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INTERNATIONAL SYMPOSIUM
ON
OPERATOR THEORY OF NETWORKS
AND SYSTEMS
Volume 2 EDITORS:
N. LEVAN R. SAEKS
AUGUST 17-19, 1977
SPONSOR ED BY
TEXAS TECH. UNIVERSITY U.S. AIR FORCE OFFICE
OF SCIENTIFIC RESEARCH UNDER GRANT 77·3382
Hilton Inn -lubbock, Texas
OPERATOR THEORY OF NETWORKS AND SYSTEMS. INTERNATIONAL SYMPOSIUM ON.
Vol. 1. Concordia University, Montreal, Canada. August 12-14, 1975. 153 pages. Paper ......... :....................................... $22.00
CONTENTS (partial):
• Structure Result for Nonlinear Passive Systems • Frequency Response Methods in Multivariable Infinite Dimensional
Linear Systems • A Walsh Operational Matrix for Solving Variational Problems • The Feedback Interconnection of Multivariable Systems: Simplifying
Theorems for Stability • Linear Hilbert Networks Containing Finitely Many Nonlinear Elements • Linear Network Synthesis Using Iteration Methods • A Note on the Nagy-Foias Lossy and Lossless Space • An Output Control Problems Containing Input Derivatives • Contractive Transfer Ratios of Operator Network
(Standing Orders Accepted)
Additional Copies Available From:
WESTERN PERIODICALS COMPANY 13000 RAYMER STREET
NORTH HOLLYWOOD, CALIFORNIA 91605
Copyright © 1977 by Western Periodicals Company 13000 Raymer Street North Hollywood, California 91605
ii
PRE F ACE
The International Symposium on the Operator Theory of Networks and Systems is a biannual
conference which is guided, ~rom year to year, by a Steering Committee composed of
representatives from a number of universities. The current symposium is the second in
the series, and the first which Texas Tech University has had the honor of hosting.
Although originally conceived as a forum for the presentation of research results in the
area of operator theoretic techniques applied to network and system theory, the scope
of the symposium has grown considerably over the years. The present volumn therefore
encompasses research spanning the entire field of Mathematical Ne~work and Systems
Theory. Mathematical tools include algebraic techniques, algebriac topology, and
differential geometry in addition to the operator theoretic and functional analytic
techniques to which the symposium was originally directed.
The proceedings clearly convey the international interests in the area of Mathematical
Network and Systems Theory. Contributions of Scientists from three continents - Asia,
Eroupe, and North America are presented.
The Symposium committee is please to acknowledge the support of the Department of
Electrical Engineering, the College of Engineering, and the Graduate School of Texas
Tech University. Moreover, the symposium could not have been conducted without the
financial support of the U. S. Air Force Office of Scientific Research.
III
SECOND INTERNATIONAL SYMPOSIUM ON THE OPERATOR THEORY OF
NETWORKS AND SYSTEMS
STEERING COMMITTEE
C. A. Desoer University of Calif. at Berkeley Berkeley, CA.
J. W. Helton University of Calif. at San Diego La Jolla, CA.
N. Levan University of Calif. at Los Angeles Los Angeles, CA.
W. A. Porter Louisiana State University Baton Rouge LA.
R. Saeks Texas Tech University Lubbock, TX.
A. H. Zemanian State university of New York Stony Brook, N.Y.
R. W. Newcomb (Chairman) University of Maryland College Park, MA.
SYMPOSIUM ORGANIZING COMMITTEE
R. M. DeSantis (Program Chairman) University of Montreal
N. Levan (Co-Chairman) University of Calif. at Los Angeles Los Angeles, CA. Montreal Quebec
Canada
R. Saeks (Co-Chairman) Texas Tech University
Lubbock, TX.
IV
TAB LEO F CON TEN T S
1: COLLOQUIUM LECTURE
"Input-Output Properties of Interconnected Systems: Part I"
C. A. Desoer, Univ. of Calif. at Berkeley ----------------------
2: GENERAL SESSION I
"Bernstein Systems for Approximation and Realization"
W. A. Porter, Louisiana State Univ. ----------------------
"A Linear Systems Theory in Multidimensional Time"
A. V. Balakrishnan, UCLA ---------
"Optimal Control in Hilbert Space" M. Steinberger, M. Schumitzky, and L. M. Silverman, Univ. of Southern Calif. ---------------
"Solvability and Linerzation of Monotone Hilbert Networks"
V. Dolezal, SUNY at Stony Brook ----------------------------
4: SESSION ON FUNCTION ANALYTIC TECHNIQUES
"Time-Varying Input-Output Systems Whose Signals are Banach-Space-Valued Distributions"
A. N. Zemanian, SUNY at Stony Brook ----------------------------
"Causality and C Operators" A. Feintuch, °Ben Gurion Univ. ----------------------------
"Wiener-Hopf Techniques in Resolution Space"
L. Tung, and R. Saeks, Texas Tech Univ. -----------------------
"Approximate Controllability and Weak State Stabilizability"
C. Benchimol, UCLA ---------------
"A Modified Discrete Convolution Operator for Simulation of Linear Continuous Systems"
H. B. Kekre and D. B. Phatak, Indian Inst. of Tech. at Bombay
5: SESSION ON DYNAMICAL SYSTEMS
"Differential Systems on Alterna-tive Algebras"
R. W. Newcomb, Univ. of
2
6
.7
13
17
21
28
34
35
__ Maryland -------------------------- 36
v
"Lagrangians with Integrals: An Approach to the Variational Theory of Dissipative Networks"
V. M. Fatic, Tri-State Univ., and W. A. Blackwell, Virginia Polytechnic Inst. and State 38 Univ. -----------------------------
"An Estimation of Parameters in Parabolic Equation with Spacially-Varying Coefficients"
Z. Jacyno, Univ. of Quebec in 56 Montreal --------------------------
"The Singularity Expansion Method _ in Electromagnetic Scattering" ~9'l
D. R. Wilton, Univ. of Miss. ------~.
"Variational Principels for Mechanical and Structural Systems with Applications to Optimality of Design"
V. Komkov, Texas Tech Univ. -------- 63
"A Numerical Calculation Method for Simultaneous Ordinary Dif-ferential Equation of High Order by the Momentary Diagonalized Modal Property"
S. Azuma, Ibaraki Univ., and B. F. Womack, Univ. of Texas at Austin -------------------------- 66
6: SESSION ON CONTROL
"On Structurally Stable Nonlinear Regulation with Step Inputs"
W. M. Wonham, Univ. of Toronto ----- 72
"Frequency Domain Stability for a Class of Partial Differential Equations"
D. Wexler, Univ. Notre Dame de la Paix ------------------------- 76
"Densensitizing Observer Design for Optimal Feedback Control"
M. M. Missaghie, Sentrol Systems Ltd. _______________________________ 81
"Grassman Manifolds and Global Properties of the Riccati Equation"
C. Martin, NASA/AMES Research Center _____________________________ 82
"Generalized Operator and Optimal Control"
S. M. Yousif, Calif. State 86 Univ. at Sacramento ----------------
"Absolute Invariant Compensators: Concepts, Properties and'Applications"
R. M. DeSantis, Univ. of Montreal --------------------------- 90
8: GENERAL SESSION II
"Stability Tests for One, Two, and Multidimensional Linear Systems"
E. I. Jury, Univ. of Calif. at Berkeley _______________________ 91
"A Darlington Realization Theory of Optimal Linear Predictors"
P. DeWilde, T.H. Delft.
"Operator Theory Techniques for Finite Dimensional Problems"
92
J. W •. Helton, Univ. of Calif. at San Diego ______________________ 96
"The Relation Between Network Theory, Vector Calculus and Theoretical Physics"
F. H. Branin, IBM Corp. ----------- 97
9: SESSION ON LINEAR NETWORKS AND SYSTEMS
"Detached Coefficients Represen-tation and Degree Functor of a Polynomial Matrix with Applica-tion to Linear Systems"
Y. S. Ho and P. H. Roe, Univ. of Waterloo _______________________ 103
"A Representation of Impedance Function in Terms of the Poles and Zeros for Transmission Lines"
F. Kato and M. Saito, Univ. of Tokyo --------------------------109
"Evaluation of Constituent Matrices of an Analytic Matrix Function"
F. C. Chang and S. R. Pulufani, Alabama A&M univ. -----------------113
"On the Losless Scattering Matrix Synthesis via State Space Techniques"
A. L. Dobruck and M. S. Piekarski, Wroclaw Technical Univ. -----------116
"Algebraic Characterizat:.ion of Matrices whose Multivariable Charac-teristic Polynomial is Hurwitzian"
M. S. Piekarski, Wroclaw Tech. Univ. -----------------------------121
10: SESSION ON OPERATOR METHODS
"Contraction Operator of Class C and the Structure of a class of 0 Infinite Dimensional System"
D. Hedberg, Hughes Aircraft Co., and N. Levan, UCLA -----------127
"Discrete-Time System Operators on Resolution Sets of Sequences"
R. J. Leake and B. Swanimathan, Univ. of Notre Dame ---------------128
VI
"Some Aspects in a Theory of General Linear Systems"
R. H. Foulker, Youngstown State Univ. -----------------------
"Certain Aspects of Inverse Filters"
V. P. Sinha, Indian Inst. of Tech. at Kanpui, and H. S. Sekhon, Punjab Argicultural Univ. -----------------------------
"Adaptive Antenna Polarization Schemes for Clutter Suppression and Target Identification"
G. Ioannids and D. Hammers, ITT Gilfillan Inc. ----------------
"On Limitations Based on Properties of the Strum-Liouville Operators in the Synthesis Procedures of Non-uniform Lines"
Z. Trazaska, Inst. of the Theory of Elec. Engrg. and Elec. Measure-ments of Warsaw -------------------
11: SESSION ON NONLINEAR SYSTEMS
"A Theory of Best Appro~imation of Nonlinear Functionals and Operators by Volterra Expansions"
133
134
135
'140
L. V. Zyla and R.J.P. de Figueiredo, Rice Univ. - ____________ 143
"Continued Fraction Describing Functions for Bilinear and Multiplicative Nonlinear Systems"
C. F. Chen, Univ. of Houston, and R. E. Yates, u.S. Army Research Lab. at Redstone Arsenal ____________________________ 144
"Nonlinear Analysis of Gyrator Networks: A Numerical Example"
M. B. Waldron and M. A. Smithers, Univ. of Houston ___________________ 148
"Lie Series and the Power System Stability Problems"
R. K. Bansal and R. Subramanina, Pubjab Argicultural Univ. ---------- 152
"A Study of Varying Efficiency Multiserver Queue Models"
H. B. Kekre, R. D. Kumar, and H. M. Srivastava, Indian Inst. of Tech. at Bombay ----------------- 156
"Modular Design of the Network which Realizes Original jProgram"
S. P. Kartashev,. Univ. of Nebraska, S. 1. Kartashev, DCA Assoc. ________ 161
INPUT-OUTPUT PROPERTIES OF INTERCONNECTED
SYSTEMS
C.A. Desoer Dept. of Electrical Engineering and Computer Science
University of California Berkeley, Ca. 94720
ABSTRACT
The purpose of this survey paper is to review some recent results on the input-output properties of both linear and nonlinear inter-connected systems. The results presented deal primarily with quali-tative system properties such as stability and parameter sensitivity with emphasis being placed on the robustness of these properties.
