Title: Realization theory of infinite-dimensional linear systems
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Title: Realization theory of infinite-dimensional linear systems
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Publication Date: 1978
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REALIZATION THEORY OF
INFINITE-DIMENSIONTAL LINEAR SYSTEMS










By
YUTAKA YAMAMOTO

















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IT PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PiILOSOHIY




UNIVERSITY OF FLORIDA
1978










ACKNOWLEDGEMENTS


I wish to express my sincere gratitude to all those who contributed
in various degrees toward the completion of this work.

I am particularly grateful to Professor R. E. KAIMAN, the chairman
of my supervisory committee, for his constant guidance in developing
scientific discipline in active research areas. It is also he who
originally motivated the research problem in realization theory of
infinite-dimensional linear systems; the key idea of this work, topolo-
gical observability, was also encountered thanks to his strong emphasis
on good understanding of concrete examples. Without the financial
support which he arranged for me during the past four years and without
the stimulating environment of the CENTER FOR MATHEMATICAL SYSTEM THEORY,
this work would not exist today.

Discussions with Dr. E. D. SONTAG have had a great influence on this
work. I deeply appreciate his friendship and his interest.

No research can be done without basic knowledge of what has already
been done in the field. In this respect the discussions with Professors
R. W. BROCKETT, S. K. MITTER, E. W. KAMEN, and others, are most appreciated.

Of course, there would be no research today were it not for the
long-term love and encouragement of a few close people. I would like
to thank all of my friends who assisted me in various ways from time to
time. Of all, I am most indebted to A. MASON and R. SMITH who gave many
useful comments on the final draft. But among all, I am most grateful to
my parents who have been a constant source of encouragement during the
past four years. To them I dedicate this work.

This research was supported in part by US Army Research Grant DAA29-
77-G-0225 and US Air Force Grant AFOSR 76-5034 Mod. B through the Center
for Mathematical System Theory, University of Florida, Gainesville,
FL 32611, USA.










TABLE OF CONTENTS


ACKNOW EDGE ENTS . . . . . . . . . . . .. ii

ABS RACT . . . . . . . . . . . . . . iv

CHAPTER

I. INTRODUCTION . . . . . . . . . . . 1

II. INPUT/OUTPUT MAPS . . . . . . . . . 11

1. Input Space . . . . . . . . . 11
2. Output Space . . . . . . . . ... .. 13
3. Input/Output Maps . . . . . . . . .. 16
4. The Space of Laurent Functions . . . . ... 22
5. Extended Linear Input/Output Maps . . . ... 22

III. REALIZATION THEORY . . . . . . . . ... ... 27

6. Systems .. . . .. . . . . . . . 27
7. Realizations and Factorizations of Input/Output Maps. 37
8. Topological Observability . . . . . . . 43
9. Existence and Uniqueness of Canonical Realizations 50
10. Realizations in the Working Mode . . . .. 52
11. Compatible Systems and Differential Equations . .. 55

IV. CANONICAL REALIZATIONS WITH HILBERT STATE SPACES . . .. 62

12. Topological Observability in Bounded Time . . .. 62
13. Necessary and Sufficient Conditions
for a Hilbert Space Canonical Realization . . 64
14. An Example of a Topologically Observable System . 67

APPENDIX . . . . . . . . ... . . . . .. . 75

REFERENCES . . . . . . . . ... . . . . . 80

BIOGRAITTICAL SKETCH . . . . . . . . ... .... 84


iii










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



REALIZATION THEORY OF
INFINITE-DIMENSIONAL LINEAR SYSTEMS



By
YUTAKA YAMAMOTO

August, 1978





Chairman: Dr. R. E. Kalman
Major Department: Mathematics

This work studies the problem of realization of constant linear input/
output maps, which do not necessarily possess a finite-dimensional reali-
zation. A class of constant linear input/output maps is introduced. This
class is then characterized as the family of continuous linear maps whose
weighting patterns are measures. The natural state-space representation
(realization) of such constant linear input/output maps is studied, and a
new notion of observability, topological observability, is introduced. It
is then seen that topological observability enables us to prove the existence
and uniqueness of canonical (quasi-reachable and topologically observable)
realizations. It is also shown that a certain subclass of realizations
admits a functional-differential equation description. Necessary and
sufficient conditions that the state space of a canonical realization be a
Banach (or Hilbert) space are obtained. A new notion, topological observa-
bility in bounded time, plays a central role in deriving such conditions.
A thorough study of a concrete example of such systems is given.









CHAPTER I. INTRODUCTION


In the present work we study the problem of realization of linear
constant (shift-invariant) continuous-time input/output maps, which do not
necessarily possess a finite-dimensional realization. In this introduction
we confine ourselves to scalar (single-input/single-output) input/output
maps. Multi-input/multi-output cases will be discussed in the main text.

Throughout this work we fix a field k, either R or C, and every
space is a locally convex Hausdorff topological vector space over k.

Let us start our discussion with some intuitive idea as to what
realization theory is. The problem of realization may be viewed as an
idealized way of formulating the problem of scientific model building.
Frequently we encounter the situation in which a system (or a plant) is
primarily defined only in terms of its external behavior. The objective
of realization theory is to construct an "internal model" of the given
external behavior so that the behavior of the system can be better studied
on the basis of the model rather than the given external behavior. Here
by an internal model (or a system) we mean some mathematical object S
which is a dynamical system in the following modern sense:

(i) Z accepts inputs and produces outputs. The present output is
produced as a function of the (present) "state" of E, which represents
the past history of inputs. (Classical dynamical systems, based on
Newton's laws, do not have this property which is essential to treat the
scientific problems of the 20th century. Note that in the mathematical
literature, dynamicall system" unfortunately usually means just a "classical
dynamical system.")

(ii) Each application of an input may alter the state of Z; this
state-transition is governed by causal (i.e., the present state does not
depend on the future values of inputs) and deterministic rules. (We do
not consider probabilistic dynamical systems in this work.)

(iii) The notion of "state" is the basic concept which needs to be
axiomatized for a deeper mathematical study. (In physics one talks about
"phase," as a very narrow special case of what in system theory is known
as the "state.")









Of course, this is not a definition by any means. But before proceed-
ing to a technical definition, we must clarify the meaning of "external
behavior."

Intuitively external behavior of a system refers to the correspondence
between inputs and outputs, i.e., whenever an input is applied to the
system we can tell what the corresponding future output will be. Here we
have implicitly made the assumption that the system is causal, i.e., the
present value of an output does not depend on the future values of inputs.
It is this property that implies the existence of an (internal) state
(see KAIMAIN, FALB, and ARBIB [1969, Comment (10.2.4)]). Let us now make
another basic assumption that the system we are concerned with is constant
(this terminology is due to KALMAN; see, for instance, KALMAN [1968]), i.e.,
defining relations (structure) of the system do not depend on time. This
property guarantees that if we execute the same experiment on the system
(i.e., apply an input) under the same experimental conditions, but at a
different time, we will obtain the same response. Hence the property of
"constancy" enables us to carry out multiple experiments without the need
to worry about structural changes of the system due to time.

It is reasonable to represent each input (output) as a k-valued
function on some time set T. If T = Z, the integers, the system is
said to be discrete-time, and if T = R, the reals, the system is said
to be continuous-time. We also assume that every input has a bounded
support, i.e., is identically zero except on a bounded interval. This
requirement is made because, in principle, we can apply inputs only for
bounded time.

Since our system is assumed to be constant, we may, without loss of
generality, call the present time 0. Then what we call external behavior
(or an input/output map) is the following correspondence between inputs and
outputs: There is a rule to produce a future output whenever we are given
an input having support in the past (i.e., (- -, 0]). Let 0 be a given
space of past inputs, and r a given space of future outputs. Suppose
that 0 and r are vector spaces over k. Then an input/output map
f: ->e is said to be linear iff f is a linear map. (We use "iff" as
the common abbreviation for "if and only if.") One can easily see that
causality is already built into the above setting.










REMARK. We do not, however, claim that this is the most general
formulation of external behavior. Indeed, inputs may be applied while
outputs are being observed in practical situations (see Section 5). Or
an output may vary depending not only on known inputs but also on unknown
inputs generated by the environment in which the system is situated.
But these questions are irrelevant (as far as theoretical studies are
concerned) if the system under consideration is constant and linear.
For a more thorough discussion on this matter, see KAIMAN, FALB, and
ARBIB [1969, Chapters 1 and 10].

Thus our problem is now reduced to finding "good" input and output
spaces. When the system is discrete-time, we have a very natural choice:
S:= sequences with bounded support contained in (- -, O] F Z),
r := (sequences which vanish on (- n, 0] Qz). By using the correspond-
ence: t o zt (t Z), one can easily verify the isomorphisms (as
vector spaces over k): n k[z], r z k[[z-l]] where z and z-
denote indeterminates. In both spaces the left shift operator is repre-
sented by the left multiplication by z. Then the requirement that an
input/output map f: 0 -> be constant is very simply expressed as
zf(c) = f(zs). This idea is due to KALMAN [1965] and is now the standard
in realization theory of discrete-time systems (one can also see its
recent evolution in systems over rings and nonlinear systems:
ROUCHALEAU [1972]; SONTAG [19761).

But despite these successful predecessors, it seems to be a major
problem to give a good framework for continuous-time input/output maps.
Furthermore, topological considerations become essential in this case.
Let us briefly examine some historical background (we do not intend to
make this review very complete).

KALMAN and HAUTUS [1972] used the setting: Q := E( .,0], the space
of distributions with bounded support contained in (- -, 01, P := E0,
the space of C-functions on [0, o), and f (input/output map) is
a continuous linear map from Q to P. With this setting they success-
fully derived a differential equation description of a realization. In
this case, the input/output map f is described by a C impulse
response function.










KAMEN [1975, 1976] defined an input/output map as a continuous linear
map f: D' ->D' that satisfies f(l()I -) =f(m2)_(, whenever
+ -' ) 2 (-,)
(_ ,z) = 2 (_, ) (D' denotes the space of distributions with
support bounded on the left). He extended the module theoretic treatment
of systems, initiated by KAIMAN and HAUTUS [1972] for continuous-time
systems, to a very large class of input/output maps. In this case, the
input/output map f is described by a distribution weighting pattern
whose support is in [0, ).

MATSUO f[1978] proposed the choice: n := Mc(- m, 0], the space of
Radon measures with compact support contained in (- m, 0], and
r := C[O, m).

Note that the above mentioned settings are given in such a way that
the input space contains Dirac's "delta function" 6 This is a definite
o
advantage for clarifying certain system-theoretic concepts such as
impulse response functions. But at the same time the following question
arises: Can these ideal inputs (such as 5o) be applied to the system
while the system is in the working mode? As is pointed out in KAIMAN
and HAUTUS [1972], this question is related to the very delicate problem
of whether the truncation of inputs at an arbitrary time t is well-
defined. This is the kind of difficulty that one never encounters in
discrete-time systems.

On the other hand, a great many authors took the viewpoint that the
input/output map is defined by a weighting pattern (or an impulse
response function) so that the output is expressed as a convolution of
the weighting pattern with the input; see, for example, BARAS, BROCKETT,
and FUHRMANN [1974]; BARAS and BROCKETT [1975]; BROCKETT and FUHRMANN
[19761, etc. Unfortunately the question, "What type of function (or
distribution) should be considered as an input (or output)?" is then
left out; without specifying the input and output spaces no satisfactory
theory is possible.

