Time-Variant Systems and Interpolation

308 Pages·1992·9.55 MB·English

by  M. A. Kaashoek (auth.), I. Gohberg, I. Gohberg (eds.)

OT56
Operator Theory: Advances and Applications
Vol. 56
Editor:
I. Gohberg
Tel Aviv University
Ramat Aviv, Israel
Editorial Office:
School of Mathematical Sciences
Tel Aviv University
Ramat Aviv, Israel
Editorial Board:
A. Atzmon (Tel Aviv) M. A. Kaashoek (Amsterdam)
J. A. Ball (Blacksburg) T. Kailath (Stanford)
L. de Branges (West Lafayette) H. G. Kaper (Argonne)
K. Clancey (Athens, USA) S. T. Kuroda (Tokyo)
L. A. Coburn (Buffalo) P. Lancaster (Calgary)
R. G. Douglas (Stony Brook) L. E. Lerer (Haifa)
H. Dym (Rehovot) E. Meister (Darmstadt)
A. Dynin (Columbus) B. Mityagin (Columbus)
P. A. Fillmore (Halifax) J. D. Pincus (Stony Brook)
C. Foias (Bloomington) M. Rosenblum (Charlottesville)
P. A. Fuhrmann (Beer Sheva) J. Rovnyak (Charlottesville)
S. Goldberg (College Park) D. E. Sarason (Berkeley)
B. Gramsch (Mainz) H. Widom (Santa Cruz)
J. A. Helton (La Jolla) D. Xia (Nashville)
Honorary and Advisory
Editorial Board:
P. R. Halmos (Santa Clara) M. S. Livsic (Beer Sheva)
T. Kato (Berkeley) R. Phillips (Stanford)
P. D. Lax (New York) B. Sz.-Nagy (Szeged)
Birkhauser Verlag
Basel· Boston· Berlin
Time-Variant Systems
and Interpolation
Edited by
1. Gohberg
Springer Basel AG
Editors' address:
1. Gohberg
Raymond and Beverly Sackler Faculty of Exact Sciences
School of Mathematical Sciences
Tel Aviv University
69978 Tel Aviv, Israel
Deutsche Bibliothek CataJoging-in-Publication Data
Time-variant systems and interpolation / ed. by 1. Gohberg. -
Basel ; Boston ; Berlin : Birkhăuser, 1992
(Operator Thcory ; VoI. 56)
ISBN 978-3-0348-9701-3 ISBN 978-3-0348-8615-4 (eBook)
DOI 10.1007/978-3-0348-8615-4
NE: Gochberg, Izrail' [Hrsg.]; GT
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© 1992 Springer Basel AG
Originally published by Birkhăuser Verlag Basel in 1992
Softcover reprint of the hardcover 1s t edition 1992
ISBN 978-3-0348-9701-3
v
Table of Contents
Editorial Introduction . vii
J.A.Ball, 1. Gohberg and M.A.Kaashoek
Nevanlinna-Pick interpolation for time-varying input-output maps:
The discrete case 1
O. Introduction . . . . . . 1
1. Preliminaries . . . . . . 4
2. J-Unitary operators on £2 17
3. Time-varying Nevanlinna-Pick interpolation 27
4. Solution of the time-varying tangential Nevanlinna-Pick interpolation problem 34
5. An illustrative example 41
References . . . . . . . . . 50
J.A.Ball, 1. Gohberg, M.A.Kaashoek
Nevanlinna-Pick interpolation for time-varying input-output maps:
The continuous time case . . 52
O. Introduction . . . . . . . 52
1. Generalized point evaluation 55
2. Bounded input-output maps 62
3. Residue calculus and diagonal expansion 65
4. J-unitary and J-inner operators 68
5. Time-varying Nevanlinna-Pick interpolation 76
6. An example 85
References . . . . 88
A.Ben-Artzi, 1. Gohberg
Dichotomy of systems and invertibility of linear ordinary differential
operators . . . 90
1. Introduction . . . . . . . . . . . . . . . . . 90
2. Preliminaries . . . . . . . . . . . . . . . . . 94
3. Invertibility of differential operators on the real line 95
4. Relations between operators on the full line and half line 102
5. Fredholm properties of differential operators on a half line 106
6. Fredholm properties of differential operators on a full line 110
7. Exponentially dichotomous operators 113
8. References . . . . . . . . . . . . 118
A.Ben-Artzi and 1. Gohberg
Inertia theorems for block weighted shifts and applications 120
1. Introduction . . . . . . . . . . . . . . . . . 120
2. One sided block weighted shifts . . . . . . . . . 121
3. Dichotomies for left systems and two sided systems 131
4. Two sided block weighted shifts. . . . . . . . . 139
VI
5. Asymptotic inertia 147
6. References . . 152
P.Dewilde, H.Dym
Interpolation for upper triangular operators 153
1. Introduction . . . . . . . . . . . 154
2. Preliminaries . . . . . . . . . . . 164
3. Colligations & characteristic functions 168
4. Towards interpolation . . . . . . . 177
5. Explicit formulas for e ..... . 193
6. Admissibility and more on general interpolation. 203
7. Nevanlinna-Pick Interpolation 210
8. Caratheodory-Fejer interpolation 215
9. Mixed interpolation problems. . 224
10. Examples ......... . 226
11. Block Toeplitz & some implications 245
12. Varying coordinate spaces 251
13. References . . . . . . . . . 259
1. Gohberg, M.A.Ka ashoek, L.Lerer
Minimality and realization of discrete time-varying systems 261
Introduction . . . . . . . . 261
1. Preliminaries . . . . . . . . . . 264
2. Observability and reachability 268
3. Minimality for time-varying systems 271
4. Proofs of the minimality theorems 274
5. Realizations of infinite lower triangular matrices 278
6. The class of systems with constant state space dimension 285
7. Minimality and realization for periodical systems 292
References . . . . . . . . . . . . . . . . . . . . . 295
VII