BERNSTEIN SYSTEMS FOR APPROXIMATION AND REALIZATION*
William A. Porter**
1. INTRODUCTION
This summary deals with the approximation of non-
linear systems by polynomic operators. For per-
spective it is helpful to consider the familiar
Volterra series expansion on L2 given by
p(x) ,= kO + S k~ (a)x(a)da +Hk2 (a,B)x(a)x(B)dadB + J~!k3(a,B,y)x(a)x(B)x(Y)dadBdY + ...
where the kernels kO' kl, ••. ,kn satisfy properties
suitable to an operator on L2
. For the obvious
reasons we refer to each term on the right hand
side as a power function. If the number of terms
is finite then p is said to be a polynomic opera-
tor. Our interest in polynomic operators centers
on their use as approximates of the more general
nonlinear functions on L2
•
The relevant lit~rature may be grouped into two
subcategories. -First-there is the analytic theory
in which f is assumed to have derivatives (Frechet
or Gateau) of all orders and p arises as a power
series expansion on a bounded domain. This line of
devElopment was initiated by Volterra [1] and was
first e.pplied in a systems setting by Weiner and in
ensuing years several others including [2], [3],
and [0]. Mere recently [5], [6], [7], [8] have
investigated the Volterra expansion of solutions to
nonlinear differEntial equations with current
emphasis on computation and convergence problems.
The analytic theory identifies the power func-
tions with the deri vat:iv'~s of the system function.
f, to be approximated. The requisite differenti-
ability is a severe condition, however, a fringe
benefit accrued is that the causality of the power
functions is dictated by the causality of f.
In an independent line of development the polynomic
approximation problem has been approached as a
generalization of the classic Weierstrass result.
In this setting the function, f, to be approximated
need not be differentiable. Emphasis is placed on
uniformly approximating f, by the polynomial p,
over an arbitrary compact set. In this approach
the causality issue is less trivial. The computa-
tion of the power functions, which no longer repre-
sent derivatives, has here to fore been obsure. In
summary we introduce the concept of a Bernstein
differential system. This system provides one con-
structive realization of the Weierstrass approach.
An incidental bonus of the Bernstein system is a
pseudo sampling-theorem for systems. In short,
given an arbitrary system with a prescribed contin-
uity modulus, the density with which one must input-
output s~~ple in order to be able to approximately
reconstruct the system is established.
For efficiency of presentation we shall in many
cases be overly restrictive in the assumptions made,
for example we consider only Hilbert spaces. Also
the development is purposely c0nstrained so as to
ilse classical results on the Bernstein polynomials.
*Supported in par .. by the United States Air Force Office of Scientific Research, Grant No. 77-0352.
**Department of Electri_al Engineering, Louisiana State University, Baton Rouge, Louisiana.
2
In closing this introduction it is noted that the
admittance characteristic of the enhancement mode
MOSFET transistor provides an almost exact square
law (see [9]). It is easily shown that squaring
devices can be used to construct general polynomic
operators of the type necessary to realize the
Bernstein system. Thus it appears that the topic
of polynomic system approximation may have a ready
practicality in terms of microcircuit technology.
2. WEIERSTRASS APPROXIMATION
As the technical work of this study deals primar-
ily with the Weierstrass approximation it is use-
ful to comment in somewhat more detail on the
existing literature. In this regard we cite first
the original contribution of Weierstrass [10]
whose fundamental result in contemporary form
reads as follows.
Let f be an arbitrary continuous function on R,
the real line. Let D cR be an arbitrary compact
set. The~for every E > 0 there exists a finite
polynomial p, such that sup {If(x) - p(x)l:
XED} < E. The Weierstrass result, over the years,
has drawn the attention of several distinguished
mathematicians including Frechet [11], Bernstein
[12] and Stone [13] who investigated the relation-
ship to power series expansions, constructive
methods for finding the polynomial and extended
the result to Rn among other things.
More recently Prenter [14] considered a real sep-
arable Hilbert space H, and showed that if K c:: H
is compact, £ > 0 and f continuous on H then there
exists a finite polynomic operator p, such that
~~~llf(X) - p(x)11 < E
In a similar effort Prenter [15] and Ahmed [16]
were able to use normed linear spaces.
The Prenter result [14], for example, states that
if f is a continuous function on real L2
(a,b) then
on every compact subset there exists a finite
number of kernels kO' kl, ... ,kn such that p of
equation 1 is an E-approximat~on for f. Actually
we would suspect more, namely, that if f is causal
then each kernel ki is causal, tha: is for instance
3
0, T > t. More generally, can a caus-
ality structure be superimposed on the function
and its approximation? This question is answered
affirmatively in [17] and [18].
The setting for [17] and [18] is a Hilbert resolu-
tion space* {H,pt } where H is real and separable.
The set K c:: H is always compact. The sets: C, SC,
M, C(K), and P denote the causal, strictly causal,
memoryless, continuous on K c: H, and polynomic
functions, respectively, on {H,pt }. For brevity
we shall say that P is dense in C(K) in the sense
of Prenter's theorem.
The results of [17] include the following. The set
P" SC is dense in elK) n SC. In L2 the stronger result that pO SC is dense in C(K) A C is also established. This last result does not abstract.
In t2 it is known [18] that P n SC is not dense in C(K) " c.
All of the above results are nonconstructive in
that they give no clue as to finding the polynomic
approximate of a given function. On the real line,
however, several constructive forms of the Weier-
strass result do exist and the Bernstein polyno-
mials constitute one of the more intriguing
approaches to such constructions. In the study we
develop a generalization of the Bernstein poly-
nomials to real Hilbert space. Using a causal data
interpolation scheme identification of the p E P~SC
that approximates f E C(K)~C results.
To be more explicit let {H,P}be any Hilbert resolu-
tion space. The set QCH is compact and E > 0 is
arbitrary, ~ denotes a tuplet of indicies. fC .!l
denotes a causal polynomial constructed explicitely.
We summarize our first result in the following
theorem
Theorem (1) If causal f is bounded on n then for each X£Q and £>0 there exists .!l such that
I If(x) - fC(x)1 I < E at every continuity point n
of f. If f is continuous on n then .!l exists such that ~~~I If(x) - f~(x)1 I < E.
The explicit nature of theorem (2) has answered the
central theoretical questions in rather complete
form. It is of interest, however, to give a real-
ization of the causal Bernstein function, fC
, in ~
state variable form. For this attention was fo-
cused on the real 12 [a,d] for O
REFERENCES
[1] Volterra, V., Theory of Functionals, Dover Publications Inc., New York, 1959.
[2] Bedrosian, E. and S. O. Rice, "The Output Properties of Volterra Systems Driven by Harmonic and Gaussian Inputs," Proc. IEEE, Vol. 59, pp. 1688-1707, 1971.
[3] Parente, R. B., "Nonlinear Differential Equations and Analytic Systems Theory," SIAM J. Appl. Math., Vol. 18, pp. 41-66,1970.
[4] Van Trees, H. L., "Functional Techniques for the Analysis of the Nonlinear Behavio~ of Phase-Locked Loops," Proc. IEEE, Vol. 52, pp. 894-911.
[5] Brockett, R. W., Volterra Series and Geo-metric Control Theory, Automatica, Vol. 12, 1976.
[6] Bruni, C., DiPillo, G., Koch, G., "On the Mathematical Models of Bilinear Systems," Ricerche di Automatica, Vol. 2, pp. 11-26, 1976.
[7] Krener, A. H., "Linearization and Bilinear-ization of Control Systems," Proc. 1974 Allerton Conf. on Cir. and Sys. Th., pp. 834-843, 1974.
[8] Gilbert, E. G., "Volterra Series and the Re-sponse of Nonlinear Differential Systems," Trans. Conf. on Inf. Sci. and Systems, Johns Hopkins University, March 1976.
[9] R. S. Cobbold, "Theory and Applications of Field-Effect Transistors," Wiley-Interscience, New York, 1970, Section 7.1.3.
[10] K. Weierstrass, "Uber die Analytische Dar-shell-bankeit Sogenannter Willkurlicher Funk-tionen Reeler Argumente," Math. Werke, III Bd., 1903.
[11] M. Frechet, "S~r les Fonctionelles Continues," Ann. de l'Ecole Normale Sup., Third Series, Vol. 27, 1910.
[12] S. Bernstein, Demonstrqtion du Theoreme de Weierstrass, fondee sur Ie calcul des prob-abilities, Com. Soc. Math., Kharkow, (2), 13, 1912-13.
5
[13] M. H. Stone, "The Generalized Weierstrass Approximation Theorem," Math. Mag., Vol. 21, pp. 167-183, 1948.