We now propose a choice of the input space and the output space. Our
requirements are (i) 0 consists of functions such that (a) the very




5





difficult problem of truncation does not occur, (b) these functions
can actually be applied to the system in the working mode, and (ii) P
also consists of functions. Of course one drawback is that we cannot
have the delta function as an input due to (i) (because 6 is merely
a distribution or a measure). But under a certain regularity hypothesis
on the input/output map, the input space can be indeed extended to a
larger space so that this "ideal" input space contains 6 (Proposition
(3.1 )).

Fix any T > 0. We take the input space and the output space on the
interval O[, T as L2 ,Tl (If we consider inputs in the past we
inter 0, O as LOT]'
take L instead.) Since we can apply inputs only in bounded time
L[_T,O]0
it is reasonable to take := T L2T On the other hand, since
T>0 [-T,O1 O
outputs may well continue for an indefinitely long time, we simply take
I to be the set of all locally L2-functions.

We must topologize 2 and r. We make an intuitive additional
requirement: the topologies of and r are so determined that the
knowledge of the behavior of the system on each bounded interval [0, T]
is enough to know the behavior of the system in an infinitely long time
period. For example, let f be a linear map from 0 to r. One may
ask the question: Is it enough to know that f is continuous on each
2
-, so as to conclude that f is continuous on Q? This is a very
L -T, 01
reasonable question since we can apply inputs only in bounded time.
Another question could be the following: Is it enough to know that
-2
f()l [O,T] is small in each L,T] so as to conclude that f(M)
itself is small in P? This is also a reasonable question because one
can observe outputs only in a finite time period (no matter how long it
may be).

Fortunately there are well-known mathematical tools to handle these
requirements, namely inductive and projective limits (cf. Appendix).
2 2
We write 0i = im L[-n0] (inductive limit) and P = lim L2,
We write 2 lim [ [0,n]
projectivee limit). Then a linear map f: ->I' is continuous iff for
2 2
each m, n the map f is continuous from L[ -m0 to L, Then
ur questions are affirmative answered.
our questions are affirmatively answered.











Each space P (and F) is equipped with a family of shift operators

(ot]t>O ((&t O0) defined by (Utm)(T) := (T + t) if T 5 t and
(rto)() : 0 if t < T < 0 (( 7)(T) := 7(T + t)).

It may be a reasonable idea to define an input/output map as a
continuous linear map f: 0 --F which commutes with shifts, i.e.,
ctf = fet for all t > 0. But unfortunately this causes a tremendous
problem in defining a system equation. Hence we impose one additional
condition: f sends C (- 0 0), the space of continuous functions
---- o ------------
with compact support in (- 0, 0), to C[O, m), the space of continu-
ous functions on [0, m), and this correspondence must be continuous
with respect to the topologies of C (- m, 0) and C[O, m) (cf.
Section 5). Then it turns out that f is precisely the map given by a
convolution of C 0 with a measure P, i.e., f(m) = P*m (Theorem
(5.12)). The measure [i is called the weighting pattern of f.

The class of input/output maps as defined above is indeed very large.
It is desirable that any input/output map can be realized by a system.
Thus our question is the following: What is a good definition of systems
so that (i) any input/output map can be realized (ii) systems still have
a "nice" structure? Let us consider what properties we wish to require
of our state spaces. For example, let f be the input/output map given
by f(c) = exp (exp t)*o. Then f does not have a realization in the
category of Banach spaces in any usual sense, i.e., described by a
functional-differential equation. Intuitively the reason is that the
weighting pattern exp (exp t) grows too rapidly whereas a semigroup
in a Banach space can grow only with exponential order. Hence the
requirement that the state space be Banach may be too restrictive. (Or
exp (exp t) should not be a weighting pattern.)

We now define a linear constant system Y as follows: 7 consists
of three objects X, cp, H along with the conditions

(a) The state space X is a complete locally convex Hausdorff
space:









2
(b) rp(t, *): XX Lt) X is a continuous linear map such
that q(0, x, u) = x, p(t + s, x, u) = c(t, c(s, x, u), odu)
(c = left shift operator, see (4.1)) for all t, s > 0, x E X, and
2
u GL ,ot
[ E Lo,t+s);
(c) H: Do(H) -)k is a densely defined linear (not necessarily
continuous) map such that ((t, x, 0) E Do(H) for all x E Do(H).

(There are other technical requirements on c and H, but we shall
postpone the discussion until the main text.)

The major difference of this definition from the classical one
(KAIMAN, FALB, and ARBIB [1969, Chapter 1]) is (c), i.e., the readout
(output) map H is not necessarily defined on the whole state space X.
This modification is motivated by the following example.

Consider an insulated uniform rod with unknown temperature distri-
bution. With proper normalization the temperature v(t, !) satisfies


(8a/t)v(t, 0) = (82/2)v(t, ), t > 0, 0 < < < 1,

(av/a5)(t, o) = (v/W)(t, 1) =0.

We want to observe y(t) := v(t, 0). It is a standard technique in the
theory of the integration of the equation of evolution to take
X = L2 and integrate the equation in this function space. In this
(0,1)
case the equation is transformed into the form (by letting
x(t)(0) := v(t, 0))


Sx(t) = (2a 2)x(t), x(t)(-) E L ,l).

See, for example, YOSHIDA [1971, Chapter XIV]. But the output equation
y(t) = v(t, 0) = x(t)(0) is not well-defined for every x(t) C L 0,1)
The fact is that y(t) = I-x(t) makes sense only on a dense subspace
Do(H) (for example, take Do(H) := C[0,1]). HELTON [1976] points out
the same type of phenomenon with an example of a transmission line.










Note that H induces a correspondence x >vho(x)(t) := Hp(t, x, 0)
because of the requirement cp(t, x, 0) E Do(H) for all t > 0. Even
though H itself is not continuous, it is quite possible that h
o
gives a continuous correspondence from Do(H) to p. (One can make sure
that this is indeed the case with the previous example.) In addition to
the conditions (a), (b), (c) on systems, we require the continuity of
h as a part of the definition. Since ho is continuous, it has a
continuous extension h : X ->r because D (H) is dense in X. We
call h the observability map of E; E is said to be observable iff
h is one-to-one.

Now take any T > 0, and let XT := {((T, 0, u): u E L O,T).
is the set of all elements reachable from 0 at time T with applica-
tion of a suitable input. Let XY := XT, and call X the reachable
set of E. We say that the system E is (exactly) reachable iff X = X,
quasi-reachable iff X is dense in X. Also, the system E is said
R
to be weakly canonical iff it is quasi-reachable and observable.

One of the basic problems in realization theory is to associate a
unique realization to the external behavior (the input/output map) with-
out assuming a priori information not implied by the external behavior.
In order that a realization be uniquely associated to the input/output
map, it must not contain any redundant part in the state space. This
requirement easily implies that this uniquely associated system must be
at least quasi-reachable and observable, i.e., weakly canonical. One
can easily construct a weakly canonical realization for any input/output
map (Proposition (7.18)). But is such a realization unique? If the
system is finite-dimensional, this is true, as a consequence of the
classical result by KALMAN (see KAIMAN, FALB, and ARBIB [1969, Chapter
10]), since quasi-reachability coincides with reachability in this case,
thereby yielding the implication "weakly canonical" = "canonical" in the
classical sense. For infinite-dimensional systems, however, BARAS,
BROCKETT, and FUHRMANN [1974] gave a counterexample, by proving the
existence of two nonisomorphic weakly canonical systems having the same
external behavior.










Many attempts have been made toward proving the uniqueness of
"canonical" realizations: BENSOUSSAN, DELFOUR, and MITTER [1975, 1976];

BROCKETT and FUHRMANN [1976]; HELTON [1974]; MATSUO [1978], etc. We
shall propose yet another approach to the problem.

The intuitive idea of observability is that any two different initial
states are distinguishable by application of a certain "suitable
procedure" to future outputs; this is pointed out in KALMAN [1968,
Chapter 10]; SONTAG [1976]; SONTAG and ROUCHALEAU [1976]. Since we are
interested in topological aspects of the problem, it is reasonable to
demand that this "suitable procedure" mean that initial states can be
determined continuously from observation data. In other words, the initial
state determination procedure must be well-posed. Indeed, if the initial
state determination procedure is not well-posed, it may occur that we
identify two quite different initial states as the same, possibly due to
observation errors. We say that a system is topologically observable
iff its initial state determination procedure is well-posed.

We shall say that a system Z is canonical iff it is quasi-reachable
and topologically observable. One of the main results of this work is
the claim: Every input/output map f admits a canonical realization Zf,
and any other canonical realization of f is isomorphic to Yf
(Theorems (9.2), (9.4)).

There are very many questions on canonical realizations that one
might want to ask. Is a canonical realization nice enough so that it
admits a differential equation description (at least for rather smooth
inputs)? Is the character of the state space "nice," such as Hilbert or
Banach or metrizable at least?

Let us discuss the first question. In order that the system be
described by a differential equation, the weighting pattern of f must
be smooth enough. We assume that the weighting pattern is actually a
locally absolutely continuous function whose derivative belongs to L
on each bounded interval. With this assumption, we have the following
result: If f is an input/output map as described, then the canonical










realization admits a differential equation description, i.e., there exists
a densely defined closed linear operator F: D(F) -X and G C D(F)
such that

(a) F generates a strongly continuous semigroup (T(t))t>O
in X;

(b) the differential equation


|x(t) = Fx(t) + Gu(t)


with the initial condition x(O) = x D(F) admits a unique solution:
x(t) = 0(t)x + t(t T)Gu(T)dT at least for uniformly continuous u;

(c) q(t, x, u) = x(t) = (t)x + I t(t T)Gu(T)dT for all
x C D(F) and uniformly continuous u.

Let us consider the character of the topology of canonical realiza-
tions. We can easily prove that every canonical realization has a Frdchet
(metrizable and complete) space as a state space. But in general the
state space of a canonical realization cannot be a Banach space. In
Chapter IV we give necessary and sufficient conditions that the state
space of a canonical realization be a Banach space. A new notion,
topological observability in bounded time, plays a crucial role. A
concrete example will be discussed also in this chapter.









CHAPTER II. INPUT/OUTPUT MAPS


In this chapter we give the definitions of an input space, output
space and linear input/output maps. We shall prove that there is a one-
to-one correspondence between a Radon measure on (O, m) and a linear
input/output map. This uniquely associated Radon measure is called
the weighting pattern of a linear input/output map and plays a crucial
role in later chapters. We shall also give definitions of the space of
Laurent functions and extended linear input/output maps. The space of
Laurent functions is an analogue of the Laurent series in the continuous-
time context; for the Laurent series see HAUTUS and HEYMANN [1978].

1. Input Space.

2
Let L be the set of all k-valued Lebesgue square integrable
[-n,O]
functions on the interval [- n, O] (n = a positive integer). As is well-
2
known, Ln, 0 is a Hilbert space with the following norm:



(1.1) 11 ,] (t)2dt1/2, cp L2[-n
-n


Clearly we may identify the space L[n,0] with the space of all functions
defined on (- c,0] which vanish outside of [- n, 0] and belong to
L when restricted on [- n, 0]. In the sequel we shall also denote
[-n,O] 2
this space by L n,]; if a precise distinction is necessary, we shall
denote it by L (the subscript 0 denotes that each member of
o,[-n,o]
the space has compact support).

