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LINEAR STATE/SIGNAL SYSTEMS
The authors explain in this work a new approach to observing and controlling linear
systems whose inputs and outputs are not fixed in advance. They cover a class of linear
time-invariant state/signal systems that is general enough to include most of the standard
classes of linear time-invariant dynamical systems, but simple enough to make it easy to
understand the fundamental principles. They begin by explaining the basic theory of
finite-dimensional and bounded systems in a way suitable for graduate courses in systems
theory and control. They then proceed to more advanced infinite-dimensional settings,
opening up new ways for researchers to study distributed parameter systems, including
linear port-Hamiltonian systems and boundary triplets. They include the general
nonpassive part of the theory in continuous and discrete time, and provide a short
introduction to the passive situation. Numerous examples from circuit theory are used to
illustrate the theory.
Encyclopedia of Mathematics and Its Applications
This series is devoted to significant topics or themes that have wide application in
mathematics or mathematical science, and for which a detailed development of the
abstract theory is less important than a thorough and concrete exploration of the
implications and applications.
Books in the Encyclopedia of Mathematics and Its Applications cover their subjects
comprehensively. Less important results may be summarized as exercises at the end of the
chapters. For technicalities, readers can be referred to the bibliography that is expected to
be comprehensive. As a result, volumes are encyclopedic references or manageable guides
to major subjects.
Published online by Cambridge University Press

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
All the titles listed below can be obtained from good booksellers or from Cambridge
University Press. For a complete series listing, visit
www.cambridge.org/mathematics.
133 P. Kornerup and D. W. Matula Finite Precision Number Systems and Arithmetic
134 Y. Crama and P. L. Hammer (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering
135 V. Berthé and M. Rigo (eds.) Combinatorics, Automata and Number Theory
136 A. Kristály, V. D. Rădulescu and C. Varga Variational Principles in Mathematical Physics,
Geometry, and Economics
137 J. Berstel and C. Reutenauer Noncommutative Rational Series with Applications
138 B. Courcelle and J. Engelfriet Graph Structure and Monadic Second-Order Logic
139 M. Fiedler Matrices and Graphs in Geometry
140 N. Vakil Real Analysis through Modern Infinitesimals
141 R. B. Paris Hadamard Expansions and Hyperasymptotic Evaluation
142 Y. Crama and P. L. Hammer Boolean Functions
143 A. Arapostathis, V. S. Borkar and M. K. Ghosh Ergodic Control of Diffusion Processes
144 N. Caspard, B. Leclerc and B. Monjardet Finite Ordered Sets
145 D. Z. Arov and H. Dym Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations
146 G. Dassios Ellipsoidal Harmonics
147 L. W. Beineke and R. J. Wilson (eds.) with O. R. Oellermann Topics in Structural Graph Theory
148 L. Berlyand, A. G. Kolpakov and A. Novikov Introduction to the Network Approximation Method
for Materials Modeling
149 M. Baake and U. Grimm Aperiodic Order I: A Mathematical Invitation
150 J. Borwein et al. Lattice Sums Then and Now
151 R. Schneider Convex Bodies: The Brunn–Minkowski Theory (Second Edition)
152 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions (Second Edition)
153 D. Hofmann, G. J. Seal and W. Tholen (eds.) Monoidal Topology
154 M. Cabrera García and Á. Rodríguez Palacios Non-associative Normed Algebras I: The Vidav–Palmer
and Gelfand–Naimark Theorems
155 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables (Second Edition)
156 L. W. Beineke and R. J. Wilson (eds.) with B. Toft Topics in Chromatic Graph Theory
157 T. Mora Solving Polynomial Equation Systems III: Algebraic Solving
158 T. Mora Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond
159 V. Berthé and M. Rigo (eds.) Combinatorics, Words and Symbolic Dynamics
160 B. Rubin Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis
161 M. Ghergu and S. D. Taliaferro Isolated Singularities in Partial Differential Inequalities
162 G. Molica Bisci, V. D. Radulescu and R. Servadei Variational Methods for Nonlocal Fractional Problems
163 S. Wagon The Banach–Tarski Paradox (Second Edition)
164 K. Broughan Equivalents of the Riemann Hypothesis I: Arithmetic Equivalents
165 K. Broughan Equivalents of the Riemann Hypothesis II: Analytic Equivalents
166 M. Baake and U. Grimm (eds.) Aperiodic Order II: Crystallography and Almost Periodicity
167 M. Cabrera García and Á. Rodríguez Palacios Non-associative Normed Algebras II: Representation Theory
and the Zel’manov Approach
168 A. Yu. Khrennikov, S. V. Kozyrev and W. A. Zúñiga-Galindo Ultrametric Pseudodifferential Equations
and Applications
169 S. R. Finch Mathematical Constants II
170 J. Krajíček Proof Complexity
171 D. Bulacu, S. Caenepeel, F. Panaite and F. Van Oystaeyen Quasi-Hopf Algebras
172 P. McMullen Geometric Regular Polytopes
173 M. Aguiar and S. Mahajan Bimonoids for Hyperplane Arrangements
174 M. Barski and J. Zabczyk Mathematics of the Bond Market: A Lévy Processes Approach
175 T. R. Bielecki, J. Jakubowski and M. Niewȩgłowski Structured Dependence between Stochastic Processes
176 A. A. Borovkov, V. V. Ulyanov and Mikhail Zhitlukhin Asymptotic Analysis of Random Walks: Light-Tailed
Distributions
177 Y.-K. Chan Foundations of Constructive Probability Theory
178 L. W. Beineke, M. C. Golumbic and R. J. Wilson (eds.) Topics in Algorithmic Graph Theory
179 H.-L. Gau and P. Y. Wu Numerical Ranges of Hilbert Space Operators
180 P. A. Martin Time-Domain Scattering
181 M. D. de la Iglesia Orthogonal Polynomials in the Spectral Analysis of Markov Processes
182 A. E. Brouwer and H. Van Maldeghem Strongly Regular Graphs

EN C Y C L O P E D I A O F MA T H E M A T I C S A N D I T S AP P L I C A T I O N S
Linear State/Signal Systems
DAMIR Z. AROV
South Ukrainian National Pedagogical University
OLOF J. STAFFANS
Åbo Akademi University, Finland

University Printing House, Cambridge CB2 8BS, United Kingdom
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It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781316519677
DOI: 10.1017/9781009024921
c
Damir Z. Arov and Olof J. Staffans 2022
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2022
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Arov, Damir Z., author. | Staffans, Olof J., 1947– author.
Title: Linear state/signal systems / Damir Z. Arov, South Ukrainian
National Pedagogical University, Olof J. Staffans, Åbo Akademi
University, Finland.
Description: Cambridge, United Kingdom ; New York, NY : Cambridge
University Press, 2022. | Series: Encyclopedia of mathematics and its
applications | Includes bibliographical references and index.
Identifiers: LCCN 2021058586 (print) | LCCN 2021058587 (ebook)
| ISBN 9781316519677 (hardback) | ISBN 9781009024921 (ebook)
Subjects: LCSH: Linear systems. | Operator theory. | Linear control
systems. | BISAC: MATHEMATICS / General
Classification: LCC QA402 .A755 2022 (print) | LCC QA402 (ebook)
| DDC 003/.74–dc23/eng/20220126
LC record available at https://lccn.loc.gov/2021058586
LC ebook record available at https://lccn.loc.gov/2021058587
ISBN 978-1-316-51967-7 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.

Contents
Preface page xxi
List of Notations xxiii
1 Introduction and Overview 1
1.1 Linear Time-Invariant Dynamical Systems 1
1.1.1 State Systems 2
1.1.2 Systems That Interact with the Outside World 3
1.1.3 Input/State/Output Systems 3
1.1.4 Input/Output Systems 4
1.1.5 Classical (Sub)networks 5
1.1.6 Port-Hamiltonian Systems 5
1.1.7 Behavioral Systems 6
1.1.8 State/Signal Systems 7
1.1.9 State/Signal versus I/S/O Systems 8
1.1.10 Frequency Domain Systems 10
1.1.11 Boundary Triplets 11
1.1.12 State/Signal versus Behavioral Systems 11
1.1.13 How to Read This Book 12
1.1.14 H-Spaces 12
1.1.15 Where to Go from Here? 13
1.2 An Overview of State/Signal and Input/State/Output Systems 14
1.2.1 Input/State/Output Systems 14
1.2.2 Well-Posed I/S/O Systems 17
1.2.3 State/Signal Systems 18
1.2.4 I/S/O Representations 19
1.2.5 Similarity of I/S/O and State/Signal Systems 20
1.2.6 Input/Output Invariant Properties of I/S/O Systems 23
1.2.7 Properties of I/S/O Systems in the State/Signal Sense 25

vi Contents
1.2.8 Static Transformations of I/S/O and State/Signal
Systems 26
1.2.9 Invariant Subspaces of I/S/O and State/Signal
Systems 28
1.2.10 Interconnections of I/S/O and State/Signal Systems 29
1.2.11 External Characteristics of I/S/O and State/Signal
Systems 31
1.2.12 Restrictions, Projections, and Compressions 31
1.2.13 I/S/O and State/Signal Systems in Discrete Time 33
1.2.14 The Resolvent Matrix of an I/S/O System 35
1.2.15 The Resolvent Set and the Characteristic Bundles of
State/Signal Systems 37
1.2.16 Well-Posed I/S/O and State/Signal Systems
in the Frequency Domain 39
1.2.17 General Resolvable Frequency Domain I/S/O and
1.2.18 Dual and Adjoint I/S/O and State/Signal Systems 42
1.2.19 Passive I/S/O and State/Signal Systems 44
1.2.20 Passive Finite-Dimensional Electrical n-Ports 45
1.2.21 Some Finite-Dimensional Passive 2-Ports 51
1.2.22 Some Distributed Parameter Passive Systems 59
1.3 Notes and Comments 68
2 State/Signal Systems: Trajectories, Transformations, and
Interconnections 71
2.1 State/Signal Nodes and State/Signal Systems 71
2.1.1 The State/Signal System and Its Trajectories 71
2.1.2 Regular and Semiregular State/Signal Nodes 75
2.1.3 Kernel and Image Representations of Closed
State/Signal Nodes 80
2.1.4 Bounded State/Signal Nodes and Systems 84
2.2 Some Basic Transformations of State/Signal Nodes 87
2.2.1 Similarity of Two State/Signal Nodes 87
2.2.2 Time Reflection of a State/Signal Node 89
2.2.3 Time Rescaling of a State/Signal Node 91
2.2.4 Exponentially Weighted State/Signal Nodes 92
2.3 Properties of Trajectories of State/Signal Systems 93
2.3.1 Classical, Generalized, and Mild Trajectories 93
2.3.2 Existence and Uniqueness of Trajectories 95

Contents vii
2.3.3 Connections between Classical, Generalized, and
Mild Trajectories 100
2.4 Some Additional Transformations of State/Signal Nodes 104
2.4.1 The (P, Q)-Image of a State/Signal Node 104
2.4.2 Parts and Static Projections of a State/Signal Node 107
2.4.3 Adding Inputs and Output to a State/Signal Node 111
2.5 Interconnections of State/Signal Nodes 118
2.5.1 The Cross Product of Two State/Signal Nodes 118
2.5.2 (P, Q)-Interconnections of State/Signal Nodes 119
2.5.3 A Short Circuit Connection of State/Signal Nodes 120
2.5.4 Examples of Interconnections of State/Signal Nodes 121
2.6 Examples of Infinite-Dimensional State/Signal Systems 123
3 State/Signal Systems: Dynamic and Frequency Domain Properties 132
3.1 Signal Behaviors and Their State/Signal Realizations 132
3.1.1 Future Signal Behaviors 132
3.1.2 External Equivalence of State/Signal Systems 133
3.2 Dynamic Properties of State/Signal Systems 133
3.2.1 Controllability and Observability of State/Signal
Systems 134
3.2.2 Intertwinements of State/Signal Systems 140
3.2.3 Compressions, Restrictions, and Projections of
3.2.4 Examples of Minimal Compressions 148
3.2.5 State/Signal Systems with the Continuation Property 153
3.3 State Systems 156
3.3.1 A State System and Its Trajectories 157
3.3.2 The Homogeneous Cauchy Problem 158
3.3.3 Bounded State Systems and Uniformly Continuous
Groups 159
3.3.4 Well-Posed State Systems and Strongly Continuous
Semigroups 163
3.3.5 Transformations and Interconnections of State Nodes 169
3.3.6 Invariant Subspaces of State Nodes 171
3.3.7 Intertwinement of State Nodes 173
3.4 Frequency Domain Characteristics of State/Signal Nodes 174
3.4.1 The Characteristic Node Bundle 174
3.4.2 The Characteristic Control Bundle 178
3.4.3 The Characteristic Observation Bundle 179
3.4.4 The Characteristic Signal Bundle 180

viii Contents
3.4.5 The Characteristic Bundles of Transformed State/Signal
Systems 182
3.4.6 The Resolvent of a Regular State Node 189
3.4.7 The Resolvent Set of a State/Signal Node 191
3.5 Invariance with Respect to Similarities 192
3.6 Dual and Adjoint State/Signal Nodes and Systems 194
3.6.1 The Dual of a State/Signal System 194
3.6.2 The Duals of Some Transformed State/Signal Nodes 202
3.6.3 The Characteristic Bundles of Dual State/Signal Nodes 206
3.6.4 The Adjoint State/Signal System 207
3.7 Notes to Chapters 2 and 3 213
4 Input/State/Output Representations 217
4.1 Input/State/Output Nodes and Systems 217
4.1.1 Regular I/S/O Nodes 217
4.1.2 General I/S/O Nodes and Systems 220
4.1.3 Kernal and Image Representations of Closed I/S/O
Nodes 223
4.1.4 State, Input/State and State/Output Nodes and Systems 226
4.1.5 Input/State and State/Output Representations of
4.1.6 Free Inputs and Continuously Determined Outputs 229
4.1.7 Existence and Uniqueness of Trajectories 230
4.1.8 Bounded I/S/O Nodes and Systems 232
4.2 Input/State/Output Representations of State/Signal Nodes and
Systems 234
4.2.1 The State/Signal Node Induced by an I/S/O Node 234
4.2.2 I/O Representations of the Signal Space 236
4.2.3 General I/S/O Representations of a State/Signal Node 239
4.2.4 Semiregular I/S/O Representations of a Semiregular
State/Signal Node 241
4.2.5 Regular I/S/O Representations of a Regular State/Signal
Node 242
4.2.6 Parametrization of I/S/O Representations 246
4.2.7 Bounded I/S/O Representations of Bounded State/Signal
Nodes 248
4.2.8 Parametrization of Bounded I/S/O Representations 251
4.3 State Feedback and Output Injection Representations 256
4.3.1 State Feedback Representations 256
4.3.2 Output Injection Representations 259