[14] P. M. Prenter, "A Weierstrass Theomem for real Separable Helbert Spaces," J. Approxi-mation Theory, Vol. 3, No.4, pp. 341-351, Dec. 1970.
[15] P. M. Prenter, "A Weierstrass Theorem for Real Normed Linear Space," Bull. American Math. Soc., Vol. 75, pp. 860-862,1969.
[16] N. W. Ahmed, "An Approximation Theorem for Continuous Functions on Lo spaces," Univ. of Ottawa, Ottawa, Can., TR-73-l8, Nov. 191 .
[17] W. A. Porter, T. M. Clark, and R. M. DeSantis, "Causality Structure and the Weierstrass Theorem," J. Math. Anal. Appli., Vol. 52, No.2, Nov. 1975.
[18] W. A. Porter, "The Common Causality Structure of Multilinear Maps and their Multipower Forms," J. Math, Anal. Appl., Vol.. 52, No.2, Nov. 1975.
A LINEAR SYSTEMS THEORY IN MULTIDIMENSIONAL TIME
A. V. Balakrishnan Department of System Science
university of California Los Angeles, Ca. 90024
ABSTRACT
A theory of linear systems which do not have a time-like parameter is formulated. The input-output properties of such systems are studied and a notion of state is developed. The resultant theory is applied to the study of Markov random fields.
6
I _
M.
OPTIMAL CONTROL IN HILBERT SPACE
* * * Steinberger, A. Schumitzky and L. Silverman * Department of Electrical Engineering ':'Department of Mathematics
University of Southern California Los Angeles, California 90007
Abstract
Using the key concepts of causal factorization and state space under Nerode equivalence, we show that for a bounded, linear, strictly causal operator on Hilbert resolution spaces with quadratic cost functional, the optimal con-trol may be expressed in memoryless state feedback form provided the forcing function is expressed as an initial state. We further conjecture that there is no causal feedback which realizes the optimal control for a larger subspace of forcing functions.
I. IN:TRODUCTION
In this paper will will examine the optimal con-trol of a linear system with respect to a quad-ratic performance criterion. In particular, we will be interested in the cases in which the optimal control can be put in causal state feed-back form. This problem was solved for the case of finite dimensional differential systems of the form
input-output point of view rather than from an explicit representation for the dynamics of the plant to be controlled. That is, we follow Porter [9] in studying systems of the form
y = Tu + f
where T is a linear, bounded, strictly causal operator from one Hilbert resolution space HI to another, HZ; and f is a given forcing func-tion. Using the concept of Nerode equivalence promoted by Kalman [10] and since adopted by many other authors, we construct a state space for T. We then go on to construct a Hilbert resolution space of state trajectories. Given this space of state trajectories, we then show that if the operator (I + T*T) admits a strictly causal factorization (in the manner of Gohberg and Krein (11]) and f is generated by an initial state, then the optimal control can be realized in memoryless state feedback form. Further-more, the optimal feedback operator can be extracted fairly directly from the factorization of (I+T*T).
x Ax + Bu
y Cx
by Kalman [1] with highly successful results. Subs equent efforts have been directed toward expanding Kalman's results to cover systems described by generalized versions of the above equations (see Delfour and Mitter [Z], Lions [3], Lukes and Russell [4]. Datko [5], Prit-chard [6], and Curtain and 1critchard [7]).
Recently, we have written a paper [8] in which we attack the optimal control problem from an
This work was supported in part by the National Science Foundation under Grants ENG 76-14379, GP - Z0130 and by JSEP through AFOSR/AFSC under Contract F446Z0-7l-C-0067.
7
In this paper we will review the pertinent de-tails of [8]. Section II will contain the neces-sary mathematical background and major supporting theorems, Section III will present the actual optimal control results, and Section IV will give some brief concluding remarks.
II. MA THEMA TICAL BACKGROUND
II. 1 Resolution of the Identity
A Hilbert space H is said to be equipped with a resolution of the identity if for every t e [to' t..,] a closed subset of IR, 3: pt:H"'H 3
to (i) p u 0 Vue H
(ii)
(iii)
(iv)
to:> p u u
ptp" = p" pt = pt
t t (u l , P uZ)H = (P u l ' uZ)H
Vue H
V ,. ~ t
Furthermore, we can define the complementary projection Pt: H'" H, Pt '" I - pt from wh ich we have I = pt + Pt' motivating the term "resolution of the identity."
In the rest of this paper, we will associate with HI (HZ) the family of projections pt(i5
t).
A more complete treatment of this subject and the subject of causality can be found in the excellent article by DeSantis (13].
II. 2 Causality
In common usage, an operator is said to be causal if and only if past outputs are only effected by past inputs. In mathematical terms
we have:
Definition Z.l: An operator T is ~ iff
ptTPtu = ptTu , V t e [to' to>], u e HI
Similarly we have:
Definition 2. Z: An operator T is anticausal iff
An operator which is both causal and anticausal is called memoryless since its present output depends not on the past or the future but on the present input. Under reasonable conditions, any causal operator can be additively decom-posed into the sum of a memoryless operator and an operator which has no memoryless part. An operator which has no memoryless part is said to be strictly causal. We can define
strict causality more precisely using partitions
of the time set [to' t." ] •
Let 0 be a countable family of finite partitions of the time set [to, to>] with the following
properties:
8
(i) 11 cO VnelN n n+I
(ii) for every t e [to' to>] and e > 0 3: n, k :;I
o < t k - t < e n
t k+l
For notational simplicity let t, k - P k P n n t
t k +l ~k ~ ~ n (:, - Pk P
n t n
causality:
n
Then we can define strict
Definition Z.3: An operator T is strictly causal
iff T is causal and
lim N(n) ~k k :0 [:; T (:,
n n o
k=O
for any family as defined above. This limit is
taken in the uniform topology.
The following results are immediate:
Pt T Pt = T Pt pt T pt = T pt
(3) If S:Hr-+Hz is strictly causal and T: HZ'" H3 is causal and bounded, then ST is strictly causal.
(4) If T: HI ... H Z is causal and bounded, and S: HZ'" H3 is strictly causal, then TS is
strictly causal.
For convenience, we will hereafter call an operator T an LBSC operator if it is linear,
bounded, and strictly causal.
II. 3 The State Trajectory Space
In order to talk about state feedback in a Hil-bert resolution space setting, we first need a space of state trajectories which is a Hilbert resolution space. Such a space' is constructed
in [8].
The most basic concept underlying our construc-tion of a state space is the concept of Nerode equivalence, which, in our setting is defined as follows:
Definition 2.4: Two inputs U 1,u2 €H I are Nerode equivalent at time t with respect to operator T, written u I , T': t u 2 , iff
~ t ~ t P
t T P u
l P
t T P u 2
Clearly "T, t" is an equivalence relation over Hi' so that it is possible to form equivalence classes of HI under "T' t". This set of equi-valence classes, XT(t),' will be called the "state space of T at time t." Using a funda-mental theorem of modern algebra, Saeks [12] has shown that ESt T pt can be decomposed into the product of two mappings, kT(t) and gT(t) in such a way that the following diagram com-mutes:
H -H
lk~ ~)2 XT(t)
We then go on to form, for any LBSC operator T, a Hilbert resolution space of state trajec-tories which may be thought of informally as being of the form
x( .) where x(t) € XT(t) for each t € [to' tex> ]
We call this space XT
and define kT and gT es sentially as
(kT u)(t)
and
We prove that gT is memoryless (as has been stated by Saeks [12]), kT is strictly causal, and T = gT kT • The reader wishing a more rigorous treatment should refer directly to [8] as we have glossed over many important details here.
Having the required tools, we then go on to prove a theorem concerning the relationship of open loop control to state feedback control,
9
motivated by an analogous theorem by Hautus and Heymann [14] in the discrete time case. In order to quote this theorem, we will need a lemma due to DeSantis [13] which states that if W is LBSC, the (I+W)-I=I+V where V is LBSC. Then we have
Theor ern I: Given k T : HI -+ X T , kT an LBSC operator, and a dynamic input transformation (I+W):HI-+H I , where W is LBSC (therefore let (I+W)-I=I+V where V is LBSC).
Then there exists linear bounded memoryless state feedback F: X T -+ HI such that
v u + Fx x
and v (I+W)u iff
The above condition may be interpreted as saying that the state of V may be found as a part of the state of T for any input at any time. The systems governed by the above equations are diagramed below.
u x
( a )
u x
( b )
Figure
What Theorem I says is that when the state of V can be found in the state of T, the open loop controller of Figure I b can be implemented as a state feedback controller of the type shown in Figure lao The solution to the optimal con-trol problem will es s entially consist of deriving the appropriate open. loop controller and then showing that Theorem I applies, thus giving us the desired state feedback control.
III. THE OPTIMAL CONTROL PROBLEM
With the background given in the previous sec-
p
tion, we are in a position to state in full the optimal control problem we wish to solve.
The t Problem: Given
(1) Two Hilbert resolution spaces HI and
HZ;
(Z) An LBSC operator
(3)
(4)
find
T : HI -t HZ
t~
A forcing function f = T P u;
Z A cost functional J(u, t, f} = I P t u I H + IPt(Tu+f}l~ ; 1
Z
Uo e: P t HI :3 J(u, t, f} is minimized.
Note that the assumption f = T pt;: is equivalent to the assumption that the initial conditions of this problem are given as an initial state. In [8], we motivate the conjecture that all pro-blems of this type for which an optimal state feedback law exists can be expressed in this form by first noting that f can be decomposed
as
and that the optimal control for fZ' is identically 0, and then using Theorem I to suggest that a causal feedback solution only exists when f=Tpt;:.
The t-problem may be easily solved using the projection theorem (15] to yield
However, since T*T is neither causal nor anticausal, the causality properties of this solution are still very much in doubt. Much can be resolved, however, through the use of causal factorization theory.
