well-posed linear systems - o. staffans

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Well-Posed Linear Systems
OLOF STAFFANS
Department of Mathematics
˚
Abo Akademi University, Finland
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru,UK
First published in print format
isbn-13 978-0-521-82584-9
isbn-13 978-0-511-08208-5

© Cambridge University Press 2005
2005
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Contents
List of figures page ix
Preface xi
List of notation xiv
1 Introduction and overview 1
1.1 Introduction 1
1.2 Overview of chapters 2–13 8
2 Basic properties of well-posed linear systems 28
2.1 Motivation 28

2.2 Definitions and basic properties 34
2.3 Basic examples of well-posed linear systems 46
2.4 Time discretization 55
2.5 The growth bound 60
2.6 Shift realizations 67
2.7 The Lax–Phillips scattering model 71
2.8 The Weiss notation 76
2.9 Comments 78
3 Strongly continuous semigroups 85
3.1 Norm continuous semigroups 85
3.2 The generator of a C
0
semigroup 87
3.3 The spectra of some generators 98
3.4 Which operators are generators? 106
3.5 The dual semigroup 113
3.6 The rigged spaces induced by the generator 122
3.7 Approximations of the semigroup 128
3.8 The nonhomogeneous Cauchy problem 133
3.9 Symbolic calculus and fractional powers 140
3.10 Analytic semigroups and sectorial operators 150
v
vi Contents
3.11 Spectrum determined growth 164
3.12 The Laplace transform and the frequency domain 169
3.13 Shift semigroups in the frequency domain 177
3.14 Invariant subspaces and spectral projections 180
3.15 Comments 191
4 The generators of a well-posed linear system 194
4.1 Introduction 194

4.2 The control operator 196
4.3 Differential representations of the state 202
4.4 The observation operator 213
4.5 The feedthrough operator 219
4.6 The transfer function and the system node 227
4.7 Operator nodes 238
4.8 Examples of generators 256
4.9 Diagonal and normal systems 260
4.10 Decompositions of systems 266
4.11 Comments 273
5 Compatible and regular systems 276
5.1 Compatible systems 276
5.2 Boundary control systems 284
5.3 Approximations of the identity in the state space 295
5.4 Extended observation operators 302
5.5 Extended observation/feedthrough operators 313
5.6 Regular systems 317
5.7 Examples of regular systems 325
5.8 Comments 329
6 Anti-causal, dual, and inverted systems 332
6.1 Anti-causal systems 332
6.2 The dual system 337
6.3 Flow-inversion 349
6.4 Time-inversion 368
6.5 Time-flow-inversion 378
6.6 Partial flow-inversion 386
6.7 Comments 400
7 Feedback 403
7.1 Static output feedback 403
7.2 Additional feedback connections 413

7.3 State feedback and output injection 422
7.4 The closed-loop generators 425
Contents vii
7.5 Regularity of the closed-loop system 433
7.6 The dual of the closed-loop system 436
7.7 Examples 436
7.8 Comments 440
8 Stabilization and detection 443
8.1 Stability 443
8.2 Stabilizability and detectability 453
8.3 Coprime fractions and factorizations 465
8.4 Coprime stabilization and detection 473
8.5 Dynamic stabilization 485
8.6 Comments 502
9 Realizations 505
9.1 Minimal realizations 505
9.2 Pseudo-similarity of minimal realizations 511
9.3 Realizations based on factorizations of the Hankel operator 517
9.4 Exact controllability and observability 521
9.5 Normalized and balanced realizations 530
9.6 Resolvent tests for controllability and observability 538
9.7 Modal controllability and observability 546
9.8 Spectral minimality 549
9.9 Controllability and observability of transformed systems 551
9.10 Time domain tests and duality 554
9.11 Comments 565
10 Admissibility 569
10.1 Introduction to admissibility 569
10.2 Admissibility and duality 572
10.3 The Paley–Wiener theorem and H