If a < b (a, b = positive integers), then there exists a natural
inclusion j : L2 L 2. Clearly ja is an isomorphism of
2 a 2[-a, o] L-b, 0 ab 2
L -aOl into L[ -, In other words, the relative topology of L [-a,0
induced from L as a subspace, is identical to the original
2[-b,0] 2
topology of L[~ -, Thus Lr -a0 can be identified with a (closed)
'[ -a, 0" L[-a,O]
subspace of L-b,] whenever a < b.
2 a
Let 01 be the union of all L_2 where n runs over all positive
1 L[-n,0 2
integers. Clearly 0 consists of all L -functions with support









bounded on the left, i.e., m belongs to 1 iff there exists n such
that m belongs to L
o,[-n,O]"
Using the inclusions jab ) as given above, we can induce the
topology of the (strict) inductive limit of the sequence (L[ n,]}; see
the Appendix. We denote the space 0 with the inductive limit topology,
as = lim Ln,

(1.2) REMARK. It is, in principle, possible that we may induce a
2
different topology if we take a different sequence (L -a,0]) where
{a n is a sequence such that an -~ as n -m. But one can easily
verify that the inductive limit topology thus induced is independent of
the choice of (an).

(1.3) DEFINITION. The input space (with m input channels) is the
space := (1 )m = (lim L 2 ) When it is necessary to refer to the
-> [-n,0]
number of input channels explicitly, we write fm.

We also define the shift (operator) at: -92 for each t > 0 by

(1.4) %o(r)(T) := )(T + t) if T < t,

0 if t < T < 0.

We need to prove the following

(1.5) PROPOSITION. The family of operators (ot)t>, is a strongly
continuous (or C -) semigroup, i.e.,

(a) T =I, t a= ot+s

(b) each ot is a continuous linear map;

(c) lim Yt = a oe 0 for every t > 0 and co in , stronga
t o- -
continuity).

PROOF. Without any loss of generality we assume m (the number
of input channels) = 1. The property (a) follows easily via direct
calculation.









We prove that at is continuous. By Proposition (A.2) it suffices
2
to prove that each restriction of t on L ] is continuous for every
['t Lpa,01 2
a. For sufficiently large b, we have 0t(L~ aO]) C L[-bO]. Then by
Proposition (A.4) we need only to prove that at is continuous as a map
2 2 2
frcn LaO] to L b For every o L we havethe
[-ao] [-bO o,[-a,Ol
qualities:



-

= I m(T + t)12da (definition of 0t),
-b


f= m(5 + t) I2d,
-a-t

0

-a




Hence at is continuous.

Let us prove the strong continuity (c). We may assume that
2
It t < 1. Let l belong to L 2 a, For sufficiently large b,
S2 o,[-a,0]
(a o) belongs to L for all t such that it t < 1.
t to O,[-b,O2 L
It suffices to show that (o- o )w -> 0 in L as t t
t t1 o,-b,0]
since the relative topology induced from 1 on L2[-b,0] is precisely
the L2-topology of L2-b,0]. Now the convergence (ot t )m ->0 is
a well-known fact of measure theory; see, for instance, HEWITT and
STROMBERG [1975, IV.15.24]. 0

2. Output Space.

2 2
Let L n]2 be the space of L -functions on the interval [0, n1
l[0,n
with the norm:

(2.1) (l[0,nl= { (t)2dt)1/2.








2 2
If a > b, there exists the natural projection Tab: L[oa] L[O,b]
-ab [,a] [0,b]
defined by W7ab () := [0pjb]. This map is obviously continuous and
linear. Clearly we may identify the space L[On] with the space of
[O,nI w
all functions defined on [0, m) which vanish outside of [0, n] and
2
belong to L[n] when restricted to [0, nl. In the sequel we shall
[O, n 2
also denote this space by L[ ,n]; if a precise distinction is necessary,
we shall denote it by L o
o[O,n]
Now let L2 [0, ) := [(: [0, m) -k: P is locally L, i.e., on
every compact interval [0, a], llp [0,a] is finite). The space
L 2o[0, m) is equipped with the countable family of seminorms (11- [n] r n
loc [O,n] n=l
defined as in (2.1). This family of seminorms defines a locally con-
vex Hausdorff topology on L2[0, ").

(2.2) REMARK, In fact [o,a] is a seminorm on Lloc[O, ) for every
positive real number a. But the topology given by the family

(Hllll[,a]: a > 0, a E R} is easily seen to be the same as the topology
defined by the family {]-II[,n]

As is proven in the Appendix (Proposition (A.7)), the space Loc[0, )
is the projective limit of the sequence of spaces (L 0,n] ). We write
P := L 2[0, m) = lim L0 protectivee limit).
loc ) +_ ,n]

(2.5) DEFINITION. The output space (with p output channels) is the
space P := (rl)p = (L2 c[0, O))p. When it is necessary to refer to the
number of output channels explicitly, we write rp.

(2.4) PROPOSITION. The space P is complete.

PROOF. We assume p = 1 without loss of generality. Clearly
2
each L[O n] is complete because it is a Hilbert space. Since a projective
limit of complete spaces is complete (SCHAEFER [1971, 1.5.5]), r must
be complete. 0

(2.5) PROPOSITION. The space P is a Fr6chet space, i.e., metrizable
and complete.

PROOF. We have only to show metrizability. We again assume P = 1








without loss of generality. By KOTHE [1969, 18.2, (2)], a locally
convex space is metrizable iff its topology is generated by a countable
family of seminorms. But this is indeed the case for P O

We define the shift (operator) 3t: P -r P for each t > 0 by

(2.6) 3t(7)(T) := 7(T + t).


(2.7) PROPOSITION. The family of operators ({dtt>O is a strongly
continuous (or Co-) semigroup, i.e.,


(a) o ts t+s

(b) each ot is a continuous linear map;

(c) tlim 7 = 7t for every t > 0 and r in P.
t() to to -o -

PROOF. (a) This is obvious via direct calculation.

(b) If a + t < b, we have the following estimate:

a
IatI20o,al I( + t)12d,
a+t
St17(,)12dr,
b
S <1(,1) 12d,


[O,bl'

Thus ot is continuous.

(c) We assume It to < 1 without loss of generality.
n
(2.8) 11(3t to)7,112 jI( + t) 2(- + )2


It is well known that the right side converges to zero as t - t
(IEWITT and STROMBERG [1975, IV.15.24]). n








3. Input/Output Maps.

We shall give the definition of linear input/output maps and prove
that to every linear input/output map there is associated a unique
matrix Radon measure on (0, -), which we shall call the weighting
pattern of a linear input/output map. We need some technical preliminaries
as follows.

Let Co(- 0 0) be the space of continuous functions on (- 0)
which vanish outside some compact subset of (- m, 0). Introduce the usual
inductive limit topology lim Co[- n, 1/n] to C (- 0, 0); see TREVES
[1967, 21]. Here C [- n, 1/n] is the space of continuous functions
on (- m, 0) which vanish outside of [- n, 1/n]; its topology is
defined by the supremum norm:

(3.1) [Ip11f_ := sup (fc(t)l: n< t < 1/n).

Clearly there is the natural continuous inclusion:


jl: (Co(- o))m -n;

it is easy to verify that jl has a dense image.

Now let C[O, 0) be the space of continuous functions on [0, m).
Its topology is given by a countable family of seminorms:

(3.2) lIIjJlJ := sup (jcp(t)j: 0< t < n}, n = 1, 2, ...

There is the natural continuous inclusion:


j2: (C[O, )P rP;

it is also easy to verify that j2 has a dense image.

We are now ready to give

(3-3) DEFINITION. A constant (shift-invariant) linear input/output map is a
continuous linear map f: 0 ->P such that








(i) the diagram



? 1
n f pr

commutes for every t > 0;

(ii) f((Co(- c, 0))m) c (C[O, m))P;

(iii) f is also continuous as a map from (Co(- 0 0))m to
(C[O, m))P.

(3.4) REMARK. The spaces (Co(- 0 0))m and (C[0, m))P are shift-
invariant, i.e., ot((Co(- m, 0))) C (Co(- O))m and
at((C[O, m))p) C (C[O, -))p for all t > 0.

(3.5) EXAMPLE. Let = (Mij)ij (i = ..., p, J = 1, ..., m) be a
matrix whose ij-entry is a Radon measure (we shall abbreviate this as
a matrix Radon measure) on (O, c). Consider a linear map f: 2 -oP
given by
m
(3.6) f() li(t) := j~1 w.(t T)dij(),
m o
= jL mCJ(T)daij(t T),

where f(OI)li and c. denote i-th and j-th entry of f(w) and u,
1 i3
respectively. We also write (3.6) as
0
(3.7) f(') = J '(t T)d1'(-) = J ()()d(t ),
t -0

for simplicity of notation.

(3.8) PROPOSITION. The linear map f given by (3.6) is a constant linear
input/output map.








PROOF. Clearly it suffices to prove the statement for the case
m p = 1. Observe that we need to integrate only on a bounded interval
for each t in (3.6) since each u in D has bounded support.

We note from L. SCHWARTZ [1966, 6.11 that f(u)(t) exists for almost
every t > 0 and f(w) belongs to L2c[O, r) (= F) for each w in D.
Furthermore, if (o belongs to L2 ,, then we have the estimate:
o,[-a,O]'

(3.9) llf()l[o,b] [0,a+b] [-a,O]
a
where IIlcn0,]r denotes I IJdl; for a proof, see DIEUDONNE [1970, 14.9.2].
[LUci0 2 1 p
Hence f is a continuous (and obviously linear) map from L -a, to
[-a,O]
for every a. By Proposition (A.2) it follows that f is a continuous
linear map from 21 to r1. It is easy to verify, via direct calculation,
that 5tf = foyt for all t > 0.

We again note from L. SCHWARTZ [1966, 6.1] (see also DIEUDONNE
[1970, 14.9.21) that f(w) is a continuous function of t whenever w
belongs to Co(- -, 0). Thus f(Co(- 0 0)) CC[O, m).

Let m be an element of Co[- a, 1/a] (a > 0). We have the
following estimate:

1/a
(3.10) sup If(w)(t)j = (-)d(t )
o
-1/a
< sup lV(()1Hdv(t )I,
o
a+b
< o sup l()) I]dP(-i) for all b > 0.
-a

Therefore f is also continuous as a map from C [- a, 1/a] to
C[O, -) for each a > 0. In view of the inductive limit topology of
Co(- c, 0), this proves the continuity of f as a map from C (- c, 0)
to C[O, m) (see Proposition (A.2)). O

When a linear input/output map is given by a matrix Radon measure [
as in (3.6) (or (3.7)), we call pi the weighting pattern of a constant
linear input/output map f.










(3.11) REMARK. If a measure 4 is given by d' = A(r)dv (dt = the
Lebesgue measure) by some function A, then (3.7) coincides with the usual
o
convolution / A(t T)o(T)dr; see L. SCHWARTZ [1966, 6.1]; we shall also
call A(T) the weighting pattern of f with slight abuse of language.

We ask the converse question: Can every constant linear input/output
map f be written as (3.7) for some matrix Radon measure p? The next
theorem states that this is indeed the case.

(3.12) THEOREM. A linear map f: 0 ->P is a constant linear input/output
map iff there exists p, a unique matrix Radon measure on (0, m), such
that (3.7) ((3.6)) holds.

PROOF. We already proved the sufficiency part in Proposition (3.8).
We prove the necessity for the case m = p = 1; the general case follows
from this by considering each factor f(j)li..

Let C (0, m) be the space of continuous functions on (0, m)
which vanish outside of some compact subset of (0, m). We introduce the
same topology to Co(0, m) as is done for C (- m, 0). Consider the
linear map -: cp given by $(t) := p(- t). Clearly gives an
isomorphism between C (- m, 0) and C (O, m), and, furthermore,
q = C for every p.

Consider the following linear functional p on Co(O, ):

(5.13) Id(c) := f(C)(O).