We will now assume that (I+T*T) admits a strictly causal factorization; that is, we will as sume that there exists an LBSC operator V
such that
,-1+ T T
(therefore let (1 + V}-l = I + W where W is LBSC.) Then we have
-Pt(V + V* + V'''V} pt~ _p (V + V '''V) pt~
t
(V'" is anticausal)
t~
PT(I+V)UO
= -Pt YP u
(by multiplying by Pt(I+ W"-))
For notational Simplicity, we define
~
as the operator which maps the forcing input u
onto its optimal control uO •
Next we will prove that the (I + W) derived from factorization theory is exactly the dynamic input transformation needed to apply Theorem 1 to the optimal control problem.
10
Consider the feedback system of Figure Z
t~ P u
t~ u+P u
Figure Z
along with its associated equation
u t~
-Pt
V(u+ P u)
t~ (I + P
t V) u -Pt
Y P u
but, since ue:PtH I , we have
t~ ~
P t u::: - (I + ~Nl P t V P u = Pt@(t) u
Thus the above system solves the t-problem for t~
f=TP u.
The following lemma will be found to be the central reason for the use of the notion of "state" in the solution of the optimal control
problem.
Lemma 1: Assume the strictly causal factori-zati';,n 1+ T*T = (1+ V'''}(I + V). Then V t e: [to' tee ];
ul T,t uZ-'ul V':t uZ·
Proof: t t
Pt
T P ul
= P t T P U z ~, t ," t
Pt
T Pt
T P u l = P t T P t T P Uz
P T*T pt u = P T'~T pt u tIt 2
(T* is anticausal)
'" * t * * t Pt(VtV tV V)P ul
= Pt(VtV tV V)Pu2
* t * t Pt(V tV V) P u l = Pt(VtV V) P u 2
(V* is anticausal)
~ t * t Pt(ItV~)PtVP u l = Pt(ItV )PtVP u 2 or, multiplying by Pt(I t W*),
t t P
t V P u
l = P
t V P u
2
Further insight can be gained by multi-plying the above equation by (I t W) to get the equation
Although it is not used in the theorem to follow, it is the heart of the concept. In words, this equation states that the initial state carries all the infor mation nec es sary for the derivation of the subsequent optimal control. This is exactly why state feedback optimal control is possible.
Note, ~ow, that by Lemma 1 and the fact that (I t W)-l = (I t V), Theorem 2 applies to the dynamic input transformation (I t W). Thus we have proved the following theorem:
Theorem 2: For any system of the form
y = Tu t f
wher e T: HI" H 2 is a bounded, linear, strictly causal operator on Hilbert resolution spaces, and I + T*T admits a strictly causal factoriza-tion, there exists a bounded· memoryless linear
t~ state feedback F: 'X T" H; such that V f = T P u, ~ e HI' t e [to,tCDj, the control Uo whic~ mini-mizes J(u, t, f) = \ :t(T P t u + f)\ H2 t \ P t U\H l over the clas s of admls sable controls u e P t H 1 satisfies
Review of Proof:
(1) P Uo minimizes J(u, t, f) over the class of admissable controls u e PtH l where f = T ptu:
iff PtUO = Pt@(t)U:.
(2) PtuO=!\@(t)\I: if£~PtUO=-PtV(PtUot pt \1:) where I t T~T = (I t V~)(I + V).
(3) Using Theorem 1 and Lemma I, there exists F linear, bounded and memoryless such that -V = FkT; 1. e., the following diagram com-mutes:
II
where F = -gv h •
IV. CONCLUSIONS
Using the key concepts of causal factorization and state space under Nerode equivalence, we showed that for a bounded, linear, strictly causal operator on Hilbert resolution spaces with quadratic cost functional, the optimal control may be expressed in memoryless state feedback form provided the forcing function is expressed as an initial state. We further conjecture that there is no causal feedback which realizes the optimal control for a larger subspace of forcing functions.
[1]
[2]
[3]
[4]
[5]
[6]
(7]
BIBLIOGRAPHY
R. E. Kalman, "Contributions to the Theory of Optimal Control:' Bol. Soc. Mat. Mexi-~, 5 (1960), pp. 102-119.
M. C. Delfour and S. K. Mitter, "Controlla-bility, Observability and Optimal Feedback Control of Affine Hereditary Differential Systems,"SIAM J. Contr., 10 (1972), pp. 298-327.
J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Springer-Verlag, Berlin, 1971.
D. L. Lukes and D. L. Russell, "The Quad-ratic Criterion for Distributed Systems," SIAM J. Contr., 7 (1969), pp. 101-121.
R. Datko, "A Linear Control Problem in Abstract Hilbert Space," J. Diff. Equ., 9 (1971), pp. 346-359.
A. J. Pritchard, "Stability and Control of Distributed Parameter Systems Governed by Wave Equations," IFAC Conference on Distributed Parameter Systems, Banff, Canada, 1971.
R. Curtain and A. J. Pritchard, "The In-finite-Dimensional Riccati Equation for Systems Defined by Evolution Operators," SIAM J. Contr., 14 (1976), pp. 951-983.
[8] M. Steinberger, A. Schurnitzky and L. M. Silverman," Optimal Causal Feedback Control of Linear Infinite Dimensional Systems," submitted to SIAM J. Contr.
[9] W.A. Porter, "A Basic Optimization Problem in Linear Systems," ~ Sys. Theory, 5 (1971), pp. 20-44.
(10] R. E. Kalman, "Lectures on Controllability and Observability," CIME Seminar on Controllability and Obs ervability, Bologna,
Italy (1968).
[11] I. C. Gohberg and M. G. Krein, "On the Factorization of Operators in Hilbert Space," Am. Math. Soc. Trans., Ser. 2, 51 (1966), pp. 155-188.
[12] R. Saeks, "Resolution Space Operators and Systems," Springer-Verlag, New York, 1973.
[13J R. M. DeSantis, "Causality Theory in Systems Analysis," IEEE Proc., 64 (1966), pp. 155-188.
[I4] M.l... J. Hautus and M. Heymann, "Linear Feedback - An Algebraic Approach," Center for Mathematical System Theory, UniverSity of Florida, 1976.
[15J D. G. Luenberger, Optimization by Vector Space Methods, Wiley, New York, 1969.
12
., - / I
SOLVABILITY AND LINEARIZATION OF MONOTONE HILBERT NETWORKS
Vaclav Dolezal State University of New York at Stony Brook
Stony Brook, New York
Abstract
Sufficient conditions for solvability of a class of monotone Hilbert networks are given. Moreover, a linearization of a nonlinear monotone Hilbert network in a neighborhood of an operating point is discussed.
1. INTRODUCTION
In the first part of the paper we give sufficient
conditions for solvability of a Hilbert network
some elements of which are described by monotone
operators defined only on subsets of the underly-
ing Hilbert space. As a special case we consider
a finite nonlinear LRC-network, whose inductors
are linear, time-varying.
In the second part we discuss a linearization of
a nonlinear monotone Hilbert network in a given
neighborhood of an operating point, which is sub-
optimal in a certain sense. An estimate for the
difference between the exact and approximate
current distribution is established.
2. SOLVABILITY
A Hilbert network ~ is called solvable, if for any
excitation by EMF and/or current sources there
exists in ~ a current distribution obeying
Kirchhoff laws. Known effective results concern-
ing solvability ([1] or ThEDrems 4, 5 in [2J) make
the assumption that the operators describing
\3
network elements are defined on the entire under-
lying Hilbert space H. Consequently, these re-
sults are unapplicable, if the network contains
d1fferentiators.
The theorem given below attempts to fill this gap.
It is based on properties of maximal monotone
operators [3J, [4J and has the following physical
interpretation:
If all elements in a (finite or infinite) monotone
network ~ which are described by operators defined
on the entire space are removed and a unit resis-
tor is inserted in every branch, and if the net-~
work ~l thus obtained is solvable, then ~ is also
solvable.
To avoid repetition of definitions, we will use
the notation and concepts introduced in the survey
paper [2J.
Theorem 2.1. Let H be a real Hilbert space, let G
be a locally finite oriented graph having
c2
:s; ~O branches, and let D CH c2 , N; n D f ¢.
A C2 zl:n-H is a monotone operator,
A C2 C2 Z2: H - H is a hemicontinuous operator
such that A A IP
(Z2Xl - Z2x2' Xl - x2> ~ c II xl - x21 for all x~,
and p >1,
x e HC2 with some fixed 2
(2.1 )
c >0
(iii) the Hilbert network 711 = (21
+I,G) possesses C2
a solution for any e e H
Then, for any e eHc2 , the network ~= (Zl +Z2,G) possesses a unique solution in B corresponding to
e.
Using Theorem 2.1, we can prove a solvability re-
sult for a finite LRC-network with linear time-
varying inductors, whose underlying space is the
real space 12
[0,TJ (we will write 12 in the
sequel).
Definition. Let G be a finite oriented graph hav-
ing c2
branches, and let 1 S k S c2
•
(i) there exist loops sl, s2,
Assume that k
(ii )
•• , S such
that, for each m = 1, 2, ., k, the loop
Sm contains b and does not contain any m
other branch in the set tbl
, b2
, ••• , bk }, t(t) is a symmetric k x k matrix having a
continuous derivative t'(t) on [O,TJ such
that t(t) and t'(t) is positive definite
and positive semi-definite, respectively,
for each t e: [0, T J, A C c
(iii) T· 1 2 -1 2 is an operator • . 2 2A k c -k c
Let the operator 1 : K x 1 2 - 1 2 be defined by 2 2
(Lx)(t) = (1(t)x(t)} I, (2.2) where 1(t) = Lr t(t) OJ is a c x c matrix, and K
° ° 2 2 is the space of all absolutely continuous functions
on [O,TJ. Then the Hilbert network 11 = (L+T,G) will be called 1-proper.
Clearly, this definition requires that all induc-
tors (and possible mutual couplings) are confined
only to the branch~s bl
, b2
, ••• , bk
• Also note
that the operator T describes the behavior of both the resistors and the capacitors in the network.
