Well posed linear system

www.pdfgrip.com

This page intentionally left blank


www.pdfgrip.com

Encyclopedia of Mathematics and Its Applications
Founding Editor G. C. Rota
All the titles listed below can be obtained from good booksellers or from
Cambridge University Press. For a complete series listing visit
/>88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
100.
102.

Teo Mora Solving Polynomial Equation Systems, I
Klaus Bichteler Stochastic Integration with Jumps
M. Lothaire Algebraic Combinatorics on Words
A. A. Ivanov & S. V. Shpectorov Geometry of Sporadic Groups, 2
Peter McMullen & Egon Schulte Abstract Regular Polytopes
G. Gierz et al. Continuous Lattices and Domains

Steven R. Finch Mathematical Constants
Youssef Jabri The Mountain Pass Theorem
George Gasper & Mizan Rahman Basic Hypergeometric Series, 2nd ed.
Maria Cristina Pedicchio & Walter Tholen Categorical Foundations
Enzo Olivieri & Maria Eulalia Vares Large Deviations and Metastability
R. J. Wilson & L. Beineke Topics in Algebraic Graph Theory


www.pdfgrip.com


www.pdfgrip.com

Well-Posed Linear Systems
OLOF STAFFANS
Department of Mathematics
˚ Akademi University, Finland
Abo


www.pdfgrip.com

cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521825849
© Cambridge University Press 2005

This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2005
isbn-13
isbn-10

978-0-511-08208-5 eBook (NetLibrary)
0-511-08208-8 eBook (NetLibrary)

isbn-13
isbn-10

978-0-521-82584-9 hardback
0-521-82584-9 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 book, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.


www.pdfgrip.com

Contents

List of figures
Preface
List of notation

page ix

xi
xiv

1
1.1
1.2

Introduction and overview
Introduction
Overview of chapters 2–13

1
1
8

2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

Basic properties of well-posed linear systems
Motivation
Definitions and basic properties
Basic examples of well-posed linear systems

Time discretization
The growth bound
Shift realizations
The Lax–Phillips scattering model
The Weiss notation
Comments

3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10

Strongly continuous semigroups
Norm continuous semigroups
The generator of a C0 semigroup
The spectra of some generators
Which operators are generators?
The dual semigroup
The rigged spaces induced by the generator
Approximations of the semigroup
The nonhomogeneous Cauchy problem
Symbolic calculus and fractional powers
Analytic semigroups and sectorial operators

v

28
28
34
46
55
60
67
71
76
78
85
85
87
98
106
113
122
128
133
140
150


www.pdfgrip.com
vi

Contents

3.11
3.12
3.13
3.14
3.15

Spectrum determined growth
The Laplace transform and the frequency domain
Shift semigroups in the frequency domain
Invariant subspaces and spectral projections
Comments

164
169
177
180
191

4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11

The generators of a well-posed linear system
Introduction
The control operator
Differential representations of the state
The observation operator
The feedthrough operator
The transfer function and the system node
Operator nodes
Examples of generators
Diagonal and normal systems
Decompositions of systems
Comments

194
194
196
202
213
219
227
238
256
260
266
273

5
5.1
5.2

5.3
5.4
5.5
5.6
5.7
5.8

Compatible and regular systems
Compatible systems
Boundary control systems
Approximations of the identity in the state space
Extended observation operators
Extended observation/feedthrough operators
Regular systems
Examples of regular systems
Comments

276
276
284
295
302
313
317
325
329

6
6.1
6.2

6.3
6.4
6.5
6.6
6.7

Anti-causal, dual, and inverted systems
Anti-causal systems
The dual system
Flow-inversion
Time-inversion
Time-flow-inversion
Partial flow-inversion
Comments

332
332
337
349
368
378
386
400

7
7.1
7.2
7.3
7.4

Feedback
Static output feedback
Additional feedback connections
State feedback and output injection
The closed-loop generators

403
403
413
422
425


www.pdfgrip.com
Contents

vii

7.5
7.6
7.7
7.8

Regularity of the closed-loop system
The dual of the closed-loop system
Examples
Comments

433
436

436
440

8
8.1
8.2
8.3
8.4
8.5
8.6

Stabilization and detection
Stability
Stabilizability and detectability
Coprime fractions and factorizations
Coprime stabilization and detection
Dynamic stabilization
Comments