Since f is a continuous linear map from Co(- m, 0) to C[O, m) by
hypothesis, P is a continuous linear form on C (0, m), i.e., I- is
a Radon measure on (0, c). Write cp(T)dl-(T) instead of p(p). Then
it follows that









f(u)(t) = (otf(w))(O) = (f(oat))(O) (f commutes with shifts),










0 t
= I (cto4(-. )di(-),






= Cw(t T)dlJ(T) for all C in C (- -, 0).
t

Hence I satisfies (3.6) for w in C (- 0 0). But since C (- 0, 0)
is dense in 91, this equality must hold for every c in 1 .

If T is another Radon measure on (0, w) that satisfies (3.6),
then we must have


ScpQ(T)dp(T) = J $(- T)dp(T) = f(P)(0) (by (3.6)),
0 0

o o
= 5 '(- r)dT;(r) = cp(r)dTl(T),
0 0

for all cp E Co(0, m). Thus [ = I i.e., P is unique. O

There are certain occasions that we want to have the "delta function"
B in the input space. Unfortunately, our input space n does not
contain o Now we show one way of introducing 50 into our framework.

Let f be a linear input/output map with the weighting pattern P.
We assume that du = A(T)dT (i.e., dIrij =Aij(T)dT) and A is of class
Cr (r > 0). What we try to show is that if A is regular enough, say
C then there exists a (unique) continuous extension f of f to an
"ideal input space" that contains 6 To be more precise, let
r ,0 be the space of Cr-functions on (- 0, 0] with the topology
.-{ 0,0J
of uniform convergence on each compact interval for each derivative of
rI
order less than or equal to r. Now let E(-m,0] be the dual space of
Snamel the et of all continuous line forms n ith
Er namely the set of all continuous linear forms on E(-O]' with
-(- ,0]'(= 0










the topology of compact convergence, that is, the topology of uniform
r r'
convergence on each compact set of E(-,0]. It is well known that E(-,0]
is the space of distributions of order < r with compact support contained
in (- 0, 0]. Thus E5 belongs to Er(,0]. It is also easy to see that
ri
Q is contained in E -,01 as a dense linear subspace. Further, each
E( 0] is equipped with shifts (ot>0,)t which are natural extensions of
'1 wr m (r) if
{ot)t>, in We denote the space T( ,O]) by (or ', if
we must specify the order). When r = m, the space E-,0] (= E(- ,O)
is the input space considered in KALMAN and HAUTUS [1972]. We now claim

(5.14) PROPOSITION. Let f be a constant linear input/output map given
by the weighting pattern di- = A(T)d' for some Cr-function A (r > 0).
Then there exists a continuous extension f: r) ->r of f such that
f f = fo for all t > O.
t t

SKETCH OF PROOF. We may assume m = p = 1 without loss of
generality. Now in view of Remark (3.11), we must define f(m) as the
convolution of in E(-,O] with a fixed element A, i.e.,
f(w)(t) := cn(A(t (*)), where the right side denotes the value of the
distribution & evaluated at the Cr-function T i->A(t T). This linear
map is a continuous linear map from (-,] to C[O, "); see, for
instance, L. SCHWARTZ [1966, 6.4, Theoreme 12 and the succeeding remark];
it is easy to modify the proof. Since the topology of C[O, m) is finer
than that of r, this map f gives a continuous correspondence from
E -, to P. It is also known that f commutes with shifts; see
L. SCHWARTZ [1966, 6.3.9]. O

(3.15) PROPOSITION. Let f be a constant linear input/output map with
the weighting pattern di = A(r)dr, where A = (Aij) is of Cr-class.
Then Aij = (5 oj) li, where 5o denote the element having the only
o0j ---- o03
nonzero term 5 in the j-th position.
o

PROOF. From the proof of Proposition (3.14), we have
(J) i(t) = (Aij* |j)(t) for all CIj in E ,]. Then
(5 olj)|i(t) = (Aijo*5 j)(t) =Aij(t) because 50 is the unit element
with respect to convolution. O








4. The Space of Laurent Functions.

Let A :=- 1 prl X X r1 with the direct sum (or product) topology.
We call A the space of Laurent functions, and its element A a Laurent
function. Since every element A of A can be uniquely represented as
S= ( + y, mc D 7 C r A can be regarded as a locally L2-function
with its support bounded on the left. Note that cu(O) and y(0) may
well be different, but this does not cause any difficulty since the point
0 is of measure 0.

Let 7Tn: Am -> Y r: Ap --r be the projections, and let j.: n ->A
and j : P ->A be the inclusion maps. Consider the following families of
linear maps:


(4.1) ( )(r) : (T t);


(4.2) (o-A)(T) := A(T t), t > 0, h C Aq, (q = integer).


(4.5) PROPOSITION. The families (At1t>O and ot rt>O are strongly
continuous semigroups in A. Furthermore, Tratj = (T and
irpt j = at for all t > 0.

SKETCH OF PROOF. It is easy to see that eo = I and ot+s
t's, and ro t+s -t
ts 0 i t+s t
To prove the continuity of at (and t) it suffices to prove
that A (a~) is continuous on (L2 -n-] 9 P ) for all n. But this
can be done in exactly the same way as in Propositions (1.5) and (2.7).

The strong continuity, lim o() = o()) (or tlimo (1)
r t-> to t 0 t-> to
tro(\)), follows directly from the fact that this is true on each
bounded interval [- a, b]; see HEWITT and STROMBERG [1975, IV.13.24].
A. A.
The qualities vrCtJn = rt and 7rotjD = Ot follow from
direct calculation. O

5. Extended Linear Input/output Maps.

In Section 3 we defined linear input/output maps. It is assumed
that every input "terminates" at 0, i.e., supp m C (- m, 01 for








every ( in P and every output "starts" at 0, i.e., supp f(m) C [O, 0 ).
In other words, we observe outputs only after the application of inputs.
This idea has been highly successful in realization theory in the sense
that "causality" is already built into the framework (KALMAN, FALB, and
ARBIB [1969, Chapter 101). Of course in the actual dynamic mode of systems,
usually outputs must be observed while inputs are applied. Input/output
behavior in this sense is usually described by the Laurent series in the
discrete-time case; see HAUTUS and HEYMANN [1978]. We show a counterpart
in the continuous-time case in this section.

Let C[a, m) be the space of continuous functions on (- c, t) which
vanish outside of [a, m). This space is a Fr6chet space with countable
seminorms:

sup (p(t)l: a < t < n), n = 1, 2, ...

Let C (- m, m) := Ua C[a, m), and introduce the inductive limit
topology. Since each inclusion: C[a, m) ->A is clearly continuous, the
induced inclusion: C (- c, c) --A is continuous; it is easy to see that
this inclusion has a dense image.

(5.1) DEFINITION. A continuous linear map f: Am -A is a (strictly
causal) extended constant linear input/output map iff
(i) o = a- =Tor, for all t> 0;

(ii) if XlI(-t,t) =2 (_.-t), then Y(hl)I(_M t] =(2) (.,t]
(strict causality) for continuous 7i and h2;

(iii) f((C+(- -, w))m) C (C+(- 2, ,));

(iv) f is continuous as a map from (C (- m, m))m to
(C+(- -, c))P.

REMARK. Property (ii) clearly implies that if

l (-,t) = 2( t) then f(Al) ( t) = f(2)1 (,t) for all A
in A .

Our objective here is to show that every strictly causal extended constant
linear input/output map is necessarily derived from a constant input/output map









and vice versa. Indeed, we now prove

(5.2) THEOREM. For any constant linear input/output map f there is a
unique strictly causal extended constant linear input/output map I
such that the diagram







0 f
a -f r


commutes. Conversely, if f is a strictly causal extended constant linear
input/output map, then f := 7rrfj is a constant linear input/output map.

PROOF. Assume m = p = 1 for simplicity of notation. Let f
be a linear input/output map with the weighting pattern p. Define a
linear map T: A ->A by

t o
(5.5) T(A)(t) := I (-r)da(t T) = I A(t T)dn( ).
-oo 0

Note that if A(T) = 0 for T < a, then f(A)(t) = 0 for t < a. This
implies that f is strictly causal by linearity of f. Note also that the
integral (5.5) is evaluated only on a bounded interval, hence is well-defined
for almost all t by L. SCHWARTZ [1966, 6.1]. Further, if t runs over a
compact set, then only the values of A on a compact set contribute to
the integral (5.3). Hence T(A) is locally L2 by DIEDONNE [1970, 14.9.2];
f(A) is a continuous function if A is continuous, again by the same
reference. Thus f(C+(- m)) C+(- t, m).

We denote by L1oc[a, -) the space of all locally L -functions
on (- m, m) which vanish outside of [a, m). The space L oc[a, m)
is topologized by the countable family of seminorms:


Ikli an]:= { j (t) 2dt)1/2, n = 1, 2, ...
[a,n] a

Clearly A = U L2 [a,).
-a






Now let \A belong to L 2[a, -). Then T(\A)(t) = 0 for t < a.
Furthermore,
( ) l(h)fi = rl Ie(b)(t)12dt)1/2< I (5.4) |(A) ,b] = f2At/2 < ([0,b-a] [a,b]
b-a
(IHI [,b := f Idll) as can be shown in the same way as (3.9). Thus
[O,b-aj o 2 2
f is continuous as a map from Llo [a, ) to Lloc[a, -). Since A
2 (Proposition (A 10)),
can be regarded as the inductive limit of Loc[a, ) (Proposition (A.10)),
this fact establishes the continuity of f: A -A by Proposition (A.2).
Now let \ belong to C[a, -). Then

(5-5) a:b af(7b(t) [ UOb-al


This can be proved in exactly the same way as in (3.10). Hence f is
continuous as a map from C[a, m) to C[a, m). In view of the inductive
limit topology of C+(- m, m), this implies that f is continuous as a
map from C (- m, o) to C (- m, m) (see Proposition (A.2)). Clearly
commutes with shifts. Therefore f is a strictly causal extended
linear input/output map. The property 7rfji = f is obvious via direct
calculation.
We must prove uniqueness. Let fl and f2 be two strictly causal
extended linear input/output maps such that f = 7Tyflj = 'TIr2j0. Take
any continuous A in A. Then

(5.6) fTl()(t)= (oo (A))(0),

= 71(o)(0),

T (j aB)(0) (strict causality),

= fl(jiTh t7A)(0),

= f(v-ot )(o) (711,lj = f),

= )(,n ^0)'











= f2(A)(t) for all t > 0.


Similarly, fl(h)(t) =2()(t) for t < 0. Hence fl = f2 on C (- m, m).
Since C+(- m, m) is dense in A, fl must be equal to f2.

Conversely, let f be a strictly causal extended linear input/
output map. Clearly f := 7Tfji is a continuous linear map from 0 to
P, and it commutes with shifts because fat = T9fj = 7T -fj =
9 t = t 1 yio
rtjrppfj" n = trTfjn = Itf. It is also clear that f maps Co(- 0)
into C[O, m) and is continuous with respect to the corresponding
topologies. O

(5.7) REMARK. The condition that linear input/output maps send continuous
functions to continuous functions requires, roughly speaking, that linear
input/output maps do not "differentiate." In other words, if is the
weighting pattern of a linear input/output map f, then P must not
contain terms such as 5', "... etc.
a' aI









CHAPTER III. REALIZATION THEORY


We shall start by defining linear systems, objects which are of our
loving concern. In Section 6, we also study some basic notions such as
quasi-reachability, observability, morphisms between systems, etc. In
Section 7, we define a realization and a factorization of a linear input/
output map. The notion of factorization of a linear input/output map is
a convenient tool to handle the problem of existence and uniqueness of
canonical realizations.