Contents ix
4.4 Basic Transformations of Input/State/Output Nodes 261
4.4.1 Similarity of I/S/O Nodes 261
4.4.2 Time Reflection of an I/S/O Node 263
4.4.3 Time Rescaling of an I/S/O Node 264
4.4.4 Exponentially Weighted I/S/O Nodes 265
4.5 Properties of Trajectories of Input/State/Output Systems 267
4.5.1 Basic Properties of the Sets of Classical
and Generalized Trajectories 267
4.5.2 Solvability and the Uniqueness Property 268
4.5.3 Connections between Classical, Generalized,
and Mild Trajectories 270
4.6 Some Simple Input/State/Output Examples 273
5 Input/State/Output Systems: Dynamic and Frequency Domain
Properties 276
5.1 Additional Transformations of Input/State/Output Nodes 276
5.1.1 Adding a Feedthrough Term to an I/S/O Node 276
5.1.2 Modifying Inputs and Outputs of an I/S/O Node 277
5.1.3 The (P, R, Q)-Image of an I/S/O Node 278
5.1.4 Parts and Static Projections of an I/S/O Node 281
5.1.5 Static Output Feedback 285
5.1.6 Adding Inputs and Outputs to an I/S/O Node 287
5.1.7 A Second Look at State Feedbacks and Output
Injections 297
5.2 Interconnections of Input/State/Output Nodes 303
5.2.1 The Cross Product of Two I/S/O Nodes 303
5.2.2 (P, R, Q)-Interconnections of I/S/O Nodes 305
5.2.3 A Short Circuit Connection of I/S/O Nodes 306
5.2.4 T-Junctions, Sum Junctions, and Difference
Junctions 307
5.2.5 Parallel and Difference Connections 311
5.2.6 Cascade Connections 313
5.2.7 Dynamic Feedback 316
5.2.8 Examples of I/S/O Interconnections 317
5.3 Realizations of Input/Output Behaviors 318
5.3.1 Future I/O Behaviors 318
5.3.2 External Equivalence of I/S/O Systems 319
5.4 Dynamic Properties of Input/State/Output Systems 320
5.4.1 Controllability and Observability of I/S/O Systems 320
5.4.2 Intertwinements of I/S/O Systems 325

x Contents
5.4.3 Compressions, Restrictions, and Projections of I/S/O
Systems 326
5.4.4 I/S/O Systems with the Continuation Property 330
5.5 Frequency Domain Characteristics of Input/State/Output Nodes 331
5.5.1 The Characteristic Node Bundle of an I/S/O Node 331
5.5.2 The I/S/O Resolvent Matrix of an I/S/O Node 334
5.5.3 Resolvability of Transformed I/S/O Nodes 338
5.5.4 Frequency Domain I/S/O-Admissible I/O
Representations 341
5.6 The Correspondence between State/Signal and Input/State/Output
Notions 343
5.6.1 I/O Invariant Notions 343
5.6.2 Properties of I/S/O Systems in the State/Signal Sense 349
5.7 Adjoint and Dual Input/State/Output Nodes and Systems 351
5.7.1 The Adjoint and the Dual of an I/S/O Node 351
5.7.2 Adjoint and Dual I/S/O Representations 354
5.7.3 I/S/O Lagrange Identities 355
5.7.4 Properties of Adjoint and Dual I/S/O Nodes and
Systems 359
5.7.5 The Adjoints and Duals of Some Transformed I/S/O
Nodes 360
5.7.6 The Adjoints and Duals of Some Interconnected
I/S/O Nodes 364
6 Bounded Input/State/Output Systems in Continuous and Discrete
Time 370
6.1 Bounded State Operators and Nodes 370
6.1.1 The Spectral Radius of a Bounded State Operator 370
6.1.2 Invariant Subspaces of Bounded State Operators
and Uniformly Continuous Groups 372
6.1.3 Parts and Projections of Bounded State Operators 373
6.1.4 Parts and Projections of Uniformly Continuous
Groups 375
6.1.5 Intertwinements of Bounded State Operators and
Uniformly Continuous Groups 377
6.1.6 Compressions of Bounded State Operators and
6.1.7 The General Structure of a Compression of a
Bounded State Operator 385

Contents xi
6.1.8 The Adjoints of Bounded State Operators and
6.2 Static Properties of Bounded Input/State/Output Nodes 393
6.2.1 Transformations of Bounded I/S/O Nodes 393
6.2.2 Interconnections of Bounded I/S/O Nodes 403
6.2.3 The I/S/O Resolvent Matrix of a Bounded I/S/O Node 406
6.3 Dynamic Properties of Bounded Input/State/Output Systems 407
6.3.1 Strongly Invariant and Unobservably Invariant
Subspaces 407
6.3.2 External Equivalence of Bounded I/S/O Systems 415
6.3.3 Intertwinements of Bounded I/S/O Systems 417
6.3.4 Restrictions and Projections of Bounded I/S/O Systems 421
6.3.5 Compressions of Bounded I/S/O Systems 423
6.3.6 The General Structure of a Bounded I/S/O Compression 428
6.3.7 Compressions into Minimal Bounded I/S/O Systems 435
6.4 The Adjoint and the Dual of a Bounded Input/State/Output
Node 439
6.5 Discrete Time Input/State/Output Systems 444
6.5.1 Introduction to Discrete Time I/S/O Systems 444
6.5.2 Properties of Discrete Time I/S/O Systems 445
6.5.3 Time Reflection of Discrete Time I/S/O Systems 448
6.5.4 Power Weightings of Discrete Time I/S/O Systems 449
6.5.5 Frequency Domain Shifts of Discrete Time I/S/O
Systems 450
6.5.6 Stable Discrete Time I/S/O Systems 451
6.5.7 Connections between Continuous and Discrete Time
I/S/O Properties 453
6.5.8 Dynamic Notions for Bounded I/S/O Nodes 454
6.6 Bounded Input/State/Output Realizations 456
6.6.1 Analyticity at Infinity of the I/S/O Resolvent Matrix 456
6.6.2 Existence of a Bounded I/S/O Realization 457
7 Bounded State/Signal Systems in Continuous and Discrete Time 460
7.1 Static Properties of Bounded State/Signal Nodes 460
7.1.1 The I/S/O-Bounded Resolvent Set of a Bounded
State/Signal Node 460
7.1.2 Transformations of Bounded State/Signal Nodes 462
7.1.3 Resolvability of Transformations of State/Signal
Nodes 476
7.2 Dynamic Properties of Bounded State/Signal Systems 481

xii Contents
Subspaces 481
7.2.2 External Equivalence of Bounded State/Signal Systems 489
7.2.3 Intertwinements of Bounded State/Signal Systems 490
7.2.4 Restrictions and Projections of Bounded State/Signal
Systems 494
7.2.5 Compressions of Bounded State/Signal Systems 496
7.2.6 The General Structure of a Bounded State/Signal
Compression 498
7.2.7 Compressions into Minimal Bounded State/Signal
Systems 504
7.2.8 Bounded State/Signal Realizations 505
7.3 The Dual and the Adjoint of a Bounded State/Signal Node 506
7.4 Discrete Time State/Signal Systems 510
7.4.1 Introduction to Discrete Time State/Signal Systems 510
7.4.2 Properties of Discrete Time State/Signal Systems 511
7.4.3 Time Reflection of Discrete Time State/Signal Systems 513
7.4.4 Power Weightings of Discrete Time State/Signal
Systems 513
7.4.5 Frequency Domain Shifts of Discrete Time State/Signal
Systems 514
7.4.6 Stable Discrete Time State/Signal Systems 515
7.4.7 Connections between Continuous and Discrete Time
State/Signal Properties 515
7.4.8 Dynamic Notions for Bounded State/Signal Nodes 516
8 Semi-bounded Input/State/Output Systems 521
8.1 C0 Semigroups and Well-Posed State Systems 521
8.1.1 On the Resolvents of Generators of C0
Semigroups 521
8.1.2 The Inhomogeneous Cauchy Problem 524
8.1.3 Invariant Subspaces of C0 Semigroups 530
8.1.4 Parts, Projections, and Restrictions of Single-Valued
Resolvable Main Operators 530
8.1.5 Parts and Projections of C0 Semigroups 533
8.1.6 Intertwinements of C0 Semigroups 535
8.1.7 Compressions of C0 Semigroups 536
8.1.8 The General Structure of a Compression of a C0
Semigroup 539

Contents xiii
8.1.9 The Adjoint of a C0 Semigroup 542
8.2 Semi-bounded Input/State/Output Systems 544
8.2.1 Introduction to Semi-bounded I/S/O Systems 544
8.2.2 Transformations of Semi-bounded I/S/O Nodes 547
8.2.3 Interconnections of Semi-bounded I/S/O Nodes 551
8.2.4 The I/S/O Resolvent Matrix of a Semi-bounded I/S/O
Node 552
8.3 Dynamic Properties of Semi-bounded Input/State/Output
Systems 553
Subspaces 553
8.3.2 External Equivalence of Semi-bounded I/S/O Systems 559
8.3.3 Intertwinements of Semi-bounded I/S/O Systems 559
8.3.4 Restrictions and Projections of Semi-bounded I/S/O
Systems 562
8.3.5 Compressions of Semi-bounded I/S/O Systems 563
8.3.6 The General Structure of a Semi-bounded I/S/O
Compression 565
8.3.7 Compressions into Minimal Semi-bounded I/S/O
Systems 570
8.4 The Adjoint of a Semi-bounded Input/State/Output Node 572
9 Semi-bounded State/Signal Systems 576
9.1 Static Properties of Semi-bounded State/Signal Nodes 576
9.1.1 Introduction to Semi-bounded State/Signal Nodes
and Systems 576
9.1.2 The I/S/O Semi-bounded Resolvent Set of a
Semi-bounded State/Signal Node 580
9.1.3 Transformations and Interconnections
of Semi-bounded State/Signal Nodes 581
9.2 Dynamic Properties of Semi-bounded State/Signal Systems 581
Subspaces 581
9.2.2 External Equivalence of Semi-bounded State/Signal
Systems 584
9.2.3 Intertwinements of Semi-bounded State/Signal Systems 585
9.2.4 Restrictions and Projections of Semi-bounded
9.2.5 Compressions of Semi-bounded State/Signal Systems 588

xiv Contents
9.2.6 The General Structure of a Semi-bounded State/Signal
Compression 589
9.2.7 Compressions into Minimal Semi-bounded State/Signal
Systems 593
9.3 The Adjoint of a Semi-bounded State/Signal Node 594
10 Resolvable Input/State/Output and State/Signal Nodes 599
10.1 Resolvable State Nodes 599
10.1.1 Linear Operator-Valued Pencils 599
10.1.2 The Resolvent of a State Node 601
10.1.3 The Interpolation Space of a Semiregular State Node 606
10.1.4 The Extrapolation Space of a Regular Resolvable
State Node 607
10.1.5 The Duals of the Interpolation and Extrapolation
Spaces 610
10.1.6 The Interpolation and Extrapolation Spaces of a
Semigroup Generator 613
10.2 Resolvable Input/State/Output Nodes 614
10.2.1 Resolvability of an I/S/O Node 615
10.2.2 Kernel and Image Representations of the I/S/O
Resolvent Matrix 618
10.2.3 The I/S/O Resolvent Identity 620
10.2.4 Representations of the System Operator 624
10.2.5 Semiregular and Regular Resolvable I/S/O Nodes 628
10.2.6 The Observation and Control Operators of a Regular
Resolvable I/S/O Node 631
10.2.7 Some Examples of Regular Resolvable I/S/O Nodes 637
10.2.8 Resolvability of Transformed I/S/O Nodes 640
10.2.9 Resolvability of Interconnected I/S/O Nodes 647
10.2.10 The Resolvent Family of Bounded I/S/O Nodes 652
10.2.11 A Finite-Dimensional Nonregular Resolvable I/S/O
Node 653
10.2.12 The Adjoint and the Dual of a Resolvable I/S/O Node 655
10.3 Resolvable State/Signal Nodes 658
10.3.1 On the Resolvent Set of a Closed State/Signal Node 658
10.3.2 Frequency Domain I/S/O-Admissible I/O
Representations 662
10.3.3 Resolvability of Transformed State/Signal Nodes 669
10.3.4 The Resolvent Family of Bounded State/Signal Nodes 673

Contents xv
10.3.5 The Dual and the Adjoint of a Resolvable State/Signal
System 674
11 Frequency Domain Input/State/Output Systems 679
11.1 Frequency Domain Input/State/Output Systems 679
11.1.1 Introduction to Frequency Domain I/S/O Systems 679
11.1.2 Frequency Domain Controllability and Observability 681
11.1.3 Frequency Domain Invariance 682
11.1.4 The Frequency Domain Behavior and External
Equivalence 689
11.1.5 Frequency Domain Intertwinements 690
11.1.6 Frequency Domain Compressions, Restrictions,
and Projections 696
11.1.7 Resolvable Frequency Domain Compressions,
Restrictions, and Projections 698
11.1.8 The General Structure of a Resolvable Frequency
Domain Compression 704
11.1.9 Compressions into -Minimal I/S/O Systems 712
11.1.10 Results for Connected Frequency Domains 715
11.2 The Adjoint and the Dual of a Frequency Domain
Input/State/Output System 725
11.2.1 Frequency Domain Lagrange Identities 726
11.2.2 Properties of Adjoint and Dual Frequency Domain
I/S/O Systems 728
11.3 Frequency Domain Notions for -Resolvable Input/State/Output
Nodes 730
11.3.1 Dynamic Properties of the Resolvent Family of
Bounded I/S/O Nodes 730
11.4 Resolvable Frequency Domain State Systems 733
11.4.2 Frequency Domain Intertwinements and
Compressions 734
11.4.3 Results for Connected Frequency Domains 738
11.4.4 Frequency Domain Duality 740
12 Frequency Domain State/Signal Systems 743
12.1 Frequency Domain State/Signal Systems 743
12.1.1 Introduction to Frequency Domain State/Signal
Systems 743

xvi Contents
12.1.2 Separately and Jointly I/S/O Admissible Frequency
Domains 745
12.1.3 Frequency Domain Controllability and Observability 747
12.1.5 The Frequency Domain Behavior and External
Equivalence 752
12.1.7 Frequency Domain Compressions, Restrictions,
and Projections 761
12.1.8 Resolvable Frequency Domain Compressions,
Restrictions, and Projections 763
12.1.9 The General Structure of a Resolvable Frequency
Domain Compression 769
12.1.10 Compressions into -Minimal State/Signal Systems 773
12.2 Local Frequency Domain Notions 775
12.2.1 Local Frequency Domain Notions for -Resolvable
12.2.2 Connected Frequency Domains 783
12.3 The Dual and the Adjoint of a Frequency Domain State/Signal
System 793
12.3.1 Frequency Domain Lagrange Identities 794
12.3.2 Properties of Dual and Adjoint Frequency Domain
12.4 Frequency Domain Notions for -Resolvable State/Signal
Nodes 798
12.4.1 Dynamic Properties of the Resolvent Family of
Bounded State/Signal Nodes 798
13 Internally Well-Posed Systems 802
13.1 Internally Well-Posed Input/State/Output Systems 802
13.1.1 Basic Definitions and Properties 802
13.1.2 Transformations and Interconnections 804
13.2 Frequency-Domain Internally Well-Posed Input/State/Output
Systems 805
13.2.3 Frequency-Domain Restrictions, Projections,
and Compressions 807
13.2.4 The General Structure of ρ+∞()-Compressions 809