443
443
453
465
473
485
502

9
9.1
9.2

9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11

Realizations
Minimal realizations
Pseudo-similarity of minimal realizations
Realizations based on factorizations of the Hankel operator
Exact controllability and observability
Normalized and balanced realizations
Resolvent tests for controllability and observability
Modal controllability and observability
Spectral minimality
Controllability and observability of transformed systems
Time domain tests and duality
Comments

505
505
511
517
521
530
538

546
549
551
554
565

10
10.1
10.2
10.3
10.4
10.5
10.6

Admissibility
Introduction to admissibility
Admissibility and duality
The Paley–Wiener theorem and H ∞
Controllability and observability gramians
Carleson measures
Admissible control and observation operators for diagonal
and normal semigroups
10.7 Admissible control and observation operators for
contraction semigroups
10.8 Admissibility results based on the Lax–Phillips model
10.9 Comments

602
610
613

11
11.1
11.2
11.3

616
616
628
636

Passive and conservative scattering systems
Passive systems
Energy preserving and conservative systems
Semi-lossless and lossless systems

569
569
572
576
583
591
598


www.pdfgrip.com
viii

Contents

11.4
11.5

Isometric and unitary dilations of contraction semigroups
Energy preserving and conservative extensions of
passive systems
11.6 The universal model of a contraction semigroup
11.7 Conservative realizations
11.8 Energy preserving and passive realizations
11.9 The Spectrum of a conservative system
11.10 Comments

643

12
12.1
12.2
12.3
12.4
12.5

Discrete time systems
Discrete time systems
The internal linear fractional transform
The Cayley and Laguerre transforms
The reciprocal transform
Comments

696
696

703
707
719
728

A.1
A.2
A.3
A.4

Appendix
Regulated functions
The positive square root and the polar decomposition
Convolutions
Inversion of block matrices

730
730
733
736
744

Bibliography
Index

750
767

655
660

670
677
691
692


www.pdfgrip.com

Figures

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
3.1
3.2
3.3
7.1
7.2
7.3
7.4
7.5
7.6
7.7

7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16

Regular well-posed linear system
Well-posed linear system
Cross-product (the union of two independent systems)
Cross-product in block form
Sum junction
Sum junction in block form
T-junction
T-junction in block form
Parallel connection
Parallel connection in block form
The sector δ
The sector θ,γ
The path in the proof of Theorem 3.10.5
Static output feedback connection
Positive identity feedback
Negative identity feedback
Flow-inversion
Another static output feedback
A third static output feedback
Output feedback in block form

Cancellation of static output feedback
Cascade connection through K
Cascade connection in block form
Dynamic feedback
Dynamic feedback in block form
Partial feedback
State feedback
Output injection
Original system with one extra output
ix

page 29
38
52
52
53
53
54
54
55
55
150
151
153
404
409
410
412
413
413

415
417
417
418
419
419
421
422
423
426


www.pdfgrip.com
x

7.17
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13

Figures

Closed-loop system with one extra output
The extended system
Right coprime factor
Left coprime factor
Cancellation of state feedback
Dynamic stabilization
Dynamic stabilization
Equivalent version of dynamic stabilization
Second equivalent version of dynamic stabilization
Third equivalent version of dynamic stabilization
Youla parametrization
Youla parametrized stabilizing compensator
Youla parametrized stabilizing compensator
Youla parametrized stabilizing compensator

426
453
454
455
456
485
494
495
497
498
499
500
501

501


www.pdfgrip.com

Preface

This main purpose of this book is to present the basic theory of well-posed
linear systems in a form which makes it available to a larger audience, thereby
opening up the possibility of applying it to a wider range of problems. Up to
now the theory has existed in a distributed form, scattered between different
papers with different (and often noncompatible) notation. For many years this
has forced authors in the field (myself included) to start each paper with a long
background section to first bring the reader up to date with the existing theory.
Hopefully, the existence of this monograph will make it possible to dispense
with this in future.
My personal history in the field of abstract systems theory is rather short but
intensive. It started in about 1995 when I wanted to understand the true nature
of the solution of the quadratic cost minimization problem for a linear Volterra
integral equation. It soon became apparent that the most appropriate setting
was not the one familiar to me which has classically been used in the field of
Volterra integral equations (as presented in, e.g., Gripenberg et al. [1990]). It
also became clear that the solution was not tied to the class of Volterra integral
equations, but that it could be formulated in a much more general framework.
From this simple observation I gradually plunged deeper and deeper into the
theory of well-posed (and even non-well-posed) linear systems.
One of the first major decisions that I had to make when I began to write
this monograph was how much of the existing theory to include. Because of
the nonhomogeneous background of the existing theory (several strains have
been developing in parallel independently of each other), it is clear that it is