Theorem 2.2. Let G be a finite oriented graph
14
having c2
branches, let lSk S C2
and let j , j , A c c l 2
••• , jk be real numbers. Let T: 1/-122 be a
hemicontinuous operator such that, for some c > 0
and p >1, we have
A A liP (TXl - TX2 , Xl - x2) ~ c Ilxl - x2 C2 A A A for all Xl' x
2e:1
2 , and let 71=(1+T,G) be 1-
c A proper. Then, for any ee:1
22 , '1lpossesses a
unique solution i = [i l corresponding to e n-o k c2-k 0 () 0 1 e:K X1
2 and 1 0 =J for m=l, 2, •.
m m
such 1h1:t
• , k.
Observe that Theorem 2.2 has the following
physical interpretation: if a network '1l satis-
fies the hypothesis, then for any EMF's el
, e2
,
••.• ' ec e: 12 and any values jl' j2' ••. , jk
there exi~ts in ~ a unique current distribution i
l, i
2, ••• , i e 12 such that, for m = 1, 2,
c2 ., k, im is absolutely continuous and satis-
fies the initial condition i (O)=j • m m
3. 1INEARIZATION
The problem dealt with in this part resembles the
small-signal analysis of nonlinear networks [5J,
but differs from it in several aspects. While in
the small-signal analysis a "strictly local" ap-
proximation (Frechet derivative) is used for ob-
taining an approximate solution, in the present
approach we use a certain "global" approximation.
Also, the Hilbert network setting is, of course,
more general.
To explain the underlying idea, consider a Hilbert
network ~= (i,G). Assume that we know the current
distribution iO in '1l corresponding to some EMF
vector eO (operating point), and that we seek dis-
tributions ie*' which correspond to excitations
eO +e*, where the e*'s satisfy the inequality
Ile*!1 Sr with some given r >0. To find approxima-
tions to the ie*'s, we linearize the network '1l
in a vicinity of iO
• More specifically, defining ,... I ,. I '" A
the operator Z by Z S = Z( iO +S) - ZiO' we assume
that we can find a linear operator Zo which satis-
fies the inequality liz 'e: -zosii sail e: lion a certain
ball centered at the origin, where a >0 is not
* t~~ large. If ie* is the s~lution of the linear netw~rk ~o= (Zo,G) corresp~nding to e*, we will
take io + i* * as an appr~ximati~n to the solution A e
i * ~f 'TI. e
The overall relative error
A = sup II i *- (iO+i** )11·lle* i1-1 r lIe*ll~r e e
e*f 0
(3.1)
depends, of course, ~n the choice ~f ZO' and con-
sequently, on the constant a. However, in speci-
fic cases ~f networks it is usually not difficult
to c~nstruct Zo s~ that, for a given r >0, a is
small en~ugh. In this context, let us emphasize
the fact that taking the Frechet derivative of
Z at iO f~r Zo (a counterpart of the small-signal
analysis) need not lead to the smallest values of
a.
It turns out that if Z is strongly monotone, we
can give a simple upper bound for Ar' which is
roughly prop~rti~nal to a for a small.
The theorems that follow use again the notation
introduced in [lJ or [2J. Without loss of gen-
erality we assume that the operating point eO and
the c~rresponding solution iO of ~ are zero. Also
note that Z is not required to be single-valued.
Theorem 3.1. Let H be a real Hilbert space, let
~ = (Z ,G) be a Hilbert network with Z being 'a set c
mapping defined on a nonempty subset D CH 2 such
that 0 € D, and let r >0. Furthermore, let A c c
F=X*(NAnD)CH 0, and let W :F ... e(H 0) be defined a A*A.A
by W = X zt.. Assume that
(i) there exists b >0 such that
(Yl -Y2' xl -x2)~b Ilxl - x2112
for all xi €F, Yi €WXi ' i=l, 2,
(ii) there exists a linear bounded operator
(3.2)
c c ZO:H 2"' H 2 and a constant a with O
Spaces and Some Applications to Nonlinear Partial Differential Equations, Contribu-tions to Nonlinear Functional Analysis, E. Zarantonello (ed.), Acad. Press, 1971, pp. 101-156.
[4J R. T. Rockafellar, On the Maximality of Suma of Nonlinear Monotone Operators, Tran& Amer. Math. Soc., 149 (1970), pp. 75-88.
[5J B. Feikari, Fundamentals of Network Analy-sis and Synthesis, Prentice-Hall, Inc., 1974.
16
j
TIME-VARYING INPUT-OUTPUT SYSTEMS WHOSE SIGNALS
ARE BANACH-SPACE-VALUED DISTRIBUTIONS
A. H. Zemanian State University of New York
Stony Brook , N. Y. 11794
Abstract
The composition of an operator-valued distribution and a Banach-space-valued distribution is established by extending Schwartz's kernel theorem in such a fashion that it has the form of the Cristescu-Marinescu composition of scalar distributions. This provides a representation for many linear continuous time-varying systems.
1. INTRODUCTION
The idea of a time-varying Banach system, which
was defined and analyzed in a prior work [10], led
to a study of composition operators acting on
spaces of distributions that take their values in
Banach spaces. Two types of composition were con-
sidered in [10]. The first, which was called
"composi tion • " makes use of Schwartz's kernel
theorem [8], [11] and its extension to Banach-
space-valued distributions. It provides an explic-
it representation n = f. for every continuous linear mappingnof £leA) into [t); B] by means of
the composition product f-v, where .f is a distri-
bution on the real plane taking its values in
[A; B] and v~D(A). Here, A and B are complex
Banach spaces, [A; B] is the space of continuous
linear mappings of A into B,1)(A) is the space of
infinitely differentiable (i.e., smooth) A-valued
functions of compact support on the real line
supplied with its customary topology, 1:> = D(C), C being the complex plane, and PD; B] is the space of B-valued distributions equipped with the topo-
logy of uniform convergence on the bounded sets
of 1). A shortcoming of this representation for n is that it is defined only for certain suitably
restricted continuous functions v and not for
singular distributions v.
17
The second type of composition, which was called
"composition 0" in [10], is an extension to Banach-
space-valued distributions of a composition product
first introduced by Cristescu [2] and subsequently'
developed by Cristescu and Marinescu [3], Sabac
[7], Wexler [9], Cioranescu [1], Pondelicek [6],
and Dolezal [4]. In contrast to composition. ,-
not all continuous linear mappings ofi)(A) into
[D; B] can be represented by a compositiono opera-
tor. However, composition 0 has the virtue that,
when it does exist, it can be applied to singular
distributions.
The present work is aimed at this gap between com-
posi tion. and composition o· . A formula is given
for extending composition. , which we henceforth
refer to simply as "composition", onto singular
Banach-space-valued distributions by using
Cristescu's form of composition. The idea is as
follows. For the sake of simplicity let us con-
sider the case where f is a scalar distribution on
the real plane and v and ~ are members of~. Let
us also assume that f is of finite order so that
f(t,x) = (-D )i(_D )rh(t,x) where h is a continuous t x
function on the real plane and Dt
= J /d t. (These
assumptions will not be imposed subsequently.)
Then, by the composition arising from Schwartz's
£i£ ;is
kernel theorem, we formally write
(t)D:V(x)dtdX
= S(D~V(x)Jh(t,X)D!q,(t)dtdX
=
= (v(x) '(Yx(t) , Ht»>
(1.1)
where Yx is a mapping of!) into C and hopefully a
distribution depending on the parameter x. The
important thing is that the right~hand side of
(1.1) is precisely Cristescu's form of composition,
and therefore, if it turns out that (Yx,q,) is a
sufficiently smooth function of x, then (1.1) pro-
vides a means of extending the operator f· onto
some space of singular distributions v. This
paper gives a rigorous presentation of this idea
and overcomes the complications that arise when
the assumptions that f is of finite order and
that f and v are scalar-valued are dropped.
The present work extends the discussion in [10] in still other ways. For instance, this discussion
encompasses the larger class of composition oper-
ators f - that are defined only on finite-order
distributions v [5], in contrast to the smaller
class considered in [10] of operators defined on infinite-order distributions v. In addition, our
present results imply an estimate on the order of
the resulting composition product, something that
was not available in [10]. Another generalization is that distributions on multidimensional euclid-
ean spaces are now allowed, whereas [10] was re-stricted to distributions v on the real line.
It should also be pointed out that, although
Schwartz's extensive work [8] discusses the com~
position product f-v of vector-valued f and
scalar-valued v (see pages 124-126 of volume 7 in
[8]), it does not discuss the problem attacked
herein, namely, that of defining f.v, where f is
an operator-valued distribution and v is a Banach-
space-valued distribution, both of which may be
singular and of arbitrarily large order.
2. NOTATIONS
The notations used in this work are the same as
those of [11]. The reader should refer to that
18
work (especially to Section 1.2 and the Index of
Symbols) for the definitions of any symbols not
defined herein. There is one exception however.
In this work we will assign a more specialized
meaning to the symbol RS as follows. m will always e denote an ordered s-triple {ml ,m2,···,ms }' each
component of which is either a nonnegative integer
or 00 For example, {2,~,0} is such a 3-tuple
RS denotes the collection of such s-tuples e
Throughout this work we will always have m~R: and . n JERe'
pS is s-dimensional real euclidean space. K will
be a compact interval in RS , and K will be its in-
terior. If HRI or T=CO~ the notation [T] denotes
the s-tuple all of whose compontents are equal to
T. However, [0] is denoted simply by O. A and B always denote complex Banach spaces with norms
V-itA and f{ '/IB
respectively. If U and V are
two topological linear spaces, the symbol [U; V] denotes the linear space of all continuous linear
mappings of U into V. Unless the opposite is
explicitly indicated, we always assign to [U; V] the topology of uniform convergence on the bound-
ed sets of U, which we call the "bounded topology".
Thus, for instance, [A; B] is assigned its opera-
tor-norm topology.
Let q, be a function from RS into some
Banach space. When we say that q, is continuous
or has a derivative, it will always be understood
that the continuity and derivative are with re-
spect to the norm topology of the Banach space.