It is well known (BARAS, BROCKETT, and FUHRMANN [1974]) that a weak
notion of canonicity, namely quasi-reachability plus observability, does
not, in general, lead us to the uniqueness of canonical realizations. A
new notion of observability, which we call topological observability, is
introduced in Section 8. We shall then prove the desired existence and
uniqueness theorem in Section 9 as a direct consequence of topological
observability; a counterexample by BARAS, BROCKETT, and FUHRMANN [1974]
is discussed in order to illustrate the theorem.

In Section 10, we prove that a realization indeed produces outputs
even while L2-inputs are applied. This is not necessarily guaranteed
by our definition of linear systems. In Section 11, we turn our attention
to differential equation descriptions of linear systems, and prove that a
canonical realization is described by a functional differential equation
if the weighting pattern of the input/output map is sufficiently smooth.
The notions of compatible and smooth systems are introduced.


6. Systems.

(6.1) DEFINITION. A linear (constant, continuous-time) system
(with m-input, p-output channels) is a triple E = (X, c, I) which
satisfies the following conditions:

(a) X is a complete locally convex Hausdorff space;

(b) for each fixed t > 0

X X (L2 t) m -X: (x, u) ,-c(t, x, u)
[Ot)









is a continuous linear map (when s < t, we denote cp(s, x, ui[,s))
by w,(s, x, u));

(c) for every t, s > O, c satisfies

p(t + s, x, u) = c(t, c(s, x, ul[os)), a[ot))

for all x in X, u in L t) (for see Section 4), and

2
cp(0, x, u) = x for all x in X and u in L[ot
-- -- [O,t)'

(d) tlj p((t, x, 0) = c(to, x, 0) for all x in X;
(e) H is a densely defined (not necessarily continuous) linear
operator: D(H) -k kP;
(f) there exists a dense subspace Do(H) C D(H) such that
C(t, x, 0) belongs to D (H) for all t > 0 and x in D (H);

(g) for every t > 0, w(t, 0, u) belongs to Do(H) for every
continuous function u such that u(0) = u(t) = 0;
(h) under the same hypothesis on u as in (g), o(s,, u)
belongs to D(H) for every 0 < s < t (but not necessarily to Do(H));

(i) the correspondence: Co[O, t] ->kp given by u Ho(t, 0, u)
is continuous with respect to the topology of uniform convergence on
C [O, t] for every t > 0;

(j) there exists a continuous linear map h : D(H) -> such
that (i) h (x) is continuous on [0, E) for some e > 0, (ii)
h (x)(0) = Hx, (iii) h (x) is continuous on [0, -) if x belongs to
Do(H), and (iv) ho(x)(t) = Hcp(t, x, 0) if x belongs to Do(H).
We call X the state space, cp the state-transition map, H
the readout (output) map of Z. We also call cp(t, x, u) the state
resulting at time t from the initial state x under the action of
inut u.

(6.2) REMARK. It is easy to see that the first and the second conditions










of (j) imply the right-continuity of ho(x) on [0, m) for all x in
Do(H). But the left-continuity is not necessarily guaranteed.

Even though Definition (6.1) may appear overly involved, the only
difference of this definition from the classical one given in KALMAN,
FALB, and ARBIB [1969, 1.1] is the fact that H maybe neither every-
where defined nor continuous. In fact, if H were continuous it would
have a continuous extension to the whole space X, and Conditions (f)
to (j) would become redundant. But as pointed out in Chapter I,
requiring that H be continuous would result in excluding many interest-
ing examples (see also Section 14).

At any rate, we must ask for some type of continuity on initial state/
output correspondence; otherwise no study could be made on topological
aspects of systems. Thus we require that h : D(H) ->r exist and be
continuous. Conditions (g) to (i) require that H behave "nicely" with
respect to continuous inputs.

(6.5) REMARK. Causality, i.e., p(t, x, ul) = cp(t, x, u2) whenever
ull[Ot) = u21 [Ot), is built into the definition. Indeed p(t, x, u)
c(t, x, ul [o~t)) = p(t, x, u2[O,t)) = cp(t, x, u2) by definition (see
(b) of Definition (6.1)).

We now investigate some direct consequences of Definition (6.1).

(6.4) PROPOSITION. Let Z = (X, ao, H) be a linear system. Then there
exists a continuous linear map h : X -> such that h (x) = ho(x) =
Hq(-, x, 0) for all x E D(H).

PROOF. We already know that h : D(H) F exists and is continu-
ous by definition. Since D(H) is dense in X and P is complete (
Proposition (2.4)), h must have a unique continuous extension
hE: X ->r such that h ID(H) = h.

(6.5) PROPOSITION. Let = (X, cp, H) be a linear system. Then there
exist a strongly continuous semigroup (O(t))t>O and a continuous linear
nap g : 0 -+X such that








(6.6) p(t, x, u) = D(t)x + g (vnr u)
2
for all t > 0, x E X, u C Lo ,, where u is regarded as an element
-- n- [ot)'
of A.

PROOF. For each fixed t > 0, p(t, -, 0): X -4X is a continuous
linear map by Condition (b) of Definition (6.1). Write O(t)x for
cp(t, x, 0). Then

(6.7) (t + s)x = p(t + s, x, 0),
= cp(t, p(s, x, 0), 0),

= ((t, a(s)x, o),
= 0(t)O(s)x,

and

(6.8) 0(O)x = cp(O, x, 0) = x for all x in X,

by Condition (c) of Definition (6.1). Furthermore,

(6.9) tlim (t)x = lim cp(t x, ) = p(to, x, 0) = 0(to)x
t- to t- to

for all x in X by Condition (d) of Definition (6.1). Thus (O(t)]t
is a strongly continuous semigroup.

Now let Cw be an element of P with its support contained in
[- a, 0]. Define g by

(6.10) g 1() := (C, 0, arQt ),

where j : 0 ->Am is the inclusion and a is a right shift operator in
Am (see Section 4). We show g (c) is well-defined. Indeed, if P > a,
we obtain

cp(P~, O, ja) = cp(a, (p a, or, ll[o,pa)), -P_(a[roa))

= p(a, p(P a, 0, 0), OinIroI a)








= p(a, 0, I[o, '

= gE(c),

because p(s, 0, 0) = 0 by linearity of cp. Hence g(m) is independent
of the choice of a as long as supp c C [- a, 0]. Thus g is well-defined
on the whole input space Q. Furthermore, since q(P(, 0, -) is continuous
S 2 m
for each fixed a > O, g must be continuous on each (L [ ,) Since
-2 E [-a, ]
n is the inductive limit of {(L _,a O]) M g must be continuous on 2
by Proposition (A.2).

Now since ((t, *) is linear, one obtains

p(t, x, u) = q(t, x, 0) + c(t, 0, u),
= 0(t)x + g ('Vtu),


because a Jr u = u if u belongs to (L ,t) M. O
t [O-t)

10(t))t>O is called the semigroup associated with E.

(6.11) DEFINITION. Let E = (X, H) be a linear system. The reach-
ability map (of E) is g given by (6.10), and the observability map
(of Y) is h given in Proposition (6.4). The reachable subspace of
E is X := g (0). The system E is quasi-reachable iff XR is dense
in X, (exactly) reachable iff XR = X, and observable iff h is one-
to-one; E is weakly canonical iff it is both quasi-reachable and observable.

(6.12) PROPOSITION. For every t > 0, g = (t)g and th =h (t).

PROOF. Let C be an element of 0 with its support contained in
[- a, 0]. By definition (see (1.4)) supp a( C [-a t, -t]. Then we
obtain


r r.
co = cp( + t, O, %-O jb)+t (tj


= cp(a + t, a, 2ro) ( +tc = a
=(a+t t =0t








= cp(t, cp(a, 0, crj ), 0),

= 'p(t, g (w), 0) (definition of g),
= ((t)gE(m) (definition of T(t)).

Take any element x in Do(H). Then we obtain


(hx))(s)= hZ(x)(s + t),

= HD(s + t)x (Proposition (6.4), definition of N(t)),

= HK(s)('(t)x),

= h (0(t)x)(s).

Hence th2 = h (t) on Do(H) for all t > 0. Since D (H) is dense
in X, the conclusion follows from the continuity of h O

The following Proposition gives a dual characterization of quasi-
reachability and observability.

(6.15) PROPOSITION. Suppose that E = (X, cp, H) is a linear system with
a reflexive state space X. Consider the following statements:

(a) Y is quasi-reachable;
(b) the adjoint (g )': X' -> is one-to-one;

(c) Z is observable;
(d) the adjoint (h )': F' ->X' has a dense image.

Then we have the equivalence (a)<=>(b) and (c)<>(d).

PROOF. [(a)<> (b)3 We quote from TREVES [1967, Corollary 5 to
Theorem 18.1] that g has a dense image iff (g )' is one-to-one. Thus
(a) is equivalent to (b).

[(c)" (d)] Since X and r are reflexive (r is easily seen
to be reflexive as a projective limit of reflexive spaces ((Ln] m
see SCIAEFER [1971, IV.4.4 and IV.4.51), h can be identified with
(h )": X" ->r". Then, as before, (h )' has a dense image iff (h )" is
one-to-one. Since (h )" is identified with h the conclusion
follows. O









The following lemma will be useful later.

(6.14) IEMMA. Let o be a dense subspace of Q. A linear system
x =(X, cp, H) is quasi-reachable iff g (no) is dense in X.

PROOF. Trivial. O

Let us now give the definition of a morphism between two systems. The
following definition is a modification to the present context of the
standard one.

(6.15) DEFINITION. Let l1 = (Xl' p1 H1) and Z2 = (X2, 2 2, H2) be
linear systems. A morphism from Z1 to E2 is a continuous linear map
T: X ->X2 such that

(i) cp2(t, Txl, u) = Tpl(t, xl, u) for all t > O, x1 E Xl,
and u EL t)
[Ot)'
(ii) there exists a dense subspace M1 of Do(H ) such that
TM1 C D(H2) and H2T =H on M1.

We say that El is isomorphic to E2 iff T is a homeomorphism
(isomorphism).

(6.16) REMARK. It is easy to see that the identity and the composition
of two morphisms are morphisms. Note also that if T is a homeomorphism,
-1
then the inverse T is automatically a morphism. For, if P2(t, Txl, u) =
Tql(t, xl, u), then Cl(t, T-lx2, u) = T- 2(t, x2, u) where x2 = Txl.
And if M1 is a dense subspace of D (H ) such that TM1 CDo(H2), then
M2 := TM1 is a dense subspace of Do(H2) because T is a homeomorphism.
Moreover, T- M2 = M C Do(H ), and if x2 = Tx belongs to M2, then
-1 -12 1 0 1 2 1 2
H1T x2 = HIT Txl = H x = H2Tx = H2x2 because xI belongs to M1.
-1
Thus T is a morphism.

We shall now investigate how the notion of morphisms is described in
terms of reachability and observability maps. The first condition of
Definition (6.15) clearly implies that Tgyl(m) = Tcp (a, 0, U jiU) =
c2(a, 0, c j%) = g 2(w) for every m with supp m C [- a, 0]. If the
2,Q.i








system El is quasi-reachable, we have the converse.