∞
576
10.4 Controllability and observability gramians 583
10.5 Carleson measures 591
10.6 Admissible control and observation operators for diagonal
and normal semigroups 598
10.7 Admissible control and observation operators for
contraction semigroups 602
10.8 Admissibility results based on the Lax–Phillips model 610
10.9 Comments 613
11 Passive and conservative scattering systems 616
11.1 Passive systems 616
11.2 Energy preserving and conservative systems 628
11.3 Semi-lossless and lossless systems 636
viii Contents
11.4 Isometric and unitary dilations of contraction semigroups 643
11.5 Energy preserving and conservative extensions of
passive systems 655
11.6 The universal model of a contraction semigroup 660
11.7 Conservative realizations 670
11.8 Energy preserving and passive realizations 677
11.9 The Spectrum of a conservative system 691
11.10 Comments 692
12 Discrete time systems 696
12.1 Discrete time systems 696
12.2 The internal linear fractional transform 703
12.3 The Cayley and Laguerre transforms 707
12.4 The reciprocal transform 719
12.5 Comments 728
Appendix 730

A.1 Regulated functions 730
A.2 The positive square root and the polar decomposition 733
A.3 Convolutions 736
A.4 Inversion of block matrices 744
Bibliography 750
Index 767
Figures
2.1 Regular well-posed linear system page 29
2.2 Well-posed linear system 38
2.3 Cross-product (the union of two independent systems) 52
2.4 Cross-product in block form 52
2.5 Sum junction 53
2.6 Sum junction in block form 53
2.7 T-junction 54
2.8 T-junction in block form 54
2.9 Parallel connection 55
2.10 Parallel connection in block form 55
3.1 The sector 
δ
150
3.2 The sector 
θ,γ
151
3.3 The path in the proof of Theorem 3.10.5 153
7.1 Static output feedback connection 404
7.2 Positive identity feedback 409
7.3 Negative identity feedback 410
7.4 Flow-inversion 412
7.5 Another static output feedback 413
7.6 A third static output feedback 413

7.7 Output feedback in block form 415
7.8 Cancellation of static output feedback 417
7.9 Cascade connection through K 417
7.10 Cascade connection in block form 418
7.11 Dynamic feedback 419
7.12 Dynamic feedback in block form 419
7.13 Partial feedback 421
7.14 State feedback 422
7.15 Output injection 423
7.16 Original system with one extra output 426
ix
x Figures
7.17 Closed-loop system with one extra output 426
8.1 The extended system 453
8.2 Right coprime factor 454
8.3 Left coprime factor 455
8.4 Cancellation of state feedback 456
8.5 Dynamic stabilization 485
8.6 Dynamic stabilization 494
8.7 Equivalent version of dynamic stabilization 495
8.8 Second equivalent version of dynamic stabilization 497
8.9 Third equivalent version of dynamic stabilization 498
8.10 Youla parametrization 499
8.11 Youla parametrized stabilizing compensator 500
8.12 Youla parametrized stabilizing compensator 501
8.13 Youla parametrized stabilizing compensator 501
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
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 mini-
mization 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 non-
well-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-well-
posed) systems generated by arbitrary system nodes.
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 stan-
dard Hilbert space setting, and connections are explored between well-posed
linear systems, Fourier analysis, and operator theory. In particular, the admissi-
bility 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

primarily been carried out at the Mathematics Institute of
˚
Abo Akademi, which
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.
Notation
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
.
RR:= (−∞, ∞).
R
+
, R
+
R
+
:= (0, ∞) and R
+
:= [0, ∞).
R
−
, R
−
R
−
:= (−∞, 0) and R
−
:= (−∞, 0].
T The unit circle in the complex plane.
T
T
The real line R where the points t + mT, m = 0, ±1, ±2,

are identified.
Z The set of all integers.
Z
+
, Z
−
Z
+
:={0, 1, 2, } and Z
−
:={−1, −2, −3, }.
jj:=
√
−1.
0 The number zero, or the zero vector in a vector space, or the
zero operator, or the zero-dimensional vector space {0}.
1 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
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-well-
posed 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 Def-
initions 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.
τ
t
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.
γ
λ
The time compression or dilation operator (γ
λ
u)(s):= u(λs).
Here λ>0.
π
J
(π
J

u)(s):= u(s)ifs ∈ J and (π
J
u)(s):= 0ifs /∈ J. Here
J ⊂ R.
π
+
,π
−
π
+
:= π
[0,∞)
and π
−
:= π
(−∞,0)
.
R
The time reflection operator about zero: (
R
u)(s):= u(−s)
(in the L
p
-case) or (
R
u)(s):= lim
t↓−s
u(t) (in the Reg-case).
See Definition 3.5.12.
R