Contents xvii
13.3 Internally Well-Posed State/Signal Systems 812
13.3.1 Basic Definitions and Properties 812
13.3.2 Frequency-Domain Compressions of Internally
Well-Posed State/Signal Systems 813
14 Well-Posed Input/State/Output Systems 816
14.1 Basic Properties of Well-Posed Input/State/Output Systems 816
14.1.1 The Definition of a Well-Posed I/S/O System 816
14.1.2 Alternative Conditions for Well-Posedness 819
14.1.3 The Fundamental I/S/O Solution of a Well-Posed
I/S/O System 825
14.2 The Growth Bound of a Well-Posed Input/State/Output System 833
14.2.1 The Growth Bound of a Well-Posed I/S/O System 833
14.2.2 Stable I/S/O Systems 838
14.3 Resolvability of Well-Posed Input/State/Output Systems 842
14.3.1 Well-Posed I/S/O Systems Are Resolvable 842
14.3.2 Growth Estimates for the I/S/O Resolvent Matrix 847
14.4 Realizations of Shift-Invariant Causal Linear Operators 850
14.4.1 Shift Invariant Causal Linear Operators 850
14.4.2 Realizations of Shift Invariant Causal Linear Operators 852
14.4.3 Toeplitz and Hankel Operators 853
14.5 Transformations and Interconnections of Well-Posed
Input/State/Output Systems 857
14.5.1 Well-Posedness and Stability of Transformed I/S/O
Systems 857
14.5.2 Well-Posedness and Stability of Interconnected I/S/O
Systems 864
14.5.3 Stabilizable and Detectable I/S/O Systems 867
14.6 Dynamic Properties of Well-Posed Input/State/Output Systems 869
Subspaces 869
14.6.2 Intertwinements of Well-Posed I/S/O Systems 870
14.6.3 Restrictions, Projections, and Compressions 874
14.6.4 The General Structure of a Well-Posed I/S/O
Compression 879
14.6.5 Compressions Into Minimal Well-Posed I/S/O Systems 886
14.7 Well-Posed Input/State/Output Systems in the Frequency
Domain 887
14.7.1 Time and Frequency Domain External Equivalence 888
14.7.2 Time and Frequency Domain Invariance 888

xviii Contents
14.7.3 Time and Frequency Domain Compressions and
Intertwinements 889
14.7.4 Frequency Domain Stability 891
14.8 The Adjoint of a Well-Posed Input/State/Output System 893
14.9 Scattering Passive Input/State/Output Systems 897
14.9.1 Hilbert Space I/S/O Nodes and Systems 897
14.9.2 Scattering Passive I/S/O Systems 898
14.9.3 The Internal I/S/O Cayley Transform 901
14.9.4 The Adjoint of a Passive Scattering System 904
15 Well-Posed State/Signal Systems 907
15.1 Basic Properties of Well-Posed State/Signal Systems 907
15.1.1 Basic Definitions 907
15.1.2 Well-Posedness and Stability of Transformed I/S/O
Systems 913
15.1.3 The Behaviors Induced by a Well-Posed State/Signal
System 914
15.1.4 The Past/Present and Present/Future Maps of a
Well-Posed State/Signal System 916
15.2 Stable State/Signal Systems 920
15.2.1 Stable State/Signal Trajectories 920
15.2.2 Stable State/Signal Behaviors 921
15.2.3 Stabilizable and Detectable State/Signal Systems 922
15.3 Realizations of Well-Posed Behaviors 925
15.3.1 Well-Posed Future, Past, and Two-Sided Behaviors 925
15.3.2 State/Signal Realizations of Well-Posed Behaviors 929
15.3.3 The Past/Future Map of a Well-Posed Behavior 929
15.4 Dynamic Properties of Well-Posed State/Signal Systems 930
Subspaces 930
15.4.2 Intertwinements of Well-Posed State/Signal Systems 932
15.4.3 Restrictions, Projections, and Compressions of Well-
Posed State/Signal Systems 933
15.4.4 The General Structure of a Compression 936
15.4.5 Compressions into Minimal Well-Posed State/Signal
Systems 939
15.5 Well-Posed State/Signal Systems in the Frequency Domain 940
15.6 The Adjoint of a Well-Posed State/Signal Node 942
15.7 Passive State/Signal Systems 944

Contents xix
15.7.1 Kreı̆n Spaces 944
15.7.2 The Kreı̆n Node Space of a Scattering Passive I/S/O
System 945
15.7.3 Passive State/Signal Systems 946
Appendix A Operators and Analytic Vector Bundles in H-Spaces 950
A.1 H-Spaces 950
A.1.1 Using More than One Norm in a Vector Space 950
A.1.2 Introduction to H-Spaces 952
A.1.3 Linear Operators in H-Spaces 953
A.1.4 Closed Linear Operators in H-Spaces 955
A.1.5 Complementary Projections and Coordinate
Respresentations of H-Spaces 956
A.1.6 Isomorphisms in H-Spaces 960
A.1.7 Partial Inverses of Bounded Linear Operators 961
A.1.8 Inversion of Block Matrix Operators 964
A.1.9 The Graph Norm and Graph Topology 965
A.1.10 Linear Multivalued Operators in H-Spaces 966
A.1.11 The Single-Valued and Injective Parts of a
Multivalued Operator 969
A.1.12 On the Resolvent of a Bounded Operator 970
A.2 Duality in H-Spaces 971
A.2.1 The Dual of an H-Space 971
A.2.2 The Adjoint of a Bounded Linear Operator 973
A.2.3 Duals of Product Spaces 976
A.2.4 The Duals of the Components of a Direct Sum
Decomposition 979
A.2.5 The Adjoint of a Linear Operator with Dense Domain 982
A.2.6 The Dual of a Continuous Dense Embedding 983
A.2.7 The Adjoint of a Multivalued Operator 984
A.3 Analytic Vector Bundles and Analytic Operator-Valued
Functions 988
A.3.1 The Dual Vector Bundle 992
References 994
Index 1005

Preface
The theory presented in this book arose as a product of a continued collaboration
between the two authors during the years 2003–2021. The basis for this collabo-
ration was our common interest in passive linear time-invariant input/state/output
systems theory. At the time this project started, O. Staffans was preparing a joint ar-
ticle (Ball and Staffans, 2006) with Prof. J. Ball that, in particular, explored the con-
nections between conservative input/state/output systems theory on the one hand
and some results in the behavioral theory introduced by J. Willems in the late 1980s
on the other hand. After extensive discussions on this approach, comparing it to the
theory of passive electrical networks, we understood that this opens up a new direc-
tion in the study of passive linear time-invariant systems. We called the new class of
systems that arose in this way passive state/signal systems. From the outset, it was
clear that the notion of passivity with an arbitrary supply rate fits more naturally
into the state/signal setting than in the input/state/output setting, and that the stan-
dard “diagonal transformation” of Livšic, the Potapov–Ginzburg transformation,
and the Redheffer and chain-scattering transformations have natural interpretations
as transformations between input/output resolvents of different input/state/output
representations of a passive state/signal system. We also soon discovered that virtu-
ally all the standard control theory notions such as controllability and observability,
minimality, stability, stabilizability, detectability, and well-posedness have natural
state/signal counterparts.
Our first article (Arov and Staffans, 2005) on the state/signal system was com-
pleted and submitted for publication in the fall of 2003, and it was followed by
many others. Some of the results presented in this book were obtained in collabo-
ration with Ph.D. Mikael Kulula. The bulk of the work was done during D. Arov’s
regular visits to Åbo Akademi during August–October 2003–2010 and to Aalto
University during August–October 2011–2017, with an average length of almost
three months. These visits were financed by the Academy of Finland, the Magnus
Ehrnrooth Foundation, and the Finnish Society of Sciences and Letters.
https://doi.org/10.1017/9781009024921.001 Published online by Cambridge University Press

xxii Preface
In the fall of 2009, it was decided that the theory was sufficiently mature to be
presented in terms of a book, and the writing of this book began on August 30,
2009. By the end of November 2009, a preliminary list of contents was ready. Two
significant factors in this decision were the research grant from the Academy of
Finland that relieved O. Staffans from teaching duties during the academic year
2009–2010 and the leave of absence for D. Arov for extensive periods of time from
the South Ukrainian Pedagogical University based on a joint exchange agreement
with Åbo Akademi.
The book we originally planned to write was supposed to be devoted to linear
time-invariant systems in discrete time. In 2011, we realized that it would be more
important to, instead, write a book on linear time-invariant systems in continu-
ous time, and in 2013 it was clear that it was not feasible to write only one book
on systems in continuous time. The continuous time theory contains a number of
mathematical difficulties that must first be sorted out, and this is done in the present
volume. The application of this theory to passive state/signal systems in continuous
time remains to be written down.
We thank the Academy of Finland, the Magnus Ehrnrooth Foundation, and the
Finnish Society of Sciences and Letters for their financial support, without which
this work could not have been carried out. We also thank Åbo Akademi and Aalto
University for excellent working facilities, and the South Ukrainian Pedagogical
University for giving D. Arov ample time to devote to research.
Above all, we are grateful to our wives Nataliya and Satu-Marjatta for their
constant support, understanding, and patience while this work was carried out.

Notations
Basic Sets and Symbols
C The complex plane.
C+
ω , C
+
ω C+
ω := {z ∈ C | z ω} and C
+
ω := {z ∈ C | z ≥ ω}.
C−
ω , C
−
ω C−
ω := {z ∈ C | z ω} and C
−
ω := {z ∈ C | z ≤ ω}.
C+, C
+
C+ := C+
0 and C
+
:= C
+
0 .
C−, C
−
C− := C−
0 and C
−
:= C
−
0 .
D+
r , D
+
r D+
r := {z ∈ C | |z| r} and D
+
r := {z ∈ C | |z| ≥ r}.
D−
r , D
−
r D−
r := {z ∈ C | |z| r} and D
−
r := {z ∈ C | |z| ≤ r}.
D+, D
+
D+ := D+
1 and D
+
:= D
+
1 .
D−, D
−
D− := D−
1 and D
−
:= D
−
1 .
R R := (−∞, ∞).
R+, R
+
R+ := (0, ∞) and R
+
:= [0, ∞).
R−, R
−
R− := (−∞, 0) and R
−
:= (−∞, 0].
T The unit circle in the complex plane.
N N is the set of natural numbers, i.e., N := {1, 2, 3, . . .}.
Z Z is the set of all integers, i.e., Z := {±1, ±2, ±3, . . .}.
Z+, Z− Z+ := {0, 1, 2, . . .} and Z− := {−1, −2, −3, . . .}.
j j :=
√
−1.
0 The number 0, or the zero vector in a vector space, or the zero
operator.
1 The number 1 and also the identity operator.
∗, † ∗ =

λ

λ ∈

and † =

−λ

λ ∈

.

xxiv List of Notations
Operators and Related Symbols
A, B, C, D In connection with an input/state/output system, A is usually the
main operator, B is the control operator, C is the observation
operator, and D is a feedthrough operator.
A, B, C, D Often A is the evolution semigroup, B is the input map, C is
the output map, and D is the input/output map of a well-posed
linear input/state/output system. See Definition 14.1.14.

A,
B,
C,
D Often
A is the state/state resolvent,
B is the input/state resol-
vent,
C is the state/output resolvent, and
D is the input/output
resolvent of an input/state/output node. See Definition 5.5.8.

(λ) If is a state/signal node with a characteristic node bundle

E[= E], then
(λ) is the state/signal node with generating sub-
space
E(λ), and if is an input/state/output node with a (for-
mal) input/state/output resolvent matrix
S, then
(λ) is the in-
put/state/output node with a system operator
S(λ). See Defini-
tion 5.5.8 and Lemma 10.3.3.
B(U; Y), B(U) The set of continuous linear operators from the H-space (or
topological vector space) U into the H-space (or topological
vector space) Y, respectively, from U into itself. See Notation
A.1.15.
ISO(U; Y), The set of continuously invertible linear operators mapping the
ISO(U) H-space (or topological vector space) U one-to-one onto the
H-space (or topological vector space) Y, respectively, from U
into itself. See Definition 2.1.28.
L(U; Y), L(U) The set of linear (single-valued) operators from the H-space (or
topological vector space) U into the H-space (or topological
vector space) Y, respectively, from U into itself. See Definition
A.1.13.
ML(U; Y), The set of multivalued linear operators from the H-space
ML(U) (or topological vector space) U into the H-space (or topological
vector space) U into Y, respectively, from U into itself. See
Definition A.1.51.
τt The bilateral shift operator on R: τtu(s) := u(s + t), t, s ∈ R
(this is a left shift when t 0 and a right shift when t 0).
τ∗t τ∗t = τ−t (this is a right shift when t 0 and a left shift when
t 0).