impossible to write a monograph which will be fully accepted by every worker
in the field. I have therefore largely allowed my personal taste to influence the
final result, meaning that results which lie closer to my own research interests
are included to a greater extent than others. It is also true that results which
blend more easily into the general theory have had a greater chance of being
included than those which are of a more specialist nature. Generally speaking,
xi


www.pdfgrip.com
xii

Preface

instead of borrowing results directly from various sources I have reinterpreted
and reformulated many existing results into a coherent setting and, above all,
using a coherent notation.
The original motivation for writing this book was to develop the background
which is needed for an appropriate understanding of the quadratic cost minimization problem (and its indefinite minimax version). However, due to page
and time limitations, I have not yet been able to include any optimal control in
this volume (only the background needed to attack optimal control problems).
The book on optimal control still remains to be written.
Not only was it difficult to decide exactly what parts of the existing theory
to include, but also in which form it should be included. One such decision
was whether to work in a Hilbert space or in a Banach space setting. Optimal
control is typically done in Hilbert spaces. On the other hand, in the basic theory
it does not matter if we are working in a Hilbert space or a Banach space (the
technical differences are minimal, compared to the general level of difficulty of
the theory). Moreover, there are several interesting applications which require
the use of Banach spaces. For example, the natural norm in population dynamics

is often the L 1 -norm (representing the total mass), parabolic equations have a
well-developed L p -theory with p = 2, and in nonlinear equations it is often
more convenient to use L ∞ -norms than L 2 -norms. The natural decision was to
present the basic theory in an arbitrary Banach space, but to specialize to Hilbert
spaces whenever this additional structure was important. As a consequence of
this decision, the present monograph contains the first comprehensive treatment
of a well-posed linear system in a setting where the input and output signals are
continuous (as opposed to belonging to some L p -space) but do not have any
further differentiability properties (such as belonging to some Sobolev spaces).
(More precisely, they are continuous apart from possible jump discontinuities.)
The first version of the manuscript was devoted exclusively to well-posed
problems, and the main part of the book still deals with problems that are well
posed. However, especially in H ∞ -optimal control, one naturally runs into nonwell-posed problems, and this is also true in circuit theory in the impedance
and transmission settings. The final incident that convinced me that I also had
to include some classes of non-well-posed systems in this monograph was my
discovery in 2002 that every passive impedance system which satisfies a certain
algebraic condition can be represented by a (possibly non-well-posed) system
node. System nodes are a central part of the theory of well-posed systems, and
the well-posedness property is not always essential. My decision not to stay
strictly within the class of well-posed systems had the consequence that this
monograph is also the the first comprehensive treatment of (possibly non-wellposed) systems generated by arbitrary system nodes.


www.pdfgrip.com
Preface

xiii

The last three chapters of this book have a slightly different flavor from the
earlier chapters. There the general Banach space setting is replaced by a standard Hilbert space setting, and connections are explored between well-posed

linear systems, Fourier analysis, and operator theory. In particular, the admissibility of scalar control and observation operators for contraction semigroups is
characterized by means of the Carleson measure theorem, and systems theory
interpretations are given of the basic dilation and model theory for contractions
and continuous-time contraction semigroups in Hilbert spaces.
It took me approximately six years to write this monograph. The work has
˚ Akademi, which
primarily been carried out at the Mathematics Institute of Abo
has offered me excellent working conditions and facilities. The Academy of
Finland has supported me by relieving me of teaching duties for a total of two
years, and without this support I would not have been able to complete the
manuscript in this amount of time.
I am grateful to several students and colleagues for helping me find errors and
misprints in the manuscript, most particularly Mikael Kurula, Jarmo Malinen
and Kalle Mikkola.
Above all I am grateful to my wife Marjatta for her understanding and
patience while I wrote this book.


www.pdfgrip.com

Notation

Basic sets and symbols
C
C+
ω,
C−
ω,
C+ ,
C− ,

Dr+ ,
Dr− ,
D+ ,
D− ,
R
R+ ,
R− ,
T
TT

+

Cω
−
Cω
+
C
−
C
+
Dr
−
Dr
+
D
−
D
+

R

−
R

Z
Z+ , Z−
j
0
1

The complex plane.
+
C+
ω := {z ∈ C | z > ω} and Cω := {z ∈ C | z ≥ ω}.
−
C−
ω := {z ∈ C | z < ω} and Cω := {z ∈ C | z ≤ ω}.
+
+
C+ := C+
0 and C := C0 .
−
−
C− := C−
0 and C := C0 .
+
Dr+ := {z ∈ C | z > r } and Dr := {z ∈ C | |z| ≥ r }.
−
Dr− := {z ∈ C | z < r } and Dr := {z ∈ C | |z| ≤ r }.
+
+