Thus, for example, if the Banach space happens to
be [A; B], the said continuity is with respect to
the operator-norm topology of [A; B]. Let
k = {k ,"',k } be a nonnegative integer in RS I s
(i.e., every component kv is a nonnegative in-
teger.) Any {partial) derivative
J Ikl q, -k k-d t 1 ,)t s s I k I = k 1 + ... +k s
. k Ikl of q, wlil be denoted by D q, = q, . We will also write D~ q,(t) = q,(k) (t). We shall refer to k (and not Ikl)as the order of the differential
operator Dk. The notation Ikl should not be
confused with the magnitude notation for the
members of RS
•
The support of any function or distribution f on
RS is denoted by supp f.
We will use the standard function spaces ~~lA), .DR~(A), c~~(a), and (?~(A), equipped with their
customary topologies. All of these are defined in
[10] and also in [11]. When m=oo, we drop the superscript m. Similarly, we drop the subscript
RS to write 1J meA) = nm(A) and E. meA) ;: [meA) AS R'
whenever there is no need to specify the euclidean
space RS •
Schwartz's Kernel Theorem. n is a sequentially continuous linear mapping of b,(A) into [DR"; B]
if and only if there exists an 'f € [.DR"+s (A); B] such that flv = f.v for every v~ D (A). Here, f is
R' uniquely determined byn, and conversely. The
composition product fov is defined by
= •. (A), ttRn , and xliRs . R" R
A proof of this theorem is given in [11; Chapter
4] .
3, COMPOSITION OPERATORS ON [em ;A] INTO [~j ;B]. . RS R"
Lemma 1. Given any v£[em; A], define q by . v
(q , Fe) = F(v, e> v
(3.1)
where F~[A; B] and eED. Then, there exists a
unique Vf[em+[2] ([A; B]) whose restriction to the elements of J)( [A; B]) of the form Fe, where
F€[A; B] and e~~ coincides with qv' Moreover,
supp v = supp v. In addi tion v ~V is a sequent-ially continuous linear injection of [em; A] into [Em+ [2] ([A; B]); B].
The proof of Lemma 1 is quite similar to that of
Lemma 4-2 of [10]. The assertion concerning the
supports is an easy consequence of the Hahn-Banach
theorem.
Next, let VE£m(A). Then, the same arguments as
those used for Lemma I show that there is a unique
V€[~m+[2]([A; Bl); B] which coincides with q on v
all elements of the form Fe. Furthermore, v gen-
erates a unique member of [~o; A], which we also denote by v, through the definition:
(v, ~> : J v(x)~(x)dx ~EDo. It'
19
(Here, the superscript ° in ~O denotes the s-tuple [0].) Moreover, v defines an [[A; B]; B]-valued function v' on RS by the definition:
v'(x)F = Fv(x) Fe[A; B], x~Rs. By the Hahn-Banach theorem, supp v' supp v and,
for each x, v(x)~v'(x) is injective. Some
straight-forward arguments establish
Lemma 2. Let v€~(A). Then, for every integer
k€Rs with O~k~m, we have that Dkv' exists, is continuous from RS into [[A: B]; B], and therefore
generates a regular member of [.Do ([A; B]); B].
Moreover,
k k D v'(x)F = FD vex) for every F€[A; B]. Finally, v'=v in the sense of
equali ty in [1)[2] ([A; B]); BJ, and supp v' = supp v. = supp v'. Now, let fEr .DRn,,$(A); B]. Choose any two compact
intervals IeRn and VCRS. It can be shown that
there exist a jeRn not depending on I, an e
hfe~KL ([A; B]), and two nonnegative inte~ers itRn
and n;Rs with i::;j such that, for all ~Et>i and ve.f)L (A) ,
= (i) (t)dt]v(r) (x)dx.
Assume that the inner integral has continuous de-
rivatives on L of at least order r+[2]. Then, we
mar integrate by parts Irl times to obtain
Sr,fn:rrh(t,x)+(i) (t)dt]v(x)dx
jir ~ (-1) 'rl pr x x
" Let us now assume in addition that supp vC L. By virtue of Lemma 2 and its notation, we may write
= S"v' (x)n: SIh(t,X)cl>(i) (t)dtdx
=
u
Assumption. AssuMe tl·at c{)rresllondi nr to a .given n I m£P~ f"('r
f"t.{DR" .. CA); P.] tt..ere is a j€Pe am an e Nhich the f"ollol-'inr condition!': arC' .
Causality and Co Operators
Avraham Feintuch Ben Gurion University of the 'legev
Beersheva, Israel
Abstract It i~ shown that the class of C Contractions appear in a natural way in the algebra of causal
linear operators. Stability propertigs of such operators are studied.
1. Introduction:
The appearance of linear transformations 'J
which operate On infinite dimensional Hilbert
space has become quite common in systems theory
today. This is true both in the state space
theory and the input-output theory. In particular,
a significant number of authors ([1], [Il], [10],
[11], [14]) have noticed the relevance of the
Nagy-Foias model theory for contractions in
systems theory. An important suhclass of these
operators is the family of Co contractions. These
seem to be a quite natural ,generalization of
finite dimensional operators and such basic
finite dimensional concepts such as minimal
polynomials and .Jordan c,anonical forms have a
natural extension to the infinite dimensional
case.
The usefullness of these operators and the
unilateral shi ft operator to \~hich they are
closely related has been noticed by a number of
authors. Common to all of them is a state-space
approach.
My purpose here is to show how these
operators appear in a natural way in causality
structures and play an important role in the
theory of stability of linear feedhack systems.
~fost of the results presented here have appeared
in the literature in an operator theoretic setting.
However, this is, to my knOldedge, the first time
that they are presented from the point of view of
systems theory.
2. Causality:
Causality is usually descrihed on an complex
Hilhert space II in terms of a resolution space
21
structure. This is descrihed in detail in [17].
Here \~e present a slightly different formal ism
first descrihed in [17]. This will make it simpler
to translate operator theoretic results directly
into a systems theory framework.
'" Definition 2.1: A family:~ of suhspaces of II is a nest if it is totally ordered hy inclusion. q is complete if
(i)
(ii)
.A. {®}, fL'E N
forNoc~, n N :-.ltNo
and VN NtNo
arc in N.
for any 'I t ~l, ~I wi 11 denote the subspace
V (L : L t 'N and L c ~I). with (0) = (0). If ~ #~, N is called the predecessor of ~I.
1\ Definition 2.2: A nest N is maximal if
"\ (i) ~ is complete
(ii) for il E ~, dimel e '1)< 1
Definition 2.3: A nest space is a pair (H, ~) consisting of a lIilhert space II and a maximal
nest :-.l of suhspaces of 11.
PM will denote the orthogonal projection
with range '1.
The nest space structure allows us to define
causality in an ahstract setting.
" Definition 2.4: Let (II, 'J) he a nest space. A bounded operator T on II is causal if 1':/ = 1'~ITP: I
A fOT all n E '\. T is ant i-causa 1 if Tl'q = 1':'ITP:'1 and memorvless if P.IT = TP~I for all :'f E ~'i.
rxaMple 2.1: The classical context of causality
is h'hen II = L2(-oo, co) and ~ = 1.2(-
i;
coincides with the classical definition. It should /to
be noted that if H £ ~,~I ~I. ~ests which have
this property are called continuous. Throughout
this paper we will consider continuous nests
unless stated otherwise. This is done since the
corresponding results in the discrete case are
quite easily obtained.
3. Shift Invariant Subspaces:
Let m denote measure on the unit circle n
in the complex plane normalized so that men) 1.
L 2 will denote L 2 (1jI). Every f t L2 has a Fouries
f(e iO) = ! ina which converges to f in the n"-CO c e
2 2 n L -norm. H will denote the subspace.
112 = (f t L2 : f(e iO ) = ~o c e ine) . n- n i. e. f t H2 if and only if its negative Fourier
coefficients are zero. The unilateral shift on 112
is the operator.
(Sf)(eiO) = eiof(eiS ).
. The basic facts about H2 and S can be found in
[12]. Any 112 function f has an analytic extrnsion
to the unit disc D = (z : Izl < 1) whose value. at
z will be denoted by fez).
The invariant subspaces of S were described in a
well-known theorem of Beurlinp,. For this we need
the notion of an inner function.
Definition 3.1: A non-constant function q analytic
in n is called inner if Iq(z)1 ~ 1 and Iq (eiS)I=1 almost everywhere on ~
Inner functions playa major role in the theory
of Co contractions and I will therefore briefly
discuss their structure. An elegant and complete
treatment can be found in [12].
If is inner then it can be factored as
rj>fil) = exB(A) s(A), Inl = 1
\~here B(A) is the Blaschke product determined by
the zeroes of rf> inside the unit disc and SeA) is
of the form _f2~eit+A
exp "t dUt , o e -1 + A S (A)
u being a non-negati.ve finite singular measure.
It is now possible tostate Beurling's Theorem
and to di.scuss its importance for cansality.
TZZT
Theorem 3.2: Let S be the unilateral shift on H2.
Then ~I c H2 is an invariant subspace for S if and only if there exists an inner function rf> such
that ~I = rf>H2 = (rf>f : f t H2). It will be more conventient to work with S*
A the adjoint of S. Every continuous chain N of
invariant suhspaces of S* has the form
~ft = (rf>t 1I2)"l where rf>t is inner. Moreover, each rf>t is a singular inner function since the presence
of a Blaschke product would introduce a jump "-
discontinuity intoN.
The importance of such chains is that, up to
unitary equivalence, they are universal.
Theorem 3.2: [13]: Every continuous chain of
suhspaces (in a separable Hilbert space) is
unitarily equivalent to a continuous chain of
invariant subspaces of S*.
22
A
Thus given a nest space (H, N) with H
separable, infinite-dimensional we may as well 2 " 2 .l} assume that II = H and that N = {(rf>t II)' for
some chain {rf>t} of inner functions.
It is worth noting that the basic theory of
inner functions leads to the fact that
rf>l H2 c rf>2 H2 if and only if rf>2 divides 4>1' Thus
(rf>l 112)1;.:::> (4)2 112).l if and only if rf>2 divides 4>1.