List of Notations xxv
τt
+ The left shift operator on R+: τt
+u(s) := u(s + t), s ∈ R+. Here
t ∈ R+.
τ∗t
+ The right shift operator on R+: τ∗t
+ u(s) := 0, 0 ≤ s t and
τ∗t
+ u(s) := u(s − t), s ≥ t. Here t ∈ R+.
τt
− The left shift operator on R−: τt
−u(s) := 0, −t s ≤ 0 and
τt
−u(s) := u(s + t), s ≤ −t. Here t ∈ R+.
τ∗t
+ The right shift operator on R−: τ∗t
+ u(s) := u(s − t), s ∈ R−.
Here t ∈ R+.
ιI The embedding operator L
p
loc(I) → L
p
loc(R): (ιIu)(t) := u(t), t ∈
I and (ιIu)(t) := 0, t /
∈ I. Here I ⊂ R.
ι+, ι− ι+ := ι[0,∞) and ι− := ι(−∞,0].
ρI The restriction operator L
p
loc(R) → L
p
loc(I): (ρIu)(t) := u(t), t ∈
I. Here I ⊂ R. ρIιI = 1L
p
loc(I) and ιIρI = πI.
ρ+, ρ− ρ+ := ρ[0,∞) and ρ− := ρ(−∞,0].
πI The projection operator in L
p
loc(R) with range L
p
loc(I) and kernel
L
p
loc(R I): (πIu)(s) := u(s) if s ∈ I and (πIu)(s) := 0 if s /
∈ I.
Here I ⊂ R. ρIπI = ρI and πIιI = ιI.
π+, π− π+ := π[0,∞) and π− := π(−∞,0].
R R is the time reflection operator in R, i.e., (R f)(t) = f(−t),
t ∈ R. See Definition 2.2.9.
Rt
s Rt
s is the time reflection operator in the time interval [s, t], i.e.,
(Rt
s f)(v) = f(s + t − v), v ∈ [s, t].
x, x∗ The continuous linear functional x∗ evaluated at x.
E⊥ If E ⊂ X, then E⊥ = {x∗ ∈ X∗ | x, x∗ = 0 for all x ∈ E}, and if
F∗ ⊂ X∗, then (F∗)⊥ = {x ∈ X | x, x∗ = 0 for all x∗ ∈ F∗}.
A∗ The (antilinear) adjoint of the operator A.
A−∗ A−∗ = (A∗)−1 = (A−1)∗.
A|X The restriction of the operator A to the subspace X.
A ⊂ B If A, B ∈ ML(X; Y) or A, B ∈ L(X; Y) and gph (A) ⊂ gph (B),
then we say that A is a restriction of B and that B is an extension
of A, and write A ⊂ B.
dom (A) The domain of the operator A.
rng (A) The range of the operator A.
ker (A) The null space (kernel) of the operator A.
mul (A) The multivalued part of the operator A.

xxvi List of Notations
dim(X) The dimension of the space X.
ρ(A) The resolvent set of the operator A (see Definitions 3.4.27 and
10.1.3).
ρ∞(A) The unbounded component of the resolvent set of the bounded
operator A (see Notation 6.1.2).
r∞(A) The spectral radius of the bounded operator A (see Notation
6.1.2).
ρi/s/o(S) The input/state/output resolvent set of S (see Definition 5.5.8).
ρ() The resolvent set of the input/state/output or state/signal system
(see Definitions 5.5.8 and 10.3.1).
ρbnd() The union of the resolvent sets of all bounded input/state/output
representations of the bounded state/signal system (see Def-
inition 7.1.1).
ρbnd
∞ () The unbounded component of ρbnd() (see Definition 7.1.1).
ρsbd() The union of the resolvent sets of all semi-bounded input/state/
output representations of the semi-bounded state/signal system
(see Definition 9.1.9).
ρsbd
+∞() The component of ρsbd() that contains a right half-plane (see
Definition 9.1.9).
ω(A) The growth bound of the semigroup A. See (8.1.1).
TI, TIC TI stands for the set of all shift invariant operators, and TIC
stands for the set of all shift invariant and causal operators. See
Definition 14.4.1 for details.
Vector Spaces
H-space A topological vector space X that is isomorphic to a Hilbert
space, i.e., the topology in X is induced by a norm induced by
a Hilbert space inner product. See Definitions 2.1.2 and A.1.6.
B-space A topological vector space X that is isomorphic to a Banach
space, i.e., the topology in X is induced by a Banach space
norm. See Definitions 2.1.2 and A.1.6.
U Frequently the input space of an input/state/output system.
X Frequently the state space of an input/state/output or state/signal
system.
Y Frequently the output space of an input/state/output system.
W Frequently the signal space of a state/signal system.

List of Notations xxvii
X•, X◦ X• is the interpolation space and X◦ is the extrapolation space
induced by a closed operator A in X with a dense domain. See
Definitions 10.1.13 and 10.1.17.
A•, A◦ A• is the part of A in X• and A◦ is the extension of A to a closed
operator in X◦.
A•, A◦ A• is the restriction of the C0 semigroup A in X to a C0 semi-
group in X• and A◦ is the extension of AA to a C0 semigroup
in X◦.
X = X1 X2 X = X1 X2 means that X is an H-space that is the direct sum
of its two closed subspaces X1 and X2, i.e., every x ∈ X has
a unique representation of the form x = x1 + x2, where x1 ∈ X1
and x2 ∈ X2.
PZ
Y If X = Y Z, then PZ
Y is the projection in X onto Y along Z,
i.e., the range of PU
Y is Y and the kernel is U.
QZ
Y If X = Y Z, then QZ
Y x = y, where y ∈ Y is the unique vector
in Y in the decomposition x = y + z with y ∈ Y and z ∈ Z. Thus,
QZ
Y is equal to PZ
Y , reinterpreted as an operator in B(X; Y) (in-
stead of an operator in B(X)). See Definition A.1.29.
U
Y

The cross-product of the two H-spaces U and Y. Thus,
U
Y

=
U
0

0
Y

. Also denoted by U × Y.
U × Y The cross-product of the two H-spaces U and Y. Also denoted
by
U
Y

.
Special Functions
eω eω(t) = eωt for ω, t ∈ R.
log The natural logarithm.
Function Spaces
V(I; Z) Functions of type V (= Lp, C, BC, etc.) on the interval I ⊂ R
with range in Z.
Vloc(I; Z) Functions that are locally of type V, i.e., they are defined on
I ⊂ R with range in Z, and they belong to V(I; Z) for every
compact subinterval I ⊂ I.
V(I; Z) Functions in V(I; Z) with compact support.
V,loc(I; Z) Functions in Vloc(I; Z) whose support is bounded to the left.
Vloc,(I; Z) Functions in Vloc(I; Z) whose support is bounded to the right.

xxviii List of Notations
Vω(I; Z) The set of functions u for which (t → e−ωtu(t)) ∈ V(I; Z). See
also the special cases listed below.
V,ω(I; Z) Functions in Vω(I; Z) whose support is bounded to the left.
Vω,loc(I; Z) The set of functions u ∈ Vloc(I; Z) that satisfy ρI∩R− u ∈
Vω(I ∩ R−; Z).
V◦(I; Z) The closure of V(I; Z) in V(I; Z). Functions in V◦(I; Z)
“vanish at infinity.” See also the special cases listed below.
BC The space of bounded continuous functions with the sup-norm.
BC◦ Functions in BC that tend to zero at ±∞.
BCω Functions u for which (t → e−ωtu(t)) ∈ BC.
BCω,loc Continuous functions whose restrictions to R− belong to BCω.
BC◦,ω Functions u for which (t → e−ωtu(t)) ∈ BC◦.
BC◦,ω,loc Continuous functions whose restrictions to R− belong to BC◦,ω.
BUC Bounded uniformly continuous functions with the sup-norm.
BUCn Functions that together with their n first derivatives belong to
BUC.
C Continuous functions. The same space as BCloc.
Cn n times continuously differentiable functions. The same space
as BCn
loc.
Lp, 1 ≤ p ∞ See Notation 2.1.4.
L
p
loc Functions that belong locally to Lp.
L
p
Functions in Lp with compact support.
L
p
,loc Functions in L
p
loc whose support is bounded to the left.
L
p
ω Functions u for which (t → e−ωtu(t)) ∈ Lp.
L
p
ω,loc(R; Z) Functions u ∈ L
p
loc(R; Z) that satisfy ρ−u ∈ L
p
ω(R−; Z).
W1,p Functions in Lp that have a (distribution) derivative in Lp. See
Notation 2.6.1.
H∞(; X) The space of bounded analytic X-valued functions on .
Spaces of Sequences
p, 1 ≤ p ∞ Sequences z = {zn}n∈I satisfying

I|zn|
p
Z ∞. See
Notation 6.6.3.
∞ The vector space of bounded sequences z = {zn}n∈I. See
Notation 6.6.3.

1
Introduction and Overview
The class of linear time-invariant state/signal systems studied in this monograph is general
enough to include most of the standard classes of linear time-invariant dynamical sys-
tems, and at the same time, it is small enough that standard control theory notions for
input/state/output (i/s/o) systems have natural extensions to this class. This includes the
notions of controllability and observability, minimality, stability, stabilizability and de-
tectability, passivity, and optimal control. Like an i/s/o system, a state/signal system has
a state component that can be used to model energy-storing elements and energy sources
and sinks, and it also has a signal component that connects the system to the outside world
and can be used to observe, control, and interconnect state/signal systems. In this chapter,
we first discuss different mathematical approaches to the notion of a linear time-invariant
dynamical system and explain the motivation behind our state/signal approach, and then
continue with an overview of the contents of this monograph.
1.1 Linear Time-Invariant Dynamical Systems
There are many different mathematical approaches to the theory of dynamical systems.
A dynamical system describes the evolution of some quantities as a function of a time
variable, which can be discrete (i.e., the time variable takes integer values) or continuous
(i.e., the time variable takes real values). In our case, this quantity will be a vector in a
vector space, whose value changes with time. This varying value gives rise to a trajectory
of the system, which is a vector-valued function of a scalar time variable. In the most
general setting, the dynamical systems are allowed to be nonlinear and time dependent,
but this monograph is devoted to the study of linear and time-invariant systems. Linearity
means that the set of trajectories is invariant both under multiplications by scalars and
under additions of trajectories defined on the same time interval, and time invariance means
that trajectories that are shifted forward or backward in time remain trajectories of the
same system. Most of the time, we take the time variable to be continuous (defined on a
subinterval of the real line), but we also include a short discussion on bounded systems
with a discrete time variable.

2 Introduction and Overview
1.1.1 State Systems
In the simplest version of a linear time-invariant system in continuous time, the trajecto-
ries consist of a finite set of real or complex state variables that satisfy a finite system of
differential equations. (If the time variable is discrete, then this system is replaced by a
system of difference equations.) The linearity and time invariance of this system mean that
the coefficients in the system of differential equations are independent of both the state and
the time variables. We call a system of this type a (linear time-invariant) finite-dimensional
state system. It can often be rewritten in the vector form
ẋ(t) = Ax(t), t ∈ R, (1.1.1)
where x(t) is an n-dimensional real or complex vector (i.e., x(t) ∈ Rn
or x(t) ∈ Cn
), ẋ(t) is
the time derivative of x, and A is an n × n matrix for a positive integer n. This system is
well-posed (or well defined), i.e., it is true that for every initial state x0
∈ Rn
or x0
∈ Cn
and
every initial time t0 ∈ R, the system has a unique trajectory, defined on the full real line
R = (−∞, ∞) with the given initial state x0
at the given initial time t0. Due to the time
invariance of the system, the initial time is irrelevant in the sense that we can always take
t0 to be zero (by a simple time shift). Thus, the past and future evolution of such a system
is determined completely by the state x(0) at time zero.
If we replace the system of differential equations (1.1.1) with some other type of equa-
tions, such as a system of partial differential equations, or integral equations, or delay
equations, or a mixture of such equations, then the dynamics of the system become more
complicated. Such a system can often still be described by a linear first-order differential
equation (of a very general type) in an infinite-dimensional vector space X with opera-
tor (possibly unbounded or multivalued) coefficients that depend neither on the space nor
the time variable. In the sequel, we refer to X as the state space of the system. Depend-
ing on the situation, the state space X may be taken to be a Hilbert space, or a Banach
space, or an even more general topological vector space. In this monograph, we concen-
trate our attention on the case where the state space is a Hilbert space (or strictly speaking,
an H-space, as explained in Section 1.1.14). The well-posedness of a system of this type
may depend on the direction of time, i.e., a system may be well-posed in the forward time
direction without being well-posed in the backward time direction. Under suitable assump-
tions, a first-order system of this type can often be rewritten as an abstract differential
equation
ẋ(t) = Ax(t), t ∈ I, (1.1.2)
where I is a subinterval of the real line R = (−∞, ∞) and A is a linear operator from its
domain dom (A) ⊂ X into X. In some cases, equation (1.1.2) needs to be replaced with the
even more general equation
ẋ(t) ∈ Ax(t), t ∈ I, (1.1.3)
where A is a linear multivalued operator, and the inclusion ẋ(t) ∈ Ax(t) is equivalent to the
requirement that ẋ(t)
x(t) ∈ gph (A), where gph (A) is the graph of A.