D+ := D+
1 and D := D1 .
−
−
D− := D−
1 and D := D1 .
R := (−∞, ∞).
+
R+ := (0, ∞) and R := [0, ∞).
−
R− := (−∞, 0) and R := (−∞, 0].
The unit circle in the complex plane.
The real line R where the points t + mT , m = 0, ±1, ±2, . . .
are identified.
The set of all integers.
Z+ := {0, 1, 2, . . .} and Z− := {−1, −2, −3, . . .}.
√
j := −1.
The number zero, or the zero vector in a vector space, or the
zero operator, or the zero-dimensional vector space {0}.
The number one and also the identity operator on any set.

Operators and related symbols
A, B, C, D

In connection with an L p |Reg-well-posed linear system or an
operator node, A is usually the main operator, B the control
xiv

www.pdfgrip.com
Notation

xv

operator, C the observation operator and D a feedthrough
operator. See Chapters 3 and 4.
C&D
The observation/feedthrough operator of an L p |Reg-wellposed linear system or an operator node. See Definition 4.7.2.
A, B, C, D
The semigroup, input map, output map, and input/output map
of an L p |Reg-well-posed linear system, respectively. See Definitions 2.2.1 and 2.2.3.
D
The transfer function of an L p |Reg-well-posed linear system
or an operator node. See Definitions 4.6.1 and 4.7.4.
B(U ; Y ), B(U ) The set of bounded linear operators from U into Y or from
U into itself, respectively.
C, L
The Cayley and Laguerre transforms. See Definition 12.3.2.
τt
The bilateral time shift operator τ t u(s) := u(t + s) (this is
a left-shift when t > 0 and a right-shift when t < 0). See
Example 2.5.3 for some additional shift operators.
γλ
The time compression or dilation operator (γλ u)(s) := u(λs).
Here λ > 0.
πJ
(π J u)(s) := u(s) if s ∈ J and (π J u)(s) := 0 if s ∈
/ J . Here
J ⊂ R.

π+ , π−
π+ := π[0,∞) and π− := π(−∞,0) .
R
The time reflection operator about zero: (Ru)(s) := u(−s)
(in the L p -case) or (Ru)(s) := limt↓−s u(t) (in the Reg-case).
See Definition 3.5.12.
Rh
The time reflection operator about the point h. See Lemma
6.1.8.
σ
The discrete-time bilateral left-shift operator (σu)k := u k+1 ,
where u = {u k }k∈Z . See Section 12.1 for the definitions of σ+
and σ− .
πJ
(π J u)k := u k if k ∈ J and (πJ u)k := 0 if k ∈
/ J . Here J ⊂ Z
and u = {u k }k∈Z .
π+ , π −
π+ := πZ+ and π− := πZ− .
w-lim
The weak limit in a Banach space. Thus w-limn→∞ xn = x in
X iff limn→∞ x ∗ xn = x ∗ x for all x ∗ ∈ X ∗ . See Section 3.5.
∗
x, x
In a Banach space setting x ∗ x := x, x ∗ is the continuous
linear functional x ∗ evaluated at x. In a Hilbert space setting
this is the inner product of x and x ∗ . See Section 3.5.
E⊥
E ⊥ := {x ∗ ∈ X ∗ | x, x ∗ = 0 for all x ∈ E}. This is the annihilator of E ⊂ X . See Lemma 9.6.4.
⊥

⊥
F
F := {x ∈ X | x, x ∗ = 0 for all x ∗ ∈ F}. This is the preannihilator of F ⊂ X ∗ . See Lemma 9.6.4. In the reflexive
case ⊥ F = F ⊥ , and in the nonreflexive case ⊥ F = F ⊥ ∩ X .


www.pdfgrip.com
xvi
A∗
A≥0
A
0
D (A)
R (A)
N (A)
rank(A)
dim(X )
ρ(A)
σ (A)
σ p (A)
σr (A)
σc (A)
ωA
T I, TIC

A&B, C&D

Notation

The (anti-linear) dual of the operator A. See Section 3.5.