h
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
−
.
π
J
(π
J
u)
k
:= u
k
if k ∈ J and (π
J
u)
k
:= 0ifk /∈ J . Here J ⊂ Z

and u ={u
k
}
k∈Z
.
π
+
,
π
−
π
+
:= π
Z
+
and π
−
:= π
Z
−
.
w-lim The weak limit in a Banach space. Thus w-lim
n→∞
x
n
= x in
X iff lim
n→∞
x
∗

x
n
= 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 an-
nihilator of E ⊂ X . See Lemma 9.6.4.
⊥
F
⊥
F :={x ∈ X |x, x
∗
=0 for all x
∗
∈ F}. This is the pre-
annihilator of F ⊂ X
∗
. See Lemma 9.6.4. In the reflexive
case
⊥
F = F
⊥
, and in the nonreflexive case
⊥
F = F
⊥
∩ X.
xvi Notation
A
∗
The (anti-linear) dual of the operator A. See Section 3.5.
A ≥ 0 A is (self-adjoint and) positive definite.
A  0 A ≥  for some >0, hence A is invertible.

D
(
A
)
The domain of the (unbounded) operator A.
R
(
A
)
The range of the operator A.
N
(
A
)
The null space (kernel) of the operator A.
rank(A) The rank of the operator A.
dim(X ) The dimension of the space X.
ρ(A) The resolvent set of A (see Definition 3.2.7). The resolvent
set is always open.
σ (A) The spectrum of A (see Definition 3.2.7). The spectrum is
always closed.
σ
p
(A) The point spectrum of A, or equivalently, the set of eigenval-
ues of A (see Definition 3.2.7).
σ
r
(A) The residual spectrum of A (see Definition 3.2.7).
σ
c

(A) The continuous spectrum of A (see Definition 3.2.7).
ω
A
The growth bound of the semigroup A. See Definition 2.5.6.
TI, TIC T I stands for the set of all time-invariant, and TIC stands for
the set of all time-invariant and causal operators. See Defini-
tion 2.6.2 for details.
A&B, C&DA&B stands for an operator (typically unbounded) whose
domain D
(
A&B
)
is a subspace of the cross-product

X
U

of
two Banach spaces X and U , and whose values lie in a third
Banach space Z.IfD
(
A&B
)
splits into D
(
A&B
)
= X
1
˙

+
U
1
where X
1
⊂ X and U
1
⊂ U , then A&B can be written in
block matrix form as A&B = [AB], where A = A&B
|X
1
and B = A&B
|U
1
. We alternatively write these identities in
the form Ax = A&B

x
0

and Bu = A&B

0
u

, interpreting
D
(
A&B
)

as the cross-product of X
1
and U
1
.
Special Banach spaces
U Frequently the input space of the system.
X Frequently the state space of the system.
Y Frequently the output space of the system.
X
n
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
= (X
−n
)
∗

. See Remark 3.6.1.
˙
+ X = X
1
˙
+ X
2
means that the Banach space X is the direct
sum of X
1
and X
2
, i.e., both X
1
and X
2
are closed subspaces
Notation xvii
of X, and every x ∈ X has a unique representation of the form
x = x
1
+ x
2
where x
1
∈ X
1
and x
2
∈ X

2
.
⊕ X = X
1
⊕ X
2
means that the Hilbert space X is the or-
thogonal direct sum of the Hilbert spaces X
1
and X
2
, i.e.,
X = X
1
˙
+ X
2
and X
1
⊥ X
2
.

X
Y

The cross-product of the two Banach spaces X and Y . Thus,

X
Y


=

X
0

˙
+

0
Y

.
Special functions
χ
I
The characteristic function of the set I .
1
+
The Heaviside function: 1
+
= χ
R
+
. Thus (1
+
)(t) = 1 for t ≥
0 and (1
+
)(t) = 0 for t < 0.