1.1.2 Systems That Interact with the Outside World
The dynamical systems that we have considered so far are “closed” (as opposed to being
“open” in the sense of Livšic (1973)), i.e., they do not include any channels that can be
used to interconnect the system with the outside world. Such channels are needed if one
wants to monitor the system from the outside, or to guide the system to a desired state, or
to interconnect two systems with each other, and they can be created in different ways.
(i) In the input/state/output (i/s/o) approach, one adds an input channel and an output
channel to a state system of the type described in (1.1.3), through which information
can enter and leave the state system. In this approach, each trajectory has three com-
ponents, all of which are functions of the time variable t, namely a state component
x(t), an input component u(t), and an output component y(t).
(ii) In the input/output (i/o) approach, each trajectory consists of two components, namely
an input component u(t) and an output component y(t). Here, the focus of attention is
on how the output y depends on the input u. This can be thought of as a “black box”
model of an i/s/o system of the type described in (i), where the underlying state system
is not known (or ignored).
(iii) In classical network theory, one starts from a finite-dimensional state system and adds
a bidirectional (multidimensional) interaction channel that connects this state system
to the outside world and permits information to both enter and leave the state system.
This channel is not a priori split into an input channel and an output channel. In this
approach, each trajectory has two components, namely a state component x(t) and an
interaction signal w(t).
(iv) In a port-Hamiltonian system, each trajectory consists of a state component and a
signal component. The equations for the “internal dynamics” of the state component
are energy preserving, and the interaction with the surroundings takes place through
the same type of (finite- or infinite-dimensional) energy-preserving port structure as
in network theory. Dissipative systems are modeled by terminating one of the ports
with a dissipative element.
(v) In the behavioral approach, trajectories are functions with values in a “signal space,”
and the attention is focused on interactions between different parts of the signal with-
out an explicit splitting of the signals into an “input part” and an “output part.” This
can be thought of as a “black box” model of a generalized version of a network of the
type described in (iii), where the underlying state system is not known (or ignored). In
this approach, each trajectory has only a signal component and no state component.
These different types of approaches are discussed in more detail in the following sections.
1.1.3 Input/State/Output Systems
In the finite-dimensional setting, it is easy to add inputs and outputs to a state system of the
type (1.1.1) by adding input and output terms to (1.1.1) to get an i/s/o system of the form
ẋ(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),
t ∈ R, (1.1.4)

where u(t) and y(t) are p-dimensional and q-dimensional real or complex vectors, and B,
C, and D are matrices of appropriate dimensions. The same approach works well in the
infinite-dimensional setting, where x(t), u(t), and y(t) take their values in some Hilbert
spaces X, U, and Y, and the operators A, B, C, and D are bounded linear operators between
the appropriate spaces. We call the resulting system a bounded i/s/o system. We may even
relax the condition for A and only require that it is the generator of a strongly continuous
semigroup in X, but keep the assumption that B, C, and D are bounded, in which case we
end up with a semi-bounded i/s/o system. More general i/s/o systems will be encountered
later in this monograph. In some cases, either the input u or the output y is missing, in
which case we have a state/output system and an input/state system. Classical i/s/o systems
are discussed, e.g., in Kalman et al. (1969).
1.1.4 Input/Output Systems
In the i/o setting, each trajectory has two components, namely an input component u whose
values lie in an input space U and an output component y whose values lie in an output
space Y, but there is no explicit state component x. In this setting, one wants to know
how the output component y depends on the input component u. In the finite-dimensional
case, it is typically assumed that the input u and output y satisfy a finite-order system of
differential equations of the type
Pout
d
dt
y = Pin
d
dt
u, (1.1.5)
where Pout and Pin are matrix-valued polynomials with the same row dimension. Under
suitable regularity assumptions, it is possible to construct an underlying i/s/o system with
the property that to each i/o pair
y
u

satisfying the relation (1.1.5), there corresponds an
i/s/o triple
x
y
u
satisfying (1.1.4). Such an i/s/o system is called a realization of the i/o
relation (1.1.5). The state x of such an i/s/o representation is not unique, but there exists a
realization with minimal state space dimension, and all realizations with the same minimal
state space dimension are similar to each other. In addition, there also exist realizations
with a nonminimal state dimension.
In the infinite-dimensional version of an i/o system, the differential equation (1.1.5)
can be replaced with some other type of linear time-invariant relation (i.e., a linear rela-
tion between u and y, which commutes with time shifts). This relation may involve par-
tial differential operators, or integral operators, or time delays, etc. The analysis of this
more general type of i/o system is often based on the existing theory of linear opera-
tors acting on some space of functions of a time variable that are invariant under right
shifts and/or left shifts, and the properties of the shift operator in various function spaces
become important. Also, in this case, it is often possible to find some underlying i/s/o
system for which the variables u and y play the role of inputs and outputs, respectively,
but that system need not be bounded or semi-bounded (and it is, of course, not
unique).

1.1.5 Classical (Sub)networks
A classical network (or strictly speaking, subnetwork) resembles a finite-dimensional i/s/o
system in the sense that it has a state variable, and it can exchange information with the
outside world, but this interchange of information does not take place through dedicated
input and output channels. Instead, there is an interface consisting of m ≥ 1 “ports,” where
each port supports two scalar signals, so that the total number of interaction signals is
even (= 2m). In an electrical circuit, each port consists of two terminals, and the two port
variables are the current entering the port through its “positive” terminal and the voltage
between the positive and the “negative” terminals. The product of these two port vari-
ables is proportional to the power absorbed by the system through this particular port,
where a positive value means that the power is absorbed by the network, and a nega-
tive value means that the power is emitted from the network. By combining port currents
and voltages in different ways, one can group the 2m-dimensional interaction signal into
an m-dimensional input and an m-dimensional output. Some choices will lead to well-
posed i/s/o systems, meaning that for each time interval [0, T], the (final) state x(T) at
time T and the restriction of the output y to the interval [0, T] depend continuously on
the (initial) state x(0) at time 0 and the restriction of the input u to the interval [0, T].
Other combinations of port currents and voltages into an m-dimensional input and an m-
dimensional output may not lead to well-posed i/s/o systems. However, in order to con-
nect two such (sub)networks to each other, there is no need to split the port currents and
voltages into dedicated inputs and outputs; instead, one simply requires the connection to
satisfy a certain energy-preserving algebraic condition – namely that the voltages over two
connected ports are the same and that the sum of the current entering the two connected
ports must be zero (i.e., the current entering one of the connected ports must be the same
as the current leaving the other). Classical network theory is discussed in, e.g., Belevitch
(1968), Fuhrmann and Helmke (2015), Kuh and Rohrer (1967), Seshu and Reed (1961),
and Wohlers (1969).
1.1.6 Port-Hamiltonian Systems
Trajectories of a port-Hamiltonian system have both a state component and a signal com-
ponent through which the system interacts with the outside world. A port-Hamiltonian
system consists of several different components that are interconnected through an energy-
preserving structure, called a Dirac structure. Two of these components are interpreted
as “internal components,” namely an energy-preserving dynamic component and a static
dissipative component, and the interconnection to the outside world takes place through
a third part of the Dirac structure that from the outside looks like the port of a network.
In the network interpretation of a finite-dimensional port-Hamiltonian system, the state
consists of a collection of capacitors and inductors that can store potential and magnetic
energy, respectively, and energy is dissipated in resistors. The Dirac structure describes the
interconnections of these elements, the signal “flows” correspond to currents entering the
ports, and the “efforts” correspond to voltages over the ports. In an infinite-dimensional

setting where the dynamics of a port-Hamiltonian system is described with a partial dif-
ferential equation in a space domain, the signal part of the system is used to describe the
flow of energy through the boundary. A port-Hamiltonian system can be interpreted (in the
linear time-invariant case) as a special case of a passive state/signal system (a short intro-
duction to passive state/signal systems is given in Section 15.7). For an introduction and
further references to port-Hamiltonian systems, we refer the reader to Cervera et al. (2003,
2007), Le Gorrec et al. (2005), Jacob and Zwart (2012), Kurula et al. (2010), Ortega et al.
(2002), van der Schaft (2000, 2006), van der Schaft and Jeltsema (2014), van der Schaft
and Maschke (1994, 2002, 2018), and Wu et al. (2018).
1.1.7 Behavioral Systems
A behavioral system resembles an i/o system in the sense that it does not postulate the
existence of an underlying state system; on the other hand, it differs from an i/o system
in the sense that the trajectories of a behavioral system are not formally decomposed into
an input component and an output component. This resembles the behavior of the port
variables of a classical network, but there is no “port structure” imposed on the trajectories,
i.e., the dimension of the signal space W in which the values of the trajectories may be even
or odd, and there is no “power” associated with the trajectories. The easiest way to arrive
at the notion of a finite-dimensional behavioral system is to combine the p-dimensional
input u and the q-dimensional output y of a finite-dimensional i/o system into a (p + q)-
dimensional signal w, and to require this signal to satisfy a simplified version of (1.1.5),
namely
P
d
dt
w = 0, (1.1.6)
where P is a matrix-valued polynomial. Every relation of the form (1.1.5) can be put into
the form (1.1.6) by defining the interaction signal w to be the i/o pair
u
y

and taking
P =

Pin −Pout

. There also exist methods to go from (1.1.6) to (1.1.5), but since the
splitting of the interaction signal w into an input u and an output y is not unique, to each
signal relation of the type (1.1.6) there correspond infinitely many i/o relations of the type
(1.1.5). It is further possible to develop i/s/o representations of the type (1.1.4) for a behav-
ioral system by first splitting the signal w into an input and an output, and then applying
known methods for getting an i/s/o representation of the i/o relation (1.1.5). Of course, there
is now an additional free parameter in this construction: in addition to the nonuniqueness
of the state space X of the system, the splitting of the signal space W into an input space
U and an output space Y is highly nonunique. For an introduction to behavioral systems
and further references, we refer the reader to Polderman and Willems (1998), Weiland and
Willems (1991), Willems (1991, 2007), Willems and Yamamoto (2007), and Willems and
Trentelman (1998, 2002).

The connection between behavioral systems and the class of state/signal systems will be
discussed in Sections 1.1.8 and 1.1.12.
1.1.8 State/Signal Systems
The class of linear time-invariant dynamical systems that we introduce in this monograph
under the name “state/signal systems” can be interpreted as a generalization of the notion
of a classical network. As in the case of a classical network, each trajectory has two com-
ponents: a state component x(t) with values in a state space X and a signal component w(t)
with values in a signal space W. These spaces are allowed to be (finite-dimensional or)
infinite-dimensional Hilbert spaces (or more precisely, H-spaces, as will be explained in
Section 1.1.14), and as in behavioral theory, there is no extra “port” structure imposed
on the signal space W. The main difference between the classes of i/s/o systems and
state/signal (s/s) systems is that in a s/s system the interaction signal is not a priori split
into an input and an output. We mentioned earlier that a behavioral system can be in-
terpreted as a “black box” model of a generalized version of a network. A more precise
statement would be that a behavioral system can be interpreted as a “black box” model of
a s/s system.
The formal definition of a s/s system is very simple, and the same definition can be
used in the finite- and infinite-dimensional settings. It does not involve any unbounded
operators. To arrive at this definition, we take a closer look at equation (1.1.4), describing
the evolution of the trajectories of a linear time-invariant finite-dimensional i/s/o system.
This equation can be interpreted as a linear relation between the four variables x(t), ẋ(t),
u(t), and y(t), where x(t) and ẋ(t) belong to the state space X, u(t) belongs to the input
space U, and y(t) belongs to the output space Y. If we remove the distinction between the
input and the output, and consider both the input u(t) and the output y(t) to be parts of the
interaction signal w(t), then we end up with a linear relation between x(t) ∈ X, ẋ(t) ∈ X,
and the signal w(t) ∈ W, where W is the signal space. Every such linear relation can be
written in the form

ẋ(t)
x(t)
w(t)

∈ V, t ∈ I, (1.1.7)
where V is a subspace of
X
X
W
. We call V the generating subspace of the system. By a
classical trajectory of (1.1.7), we mean a pair of functions
x
w

, where x is continuously
differentiable on I, w is continuous on I, and (1.1.7) holds.
It is, of course, possible to rewrite (1.1.7) into several other equivalent forms. If V is
closed, then we can think about V as the kernel of a surjective bounded linear operator

−E M N

from
X
X
W
into an auxiliary space Y and rewrite (1.1.7) in the form
Eẋ(t) = Mx(t) + Nw(t), t ∈ I. (1.1.8)

This representation is unique up to the multiplication with a bounded linear operator with
bounded inverse from the left. We call this a kernel representation of (1.1.7). Another
possibility is to interpret V as the range of an injective bounded linear operator
K
F
L
from
an auxiliary space U into
X
X
W
, which can be used to rewrite (1.1.7) in the form
d
dt
Fv(t) = Kv(t),
x(t) = Fv(t),
w(t) = Lv(t),
t ∈ I. (1.1.9)
This representation is unique up to the multiplication with a bounded linear operator with a
bounded inverse from the right. We call this an image representation of (1.1.7). If the gen-
erating subspace V has the property that the first component ẋ(t) in (1.1.7) is determined
uniquely by the other two components x(t)
w(t) , and if we let G be the linear operator map-
ping x(t)
w(t) into ẋ(t) whose graph is equal to V, then (1.1.7) can alternatively be written in
the form
ẋ(t) = G x(t)
w(t) , t ∈ I. (1.1.10)
The domain of the operator G need not be the full space
X
W

or even dense in
X
W

, which
means that the implicit condition x(t)
w(t) ∈ dom (G) hidden in (1.1.10) creates a linear de-
pendence between x(t) and w(t).
In this monograph, we primarily use the representation (1.1.7), but certain results are
easier to prove using the representation (1.1.8), (1.1.9), or (1.1.10).
1.1.9 State/Signal versus Input/State/Output Systems
Above we described how to convert the i/s/o system (1.1.4) into a s/s system (1.1.7) by
combining the input u(t) and the output y(t) into an interaction signal w(t) = y(t)
u(t) . This
process can be reversed by splitting the signal space W of the s/s system (1.1.7) into W =
U Y, and splitting the signal w(t) accordingly into w(t) = u(t) + y(t) where u(t) ∈ U and
y(t) ∈ Y. By doing so we can rewrite (1.1.7) in the form (where we have reordered the
components, so that y(t) comes before x(t))
ẋ(t)
y(t)
x(t)
u(t)

∈ Vi/s/o, t ∈ I, (1.1.11)
where Vi/s/o is the subset of
X
Y
X
U

that we obtain from the generating subspace V in (1.1.7).
If Vi/s/o has the property that the pair ẋ(t)
y(t) in (1.1.11) is defined uniquely by x(t)
u(t) , then
we can rewrite (1.1.11) into a more familiar i/s/o form
ẋ(t)
y(t) = S x(t)
u(t) , t ∈ I, (1.1.12)

where the system operator S is the linear operator from
X
U

into
X
Y

whose graph is equal
to Vi/s/o. The advantage of the representation (1.1.12) compared with the representation
(1.1.10) is that with a suitable choice of the decomposition W = U Y, it is usually possi-
ble to guarantee that the domain of S is dense in
X
U