A is (self-adjoint and) positive definite.
A ≥ for some > 0, hence A is invertible.
The domain of the (unbounded) operator A.
The range of the operator A.
The null space (kernel) of the operator A.
The rank of the operator A.
The dimension of the space X .
The resolvent set of A (see Definition 3.2.7). The resolvent
set is always open.
The spectrum of A (see Definition 3.2.7). The spectrum is
always closed.
The point spectrum of A, or equivalently, the set of eigenvalues of A (see Definition 3.2.7).
The residual spectrum of A (see Definition 3.2.7).
The continuous spectrum of A (see Definition 3.2.7).
The growth bound of the semigroup A. See Definition 2.5.6.
T I stands for the set of all time-invariant, and TIC stands for
the set of all time-invariant and causal operators. See Definition 2.6.2 for details.
A&B stands for an operator (typically unbounded) whose
domain D (A&B) is a subspace of the cross-product UX of
two Banach spaces X and U , and whose values lie in a third
˙
Banach space Z . If D (A&B) splits into D (A&B) = X 1 +
U1 where X 1 ⊂ X and U1 ⊂ U , then A&B can be written in
block matrix form as A&B = [A B], where A = A&B |X 1
and B = A&B |U1 . We alternatively write these identities in
the form Ax = A&B x0 and Bu = A&B u0 , interpreting
D (A&B) as the cross-product of X 1 and U1 .

Special Banach spaces
U

X
Y
Xn

X n∗
˙
+

Frequently the input space of the system.
Frequently the state space of the system.
Frequently the output space of the system.
Spaces constructed from the state space X with the help of the
generator of a semigroup A. In particular, X 1 is the domain
of the semigroup generator. See Section 3.6.
X n∗ := (X ∗ )n = (X −n )∗ . See Remark 3.6.1.
˙ X 2 means that the Banach space X is the direct
X = X1 +
sum of X 1 and X 2 , i.e., both X 1 and X 2 are closed subspaces


www.pdfgrip.com
Notation

⊕

X
Y

xvii

of X , and every x ∈ X has a unique representation of the form
x = x1 + x2 where x1 ∈ X 1 and x2 ∈ X 2 .
X = X 1 ⊕ X 2 means that the Hilbert space X is the orthogonal direct sum of the Hilbert spaces X 1 and X 2 , i.e.,
˙ X 2 and X 1 ⊥ X 2 .
X = X1 +
The cross-product of the two Banach spaces X and Y . Thus,
X
X
˙ 0
Y = 0 + Y .

Special functions
χI
1+
B
eω
log

The characteristic function of the set I .
The Heaviside function: 1+ = χR+ . Thus (1+ )(t) = 1 for t ≥
0 and (1+ )(t) = 0 for t < 0.
The Beta function (see (5.3.1)).
The Gamma function (see (3.9.7)).
eω (t) = eωt for ω, t ∈ R.
The natural logarithm.

Function spaces
V (J ; U )
Vloc (J ; U )

Vc (J ; U )
Vc,loc (J ; U )
Vloc,c (J ; U )
V0 (J ; U )
Vω (J ; U )
Vω,loc (R; U )
V (TT ; U )

BC
BC0
BCω

Functions of type V (= L p , BC, etc.) on the interval J ⊂ R
with range in U .
Functions which are locally of type V , i.e., they are defined
on J ⊂ R with range in U and they belong to V (K ; U ) for
every bounded subinterval K ⊂ J .
Functions in V (J ; U ) with bounded support.
Functions in Vloc (J ; U ) whose support is bounded to the left.
Functions in Vloc (J ; U ) whose support is bounded to the right.
Functions in V (J ; U ) vanishing at ±∞. See also the special
cases listed below.
The set of functions u for which (t → e−ωt u(t)) ∈ V (J ; U ).
See also the special cases listed below.
The set of functions u ∈ Vloc (R; U ) which satisfy π− u ∈
Vω (R− ; U ).
The set of T -periodic functions of type V on R. The norm in
this space is the V -norm over one arbitrary interval of length
T.
Bounded continuous functions; sup-norm.