(6.17) PROPOSITION. Let Z = (X1, 91, H1) and P2 = (X2, p2, H2) be
linear systems. Suppose that YI is quasi-reachable. If T: X -)X is
a continuous linear map such that Tg = g2, then T satisfies
2(t, Txl, u) = Tl(t, xl, u) for all t > 0, xl E X, and u L2,t)

PROOF. Let xl be an element of X,R = g l(Q), i.e., x =
gE-(w) for some 0 in 0. Then

O2(t, Txl, u) = 2(t)Tgzl(J ) + g 2(w0ou) ((6.6)),

= 2(t)g2(o) + g (2(sr u) (by hypothesis),

= g2( ) + g 2(ttu) (Proposition (6.12)),

= g 2(ot1 + o'w) (g 2 is linear),

= Tg l(taC + t u) (by hypothesis),

= T(g 1l(t) + g 1(7rT u)],

= T( (t)g1(m) + g l(Tc, u) (Proposition (6.12)),

= Tc1(t, Xl, u) ((6.6)).

Thus 92(t, Txl, u) = Tcp(t, xl, u) for every xl in X1,R. Since
X1,R is dense in X1, the conclusion follows by the continuity of
p92(t, T(-), u) and Tpl(t, *, u). O

Now if the second condition of Definition (6.15) is satisfied, then
clearly h2T = h1l follows. Conversely, we prove

(6.18) PROPOSITION. Let 1 = (Xl, '1, H1) and 12 = (X2, p2, H2) be
linear systems. Suppose that 7Z is quasi-reachable. If T: X1 ->X2
is a continuous linear map such that Tg = g2 and h 2T = h 1, then
there exists a dense subspace M 1C Do(H1) such that T41 C Do(H2) and
H2T = H on M1.
2 1 --








PROOF. Take M1 : g1((Co(- O))m). Since (Co(- c, 0))m
is dense in n, MI must be dense in X1 by Lemma (6.14) because EZ
is quasi-reachable. By Definition (6.1) (g), M1 must be contained in
Do(H ). Since TgEl(o) = g 2()) for every w in 0, clearly
TM1 C g2((Co(- -, O))m) follows. This yields TM C Do(H2) because
g2((Co(- ", 0))m) C D (H2) again by Definition (6.1) (g).
For every x in Mi, H2Tx = Hlx follows immediately by evalu-
ating h2Tx = h lx at t = 0. O

Combining Proposition (6.17) with Proposition (6.18), we obtain the
following

(6.19) THEOREM. Let El = (X1, P1, HI) and F2 = (X2' cP2 H2) be
linear systems. Suppose that E1 is quasi-reachable. A continuous
linear map T: X1 ->X2 is a morphism iff Tgl = gF and h2T = h 1.

PROOF. Obvious from Propositions (6.17) and (6.18) and the
remarks preceding them. O

In order to illustrate Definition (6.1) we here give two examples
of systems.

(6.20) EXAMPLE (BARAS, BROCKETT, and FUHRMANN [1974]). Let X := 2
and let (gn) and (hn be 2 -sequences. Also let (An be a bounded
sequence. Define cp and H by


qc(t, (x, u) In := e n + eAn(t- u()dT;


H((xn) := 1 h x for all {x ) in
n n=l n n

where yn denotes the n-th coordinate of y. It is easy to see that
(e2, cp, H) satisfies the axiom of systems (note that H is continuous,
hence conditions (f) (j) are automatically fulfilled).

2
(6.21) EXAMPLE. Let X := Tx La1, We denote an element of X by
(x, z(*)) E Ux L- ,0]. Take








D(H) := ((x, z) C X: z is continuous on (- E, 01 for some e > 0);

Do(H) := (x, z) E X: z is continuous on [-1, 0] and z(- 1) = x).

The inclusion D (H) C D(H) is obvious. We prove that D (H) is dense
in X.
2
For each fixed x in R take M := (z L : z is
x [-1,01
continuous and z(- 1) = x). We prove Mx is dense in L-1,0. Indeed,
take an element w -1 L[-1 501. Then for every e > 0 there exists
5 > 0 such that f w(t)j dt < E. Let z be a continuous function
-1
which satisfies

-1+5
(i) I Iz(t)12dt < e,
0
-1

(ii) I w(t) z(t)I2dt < E and z(- 1 + 5) = 0.
-1+5
Such a function z clearly exists because C [- 1 + 6, 0] is known to
2 0
be dense in Lr[1 +0]. It is also clear that z satisfies
o I1+5,OJ 2
I w(t) z(t)I2dt < 3e, and hence Mx is dense in L[-1,0]. Since
D (H) =U (x, z): z E M and x C R), D (H) must be dense in X.

Define p and H as follows:

(i) If O
t z(0-t), for t-l (p(t, (x, z), u) := (x + I u(-)dT, t-l-e ).
o -x+ u(r)dT, for -l<0 o

(ii) If t>l

t t-l-6
cp(t, (x, z), u) := (x + f u(r)dT, x + u(e)d9);
o o

H(x, z) := z(0) ((x, z) belongs to D(H)).

Also consult the following figure:

x z


S-1,0]








It is now easy to check that the above defined (X, (, H) is
indeed a linear system. Note that D(H) is not 0-invariant, whereas
Do(H) is (0 = the semigroup associated with Z).


7. Realizations and Factorizations of Input/Output Maps.

We give the definitions of realizations and factorizations of input/
output maps. A factorization is not necessarily a system, hence not a
realization, in general. Our main objective in this section is to prove
that every quasi-reachable factorization is indeed a system, hence a
realization. This enables us to study the problem of existence and
uniqueness of canonical realizations in terms of factorizations.

(7.1) DEFINITION. Let f: S -> be a linear input/output map. A linear
system Z = (X, p, H) is a realization of f iff f = hg

Since the composition h g clearly gives the correspondence: "past
inputs" -> "future outputs" of the system Z, this definition is the
natural one. We have the following easy

(7.2) PROPOSITION. For every linear system E = (X, (, H), hgE is
an input/output map.

PROOF. Clearly h g : 0 -- is a continuous linear map.

Take any c) in (Co(- 0, O))m. By Condition (g) of Definition
(6.1), g (C) must belong to D (H). Then by Condition (i) of Definition
(6.1), the correspondence: co -Hg (G) is continuous in view of the
inductive limit topology of (Co(- -, O))m (see Proposition (A.2)). Thus
HgE must belong to the dual space of (C (- m, O))m. Hence it must be
represented by a matrix Radon measure P as Hg (c) = I w(T)dl(- T) =
I c(- T)di(T) (consider the definition of Radon measures; see also the
proof of Theorem (3.12)). Since Hg (C) = h g (c)(O) by Condition (j)
of Definition (6.1), the proof is complete. O

Proposition (7.2) claims that a linear system realizes some input/
output map. At the end of this section, we shall answer the converse
question, "Is every (linear) input/output map realized by a linear system?"








Definition (7.1) motivates the following

(7.3) DEFINITION. Let f: N -4P be an input/output map. A triple
(X, g, h) is a factorization of f iff

(a) X is a complete locally convex Hausdorff space equipped
with a strongly continuous semigroup ({(t),t>O;

(b) g: n ->X and h: X ->' are continuous linear maps such that
(t)g = gat and oth = hM(t) for all t> 0;

(c) f = hg.

In view of Proposition (6.5) and (6.12), a realization of an input/
output map is always a factorization. But the converse is false; see
Counterexample (7.9).

It is convenient to say that a factorization (X, g, h) is quasi-
reachable iff g(N) is dense in X, and observable iff h is one-to-one,
and weakly canonical iff it is both quasi-reachable and observable. Since
quasi-reachability and observability of systems coincide with their counter-
parts for factorizations (see Definition (6.11)), there is no possible
danger of confusion.

We now prove the following

(7.4) PROFOSITION. Every quasi-reachable factorization (X, g, h) of
a linear input/output map f is a linear system, i.e., realization. To
be precise, there exists a linear system S = (X, T, H) such that
SZ
g = g and h = h.
Z
PROOF. In view of (6.6) and the requirement g = g, we must
define p as

(7.5) p(t, x, u) := O(t)x + g(rptu),

where {Q(t)) is the semigroup associated with (X, g, h). Clearly
g = g follows from this definition. It is merely a routine to check that
9 satisfies Conditions (b) (d) of Definition (6.1).

Let Co(- m, 0] be the space of all continuous functions with







compact support in (- 0, 0] which need not vanish at 0. Define D(H)
and Do(H) by

(7.6) D(H) := g((Co(- ])m);
(7.7) Do(H) := g((Co(- -, 0))m).

Clearly D(H) contains Do(H); Do(H) is 0-invariant because (Co(- 0))m
is invariant under (at)t>0 (Remark (3.4)) and g commutes with shifts.
By quasi-reachability of (X, g, h), Do(H) (hence, a fortiori, D(H)
too) is dense in X because (Co(- m, 0))m is dense in 0. Thus Conditions
(f) and (g) of Definition (6.1) are satisfied.
Let u be a continuous function with its support contained in
[0, t]. Then for each s C [0, t], vr u belongs to (Co(- 0])m
Hence cp(s, 0, u) = g(Q~ u) belongs to D(H). Thus Condition (h) is
also satisfied.
Now define H: D(H) ->kp by

(7.8) Hx := h(x)(0).

Since x = g(m) for some c in (Co(- ), 0])m, h(x) = hg(c) = f(&) is
continuous in a neighborhood of 0 because f(Q) is given by the con-
volution of a matrix Radon measure 1 with c. Furthermore, if
x = g(co) = g(&2), then h(x) = f() = f(2), so (7.8) is well-defined.
If, further, x belongs to Do(H), then h(x) = f(m) (m e (Co(- m, 0))m
belongs to (C [o,)m by Definition (3.3); HO(t)x = h(O(t)x)(O) = Uth(x)(O) =
h(x)(t) for every x in Do(H). Hence Condition (j) of Definition (6.1)
is satisfied.
Finally, since the correspondence: (Co(- 0 0))m -4k given by
c f(m)(0) is continuous by Definition (3.5), Condition (i) is satisfied.
The equality h = h follows from the previous identity h(x)(t) = HO(t)x
on Do(H). O

The hypothesis that (X, g, h) is quasi-reachable is crucial in the
above proof. Indeed, if (X, g, h) is not quasi-reachable, we have the
following counterexample.








(7.9) COUNTEREXAMPLE. Let f be a scalar input/output map given by


f(aU) fI mU(r)e dr.


Clearly f is a linear input/output map by Proposition (3.8). Define


X := RB

o
g(c) := ( c(r)e-ds, 0);
tCO

h(xl, x2)(t) := etxl + t-1/32;


0(t)(xl, x2) := (et x, x2).


Clearly (R-, g, h) is a factorization of f; this is not quasi-reachable.
And for any state (0, x2) (x2 S 0), h(O, x2(t) = t-/x2 is not
2
continuous at 0. Hence the factorization (R g, h) cannot be a system.

We give some examples of realizations.

(7.10) EXAMPLE (BARAS, BROCKETT, and FURHMANN [1974]). Let (a n be an
1 2
1 -sequence given by an = h gn, where (gn), (hn are -sequences given
in Example (6.20). Let (An] be a bounded sequence as in Example (6.20).
Take


A(t) := a ent.
nl n

This series converges uniformly on every compact interval. Hence A(t)
is analytic. Now define

o
(7.11) f(m)(t) := A(t ')o(T)dr,


= n 7 en( (T)dT.
n=l -_ n


By Proposition (3.8), f is a linear input/output map.








Let the system Z be as given in Example (6.20). It is easy to
y ^E
see that the reachability map g and the observability map h are given
by


(7.12) gZ()) = J e-ngnm(T)dr;
n


(7.13) h({xn )(t) := hn xnent

Then we obtain


hg (-)(t) = Z h g()! ent
n=l an n

S h e-t f e-ng o)(r)dT,
n=l n n




0
n=l nn _

f a e -n(tg)r(,)d-,
n=l -m n

= f(m)(t).