. In the finite-dimensional case, after
working out all the details, one ends up with an equation of the type (1.1.4) (with R being
replaced by I). The i/s/o system (1.1.11) is called an i/s/o representation of the state/signal
system (1.1.7). Later, we shall sometimes drop the condition that ẋ(t)
y(t) in (1.1.11) is
defined uniquely by x(t)
u(t) and permit the operator S in (1.1.12) to be multivalued, in which
case (1.1.12) should be replaced with the relation
ẋ(t)
y(t) ∈ S x(t)
u(t) , t ∈ I. (1.1.13)
The close relationship between i/s/o and state/signal systems expressed by (1.1.7),
(1.1.11), and (1.1.13) makes it possible to transfer many standard system theoretic notions
for i/s/o systems to the class of state/signal systems, provided we make a small change
(with drastic consequences) in the standard definition of what one means by a classical
trajectory of an i/s/o system. In the standard finite-dimensional setting (1.1.4), if we as-
sume that x and u are continuous functions on an interval I, then it follows from (1.1.4)
that x is continuously differentiable on I and that y is continuous on I. If we replace (1.1.4)
with (1.1.12) or (1.1.13), then the continuity of y and continuous differentiability of x can
no longer be taken for granted. Instead, we therefore impose an a priori continuous dif-
ferentiability assumption on x and an a priori continuity assumption on y in (1.1.12) or
(1.1.13), in addition to the assumption that u is continuous on I. (In the finite-dimensional
well-posed case, this extra condition is redundant.) With this added smoothness condition,
there is a one-to-one correspondence between classical trajectories of (1.1.7) and those of
(1.1.12) or (1.1.13), as soon as the decomposition W = U Y is fixed. This makes it pos-
sible to transfer all the standard “dynamic” notions for i/s/o systems that can be defined
in terms of the behavior of trajectories into corresponding notions for state/signal systems.
This includes notions related to stability, stabilizability and detectability, controllability
and observability, minimality, compressions and dilation, and various transformations and
interconnections. Depending on the particular i/s/o notion, the corresponding state/signal
notion falls into one of the following two categories:
(i) In some cases, if one of the i/s/o representations of a s/s system has a particular i/s/o
property, then every i/s/o representation of has the same property, in which case
we say that the corresponding state/signal system has the analogous s/s property. For
example, the property that the system operator S in (1.1.12) is closed is of this type, i.e.,
if V has at least one representation as the graph of a closed system operator, then V is
closed, and every other system operator in a graph representation of V is also closed.
Thanks to our slightly nonstandard definition of the notion of a trajectory, also the
notions of controllability, observability, minimality, stabilizability, and detectability
are of the same type. There even exist some weak existence and uniqueness properties
(where the input and the output are treated in a symmetrical way), which belong to the
same class. We call this class of i/s/o properties i/o invariant.

(ii) If it is instead true that the s/s system has a particular s/s property as soon as at least
one of its i/s/o representations has the analogous i/s/o property (but the s/s system may
also have i/s/o representations that do not have this property), then we say that this
i/s/o property is i/o dependent.
Thus, in order to show that a particular i/s/o representation has a property of the type (i),
it suffices to show that some other i/s/o representation has the same property, and in order
to show that a state/signal system has a property of type (ii), it suffices to show that it
has at least one i/s/o representation that has the corresponding i/s/o property. For example,
the i/s/o notions of boundedness, semi-boundedness, well-posedness, and stability are i/o
dependent, i.e., every bounded, or semi-bounded, or well-posed, or stable s/s system has
at least one bounded, or semi-bounded, or well-posed, or stable i/s/o representation, but it
may also have i/s/o representations that do not have these properties. This will be explained
in more detail in Section 1.2.
1.1.10 Frequency Domain Systems
In the existing literature, nonlinear and time-dependent systems are primarily discussed in
the time domain (as we have done above). Linear time-invariant i/o and i/s/o systems also
have a rich frequency domain theory that complements the time domain theory. In this
monograph, we develop an analogous frequency domain theory for state/signal systems,
and in addition, we expand the frequency domain theory for i/s/o systems by introducing
the notion of a frequency domain trajectory of an i/s/o system. A time domain trajectory
is a vector-valued function of a time variable, whereas a frequency domain trajectory is an
analytic vector-valued function of a frequency variable. In the i/s/o setting, both the time
domain and frequency domain trajectories have an initial state, a “final” state, an input,
and an output, and in our state/signal setting, both the time domain and frequency domain
trajectories have an initial state, a “final” state, and an interaction signal. Under additional
regularity assumptions, frequency domain trajectories can be interpreted as Laplace trans-
forms of time domain trajectories in the case where the time variable is continuous, or as
Z-transforms of time domain trajectories when the time variable is discrete. A time do-
main trajectory is defined in some time interval (finite or infinite), whereas a frequency
domain trajectory is defined in an open subset of the complex plane. The choice of which
particular frequency domain to use depends on the situation at hand. For example, for
a passive discrete time system, the natural frequency domain is either the outside or the
inside of the unit disk, depending on whether we are looking for the evolution in the for-
ward or backward time direction, and for a passive continuous time system, the natural
frequency domain is either the right or the left half-plane, again depending on the direction
of time.
Basically, all the standard frequency domain i/s/o notions have state/signal counterparts,
although these counterparts often appear in a different form. For example, the standard
i/o “transfer function” or “characteristic function,” which is an analytic operator-valued
function, is replaced by an analytic vector bundle that we call the characteristic signal

bundle. This is an analytic family of subspaces parameterized by a complex variable. If an
i/s/o system is reformulated as a state/signal system, then the graph of the transfer function
becomes the characteristic signal bundle, and conversely, to the characteristic signal bundle
of a state/signal system correspond infinitely many i/o transfer functions, which differ from
each other by the way in which the interaction signal is split into an input and an output.
1.1.11 Boundary Triplets
The notion of a boundary triplet is usually not associated with the notion of a dynamical
system, but it can be interpreted as a frequency domain representation of a special class
of conservative (or passive) state/signal systems, which is closely related to the class of
port-Hamiltonian systems. The starting point in the theory of boundary triplets is a closed
densely defined symmetric operator A0 with equal deficiency indices, and the problem is to
find all self-adjoint operators A satisfying gph (A0) ⊂ gph (A) ⊂ gph

A∗
0

. One way to think
about this is that A∗
0 is a partial differential operator without any boundary conditions, A0 is
the corresponding partial differential operator subject to a “maximal” set of boundary con-
ditions, and the problem is to characterize which “intermediate” set of boundary conditions
leads to a self-adjoint operator A. In the state/signal interpretation of a boundary triplet, the
signal space consists of two copies of the “coefficient space,” and the Weyl function and
γ -field are related to our notions of an i/o and input/state resolvent, respectively. For more
details and further references, we refer the reader to Arov et al. (2012a,b), Behrndt et al.
(2009, 2020), Derkach et al. (2006, 2009), and Gorbachuk and Gorbachuk (1991).
1.1.12 State/Signal versus Behavioral Systems
As mentioned in Section 1.1.8, a behavioral system can be interpreted as a “black box”
description of a state/signal system where the underlying state system is not known (or
ignored).
As soon as one removes or ignores the state component of a state/signal system the
majority of the standard system theoretic notions, i.e., those that in one way or another
refer to the state of the system, become undefined. This is, in particular, true about all the
system theoretic notions listed in Section 1.1.9. This problem has been remedied in be-
havioral theory by introducing new alternative definitions that are in some sense related
to the original system theoretic notions, at least in the finite-dimensional case. This is, in
particular, true about the notions of “controllability” and “observability” used in finite-
dimensional behavioral theory. Since we shall have no explicit use of these alternative
notions in this monograph, we do not describe them here in detail, but instead refer the
reader to the existing literature on behavioral systems. Suffice it to say that the behavioral
versions of these notions are less intuitive and more difficult to extend to the infinite-
dimensional case than our corresponding state/signal notions. For example, in the behav-
ioral setting, it is no longer true that minimality is equivalent to controllability and observ-
ability, and neither is it true that controllability and observability are dual properties in the

sense that a behavioral system is controllable if and only if the adjoint system is observ-
able. State/signal systems, on the other hand, have both of these properties (at least in the
well-posed case).
1.1.13 How to Read This Book
We expect the state/signal formulation of a linear time-invariant dynamical system to be of
interest not only to mathematicians interested in mathematical systems theory, functional
analysis, or operator theory, but also to the control community that faces everyday finite-
dimensional control problems in continuous and discrete time.
Those readers who are primarily interested in finite-dimensional systems (or more gen-
erally, in bounded systems in discrete or continuous time) should glance through Section
1.2 to get an overview of the contents of this book, skip most of the proofs in Chapters 2–5,
assuming that all the systems are regular and solvable and have the uniqueness and contin-
uation properties, and then continue to study Chapters 6 and 7 in detail. Readers who are
primarily interested in the class of semi-bounded continuous time systems should read, in
addition, Chapters 8 and 9. A reader with primary interest in frequency domain theory may,
after reading Chapters 6 and 7, skip Chapters 8 and 9 and go directly to Chapters 10–12.
The theory of well-posed systems is presented in Chapters 14 and 15. Before reading these
two chapters, the reader should at least glance through Section 1.2 and Chapters 2–5, read
Chapters 6 and 7, the first part of Chapter 8 (on semigroup theory), and the first part of
Chapter 10. Some results from Chapters 11–13 are also needed for a complete understand-
ing of Chapters 14 and 15, but a detailed study of these chapters is not necessary for the
understanding of the basic time domain theory of well-posed i/s/o and s/s systems.
1.1.14 H-Spaces
Throughout this monograph, we take the state space, signal space, input space, and output
space to be H-spaces. An H-space is a “topological version” of a Hilbert space, i.e., it is
a topological vector space whose topology is induced by some Hilbert space norms. This
norm is not unique, since it is possible to define many different inner products that induce
equivalent norms. Such inner products and norms are called admissible Hilbert space inner
products and norms. A reader who does not feel comfortable with the notion of an H-
space may throughout replace our H-spaces with Hilbert spaces by simply fixing some
suitable admissible inner products and norms in these spaces, and sticking to the same
inner products and norms throughout all the proofs. Our motivation for using an H-space
setting instead of a Hilbert space setting (except when we discuss passive systems) is that
we want to emphasize the fact that the results we present do not depend on the choice of
admissible inner products and norms in the H-spaces. This is especially important in the
theory of passive state/signal systems where the natural signal space is a Kreı̆n space with
a “canonical” indefinite inner product, but does not have a “canonical” (uniquely defined)
Hilbert space inner product.

1.1.15 Where to Go from Here?
The “state/signal systems story” that we present in this book is far from finished. It can
be interpreted as an alternative way of looking at “open linear time-invariant dynamical
systems” (where the word “open” is used in the sense of Livšic (1973)), and depending on
the interest of the reader, it can be developed further in many different directions.
• One obvious possibility is to apply the theory presented here to conduct a more de-
tailed study of the class of passive state/signal systems defined in Section 15.7, thereby
extending existing results on passive i/s/o systems to the state/signal setting. See, e.g.,
Arov and Staffans (2007b, 2009a,b, 2010, 2012), Arov et al. (2011, 2012a,b), Ball and
Staffans (2006), Ball et al. (2015), Kurula (2010), Malinen and Staffans (2006, 2007),
Malinen et al. (2006), Tucsnak and Weiss (2003), and Weiss and Tucsnak (2003).
• Another direction is to extend the existing theory on H2
-optimal control for i/s/o systems
to state/signal systems. See, e.g., Callier and Winkin (1992, 1999), Curtain and Op-
meer (2005, 2006), Curtain et al. (1996), Flandoli et al. (1988), Grabowski (1989, 1991),
Ionescu and Weiss (1993), Lasiecka and Triggiani (1986, 1991, 2000a,b), Mikkola (2002,
2006a, 2007), Opmeer and Curtain (2004), Opmeer and Staffans (2008, 2010, 2012,
2014, 2019), Staffans (1995, 1996, 1997, 1998a,b, 1999b), Weiss (2003), Weiss and Re-
barber (2000), Weiss (1994b, 1997), Weiss and Weiss (1995, 1997), and Zwart (1996)
for recent i/s/o studies of H2
-optimal control.
• The connection between the state/signal and i/s/o interconnections presented here and the
theory of interconnections of behavioral systems and port Hamiltonian systems needs to
be clarified. See, e.g., Aalto and Malinen (2013), Behrndt et al. (2020), Cervera et al.
(2007), Gorbachuk and Gorbachuk (1991), Kurula et al. (2010), Maschke and van der
Schaft (2005), Megawati and van der Schaft (2017), van der Schaft and Jeltsema (2014),
van der Schaft and Maschke (2002, 2018), and Willems (2007). In our terminology, each
port connection in the network or port-Hamiltonian theory amounts to taking a “part” (by
forcing the voltages over the connected ports to be equal and the currents to be opposite)
combined with a “static projection” (in order to ignore the values of the voltages over
and the currents passing through the connected port). To ensure that the interconnected
network is well-posed or minimal, one may, in addition, have to take a part and/or apply
a static projection in the state space to remove irrelevant components of the original state
space.
• Many more examples need to be worked out. Due to time and space limitations,
this monograph contains only some finite-dimensional examples plus a few infinite-
dimensional examples that are simple enough to be analyzed by elementary methods.
Additional examples can be found in the existing literature (usually in an i/s/o setting).
See, e.g., Arov and Dym (2008, 2012, 2018), Curtain and Zwart (1995, 2020), Kurula
and Zwart (2015), Lasiecka and Triggiani (2000a,b), Livšic (1973), Lions (1971), Lions
and Magenes (1972a), van der Schaft and Jeltsema (2014), Staffans and Weiss (2012),
Tucsnak and Weiss (2003, 2009), Weiss and Staffans (2013), Weiss and Tucsnak (2003),
and Weiss and Zwart (1998).