Functions in BC that tend to zero at ±∞.
Functions u for which (t → e−ωt u(t)) ∈ BC.


www.pdfgrip.com
xviii

Notation

Functions u ∈ C(R; U ) which satisfy π− u ∈ BCω (R− ; U ).
Functions u for which (t → e−ωt u(t)) ∈ BC0 .
Functions u ∈ C(R; U ) which satisfy π− u ∈ BC0,ω (R− ; U ).
Bounded uniformly continuous functions; sup-norm.
Functions which together with their n first derivatives belong
to BUC. See Definition 3.2.2.
C
Continuous functions. The same space as BCloc .
Cn
n times continuously differentiable functions. The same
space as BCnloc .
∞
C
Infinitely many times differentiable functions. The same
space as BC∞
loc .
1/ p
p
L , 1 ≤ p < ∞ Strongly measurable functions with norm |u(t)| p dt
.
∞

L
Strongly measurable functions with norm ess sup|u(t)|.
p
p
∞
L 0 = L p if 1 ≤ p < ∞, and L ∞
L0
0 consists of those u ∈ L
which vanish at ±∞, i.e., limt→∞ ess sup|s|≥t |u(s)| = 0.
p
Lω
Functions u for which (t → e−ωt u(t)) ∈ L p .
p
p
p
L ω,loc (R; U )
Functions u ∈ L loc (R; U ) which satisfy π− u ∈ L ω (R− ; U ).
p
p
L 0,ω
Functions u for which (t → e−ωt u(t)) ∈ L 0 .
p
p
p
L 0,ω,loc (R; U )
Functions u ∈ L loc (R; U ) which satisfy π− u ∈ L 0,ω (R− ; U ).
W n, p
Functions which together with their n first (distribution)
derivatives belong to L p . See Definition 3.2.2.
Reg

Bounded right-continuous functions which have a left hand
limit at each finite point.
Reg0
Functions in Reg which tend to zero at ±∞.
Regω
The set of functions u for which (t → e−ωt u(t)) ∈ Reg.
Regω,loc (R; U ) The set of functions u ∈ Regloc (R; U ) which satisfy π− u ∈
Regω (R− ; U ).
Reg0,ω
The set of functions u for which (t → e−ωt u(t)) ∈ Reg0 .
Reg0,ω,loc (R; U ) Functions u ∈ Regloc (R; U ) which satisfy π− u ∈
Reg0,ω (R− ; U ).
Regn
Functions which together with their n first derivatives belong
to Reg. See Definition 3.2.2.
L p |Reg
This stands for either L p or Reg, whichever is appropriate.
BCω,loc (R; U )
BC0,ω
BC0,ω,loc (R; U )
BU C
BU C n


www.pdfgrip.com

1
Introduction and overview

We first introduce the reader to the notions of a system node and an L p -wellposed linear system with 1 ≤ p ≤ ∞, and continue with an overview of the

rest of the book.

1.1 Introduction
There are three common ways to describe a finite-dimensional linear timeinvariant system in continuous time:
(i) the system can be described in the time domain as an input/output map D
from an input signal u into an output signal y;
(ii) the system can be described in the frequency domain by means of a
transfer function D, i.e., if uˆ and yˆ are the Laplace transforms of the
input u respectively the output y, then yˆ = Duˆ in some right half-plane;
(iii) the system can be described in state space form in terms of a set of first
order linear differential equations (involving matrices A, B, C, and D of
appropriate sizes)
x˙ (t) = Ax(t) + Bu(t),
y(t) = C x(t) + Du(t),

t ≥ 0,

(1.1.1)

x(0) = x0 .
In (i)–(iii) the input signal u takes its values in the input space U and the
output signal y takes its values in the output space Y , both of which are
finite-dimensional real or complex vector spaces (i.e., Rk or Ck for some
k = 1, 2, 3, . . .), and the state x(t) in (iii) takes its values in the state space
X (another finite-dimensional vector space).
All of the three descriptions mentioned above are important, but we shall
regard the third one, the state space description, as the most fundamental
1

www.pdfgrip.com
2

Introduction and overview

one. From a state space description it is fairly easy to get both an input/
output description and a transfer function description. The converse statement
is more difficult (but equally important): to what extent is it true that an input/
output description or a transfer function description can be converted into a
state space description? (Various answers to this question will be given below.)
The same three types of descriptions are used for infinite-dimensional linear time-invariant systems in continuous time. The main difference is that we
encounter certain technical difficulties which complicate the formulation. As a
result, there is not just one general infinite-dimensional theory, but a collection
of competing theories that partially overlap each other (and which become
more or less equivalent when specialized to the finite-dimensional case). In this
book we shall concentrate on two quite general settings: the case of a system
which is either well-posed in an L p -setting (for some p ∈ [1, ∞]) or (more
generally), it has a differential description resembling (1.1.1), i.e., it is induced
by a system node.
In order to give a definition of a system node we begin by combining the
four matrices A, B, C, and D into one single block matrix S = CA DB , which
we call the node of the system, and rewrite (1.1.1) in the form
x˙ (t)
x(t)
=S
,
y(t)
u(t)

t ≥ 0,

x(0) = x0 .