EDITORIAL INTRODUCTION
This volume consists of six papers dealing with the theory of linear time-
varying systems and time-varying analogues of interpolation problems. All papers are
dedicated to generalizations to the time-variant setting of results and theorems from oper-
ator theory, complex analysis and system theory, well-known for the time-invariant case.
Often this is connected with a complicated transition from functions to infinite dimensional
operators, from shifts to weighted shifts and from Toeplitz to non-Toeplitz operators (in
the discrete or continuous form). The present volume contains a cross-section of recent
progress in this area.
The first paper, "Nevanlinna-Pick interpolation for time-varying input-
output maps: The discrete case" of J .A. Ball, I. Gohberg and M.A. Kaashoek, general-
izes for time-varying input-output maps the results for the Nevanlinna-Pick interpolation
problem for strictly contractive rational matrix functions. This paper is based on a sys-
tem theoretic point of view. The time-variant version of the homogeneous interpolation
problem developed in the same paper, plays an important role.
The second paper, also of J.A. Ball, I. Gohberg and M.A. Kaashoek, is enti-
tled "Nevanlinna-Pick interpolation for time-varying input-output maps: The continuous
time case". The previous paper contains a time-varying analogue of the Nevanlinna-Pick
interpolation for the disk. This paper contains the time-varying analogue for the half
plane, and hence the latter results may be viewed as appropriate continuous analogues
of the results of the first paper. Here, as well as in the previous paper, all solutions are
described via a linear fractional formula.
In the third paper, "Dichotomy of systems and invertibility of linear ordinary
differential operators" of A. Ben-Artzi and I. Gohberg, are considered linear ordinary
differential operators of first order with bounded matrix coefficients on the half line and on
VIII
the full line. Conditions are found when these operators are invertible or Fredholm on the
half line. The main theorems are stated in terms of dichotomy. In the case of invertibility,
the main operator is a direct sum of two generators of semigroups, one is supported on the
negative half line and the other on the positive half line.
The fourth paper, "Inertia theorems for block weighted shifts and applica-
tions" of A. Ben-Artzi and 1. Gohberg, contains time-variant versions of the well-known
inertia theorem from linear algebra. These theorems are connected with linear time depen-
dent dynamical systems and are stated in terms of dichotomy and Fredholm characteristics
of weighted block shifts.
The fifth paper, "Interpolation for upper triangular operators" of P. deWilde
and H. Dym, treats for the time-varying case the tangential problems of Nevanlinna-Pick
and Caraththeodory-Fejer, as well as more complicated ones for operator-valued functions.
Here both the cantractive and the strictly contractive cases are considered. The description
of all solutions in a linear fractional form is given. The general case of varying coordinate
spaces is analysed. The main method is based on an appropriate generalization of the
theory of reproducing kernel spaces.
The sixth paper, "Minimality and realization of discrete time-varying sys-
tems" of I. Gohberg, M.A. Kaashoek and L. Lerer, analyses time-varying finite dimensional
linear systems with time-varying state space. A theory which is an analogue of the classi-
cal minimality and realization theory for time independent systems, is developed. Special
attention is paid to periodical systems.
1. Gohberg
Operator Theory: 1
Advances and Applications, Vol. 56
© 1992 Birkhiiuser Verlag Basel
NEVANLINNA-PICK INTERPOLATION FOR TIME-VARYING
INPUT-OUTPUT MAPS: THE DISCRETE CASE
J.A. Ball*, 1. Gohberg and M.A. Kaashoek
This paper presents the conditions of solvability and describes all solutions
of the matrix version of the Nevanlinna-Pick interpolation problem for time-varying input-
output maps. The system theoretical point of view is employed systematically. The tech-
nique of solution generalizes the method for finding rational solutions of the time-invariant
version of the problem which is based on reduction to a homogeneous interpolation prob-
lem.
O. INTRODUCTION
The simplest interpolation problem of Nevanlinna-Pick type reads as follows.
Given N different points Zl, ..• ,ZN in the open unit disc D of the complex plane and
arbitrary complex numbers Wl,"" WN, determine a function f, analytic in D, such that
(i) f(zj)=wj, j =l, ... ,N,
(ii) sUPI%I<l If(z)1 < 1.
This problem can be restated as a problem involving double infinite Toeplitz matrices
acting on .e2 (Z), namely find a lower triangular Toeplitz matrix T,
fo 0 0
(0.1) T= h [lQ] 0
h h fo
such that
* The first author thanks the Netherlands organization for scientific research
(NWO) for supporting his research.