1.2 An Overview of State/Signal and Input/State/Output Systems
There is a rich literature on linear time-invariant i/s/o systems; see, e.g., Behrndt et al.
(2020), Curtain and Zwart (1995, 2020), Fuhrmann (1981), Fuhrmann and Helmke (2015),
Jacob and Zwart (2012), Kalman et al. (1969), Lasiecka and Triggiani (1991, 2000a,b),
Nikolski (2002a,b), Popov (1973), and Staffans (2005). Although this book contains a num-
ber of new results for the same class of systems, our main purpose is to introduce and study
a different class of systems, namely the linear time-invariant state/signal (s/s) systems, and
to investigate the relationship between this class of systems and that of i/s/o systems. Both
of these classes of systems contain a state variable x in a vector space X, called the state
space. A s/s system contains an additional signal variable that makes it possible for the
system to interact with the external world (as indicated in Figure 1.1), whereas an i/s/o
system contains additional input and output variables through which it interacts with its
surroundings (cf. Figure 1.2).
1.2.1 Input/State/Output Systems
Linear time-invariant i/s/o systems can be treated in many different settings. In the simplest
case, the dynamics of the system is described by a system of differential/algebraic equations
of the type
i/s/o :
ẋ(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),
t ∈ I. (1.2.1)
Here t is the time variable, I is a subinterval of the real line R, and the functions x, u,
and y take their values in H-spaces X, U, and Y. These functions are called the state,
the input, and the output of i/s/o, respectively, and the corresponding spaces are called
the state space, the input space, and the output space. The operators A, B, C, and D in
(1.2.1) are bounded linear operators with the appropriate domain and range spaces. The
most important of these operators is the main operator A, which determines the evolution
of the state x(t) for the zero input u = 0. We call an i/s/o system of this type a bounded i/s/o
system. Some basic results on this class of systems are presented in Chapters 4 and 5, and
more details are given in Chapter 6.
w(t)

x(t)
x(t)
w(t)

∈ V
Figure 1.1 State/signal system
u(t) y(t)
x(t)
y(t) ∈ S x(t)
u(t)
Figure 1.2 Input/state/output system

We get a larger class of systems by allowing A to be unbounded, but still requiring
B, C, and D to be bounded. In order to get a “well-posed” evolution in the forward time
direction, we add the condition that the main operator A is the generator of a strongly con-
tinuous semigroup At
, t 0, of bounded linear operators, called the evolution semigroup.
The relation between this evolution semigroup and the system (1.2.1) is the following:
If u = 0 and x0 ∈ dom (A), then the first equation in (1.2.1) with I = R+
:= [0, ∞) has a
unique continuously differentiable solution x given by x(t) = At
x(0), t ∈ R+
. There also
exists a more general formula for the case where u = 0. We call an i/s/o system of this type
a semi-bounded i/s/o system, and Chapter 8 is devoted to this class of systems. Both in the
bounded and the semi-bounded cases, we use the notation i/s/o =
A B
C D

; X, U, Y

for
an i/s/o system defined by (1.2.1).
Unfortunately, typical time-invariant i/s/o systems modeled by partial differential equa-
tions with boundary control or observation are not semi-bounded in the sense that even if
it might be possible to describe the dynamics of the system with an equation of the type
(1.2.1), where A is the generator of a C0 semigroup, the operators B, C, and D need not
be bounded (or even well defined). For this reason, a more general version of (1.2.1) is
needed. Clearly, equation (1.2.1) can be rewritten in the form
i/s/o :

ẋ(t)
y(t)

= S

x(t)
u(t)

, t ∈ I, (1.2.2)
where S is the block matrix operator S =
A B
C D

. We get a much wider class of linear time-
invariant continuous time i/s/o systems by removing the assumption that S has a four-
block decomposition S =
A B
C D

, i.e., we replace S in (1.2.2) by a general linear operator
X
Y

→
X
Y

, called the system operator, and rewrite (1.2.2) in the form
i/s/o :
⎧
⎪
⎪
⎨
⎪
⎪
⎩
x(t)
u(t) ∈ dom(S),

ẋ(t)
y(t)

= S

x(t)
u(t)

,
t ∈ I. (1.2.3)
At this stage, we no longer assume that the system operator S has a block decomposi-
tion of the type S =
A B
C D

. We use the notation i/s/o = (S; X, U, Y) for an i/s/o system
defined by (1.2.3). This class of systems covers “almost all” the standard models from
mathematical physics. We call i/s/o regular if S is closed and the domain of S is dense in
X
Y

.
In the most general setting, the system operator S in (1.2.3) is allowed to be multivalued
(i.e., it is replaced by an operator relation), in which case (1.2.3) should be rewritten in the
form
i/s/o :
⎧
⎪
⎪
⎨
⎪
⎪
⎩
x(t)
u(t) ∈ dom(S),

ẋ(t)
y(t)

∈ S

x(t)
u(t)

,
t ∈ I, (1.2.4)

i.e., the equality sign “=” in (1.2.3) is replaced by an inclusion. We still use the same type
of notation i/s/o = (S; X, U, Y) for this class of systems. Input/state/output systems with
multivalued system operators already appear naturally in the finite-dimensional setting.
When a finite-dimensional electrical circuit is written in i/s/o form, some choices of input
and output signals may lead to a multivalued i/s/o relation (i.e., the input may not be “free,”
and the output may not be determined uniquely by the input and the state). This is, in
particular, true for the so-called impedance, admittance, and transmission representations
of circuits: these may be multivalued in the time domain but still be well behaved in the
frequency domain, as we show in Examples 1.2.1–1.2.7.
If we replace S by the graph of S, then both (1.2.3) and (1.2.4) can be written in the same
form:
i/s/o :
⎡
⎣
ẋ(t)
y(t)
x(t)
u(t)
⎤
⎦ ∈ gph (S), t ∈ I. (1.2.5)
By a closed i/s/o system we mean a system of the type (1.2.4) with a closed (possibly
multivalued) system operator S. Although we do present some results that are true also for
nonclosed i/s/o systems, in this book, we mainly restrict our attention to i/s/o systems that
are closed.
In the literature, it is possible to find several different definitions of what one means by a
“solution” or a “trajectory” of (1.2.1), (1.2.3), or (1.2.4). In this book, we primarily use the
notions of “classical” and “generalized” trajectories of (1.2.1), (1.2.3), and (1.2.4), which
we define as follows1
:
• A triple
x
y
u
of functions with values in
X
Y
U
defined on a time interval I is called a clas-
sical trajectory of i/s/o on I if x is continuously differentiable, u and y are continuous,
and (1.2.4) (or (1.2.1) or (1.2.3)) holds for all t ∈ I.
• A triple
x
y
u
of functions with values in
X
Y
U
defined on a time interval I is called a
generalized trajectory of i/s/o on I if x is continuous, u and y belong locally to L2
, and if
it is true for each compact subinterval I
of I that
x
y
u
can be approximated in
C(I;X)
L2(I;Y)
L2(I;U)

by a sequence of classical trajectories of on I
.2
Since the operators
A B
C D

and S in (1.2.1), (1.2.3), and (1.2.4) do not depend on the time
variable t, the above sets of classical and generalized trajectories are time-shift invariant
1 Our definitions of classical and generalized trajectories are slightly nonstandard in the following sense. In the “standard”
definition of a classical trajectory of i/s/o on I, the output y is not explicitly assumed to be continuous (it is usually implied
by some other regularity conditions), and the condition y ∈ L2(I; Y) and the convergence of yn to y in L2(I; Y) where I is a
compact subinterval of I are not standard parts of the definition of a generalized trajectory of on I (it is usually replaced by
a well-posedness condition). Our definitions of these notions are motivated by our desire to impose symmetrical regularity
conditions on inputs and outputs in order to arrive at the notion of a state/signal system.
2 The notations C(I; X) and L2(I; X) stand for the spaces of continuous and square integrable functions, respectively, on the
interval I with values in the H-space X. See Notation 2.1.4.

in the sense that if these trajectories are shifted forward or backward in time, then they are
still classical or generalized trajectories on the corresponding shifted intervals. Moreover,
the linearity of
A B
C D

and S implies that the sets of classical and generalized trajectories
are linear subspaces of the appropriate function spaces.
The special cases where I = R+
, I = R−
, or I = R in (1.2.1), (1.2.3), and (1.2.4) will be
important. The corresponding trajectories are called future, past, and two-sided trajectories
of , respectively.
1.2.2 Well-Posed Input/State/Output Systems
A special class of regular i/s/o systems is the class of well-posed i/s/o systems, studied
in Chapter 14. A well-posed i/s/o system = (S; X, U, Y) is required to be regular and
solvable (the notion of “solvable” will be defined shortly). In addition, has the property
that:
• For every x0 ∈ X and every u ∈ L2
loc(R+
; U), there exists a unique generalized future
trajectory
x
y
u
with the initial state x(0) = x0 (and input u).3
Finally, the state component x and the output component y of a generalized future trajectory
x
y
u
of are assumed to depend continuously on x0 ∈ X and u ∈ L2
loc(R+
; U) in the sense
that
• For some triple of admissible norms4
·X , ·U , and ·Y in X, U, and Y, there ex-
ists a nonnegative locally bounded function η on R+
such that all generalized future
trajectories
x
y
u
of satisfy
x(t)2
X +
t
0
y(s)2
Y ds ≤ η(t)

x(0)2
X +
t
0
u(s)2
U ds

, t ∈ R+
. (1.2.6)
We observe that the function η in (1.2.6) satisfies η(0) ≥ 1, and that η cannot tend to
zero as t → ∞ unless y vanishes identically. However, it is possible to replace (1.2.6) by
another condition (which can be shown to be equivalent to (1.2.6)): we replace the time-
dependent constant η(t) on the right-hand side of (1.2.6) by a fixed constant M ≥ 1 and
instead multiply x(t), u(t), and u(t) by an exponential e−αt
with α ∈ R to get the condition
e−αt
x(t)2
X +
t
0
e−αs
y(s)2
Y ds
≤ M

x(0)2
X +
t
0
e−αs
u(s)2
U ds

, t ∈ R+
.
(1.2.7)
3 The notation L2
loc(I; X) stands for the space of functions that belong to L2(I; X) for every compact subinterval I of I. See
Notation 2.1.4.
4 By an admissible norm in an H-space X we mean one of the (infinitely many) norms that induces the topology in X.

The infimum of all α such that (1.2.7) holds for some M ≥ 1 is the growth bound of i/s/o.
If (1.2.7) holds for α = 0 (or equivalently, (1.2.6) holds with η(t) replaced with a constant
M ≥ 1), then i/s/o is said to be stable. If (1.2.7) holds for some α 0, then i/s/o is said
to be exponentially stable.
Every well-posed i/s/o system i/s/o = (S; X, U, Y) has a main operator A of S
defined by
Ax =

1X 0

S
x
0

,
x
0

∈ dom (S), (1.2.8)
and A is the generator of a C0 semigroup. Semi-bounded i/s/o systems are well-posed.
1.2.3 State/Signal Systems
The idea behind the definition of a state/signal (s/s) system is to remove the distinction
between the “input” and the “output” of an i/s/o system. The separate input and output
spaces, U and Y are now replaced by a signal space W. In the case of an i/s/o system, both
(1.2.3) and (1.2.4) can be interpreted as linear relations between the four variables x(t),
ẋ(t), u(t), and y(t), where x(t) and ẋ(t) belong to the state space X, u(t) belongs to the input
space U, and y(t) belongs to the output space Y. If we remove the distinction between the
input and the output, and consider both of these to be parts of the interaction signal w(t),
then we end up with a linear relation between x(t) ∈ X, ẋ(t) ∈ X, and the signal w(t) ∈ W,
where W is the signal space. Every such relation can be written in the form
:

ẋ(t)
x(t)
w(t)

∈ V, t ∈ I, (1.2.9)
where V is a subspace of
X
X
W
. We denote the s/s system in (1.2.9) by = (V; X, W).
Classical and generalized trajectories of are defined in the same way as in the i/s/o
setting, with (1.2.4) (or (1.2.1) or (1.2.3)) replaced by (1.2.9). Thus, a classical trajectory
x
w

of on the interval I belongs to C1(I;X)
C(I;W)
5
and (1.2.9) holds for all t ∈ I, and the
restriction of every generalized trajectory
x
w

of to every compact subinterval I
of I
belongs to C(I;X)
L2(I;W)
and can be approximated by a sequence of classical trajectories in this
space. The subspace V of
X
X
W
is called the generating subspace of since it “generates”
the set of all classical and generalized trajectories of by means of (1.2.9), and we say
that is closed if V is closed. As before, we refer to trajectories defined on R+
, R−
, or R
as future, past, and two-sided trajectories of , respectively.
It is easy to convert an arbitrary i/s/o system i/s/o = (S; X, U, Y) into a s/s system
by simply “combining the input u and output y into an interaction signal w.” We keep the
original state x(t) and state space X, but replace the input u(t) and the output y(t) by the
signal w(t) = y(t)
u(t) , which takes its values in the signal space W :=
Y
U

. The generating
5 The notation C1(I; X) stands for the space of continuously differentiable functions on I with values in the H-space X.

subspace V of is the reordered graph of the operator S in (1.2.4), i.e.,
V =
z
x
y
u
∈
X
X
Y
U

x
u

∈ dom (S) ,
z
y

∈ S
x
u

=
1X 0 0 0
0 0 1X 0
0 1Y 0 0
0 0 0 1U

gph (S).
(1.2.10)
In particular, V is closed if and only if S is closed. This construction preserves the trajec-
tories of i/s/o in the sense that
x
y
u
is a classical or generalized trajectory of i/s/o on an
interval I if and only if
x
y
u

is a classical or generalized trajectory of on I.6
The s/s
system constructed from i/s/o in this way is called the s/s system induced by i/s/o.
1.2.4 Input/State/Output Representations
It is also possible to go in the opposite direction, i.e., to convert a s/s system = (V; X, W)
into an i/s/o system i/s/o = (S; X, U, Y) (as long as we do not care about the regularity
properties of the i/s/o system). Again we keep the original state x(t) and the original state
space X. We choose the input space U and the output space Y of i/s/o to be the compo-
nents in a direct sum decomposition W = U Y of W, take the input u(t) of i/s/o to be
the projection of the signal w(t) of onto U along Y, and take the output y(t) of i/s/o to
be the projection of w(t) onto Y along U. Observe that the ordering of the two subspaces U
and Y in the direct sum decompositon W = U + Y of W is significant in the sense that the
first component is taken to be the input space and the second component is taken to be the
output space of i/s/o. This constructions preserves the trajectories of in the sense that
x
y
u
is a classical or generalized trajectory of i/s/o on an interval I
if and only if
x
w

is a classical or generalized trajectory on I
of , where w = u + y.
(1.2.11)
The formal connection between the generating subspace V of and the system operator S
of i/s/o is as follows:
gph (S) =
⎡
⎢
⎢
⎣
1X 0 0
0 0 QU
Y
0 1X 0
0 0 QY
U
⎤
⎥
⎥
⎦ V (1.2.12)
and
dom (S) =

0 1X 0
0 0 QY
U

V, (1.2.13)
6 See also footnote on page 16.