(1.1.2)

For a moment, let us ignore the original matrices A, B, C, and D, and simply
regard S as a linear operator mapping UX into YX (recall that we denoted the
input space by U , the state space by X , and the output space by Y ). If U , X and
Y are all finite-dimensional, then S is necessarily bounded, but this need not
be true if U , X , or Y is infinite-dimensional. The natural infinite-dimensional
extension of (1.1.1) is to replace (1.1.1) by (1.1.2) and to allow S to be an
unbounded linear operator with some additional properties. These properties
are chosen in such a way that (1.1.2) generates some reasonable family of
trajectories, i.e., for some appropriate class of initial states x0 ∈ X and input
functions u the equation (1.1.2) should have a well-defined state trajectory x(t)
(defined for all t ≥ 0) and a well-defined output function y. The set of additional
properties that we shall use in this work is the following.
Definition 1.1.1 We take U , X , and Y to be Banach spaces (sometimes Hilbert
spaces), and call S a system node if it satisfies the following four conditions:1
(i) S is a closed (possibly unbounded) operator mapping D (S) ⊂
X
Y ;
1

X
U

into

It follows from Lemma 4.7.7 that this definition is equivalent to the definition of a system node

given in 4.7.2.


www.pdfgrip.com
1.1 Introduction

3

(ii) if we split S into S = SSYX in accordance with the splitting of the range
space YX (S X is the ‘top row’ of S and SY is the ‘bottom row’), then S X
is closed (with D (S X ) = D (S));
(iii) the operator A defined by Ax = S X x0 with domain D (A) = {x ∈ X |
x
0 ∈ D (S)} is the generator of a strongly continuous semigroup on X ;
(iv) for every u ∈ U there is some x ∈ X such that ux ∈ D (S).
It turns out that when these additional conditions hold, then (1.1.2) has
trajectories of the following type. We use the operators S X and SY defined in
(ii) to split (1.1.2) into
x˙ (t) = S X

x(t)
,
u(t)

t ≥ 0,

y(t) = SY

x(t)
,

u(t)

t ≥ 0.

x(0) = x0 ,
(1.1.3)

x

0
If (i)–(iv) hold, then for each x0 ∈ X and u ∈ C 2 ([0, ∞); U ) such that u(0)
∈
1
D (S), there is a unique function x ∈ C ([0, ∞); X ) (called a state trajectory)
x(t)
x(t)
∈ D (S), t ≥ 0, and x˙ (t) = S X u(t)
, t ≥ 0. If we
satisfying x(0) = x0 , u(t)

x(t)
define the output y ∈ C([0, ∞); Y ) by y(t) = SY u(t)
, t ≥ 0, then the three
functions u, x, and y satisfy (1.1.2) (this result is a slightly simplified version
of Lemma 4.7.8).
Another consequences of conditions (i)–(iv) above is that it is almost (but
not quite) possible to split a system node S into S = CA DB as in the finitedimensional case. If X is finite-dimensional, then the operator A in (iii) will
be bounded, and this forces the full system node S to be bounded, with
D (S) = UX . Trivially, in this case S can be decomposed into four bounded
operators S = CA DB . If X is infinite-dimensional, then a partial decomposition still exists. The operator A in this partial decomposition corresponds to an

extension A|X of the semigroup generator A in (iii).2 This extension is defined
on all of X , and it maps X into a larger ‘extrapolation space’ X −1 which contains X as a dense subspace. There is also a control operator B which maps
U into X −1 , and the operator S X defined in (ii) (the ‘top row’ of X ) is the
restriction to D (S) of the operator A|X B which maps UX into X −1 . (Furthermore, D (S) = ux ∈ UX | A|X B ux ∈ X .) Thus, S X always has a
decomposition (after an appropriate extension of its domain and also an extension of the range space). The ‘bottom row’ SY is more problematic, due to
the fact that it is not always possible to embed Y as a dense subspace in some
larger space Y−1 (for example, Y may be finite-dimensional). It is still true,

2

We shall also refer to A as the main operator of the system node.