Thus the system E is a realization of f.

(7.14) EXAMPLE. Define A(t) as follows:

A(t) := O if O< t < 1, A(t) := 1 if t > 1.

Let f be the input/output map having A as the weighting pattern, i.e.,


(7.15) f(m)(t) := f A(t- T)-(T)dr,

t-1
f= m(T)dT if 0 < t < 1,


J w(T))dT if t > 1.


Now let the system Z be the one given in Example (6.21). We
ca easily calculate g and
can easily calculate g and h as follows:







o -1-8
(7.16) g (c) = (f m(r)dT, I a(nT)d), 1 < e < 0;


(7.17) h (x, z)(t) = z(- t) if 0 < t 1,

(x if t > 1.

It follows that

t-1
h g (m)(t) = I c(-)dT if 0 < t < 1,

0
f m(T)dT if t > 1,

= f(c)(t).

Thus E is a realization of f.

Let us now prove the converse of Proposition (7.2): Given an input/
output map, there exists a realization; moreover, it can be taken to be
weakly canonical.

(7.18) PROPOSITION. For every linear input/output map f: 0 F, there
exists at least one weakly canonical realization.

PROOF. Define

(7.19) (i) X := im f (the closure is taken in r);

(ii) g := f;

(iii) h := j: im f ->r (the inclusion).

Then clearly (im f, f, j) is a weakly canonical factorization. Hence
by Proposition (7.4), (im f, f, j) must be a realization. According to
(7.5) and (7.8), 9( and H in the present context are given by

(7.20) cp(t, x, u) := tx + f(ro);


(7.21) Hx := x(o),


where D(H) = f((Co(- -, 0])B).








8. Topological Observability.


The two basic questions in realization theory are

(i) Given an input/output map, does there exist a "canonical"
realization?

(ii) Is a "canonical" realization unique?

Of course, one cannot answer these questions without specifying the
meaning of "canonical." Let us suppose, at the moment, that "canonical"
means "weakly canonical." By Proposition (7.18) the existence question
(i) is answered affirmatively. However, the uniqueness question (ii) is
much more delicate if the state space is not finite-dimensional, and unique-
ness does indeed fail to hold. The following counterexample is due to BARAS,
BROCKETT, and FUHRMANN [1974] (see also BROCKETT and FURRMANN [1976]).

(8.1) COUNTEREXAMPLE (BARAS, BROCKETT, and FUHRMANN [1974]). Let
p
S= (2, cp, H) be the system given in Example (6.20). By Example (7.10)
it is known that E is a realization of the input/output map f given
by (7.11).
Now assume the following conditions:

(a) {gn/n) and {nhn) again belong to .2

(b) g / 0, h / 0, for all n, and An / for n m.

Let H = (A2, 9, H) be the system defined by


$(t, xn), u) In := ent + I ent /n)u(T)dT;


H({x n) := nh x, D (H) = D(H)= A2.
n n=l n n o

By Condition (a), 2 is a system; further, Z realizes the same
input/output nap f because of (gn/n)(nh) = gnh (see the calculation
given in Example (7.10)). By the following Lemma (8.2), Z and Z are
both weakly canonical. But they are not isomorphic. In order to see this,
2 2
consider the continuous linear map T: A -> given by


T((x)n) := xn/n) for all {xn) in A2.








It is easy to check that T is indeed a morphism: S -> However, T
is not invertible, i.e., not an isomorphism. Since T is the only
possible choice as a morphism (see Lemma (8.3) below), E and E are
not isomorphic even though they are both weakly canonical realizations of
the same input/output map.

(8.2) LEMMA. Under the hypothesis (b) in Example (9.1), the systems
E and Z are weakly canonical.

PROOF. Since Z is defined by replacing (gn) and (hn) of E
by (gn/n) and {nhn), it obviously suffices to prove the statement for S.

We show observability first. Suppose that h ({x ))(t)
E h x n = 0 for all t > 0. Since Z h x ent is clearly an analytic
n=l n n n n n
function of t (the series converges uniformly on every compact interval
[-n, n]), the assumption implies that nE hnxne-ns = 0 for all s C C.
Ncte that the series n h x ee-n" is the Laplace transform of a Radon
co n=1 n n
measure = n-4 hxnhn ( n = Dirac's point measure at hn). By unique-
ness of Laplace inverse images (L. SCHWARTZ [1961, VI.2.4]) the measure
I itself must be zero. Let p.(t) be the characteristic function of the
point \ij), i.e., cp.(t) = 1 if t = A. and cp(t) = 0 if t / \j. By a
standard technique of measure theory, we see that P acts on (pj and


0 = p( Y,) = nl hnXnn (P) = h .x
n:: n n n 3 0

because 0 = 0 and A. / 7. (i / j). Since h. / O, x. must be zero.
Thus {xn} = 0, that is, E is observable.

By easy calculation, one sees that the adjoint of g is given
by the correspondence (Xn] 4n l gnxne n (t < 0). Since this
correspondence is of the same form as h it must be one-to-one. Then
by Proposition (6.13), E must be quasi-reachable. O

(8.3) LEMMA. Let E = (X, T, H) be a quasi-reachable linear system,
and E = (, H) a linear system. Suppose that T1, T2: 2 -s are
morphisms. Then T = T








PROOF. Let x = g (w) for some w in n. Then T1x = Tg () -
g (m) = T2g C() = T2x since both of T and T2 are morphisms. Thus
T = on g (9). Since g (n) is dense in X, T1 must be equal to
T2 on X. Ol

Example (8.1) shows that the notion of "weakly canonical" is too weak
to obtain the uniqueness theorem of "canonical" realizations. Therefore
we must impose a stronger condition on "canonical" realizations in order
to obtain the desired uniqueness.

The intuitive idea of observability is to determine the initial state
based on suitable input/output experiments; see KALMAN 11968, Chapter 10].
Since our systems are always linear, we can reduce the problem to identifying
the initial state by a suitable procedure based on observation data.
Observability guarantees the abstract possibility of uniquely determining
initial states from observed data.

It turns out that observability implies that initial states can also
be continuously determined from observed data for finite-dimensional
linear systems; see Example (8.8). In other words, the initial state
determination procedure is well-posed for finite-dimensional linear systems.

However, for infinite-dimensional systems (i.e., the state space is not
finite-dimensional), observability does not, in general, imply the above
mentioned well-posedness; see Example (8.10).

As explained in Chapter I, we regard the well-posedness of initial
state determination procedure as one of the basic properties of systems.
Thus we give

(8.4) DEFINITION. Let = (X, 9, H) be a system, and hE its observa-
bility map. The system H is topologically observable iff the following
statement is true:

(8.5) For every neighborhood U of 0 in X, there exists a neighbor-
hood V of 0 in r, such that (h )-1() C U.

(8.6) RIMARK. In this definition we require that the initial state
determination be a continuous procedure, i.e., a procedure which belongs
to the scope of the categorical aspects (continuity) of the problem.









An analogous definition was used successfully in the (different) category
of polynomial systems; see SONTAG [1976]; SONTAG and ROUCHALEAU [1976].

We remark that topological observability implies observability.

(8.7) PROPOSITION. If E = (X, C, H) is topologically observable, then
it is observable.

PROOF. Suppose h (x) = 0. Take a neighborhood U of 0 in X.
Then, by topological observability, there exists V, a neighborhood of 0
in P, such that (h )- (V) C U. Since h (x) = 0, x must belong to
(h )- (V) C U. Thus x must belong to every neighborhood of 0 in X.
Since X is a Hausdorff space, x must be 0. 0

(8.8) EXAMPLE. Let Z = (X, cp, H) be a finite-dimensional linear system,
i.e., dim X = n < m. Suppose that E is observable, i.e., h : X ->P is
one-to-one. Then h : X -h (X) is a bijection and hence dim h (X) = n.
By SCHAEFER [1971, 1.3.4], hE must be an isomorphism. Hence if U is a
neighborhood of 0 in X, then h (U) is a neighborhood of 0 in h (X).
By definition of the subspace topology, there exists a neighborhood V of
0 in P such that h (U) = V(h (X). Then (h )-1(V) = (h )-1(v h (x)) =
(h)-l(h (U)) C U. Thus Y is topologically observable.

We give an example of a system which is observable but not topologically
observable.

(8.9) EXAMPLE. Let E be the system given in Example (6.20). Under
the assumption (b) of Counterexample (8.1), E is observable.

Let eN be the element of 2 whose only nonzero entry is 1
at the N-th position. Clearly h (e)(t) = h eNt. Now take any T,
e > 0. By choosing sufficiently large N, we have iIh (eN )I[O,T]
lhNeeNt [0,T] < c since hn ->0 as n ,-. But leN2 = 1 for all N.
This clearly contradicts topological observability.

We now give several equivalent conditions for topological observability.

(8.10) PROPOSITION. Let E = (X, p, H) be a linear system, and let
(pu}9oA be a fundamental family of seminorms of X. The following




47


statements are equivalent:
(a) E is topologically observable.
(b) Let E be a dense subspace of X. For every U a neigh-
borhood of 0 in E, there exists V, a neighborhood of 0 in r,
such that (h )-1 () lE C U .
(c) For each a A, there exist T > 0 and a constant C > 0
such that

(8.11) p,(x) < CjIHD()xl [0,T] for all x in D (H).

(d) For each a A, there exist T > 0 and a constant C > 0
such that

(8.12) pa(x) < Clh (x)J[O,T] for all x in X.

(e) h : X -h E(X) is an isomorphism.
(f) Let X = X1 ( X, where dim X < m. The observability map
h is one-to-one, and h X2: X2 -4hE(X2) is an isomorphism.

PROOF. (a) => (b) By definition of the subspace topology, there
exists U, a neighborhood of 0 in X, such that Uo = UfE. Also, there
exists V, a neighborhood of 0 in r, such that (h )-1(V) C U. This
implies (h)-l(v) nE C U fnE = U .
(b) = (c) Take E := Do(H), and U0 := fx E Do(H): pO(x) < 1}.
Then there exists V, a neighborhood of 0 in r, such that
(h )-1() (lD(H) CUo, i.e., if x belongs to Do(H) and h (x) belongs
to V then po(x) < 1. Since (II'[IIO,T] T> is a fundamental family of
seminorms of r, there exist E > 0 and T > 0 such that V :=
(7 E F: /17r[0 T] < e} is contained in V. Then (h)-l(Vo) nD (H) C
(hE)-1(V) Do(H) CUo, i.e., if x belongs to Do(H) and llh (x)1[0,T]
IH(.-)x[OT] < E, then pa(x) < 1. It follows that if IIHO(-)xl[O,T] < c/n
then p (x) < 1/n. Therefore, if HJII(.)x [0,T] = 0, then Pa(x) = 0.
Now take C := 1/e. It is clear that if 1h (x)I[OT] = O, then
pO(x)






S:= (/I!h (x)[O, T)x. Clearly IJhE(y)l = e. Hence p (y) =
P l((E/jlh()l O,T])x) = (E/llh (x)![ ,T])pa(x) < 1. Therefore,
pa(x) < aCJh(x)Il[O,T] for all x in Do(H).
(c) =(d) Take any x in X, and let fx ) be a net on Do(H)
which converges to x. Then we have

p (x) = pc(lim x ) = lim p (x ) (continuity of p.),


= Ccllim h (x v)[OT (continuity of l-![O,T])'

= C 1h (lim xV)[OT] (continuity of h ),

= C llh(x)l[o, T]

(d) = (e) First we prove that hE is one-to-one. Let h (x) = 0
and take any a G A. By (d), we have Pc,(x) < Cj h (x) II ,T = 0. Since
X is a Hausdorff space, x must be 0. Thus h : X ->h (X) is a con-
tinuous bijection. But (8.12) clearly implies that (h )-l: h (X) ->X
is continuous.
(e) > (f) Trivial.
(f) = (e) We first claim that h (Xl OX2) X h (X ) hE (X ).
Since h : X1 O X2 h (Xl X2) is a bijection, h (X1 OX2) h (Xl) h (X2)
as vector spaces over k. Note that X2 is a closed subspace of XI X2
because it is a direct summand. Hence X2 is complete because X1 X2
is complete. So h (X2) is complete by the isomorphism: X2 L h (X2).
Thus h (X2) is a closed subspace of r, and hence a closed subspace of
hZ(X1 O X2). Furthermore the codimension of h (X2) in h (X1 OX2) is
finite since h (X1 E 2) -h h(X ) h (X ). Therefore, the direct sum
h (X1 O X2) h (X1) I h (X2) must also be topological, by SCHAEFER
[1971, I.3.51.
Since dim X (= dim h (X)) < h : XI ->h (X1) is an
isomorphism by SCHAEFER [1971, 1.3.2]. Therefore,
h : X1 X2 hZ(Xl) hC(X2) h (Xl X2 ) is an isomorphism (because
h : X2 h (X2) is an isomorphism).