It is then quite natural to use the expression
"a chain of inner functions". /'.
The identification of (H, N) with
(112, {(rf>t H2).l}) will allow as to show that the
algebra of causal operators contains many Co
operators.
4. Co Contractions:
In this section we present the relevant
material from the theory of Co contractions. No
attempt at completeness is made. A complete
treatment is given in [19, Ch. 3].
Let T be a contraction (IITII ~ 1) on H. Then
9[19], Ch. 1) there corresponds a decomposition
of II into an orthogonal sum of two subspaces
reducing T, say II = "0 III HI' such that the part
of T on "0 is unitary and the part of T on "1 is
completely non-unitary. This means that Ttlll has
no non-zero reducing subspace L for which T\L is
a unitary operator. This decomposition is unique
and 110 or HI may equal the trivial subspace (0).
We will concern ourselves with the case that
110 = (0); i. e. T is completely non-unitary. By,,"" we denote that algebra of bounded
analytic functions on n with the usual norm (or 0;,
equivalenty, the subalr,ebra of L consisting of
those functions whose negative Fourier
coefficients vanish). Then if T is'c.n.u.
(completely non-unitary) it is possihle to define 00
the operator Ijl(T) for all ¢ t II
Oefinition: Co is the class of completely non-
unitary contractions T for which there exists a
non-zero function UtII'''' such that u(T) = o. u £ I~ can he factored into u u. with u an
e 1 e outer function and u. an inner function. It is
1
shown in [19] that if u(T) = n then Hi (T) = O.
\lso if T is of class COlT has a minimal function
mT
Ivhuch divides every flinction u such that
ueT) = O. This function is determined up to a
constant factor of modulus 1. In the sequel the
spectrum of T will play an important role. This
is completely determined hy mT
. Let
m,.().) = fl(A)S(X) be the factorization of mT
•
Theorem 4.1: [19, p. 126]: Let fTIT he the minimal
function and aCT) the spectrum of the contraction
T of class Cf)' Let 5r he the set consisting of the zeroes of mT in the open unit disc n rl.nd of the compliment, in the unit circle lT of the union
of nrc! of lTon which mT is analytic. Then
aCT) = ST' The simplest examples of Co operators are
given in the following theorem.
Theorem 4.2 [19, p. 124]: Let he a non-constant
inner function and let 14 = (dlH2).L in 112. Then
the operators p~lsl~1 and S*I~1 he long to Co and
their minimal function is
~low we return to callsa 1 i tv. A.s seen ahove, 1\ . • if (II, ~) is a nest space, we Crl.n assume it is
(112 , {(¢t ,,2):-}).
the
2'" suppose ~I = (
difference of the system. It is easy to see that
(I - KF)-l exists if and only if (I - FK)-l exists
and one is causal if and only if the other is.
Stability of feedback system is usually
defined in terms of extension spaces (see [17],
p. 65). Fortunately an important theorem of
IHllems ([20]) allows us to avoid this approach.
IHllems result will be used as our definition of
stability. What is lost in the intuitiveness of
the extension spaces is more than made up for in
mathematical simplicity.
Definition 5.1:
rhe feedback system (1) is well posed and -1 -1 stable if (I - KF) (equivalently (I - FK) )
exists as a bounded operator on " and is causal on
(II, N).
Here we consider the case where the return
difference ( I - KF~ is a Co ·contraction. While
we do not have a complete solution the result to
be presented.
Definition 5.2:
A contraction r on a separable IIilhert space
is essentially unitary if hoth I - r*r and
I - TT* are compact.
I~ is worth making some remarks ahout the
operators I - r*r and I - TT* for a contraction T.
rhe square roots Dr = (I - r*T)1/2 and Dr * = (I - TT*)1/2 are called the defect operators for r, the closure of their ranges DT, Dr * are
called the defect spaces and dr = dim DT, ~* = dim Dr * are called the defect indices of T.
It is easy to see that dr = 0 characterizes the isompetric operators and dT = dr * = 0 characterizes the unitary operators. Thus, the
defect indices measure, in a sense, the deviation
of the contraction T from heing unitary.
Another way of looking at this is the
following. The unilateral shift defined in 3 is in
a certain sense a universal operator. If T is a
contraction such that rn ~ 0 strongly then T is
unitarily equivalent to a compression of some
multiplicity (possihly infinite) unilateral shift
24
to some co-invariant suhspace. ([19]).
Our restriction means that T is related to
a unilateral shift of essentially multiplicity.
In the case of essentially unitary Co contractions we always have l~el1 posedness and
stability ([17J).
Theorem 5.3: I f (I - KF)-l is hounded and I - KF
is an essentially unitary Co contraction, then the system (1) is well posed and stahle.
6. Strict Causality:
It turns out that for most desired properties
of feedhack system the assumption of causality of
K and F is not enough. Thus the concept of
causality was strenghed in various directions hy
a number of authors (see, for example [21J, [2J,
r33]). One particularly useful direction is the
concept of strict causality introduced by
De Santis, Porter and Saeks ([2J, [4J, [17J). One
of the rather surprising aspects of strict
causality is tlJat Nhile it is in some sense
natural for linear systems it turns out to be
important for non-linear systems as well ([12],
(5]) .
The "property of causality for an operator
means, in the finite dimensional case, that it has
a lONer triangular matrix representation. Strict
causality Nill reduce, in this case, to a
strictly lower triangular matrix repersentation.
We present the concept very hriefly and refer the
interested reader to [17J.
Let E he the set of projections onto the
memhers of N. By a partition P of E is meant a
finite suhset.
P = {Ei : 0 < i ~ n} of E such that
... < E = I. n
Ai will denote the projection El - Ei _l • A
partition PI is a refinment. P ~ Pl' Note that the
partitions of E form a directed set under refinment.
Let T he a hounded operator on II and P any
partition of N. Form the sum. Lp(T) =i~l Ei _l TAi
Theorem 6.1 ([6], [17]): T is causal if and only , , for any E > 0, there ,exists a partition P of such
that for any refinement PI of P
IIJJp (T) II < s: • 1
It is worth noting that this is wquivalent
to strong convergence of Lp(T) to zero; i.e.
causality = strong causality. This will not be
the case for strict causality which we define now.
Definition 6.2: Let T be a causal operator on
" n (H,N). Let Vp(T) = E ~. T~. i=l 1 1
T is strictly cousal if for any E > 0 there
exists a partition P of E such that for any refinement PI of P,
IIVp (T) II < E. 1
Intuitively this means that the elements of
the diagonal of the matrix of T with respect to
any partition are zero.
It is of interest that the property of strict
causality defines the spectrum of T.
Theorem 6.3 [6]: If is strictly causal, then T is
quasinilpotent; i.e. the spectrum of T consists
of the point {oJ.
An immediate consequence of this is that if
the open loop gain of the system (1) is strictly
causal, then the system is well posed and stable.
We now return to Co contractions and consider
the problem of classifying the strictly causal
operators of this class. As mentioned above. the
spectral properties of such operators play an
important role. These were summarized in Theorem
4.1. The only quasinilpotent Co contractions are
the nilpotent ones. Of equal interest is the case
where T has a spectrum consisting of a single
point which is not necessarily O. An examination
of the spectrum of T allows us (up to a similarity)
to two possibilities:
(i) aCT) 0
(ii) aCT) 1. In the first case the minimal function of T is
just u(t) = zn. In the second it is . z+l
u(z) = exp a(~).
Both situations are included in the next
theorem.
Theorem 6.4:(£7]) If T is caulal and I - T*T is
compact. then A - T is strictly causal if and only if aCT) = A.
At this point I would like to mention two
open prohlems in strict causality, the first in
the context of Co contractions and the second in
more general context.
(1) Can Theorem 6.4 he extended; i.e. Can the
condition I - T*T be dropped? I conjecture that
the answer in general is negative though I'm not
sure why.
(2) Suppose T is a nilpotent causal operator.
Is T strictly causal. The answer is yes ifAthe
nest fJ· is discrete (dim M 9 ~I = 1 for ~1 E N).
7. Strong Strict Causality:
Definition 7.1: Let T he a causal operator on
(H,~)
for a partition P. T is strongly strictly causal if for any E > 0 and x E II there exists an
partition P of E such that for any refinemeT.t PI
of P,
IIVp (T)xll < E. 1
Clearly strict causality inplies strong
strict causality. The converse does no~ hold.
An example that shows this as well as the fact
that strong strict callsality is a more natural
concept will he given helow. First however, I
would like to mention where this concept prove<
useful in Systems Theory.
C Causality plays a role not only in stability
prohlems hut also in state decomposition for
non-time-invariant systems (see [17]). ,\s ~':lS
shown hy Saeks ([17], p. 130) in the case ~here
T is strongly strictly causal, a minimal state
decomposition gives a complete set of invariants
for T.
Example 7.2: Let II = [2(0.00) the space of
sequences {an}oon=o such that r la 12 < 00. n=O n
For sequence (a O' aI' 8 2 , ... ) let s(ao' aI' a 2 ,
) = (0, aO' aI' ••• ). This is just the
previously mentioned unilateral shift in a
different context.
25
n Let N - {V 8 i - M } where e i is the i-th i-O n
co-ordinate vector. Then S is causal. It is not
strictly causal since a (5) • h I I z I " 1l. lIowever S is strongly strictly causal. For the matrix of S
with respect to {e } is n
/ 0 ..........
f 1 0
\ 0 0
L~t E ~ 0, and x = (aO
' al' ••• ,) • Then choose .. ~ .. uch that :.\a \2 < c.
~ n
Let P be the partition {o'. PI' ••• , PN, J} where 1
Pi is the projection on ~ ek •
n Then r ~.S~. has the matrix
i= 1 1 1
n ....... ..
o
o o
where the 0 in the upper left corner represents an ~ x 'i hlick.
If dim /I < 00 then aCT) • {O},
Note that for T a Co-contraction any
eigenvalue A of T mllst have IAI < 1. This is not always the case. One need only
take 5* which is strongly strictly anti-causal
(n dual property) and has a large point spectrum
{z I I z 1 < I}. I'le wi 11 present what we hope is a reasonable hypothesis.