www.pdfgrip.com
4

Introduction and overview

however, that it is possible to define an observation operator C with domain
D (C) = D (A) by C x = SY x0 , x ∈ D (A). The feedthrough operator D in
the finite-dimensional decomposition A = CA DB need not always exist, and it
need not be unique. However, this lack of a unique well-defined feedthrough
operator is largely compensated by the fact that every system node has a transfer
function, defined on the resolvent set of the operator A in (iii). See Section 4.7 for
details.3
The other main setting that we shall use (and after which this book has
been named) is the L p -well-posed setting with 1 ≤ p ≤ ∞. This setting can be
introduced in two different ways. One way is to first introduce a system node
of the type described above, and then add the requirement that for all t > 0, the
final state x(t) and the restriction of y to the interval [0, t) depend continuously

on x0 and the restriction of u to [0, t). This added requirement will give us an
L p -well-posed linear system if we use the X -norm for x0 and x(t), the norm in
L p ([0, t); U ) for u, and the norm in L p ([0, t); Y ) for y.4 (See Theorem 4.7.13
for details.)
However, it is also possible to proceed in a different way (as we do in
Chapter 2) and to introduce the notion of an L p -well-posed linear system without
any reference to a system node. In this approach we look directly at the mapping
from the initial state x0 and the input function (restricted to the interval [0, t))
to the final state x(t) and the output function y (also restricted to the interval
[0, t)). Assuming the same type of continuous dependence as we did above, the
relationship between these four objects can be written in the form (we denote
the restrictions of u and y to some interval [s, t) by π[s,t) u, respectively π[s,t) y)
x(t)
=
π[0,t) y

At0 Bt0
Ct0

Dt0

x0
,
π[0,t) u

t ≥ 0,

for some families of bounded linear operator At0 : X → X , Bt0 : L p ([0, t); U )→
X ,Ct0 : X → L p ([0, t); Y ), and Dt0 : L p ([0, t); U ) → L p ([0, t); Y ). If these
families correspond to the trajectories of some system node (as described earlier), then they necessarily satisfy some algebraic conditions, with can be stated

without any reference to the system node itself. Maybe the simplest way to list
these algebraic conditions is to look at a slightly extended version of (1.1.2)

3

4

Another common way of constructing a system node is the following. Take any semigroup
generator A in X , and extend it to an operator A|X ∈ B(X ; X −1 ). Let B ∈ B(U ; X −1 ) and
C ∈ B(X 1 ; U ) be arbitrary, where X 1 is D (A) with the graph norm. Finally, fix the value of the
transfer function to be a given operator in B(U ; Y ) at some arbitrary point in ρ(A), and use
Lemma 4.7.6 to construct the corresponding system node.
Here we could just as well have replaced the interval [0, t) by (0, t) or [0, t]. However, we shall
later consider functions which are defined pointwise everywhere (as opposed to almost
everywhere), and then it is most convenient to use half-open intervals of the type [s, t), s < t.


www.pdfgrip.com
1.1 Introduction

5

where the initial time zero has been replaced by a general initial time s, namely
x˙ (t)
x(t)
=S
,
y(t)
u(t)

t ≥ s,

x(s) = xs ,

(1.1.4)

and to also look at the corresponding maps from xs and π[s,t) u to x(t) and π[s,t) y
which we denote by
x(t)
=
π[s,t) y

Ats Bts
Cts

xs
,
π[s,t) u

Dts

s ≤ t.

These two-parameter families of bounded linear operators Ats , Bts , Cts , and Dts
have the properties listed below. In this list of properties we denote the left-shift
operator by
(τ t u)(s) = u(t + s),

−∞ < s, t < ∞,

and the identity operator by 1.
Algebraic conditions 1.1.2 The operator families Ats , Bts , Cts , and Dts satisfy
the following conditions:5
(i) For all t ∈ R,
Att Btt
Ctt

=

Dtt

1 0
0 0

.

(ii) For all s ≤ t,
Ats Bts
Cts Dts

=

Ats

Bts π[s,t)

π[s,t) Cts π[s,t) Dts π[s,t)

.

(iii) For all s ≤ t and h ∈ R,
t+h
At+h
s+h Bs+h
t+h
Ct+h
s+h Ds+h

=

Ats

Bts τ h

τ −h Cts τ −h Dts τ h

.

(iv) For all s ≤ r ≤ t,
Ats Bts
Cts Dts

=

Art Ars

Brt + Art Brs

Crt Ars + Crs Drt + Crt Brs + Drs

.

All of these conditions have natural interpretations (see Sections 2.1 and 2.2
for details): (i) is an initial condition, (ii) says that the system is causal, (iii)

5

By Theorem 2.2.14, these algebraic conditions are equivalent to those listed in Definition 2.2.1.