. DA-3,
DU-l, DI-l, ZE-l. ZE-2). We begin with a formulation of the basic defini
tions and notation for resolution space along with a number of examples
which are used throughout the remainder of the work. We then introduce
causal operators. which prove to be the natural operators with which one
deals in a resolution space setting, and the related concepts of anti
causal and memoryless operators. Various algebraic and topological clo
sure theorem are then formulated for these operator classes with special
emphasis being placed on causal invertibility theorems.
Our first encounter with operator valued integrals in our theory is
with the integrals of triangular truncation which define (unbounded)
projections of the space of all bounded linear operator onto the space of
causal operators. Since the causal operators are algebraically defined it
is not surprising that these integrals are independent of the operator
topology in which they are defined. To the contrary, however, if one changes
the bias on the Cauchy integrals used to define the integrals of triangular
truncation to obtain the integrals of strict triangular truncation they are
topology dependent. As such these latter integrals define various classes
of strictly causal operators depending on the operator topology employed.
Finally, the problem of decomposing a general operator into a sum or