B The Beta function (see (5.3.1)).
 The Gamma function (see (3.9.7)).
e
ω
e
ω
(t) = e
ωt
for ω, t ∈ R.
log The natural logarithm.
Function spaces
V (J ; U) Functions of type V (= L
p
, BC, etc.) on the interval J ⊂ R
with range in U.
V
loc
(J; 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 .
V
c
(J; U ) Functions in V (J ; U ) with bounded support.
V
c,loc
(J; U ) Functions in V
loc
(J; U ) whose support is bounded to the left.
V
loc,c

(J; U ) Functions in V
loc
(J; U ) whose support is bounded to the right.
V
0
(J; U ) Functions in V (J ; U) vanishing at ±∞. See also the special
cases listed below.
V
ω
(J; U ) The set of functions u for which (t → e
−ωt
u(t)) ∈ V ( J ; U ).
See also the special cases listed below.
V
ω,loc
(R; U) The set of functions u ∈ V
loc
(R; U) which satisfy π
−
u ∈
V
ω
(R
−
; U).
V (T
T
; 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 .

BC Bounded continuous functions; sup-norm.
BC
0
Functions in BC that tend to zero at ±∞.
BC
ω
Functions u for which (t → e
−ωt
u(t)) ∈ BC.
xviii Notation
BC
ω,loc
(R; U) Functions u ∈ C(R; U) which satisfy π
−
u ∈ BC
ω
(R
−
; U).
BC
0,ω
Functions u for which (t → e
−ωt
u(t)) ∈ BC
0
.
BC
0,ω,loc
(R; U) Functions u ∈ C(R; U) which satisfy π
−

u ∈ BC
0,ω
(R
−
; U).
BUC Bounded uniformly continuous functions; sup-norm.
BUC
n
Functions which together with their n first derivatives belong
to BUC. See Definition 3.2.2.
C Continuous functions. The same space as BC
loc
.
C
n
n times continuously differentiable functions. The same
space as BC
n
loc
.
C
∞
Infinitely many times differentiable functions. The same
space as BC
∞
loc
.
L
p
, 1 ≤ p < ∞ Strongly measurable functions with norm



|u(t)|
p
dt

1/ p
.
L
∞
Strongly measurable functions with norm ess sup|u(t)|.
L
p
0
L
p
0
= L
p
if 1 ≤ p < ∞, and L
∞
0
consists of those u ∈ L
∞
which vanish at ±∞, i.e., lim
t→∞
ess sup
|s|≥t
|u(s)|=0.
L

p
ω
Functions u for which (t → e
−ωt
u(t)) ∈ L
p
.
L
p
ω,loc
(R; U) Functions u ∈ L
p
loc
(R; U) which satisfy π
−
u ∈ L
p
ω
(R
−
; U).
L
p
0,ω
Functions u for which (t → e
−ωt
u(t)) ∈ L
p
0
.

L
p
0,ω,loc
(R; U) Functions u ∈ L
p
loc
(R; U) which satisfy π
−
u ∈ L
p
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.
Reg
0
Functions in Reg which tend to zero at ±∞.
Reg
ω
The set of functions u for which (t → e
−ωt
u(t)) ∈ Reg.

Reg
ω,loc
(R; U) The set of functions u ∈ Reg
loc
(R; U) which satisfy π
−
u ∈
Reg
ω
(R
−
; U).
Reg
0,ω
The set of functions u for which (t → e
−ωt
u(t)) ∈ Reg
0
.
Reg
0,ω,loc
(R; U) Functions u ∈ Reg
loc
(R; U) which satisfy π
−
u ∈
Reg
0,ω
(R
−

; U).
Reg
n
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.
1
Introduction and overview
We first introduce the reader to the notions of a system node and an L
p
-well-
posed 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 time-
invariant 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
ˆ
u and
ˆ
y are the Laplace transforms of the

input u respectively the output y, then
ˆ
y =

D
ˆ
u 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) = Cx(t) + Du(t), t ≥ 0,
x(0) = x
0
.
(1.1.1)
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., R
k
or C
k
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
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 lin-
ear 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 =

AB
CD

, which
we call the node of the system, and rewrite (1.1.1) in the form

˙
x(t )
y(t)


= S

x(t )
u(t)

, t ≥ 0, x(0) = x
0
. (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

X
U

into

X
Y

(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,orY 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 x
0
∈ 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
U

into

X
Y

;
1
It follows from Lemma 4.7.7 that this definition is equivalent to the definition of a system node
given in 4.7.2.
1.1 Introduction 3
(ii) if we split S into S =