where QY
U and QU
Y are the coordinate maps7
corresponding to the decomposition of W into
U Y and
V :=
⎧
⎨
⎩
⎡
⎣
z
x
w
⎤
⎦ ∈
⎡
⎣
X
X
W
⎤
⎦

x
QY
U w

∈ dom (S) and

z
QU
Y w

∈ S

x
QY
U w

⎫
⎬
⎭
=
⎡
⎣
1X 0 0 0
0 0 1X 0
0 IY 0 IU
⎤
⎦ gph(S).
(1.2.14)
Here IU and IY are the embedding operators IU : U → W and IY : Y → W. In particu-
lar, it is still true that S is closed if and only if V is closed. Each system i/s/o that can
be obtained from in this way (with different choices of the ordered direct sum decom-
position W = U Y) is called an i/s/o representation of . Observe that it is possible to
recover from the s/s system induced by each i/s/o representation i/s/o by identifying
{0}
U

with U and
Y
{0}

with Y. If dim W 1, then W has an infinite number of ordered
direct sum decompositions W = U Y, and thus has an infinite number of i/s/o rep-
resentations whenever dim W 1. (If dim W = 1, then has two i/s/o representations:
one with the input space U = W and the zero output space, and the other with the zero
input space and the output space Y = W.) However, without any further restrictions on
U and Y, the corresponding i/s/o representation may not have any significant regularity
properties.
By a bounded, or semi-bounded, or well-posed, or stable s/s system, we mean a s/s
system that has at least one bounded, or semi-bounded, or well-posed, or stable i/s/o
representation. Bounded s/s systems are studied in Chapter 7, semi-bounded s/s systems
in Chapter 9, and well-posed and stable s/s systems in Chapter 15. Equivalent charac-
terizations for boundedness and semi-boundedness are presented in Definition 2.1.37 and
Theorems 4.2.31 and 9.1.3.
1.2.5 Similarity of Input/State/Output and State/Signal Systems
As we mentioned above, throughout this monograph, we work in a setting where the state
space X, the input and output spaces U and Y of an i/s/o system, and the signal space
W of a s/s system are H-spaces, i.e., the topologies in these spaces are induced by some
Hilbert space norms, but we do not fix some preferred admissible Hilbert spaces inner
products in X, U, Y and W. Of course, it is possible to convert these H-spaces into Hilbert
spaces by fixing some Hilbert space inner products in X, U, Y, and W, but the Hilbert
space i/s/o and s/s systems that we obtain in this way depend on the specific choices of
inner products in X, U, Y. For example, let i/s/o = (S; X, U, Y) be an i/s/o system (the
case where = (V; X, W) is a s/s system is analogous). Let us fix two (different) sets of
7 The coordinate map QY
U is equal to the projection in W onto U along Y interpreted as an operator with the range space U
instead of W, with an analogous definition of QU
Y .

u1(t)
u2(t)
y1(t)
y2(t)
x2 = Px1
u2 = Qu1 y2 = Ry1
ẋ1(t)
y1(t) ∈ S1
x1(t)
u1(t)
ẋ2(t)
y2(t) ∈ S2
x2(t)
u2(t)
Figure 1.3 Similarity of two input/state/output systems
admissible inner products in each of the H-spaces X, U, and Y and denote the resulting
two Hilbert space i/s/o systems by j = (Sj; Xj, Uj, Yj), j = 1, 2. From a topological point
of view 1 and 2 are the same system, but from a Hilbert space point of view they
are different. However, it is still true that they are i/s/o similar to each other in the sense
defined below.
We say that two i/s/o systems
j
i/s/o = (Sj; Xj, Uj, Yj), j = 1, 2, (where Xj, Uj, and Yj
are either H-spaces or Hilbert spaces) are i/s/o similar (input/state/output similar) (cf.
Figure 1.3) if there exist continuous linear operators with continuous inverses
P: X1 → X2, Q: U1 → U2, and R: Y1 → Y2 such that
S2 =

P 0
0 R

S1

P−1
0
0 Q−1

, (1.2.15)
or equivalently,
gph (S2) =
⎡
⎢
⎢
⎣
P 0 0 0
0 R 0 0
0 0 P 0
0 0 0 Q
⎤
⎥
⎥
⎦ gph (S1). (1.2.16)
If X1 = X2 = X and P = 1X then we say that 1 and 2 are i/o similar, and if U1 = U2 = U,
Y1 = Y2 = Y, Q = 1U , and R = 1Y , then we say that 1 and 2 are state similar. The map-
ping from S1 to S2 in (1.2.15) and (1.2.16) is called an (i/s/o) (P, R, Q)-similarity transfor-
mation. To see that the two Hilbert space systems 1 and 2 in the preceding paragraph
are i/s/o similar to each other it suffices to choose P, R, and Q to be the identity operators
from X1 → X2, Y1 → Y2, and U1 → U2, respectively. These operators are continuous and
have continuous inverses since the two different (but admissible) norms in Xj, Yj, and Uj,
j = 1, 2, are assumed to be equivalent to each other.

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Title: Tessa, Our Little Italian Cousin
Author: Mary Hazelton Blanchard Wade
Illustrator: L. J. Bridgman
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Most recently updated: October 23, 2024
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*** START OF THE PROJECT GUTENBERG EBOOK TESSA, OUR
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TESSA
Our Little Italian Cousin
THE
Little Cousin Series
(TRADE MARK)
Each volume illustrated with six or more full-page
plates in
tint. Cloth, 12mo, with decorative cover,
per volume, 60 cents
LIST OF TITLES
By Mary Hazelton Wade
(unless otherwise indicated)
Our Little African Cousin
Our Little Alaskan Cousin
By Mary F. Nixon-Roulet
Our Little Arabian Cousin
By Blanche McManus
Our Little Armenian Cousin

By Constance F. Curlewis
Our Little Australian Cousin
Our Little Brazilian Cousin
Our Little Brown Cousin
Our Little Canadian Cousin
By Elizabeth R. MacDonald
Our Little Chinese Cousin
By Isaac Taylor Headland
Our Little Cuban Cousin
Our Little Dutch Cousin
By Blanche McManus
Our Little Egyptian Cousin
By Blanche McManus
Our Little English Cousin
By Blanche McManus
Our Little Eskimo Cousin
Our Little French Cousin
By Blanche McManus
Our Little German Cousin
Our Little Greek Cousin
Our Little Hawaiian Cousin
Our Little Hindu Cousin
By Blanche McManus
Our Little Indian Cousin
Our Little Irish Cousin
Our Little Japanese Cousin
Our Little Jewish Cousin
Our Little Korean Cousin
By H. Lee M. Pike
Our Little Mexican Cousin

By Edward C. Butler
Our Little Norwegian Cousin
Our Little Panama Cousin
By H. Lee M. Pike
Our Little Philippine Cousin
Our Little Porto Rican Cousin
Our Little Russian Cousin
Our Little Scotch Cousin
By Blanche McManus
Our Little Siamese Cousin
Our Little Spanish Cousin
Our Little Swedish Cousin
By Claire M. Coburn
Our Little Swiss Cousin
Our Little Turkish Cousin
L. C. PAGE COMPANY
New England Building, Boston, Mass.

Tessa
By
Mary Hazelton Wade
Illustrated by
L. J. Bridgman
Boston
L. C. Page Company
Publishers
MDCCCCVII
Copyright, 1903
By L. C. Page Company
(INCORPORATED)
All rights reserved
THE LITTLE COUSIN SERIES
(Trade Mark)
Published, July, 1903
Fifth Impression, June, 1908

Sixth Impression, November, 1909
Seventh Impression, August, 1910

Preface
Many people from other lands have crossed the ocean to make a
new home for themselves in America. They love its freedom. They
are happy here under its kindly rule. They suffer less from want and
hunger than in the country of their birthplace.
Their children are blessed with the privilege of attending fine
schools and with the right to learn about this wonderful world, side
by side with the sons and daughters of our most successful and
wisest people.
Among these newer-comers to America are the Italians, many of
whom will never again see their own country, of which they are still
so justly proud. They will tell you it is a land of wonderful beauty;
that it has sunsets so glorious that both artists and poets try to
picture them for us again and again; that its history is that of a
strong and mighty people who once held rule over all the civilized
world; that thousands of travellers visit its shores every year to look
upon its paintings and its statues, for it may truly be called the art
treasure-house of the world.
When you meet your little Italian cousins, with their big brown
eyes and olive skins, whether it be in school or on the street,
perhaps you will feel a little nearer and more friendly if you turn your
attention for a while to their home, and the home of the brave and
wise Columbus who left it that he might find for you in the far West
your own loved country, your great, grand, free America.

Contents
PAGE
I.Tessa 9
II.Rome 18
III.The Story of Æneas 38
IV.Christmas 52
V.Saint Peter's 64
VI.The Christening 75
VII.The Twins 86
VIII.The Carnival 101
IX.The Buried City 115

List of Illustrations
PAGE
Tessa Frontispiece
Beppo walked by her side 19
In the Palace Garden 33
In St. Peter's 64
Were soon in the midst of a merry crowd 106
It was a strange place 122

TESSA

CHAPTER I.
TESSA
There comes babbo! There comes babbo! cried Tessa, as she
ran down the narrow street to meet her father, with baby Francesca
toddling after her.
The man was not alone,—Beppo and the donkey were with him.
They were very tired, for it was a hard trip from the little village on
the hilltop to the great city, miles away, and back again. The donkey
was not of much help on the homeward journey, either. Poor little
patient beast! he was getting old now, and he felt that his day's
work was done when he had carried a load of nuts and vegetables
to Rome in the morning. But when he had to bring Beppo back
again, he felt a little bit sulky. So it was no wonder that he stood
quite still every few minutes and did not seem to hear his little
master scold.
Get up, Pietro, get up. We shall be late to supper, Beppo would
say, but the donkey would not move till Beppo's father used the
whip. He did not strike hard enough to hurt the poor creature,
though. Oh no, the kind man would not do that, he was too gentle.
But he must make the donkey know the whip was there, or they
would never get home.
When they had crossed the wide plain and reached the foot of
the hill, Beppo got down and walked. It was too hard on Pietro to
make him carry even a little boy now.
They came up the narrow road slowly till they reached the village.
And just as the sunset spread over the sky, and gave a glory even to
the stones, Tessa caught sight of them.

My darling Tessa, said her father. My dear little Francesca.
Tired as he was, he took the two children in his arms and hugged
them as though he had been away many days. Yet he had left them
at five o'clock that very morning.
We have good news for you, Beppo and I, he went on.
Beppo laughed till the high, pointed hat nearly fell off his head.
Oh, yes, good news, said Beppo. You cannot think what it is,
Tessa. May I tell her, babbo?
Yes, my child, his father answered.
You are to go to Rome to-morrow with babbo and me. The great
artist who buys our fruit wants to see you. He thinks he may want
you for a model. And me, too, Tessa, he wants me! He will put us
both in a picture. Babbo said you also had long hair, and that we
look much alike.
Only think, Tessa! he will pay babbo for letting him paint us. And
mother shall have a new dress, and you shall have some red
ribbons. We will all have a feast. Say, Tessa, is there a nice chestnut
cake waiting for our supper? I am so hungry.
The boy's great black eyes sparkled as he told the story. His long
hair hung down over his shoulders, under the odd pointed hat. He
was a beautiful child. It was no wonder the American artist wished
to put him in a picture.
But Tessa was beautiful, too. The artist would not be disappointed
when he saw her. Her skin was clear, but like the colour of the olives
which grew on the old tree behind her house. And now there was a
faint pink blush in her cheeks as she listened to Beppo's story.
They were very happy children, but oh, so poor, you would think
if you should visit them in the old house where they have always
lived. It is no wonder they like best to be outdoors.

The house is all of stone, and the floor is made of bricks. It seems
dark and chilly inside after leaving the glorious sunset. The plaster is
blackened with smoke and age. In some places it is broken away
from the wall and is falling down.
But there is a picture of the Christ-child hanging over the rough
table, and the children do not think of the dingy walls. It is home,
where a loving father and mother watch over them and guard them
from harm.
See! the table is spread with the simple supper. There are the
cakes made from chestnut flour mixed with olive oil, and of which
Beppo is so fond. And here is milk from Tessa's pet goat. Beppo runs
over to the stone fountain in the middle of the village and fills a
copper dish with fresh water, and the little family sit down to their
evening meal.
The mother hears the good news, and claps her hands in delight.
But what shall Tessa wear? It troubles the good soul, for Tessa has
no shoes, and both of her dresses are old and worn.
Never mind, never mind, says her husband, don't trouble
yourself about that. The artist says he does not care about the
clothes. He was much pleased with Beppo's cloak, however. He says
it will be fine in the picture. Let Tessa wear her wide straw hat and
her old clothes; that is all he asks.
But how will she manage to travel so far? The child has never
before gone such a distance from home, continued her mother.
She is not heavy. She can sit on Pietro's back between the
panniers. I will not load them heavily to-morrow, and then Pietro will
not complain. And when we come home at night, Beppo can walk, I
am sure. He may be tired, but he is a stout lad, my Beppo is. What
do you say, my boy?
Beppo was sure he could get along. He was only too glad to have
Tessa's company.

But think, babbo, he exclaimed, it is not for one day that the
artist wishes us. It is many, many, before the picture will be finished.
We can manage somehow, I am sure. I am nearly twelve years old
now, and I am getting very strong.
But what will mother do with me away all day long? said Tessa.
Who will take care of the baby while she works in the garden? And
who will help her pull the weeds?
Bruno shall watch Francesca. He will let no harm come to her,
you may be sure. Besides, she can walk alone so well now, she is
little care. As for the garden, there is not much more to do at
present. It almost takes care of itself, said the mother.
Yes, Bruno can be trusted, said the father, he is the best dog I
ever knew.
As he heard his name spoken, the sheep-dog came slowly out of
the chimney-corner. He wagged his tail as though he knew what his
master and mistress had been saying. Beppo threw him his last bit
of cake and Bruno caught it on his nose, from which it was quickly
passed into his mouth.
Dear old Bruno, said Tessa, you took care of me when I was a
baby, didn't you? Mamma, did Bruno really rock the cradle and keep
the flies off, so I could sleep?
Yes, my child; when I was very ill he would watch you all day
long. And when you began to creep, he followed you about. If you
got near the edge of a step, or any other unsafe place, he would lift
you by your dress and bring you to my side. We should thank the
good Lord for bringing Bruno to us.
The mother looked up to the picture of Jesus and made the sign
of the cross on her breast.
An hour later the whole family were sound asleep on their hard
beds.

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