1,;u f/lElTON tt1.tr# C>CPf" UL S-,,\,N bJC LA -:S.? lie..

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