Prohlems 7.3:(1) If T is strongly strictly causal
then the point spectrum of T has only one
component; Le. T has no more than one isolated
eihenvalue:
(2) If T is Co with I -T*T compact and
o(T) c {Izl I} then T is strongly strictly cousal.
(3) Can the assumption I -T*T compact be sropped?
8. Conclusion:
have tried to show that Co cOntractions
are an important class of operators for the study
of problems in linear feedback systems. Under the
assumption that the defect operators of such a
contraction are sufficiently small we have seen
that systems involving Co coptractions are well
behaved. I have also mentioned a numver of open
quest ions I~hich I feel are worth considering and
are basic to the understanding of the behavior of
linear feedback systems.
Bibliographr
[1) ? .J.S. Baras and R.W. Brockett, "H--functions
Thus n and infinite dimensional realizat ion theory", l: fli S fli x = (0, .... aN' aN+I , ... ). i= I SIAl-I .J. Control. n
It follows that I I l: fl. S fl. xl I < E. This also ;=, 1 1 dearly holds for any refinement of P.
~e note that for strongly strictly causal
operators the spectrum can be quite large. 1I0liever
the propert), that we would hope carries over from
the finite-dimensional space is that the point s
spectrum can not consist of more than one point.
This is in fact the case for the class of Co
contractions that we considered ([S»).
Theorem 7.3: Suppose T is a Co-contraction which
is strongly strictly causal. and su~h that I-T*T
i~ compact. If dim H = "". then o(T" el' z I = 1 }.
(2) R.H. DeSantis, "Causality for nonliflear
systems in llilbert space". ~Iath. Sys. tho 7, No.4
(1974), 323-337.
[3] , "Causality, strict causality,
and invertibility for systems in lIilhert resolution
space, SIAII .J. Control 12. No.3 (1974),536-553.
[4] R.H. DeSantis and N.A. Porter. "On time-
related properties of non-linear system, SIAH .J.
App!. Hath. 24, :-lo. 2 (1973)'. 188-206.
[5) --------------, "On the analysis of feedhack systems with a polynomial
plant", Int . .J. Control 21. No.1 (1975), 159-175.
26
[6] A. Felntuch, "Causal alld strictly causal operators, to appear •
[7] ------, "on stability", to appear.
Co operators and feedbaCk1 I
[8] ------, "On strong strict causality for operators", to appear.
[9] P.A. Fuhrmann, "Realization theory in Hilbert
space for a class of transfer functions, J. Funct. Anal. 18, No.4 (1975), 338-349.
[10] ------, "On realization of linear
systems and applications to some questions of
stability", Mat. Sys. Th .• 8, No.2 (1974), 132-141.
[11] J.W. IIilton, "Discrete time systems,
operator models and scattering theory" .J. Funct.
Anal. 16 (1974), 15-38.
[12] K. 1I0ffman, "Banach Spaces of Analytic
Functions", Prentice lIa11, Englewood Cliffs, N.J. 1962.
(13] T.L. Kriete, "Fourier transforms and chains
of inner functions", Duke Math. J. 40, No.1
(1973), 131-143.
(14] N. Levan, "The Nagy-Foias Operato~ ~iodels,
Networks, and Systems",
IEEE Trans. on Circuits and Systems, Vol. CAS-23,
No.6 (1976), 335-343.
[IS] W.A.Porter, "The conunon causality structure
of multilinear maps and their multipower forms",
J. Math. Analysis and Applic. 57 (1977).
[16] W.A. Porter and R.M. De Santis, "Linear
systems with mult iplicative control", Int. J.
Control 20, No.2 (1974), 257-266.
[17] R. Sacks, "Resolution Space Operators and
Systems", Lecture notes in Economics and Hath.
Systems 82, Springer-Verlag 1973.
[18] D.Sarason, "A remark on the Volterra operator,
J. Hath. Analysis and Applic. 12 (1965), 244-246.
[19] B. Sz.-Nagy and e. Foias, "lIarmonic Analysis of Operators on Hilbert Space", North-Holland,
American Elsevier, New York 1970.
[20] J.e. l1i 11 ems , "Stability, instability, invertibility and causality", SIAM J. Cont. 7
(1968), 645-671.
[21] , Analysis of Feedback Systemd, Ca,brodge, ~nT Press, 1971.
27
WIENER-HOPF TECHNIQUES IN RESOLUTION SPACE
L. Tung and R. Saeks Dept. of Electrical Engineering
Texas Tech University Lubbock, Texas 79409
I. INTRODUCTION
Wiener-Hopf filtering is a widely used
technique in certain kinds of optimization
problems. The purpose of this paper is to
formulate Wiener-Hopf filtering in abstract
spaces (reflexive Banach resolution spaces)
and to examine problems involved for the
formulation and the solving of the Wiener-
Hopf filter.
Referring to what has been done in the fre-
quency domain of the classical Wiener-Hopf
filteringl, we've found five major problems
for the formulation of Wiener-Hopf filter-
ing in abstract spaces. They are
i. Random variables in abstract spaces
ii. Causality
iii. Operator factorization
iv. Operator decomposition
v. Optimization.
These problems are briefly introduced as
follows:
i. Random process can be thought of
as a random variable which takes values in
a function space. In order to do so, we
need an adequate probability measure over
the space involved. Fortunately, this
kind of probability measure has been de-. 2 F fined over metr1c space or our pur-
poses, we assume that the space involved
is reflexive Banach space, not only be-
cause this kind of space possesses nice
properties but also because stochastic con-
cepts such as "mean" and "variance opera-
28
tion" can be defined therein. Random vari-
ables taking values in reflexive Banach
space is discussed in section II with pro-
bability measure assumed implicitly.
ii. Concepts of causality have been
introduced into Hilbert space-the so-called
Hilbert resolution space3 • In section III,
we extend the works done for Hilbert. space
to Banach space. Concepts of causality,
such as causal, anti-causal, miniphase and
maxiphase, are defined. Emphases are given
to reflexive Banach resolution space.
iii. Operators to be factorized in the
form of KK*, where K* denotes the adjoint
of K, have to be "positive"and "self-ad-
joint". These commonly-used properties
among operators on Hilbert space can be
extended to operators which map reflexive
Banach space to its dual space. Factoriza-
tion theorem is given in section IV. Fac-
tor operator K is required to be left-mini-
phase.
iv. The decomposition of operators over
Hilbert spaces is treaded in Ref. 3. For
operators over Banach spaces, this problem
is still under research. For our conveni-
ence, operators are restricted to those
which guarantee the decomposition.
v. As in the classical Wiener-Hopf
filtering, we would like to minimize the
variance of the error. However, when Wiener-
Hopf filtering is formulated in reflexive
Banach space, the variance of the error is
a positive and self-adjoint operator which
can only be minimized in the partial order-
ing of the positive operators.' This subject
is treaded in section V.
II. BANACH SPACE VALUED RANDOM
VARIABLES
The theory of Banach space valued random
variables has been studied in Ref. 2. For
our purpose, we discuss reflexive Banach
space valued random variables with proba-
bility measure over the space assumed im-
plicitly. The development follows that of
Parthasarathy (2) and Balakrishnan (4); the
reader is referred to these works for the
details.
Let p, TI denote finitely additive random
variables taking values in a reflexive
Banach space B. For such random variables,
we assume
* * * E{ I (p, x ) I} < CD , for all x £ B *
(2.1) E{(p, x )} is continuous in x*
Here E{.} denotes the expected value of a
scalar valued random variable with respect
to the probability space underlying p. For
random variables satisfy condition (2.1),
there is a unique vector mp in B satisfy-
ing
* * * * E{(p, x )} = (mp, x ), for x £ B •
mp is termed as the mean of random variable
p. As in most stochastic processes, mean
is not our prime concern. Therefore, in
the sequel we only deal with zero-mean ran-
dom variables. For such random variables,
we further assume
E{ I (p , * * ) I } x ) (TI , Y < CD , * for all x , y* £ B*
(2.2) E{ (p , x*) (TI , y*) }
is continuous in x* and y*
It can be shown that condition (2.2) implies
condition (2.1). Now let's take a look at
E{(p,x*)
E{ (p, x*)
(TI, y*)}. If we fix y*, then
(TI, y*)} is a bounded linear
29
functional on B* (so an element of B**=B).
This means that there exists a unique Py*
in B such that E{ (p, x*) (TI, y*)}
= (Py*' x*) for x* £ B*.
QPTI
= B* + B, by QPTI
y*
Define a mapping
Py*.
Hence E{ (p, x*) (TI, y*)} (QPTI y*, x*).
It can be easily proved that Q is linear. PTI Moreover, Q is
PTI bounded. Q is termed
PTI as the covariance operator of random vari-
abIes p and TI. Covariance operators satis-
fy following conditions:
i. Q(Lp) (MTI) = L QPTIM* , where Land are linear bounded operator on B.
ii. Define Qp Qpp then
iii.
iv.
QP
+TI = Qp + QPTI + QTIP + QTI •
Qp is called the variance operator
of p.
Q* in particular Q = Q * TIP , P P
Q is positive in the sense that p
(Qp y* , y*)
for all y* £ B*.
These conditions result from straight for-
ward manipulation of the defining equation
for the covariance operator. Using QPTI'
we say that p and TI are independent if
QPTI
= O.
III. BANACH RESOLUTION SPACE
By a Banach resolution space, we mean a
2-tuple, (B, BF), where B is a Banach space
and BF is the so-called resolution of iden-
tity in B, which is defined in the following:
(A) Resolution of identity
Definition 3.1. Let B be a Banach space.
By a resolution of identity, BF, in B, we
mean a family of linear bounded operators,
BF(~), on B defined for each Borel subset,
~, of the real number set R, satisfying
the followings:
i. BF (R) = IB-identity operator on B
ii. BF(~Il . BF(~2) = BF(~lM2)' for
all ~l' ~2 £ S (R)- the set of all