S
X
S
Y


in accordance with the splitting of the range
space

X
Y

(S
X
is the ‘top row’ of S and S
Y
is the ‘bottom row’), then S
X
is closed (with D
(
S
X
)
= D
(
S
)
);
(iii) the operator A defined by Ax = S
X

x
0

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

x
u

∈ 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 S
Y
defined in
(ii) to split (1.1.2) into

˙
x(t ) = S
X

x(t )
u(t)

, t ≥ 0, x(0) = x
0
,
y(t) = S
Y

x(t )
u(t)

, t ≥ 0.
(1.1.3)
If (i)–(iv) hold, then for each x
0
∈ X and u ∈ C
2
([0, ∞); U ) such that

x
0
u(0)

∈
D

(
S
)
, there is a unique function x ∈ C
1
([0, ∞); X) (called a state trajectory)
satisfying x(0) = x
0
,

x(t)
u(t)

∈ D
(
S
)
, t ≥ 0, and
˙
x(t ) = S
X

x(t)
u(t)

, t ≥ 0. If we
define the output y ∈ C([0, ∞); Y )byy(t) = S
Y

x(t)

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 =

AB
CD

as in the finite-
dimensional 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
)
=

X
U

. Trivially, in this case S can be decomposed into four bounded
operators S =

AB
CD

.IfX is infinite-dimensional, then a partial decomposi-

tion 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 con-
tains 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

X
U


into X
−1
. (Fur-
thermore, D
(
S
)
=

x
u

∈

X
U

|

A
|X
B

x
u

∈ X

.) Thus, S
X

always has a
decomposition (after an appropriate extension of its domain and also an ex-
tension of the range space). The ‘bottom row’ S
Y
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.
4 Introduction and overview
however, that it is possible to define an observation operator C with domain
D
(
C
)
= D
(
A
)
by Cx = S
Y

x
0

, x ∈ D
(
A

)
.Thefeedthrough operator D in
the finite-dimensional decomposition A =

AB
CD

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 x
0
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 x
0
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 x
0
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)byπ
[s,t)
u, respectively π
[s,t)
y)

x(t )
π
[0,t)
y

=


A
t
0
B
t
0
C
t
0
D
t
0


x
0
π
[0,t)
u

, t ≥ 0,
for some families of bounded linear operator A
t
0
: X → X, B
t
0
: L
p
([0, t); U)→

X,C
t
0
: X → L
p
([0, t); Y ), and D
t
0
: L
p
([0, t); U) → L
p
([0, t); Y ). If these
families correspond to the trajectories of some system node (as described ear-
lier), 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
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.
4
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.
1.1 Introduction 5
where the initial time zero has been replaced by a general initial time s, namely

˙
x(t )
y(t)

= S

x(t )
u(t)

, t ≥ s, x(s) = x
s
, (1.1.4)
and to also look at the corresponding maps from x
s
and π
[s,t)

u to x(t) and π
[s,t)
y
which we denote by

x(t )
π
[s,t)
y

=

A
t
s
B
t
s
C
t
s
D
t
s


x
s
π
[s,t)

u

, s ≤ t.
These two-parameter families of bounded linear operators A
t
s
, B
t
s
, C
t
s
, and D
t
s
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 A
t
s
, B
t
s
, C
t
s

, and D
t
s
satisfy
the following conditions:
5
(i) For all t ∈ R,

A
t
t
B
t
t
C
t
t
D
t
t

=

1
0
0 0

.
(ii) For all s ≤ t,


A
t
s
B
t
s
C
t
s
D
t
s

=

A
t
s
B
t
s
π
[s,t)
π
[s,t)
C
t
s
π
[s,t)

D
t
s
π
[s,t)

.
(iii) For all s ≤ t and h ∈ R,

A
t+h
s+h
B
t+h
s+h
C
t+h
s+h
D
t+h
s+h

=

A
t
s
B
t
s

τ
h
τ
−h
C
t
s
τ
−h
D
t
s
τ
h

.
(iv) For all s ≤ r ≤ t,

A
t
s
B
t
s
C
t
s
D
t
s


=

A
t
r
A
r
s
B
t
r
+ A
t
r
B
r
s
C
t
r
A
r
s
+ C
r
s
D
t
r

+ C
t
r
B
r
s
+ D
r
s

.
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.