(e) => (a) Since h : X ->h (X) is an isomorphism, h (U) is
a neighborhood of 0 in h (X) for every U, a neighborhood of 0 in
X. By definition of the subspace topology, there exists V, a neighborhood
of 0 in F, such that h (U) = Vfh (X). Then (h )-1() =
(h )-1(V h (X)) = (h --l(h(U)) C U.

The following Proposition (8.13) gives the dual characterization of
topological observability. (The proof of the sufficiency part is very
technical.)

(8.13) PROPOSITION. A system E = (X, p, H) is topologically observable
iff the following conditions (i) and (ii) are satisfied:

(i) X is reflexive.

(ii) (h )': P' ->X' is onto.

PROOF. Necessity. By Proposition (8.10) we can identify X with
h (X). Since P is a Fr6chet space (Proposition (2.5)), X is also a
Fr6chet space. As pointed out in the proof of Proposition (6.15), P
is reflexive. Hence X, regarded as a closed subspace of r, is semi-
reflexive (see SCHAEFER [1971, IV.5.7]). Since X is also barreled
(because it is a Fr6chet space), X must be reflexive; see TREVES [1967,
Proposition (36.5)1.

Take any x* E X'. Define a linear functional x* on h (X) by
(9, h5(x)) := (x*, x). By the isomorphism X Q h (X), is clearly a
continuous linear form on h (X). Then by the Hahn-Banach theorem, there
exists 7* E P' such that 7*h (X) = r*. It follows that ((h )'(7*), x) =
(7*, hE(x)) = (2*, h (x)) = (x*, x). Hence x* = (h )'(TE), i.e.,
(h )' is onto.

Sufficiency. Since r' is a strong dual of a reflexive Fr6chet
space r, it is a Ptak (B-complete) space; see HUSAIN [1965, Chapter 4,
Proposition 71. The strong dual X' is a reflexive space because X is
reflexive; hence it is a barreled space (SCHAEFER [1971, IV.5.7]). Then
we can apply Ptak's open mapping theorem (SCHAEFER [1971, IV.8.5, Corollary 1])
to (h )': r' -X', and conclude that (h )' is an open mapping because
(hl)' is onto. In other words, X' is isomorphic to the quotient space
F'/ker (h )'.










Since X and r are reflexive, it is enough to prove that
(hE)": X" ->I" is an isomorphism into P". By identifying X' with
P'/ker (h )', it suffices to prove that r': (r'/ker (h )')' -r" is an
isomorphism into r", where T: r' > F'/ker (hj)' is the canonical
projection.

We note that 1' is a DF-space as a strong dual of a metrizable
space r (KOTHE [1969, 29.3]). Then by KOTHE [1969, 29.5, (1)] we
conclude that 7' gives the isomorphism:
(r'/ker (h )')' -4*'((r'/ker (h2)')') C r". O


9. Existence and Uniqueness of Canonical Realizations

(9.1) DEFINITION. A linear system E is canonical iff it is quasi-
reachable and topologically observable.

We prove the existence of a canonical realization first.

(9.2) THEOREM. Every input/output map f has at least one canonical
realization.

PROOF. Let Hf = (im f, f Hf) be the system given in Proposition
(7.18). We have already shown that Ef is weakly canonical. So we only
need to show topological observability.

Recall that hZf, the observability map of Zf, is given by the
inclusion j: im f -P. Let U be any neighborhood of 0 in im f. By
definition of the subspace topology, there exists V, a neighborhood of 0
in r, such that V( im f = U. It follows that j-1(V) = V 7 im f = U. O

We now prove the uniqueness. We first prove the following statement,
which is a counterpart of "Zeiger's lemma" (KAIMAN, FALB, and ARBIB [1969,
Chapter 10]).

(9.3) LEMMA. Suppose that 1 = (X1, p1, H1) and Z2 = (X2, 2, H2)
are both realizations of the same input/output map f: 0 r. Suppose
also that EI is quasi-reachable and Z2 is topologically observable.
Then there exists precisely one morphism T: X X2.








PROOF. Write gl := gl g2 := g h2, h := 1, h2 := h 2 and
X1,R:= g1((). Since ZI is quasi-reachable, we need only to prove the
existence of a continuous linear map T: X1, 2 such that Tgl = 2
and h2T = hI by Theorem (6.19). The uniqueness of such T is already
known by Lemma (8.5).

Define a linear map T: X,R 2 by


T(x) := h2l(x) for x in X1 R


Since x = gl(m) for some m (if x E X1, ), hl(x) = hlgl(c) = f(o) =
h2g 2() im h2. Moreover, h2 is one-to-one, so T(x) is well-defined.
Clearly h2T = h on X1 R. And for every w in Q, Tgl(m) = h2 l(g l())=
h2 (f(m)) = h2 (h2g2(o)) = g2(0) (h2 is one-to-one). Thus Tgl = g2'
-1
By Proposition (8.10), h2 : im h2 -X2 is continuous. Hence
T is continuous as a composition of continuous linear maps. Then, since
X1,R is dense in X1 and X2 is complete, there exists a unique continuous
extension T: X1 X2 such that TIX,R = T. Clearly T satisfies
Tgl = g2 and h2T = hI because these qualities hold on X1 R, which is
dense in X1. O

We are now ready to prove

(9.4) THEOREM. Suppose that El = (Xl, 91' H1) and Z2 = (X2, '2' H2)
are two canonical realizations of the same input/output map f: n rP.
Then E, and K2 are isomorphic.

PROOF. Let T : K1 -F2 and T2: K2 1 be morphisms as given
in Lemna (9.5). Observe that T2T : -1 1 and 1X : E -41 are again
morphisms (Remark (6.16)). By the uniqueness of morphisms, T2T1 = IX1
Similarly, T1T2 = L 2. Hence T1 (T2) is an isomorphism. D

It is now clear why nonuniqueness occurs in Counterexample (8.1).
As is shown in Example (8.9), the systems discussed in Counterexample (8.1)
are not topologically observable even though they are observable.

(9.5) REMARK. One can easily check that the system given in Example
(6.21) is indeed canonical.








Recall that when the weighting pattern of an input/output map f is
of class Cr (r > 0), f has a unique continuous extension f: ( r' ,m
by Proposition (3.14). We prove that this extension of the input space
does not affect the canonical realization. To be more precise, we state

(9.6) PROPOSITION. Let f and f be as described above. Then
im f = im f.

PROOF. Since 2 is contained in (Er( 1 )m, im f is contained
in im f. Hence im f must be contained in im f. On the other hand,
every element ?(S) can be approximated by elements in im f
because n is dense in (E, ) Hence im fC im f. Since im f is
closed, im f C im f follows. O

This Proposition (9.6) (and its proof) shows that the canonical realization
2f = (im f, ~f, Hf) is determined uniquely (up to isomorphism) by its
weighting pattern [ (and the output space F, of course) irrespective
of the input space n as long as 0 is dense in ?. When = (E'( _0 )m
we obtain the uniqueness theorem for the case treated by KALMAN and HAUTUS
[1972] (modulo the difference of the output space).

(9.7) REMARK. As is clear from the proof of Lemma (9.5), topological
observability leads to the uniqueness of canonical realizations for each
fixed choice of output spaces (note that the equivalence (a)4->(e) in
Proposition (8.10) is always true).

(9.8) REMARK. Note that im f is always a reflexive Fr4chet space (see
the proof of Proposition (8.13)).


10. Realizations in the Working Mode.

In Section 7 we defined a realization of a linear input/output map f
as a linear system Z = (X, cp, H) which factors f, i.e., f = h g
This means that the outputs of the system Z are equal to those of the
external behavior only after inputs are terminated. Then a very natural
question arises: Are those outputs, one induced by f the other induced
by E, equal even while an input is being applied? One may rephrase







the question as follows: For every u in (L ,T) m, is it ture that
Hcp(t, 0, u) = T(u)(t) for almost all t in [0, T)? (The map T is the
extended input/output map: Am -A associated with f.) (We may as well
assume that the initial state x is 0 since the system is linear.)

We remark that cp(t, 0, u) does not necessarily belong to D(H),
and so HP(t, 0, u) may not be well-defined. It is guaranteed that
cp(t, 0, u) belongs to D(H) only when u belongs to (Col0, T])m
(continuous functions vanishing at 0 and T). Thus for general inputs
belonging to (L 0,T)) we must find a way of justifying the corresponding
outputs.

All these technical problems do not arise (or can be solved rather
easily) in discrete-time systems. But because of varied topological
questions these problems become much more delicate than those in the
discrete-time case. Fortunately, we can answer our questions affirmatively
as follows. We start by proving the following

(10.1) PROPOSITION. Let E = (X, 9, H) be a realization of an input/output
map f. For every u in (Co[0, T])m,

(10.2) T(u)(t) = HT(t, 0, u) for all t E [0, T).

PROOF. Since u belongs to (Co[O, T])m, Hp(t, O, u) is well-
defined for every t E [0, T) by Condition (h) of Definition (6.1). Since
u is continuous and u(O) = u(T) = O, T(u)(t) is a continuous function
of t by Theorem (5.2). Thus we obtain

T(u)(t) = (o{-(u))(o) = T( u)(o),

= (orTf(au))(0),

= (Tfj,(7rq u))(o) (causality of f),

= f(T0 u)(o) (Theorem (5.2)),

= hg Z(7 u)(O) (E is a realization of f),

= h p(t, 0, u)(O) (by (6.6)),
= Hc(t, 0, u) (by Definition (6.1) (j)). O







Thus we have assured the desired property of Z at least for continuous
inputs with compact support. In order to extend our result to L2-inputs,
we need the following lemma.
2 m
(10.3) LEMMA. Let T > 0 be fixed. For every u in (L2 ,T)),
------------------ [0,T)
p(t, 0, u) is a continuous function of t E [0, T] (with its value in X),
and, furthermore, the correspondence g: (Lo,T) m ->C(0, T; X): u "-(, 0, u)
is continuous, where C(O, T: X) is the set of all X-valued continuous
functions on [0, T] with the topology of uniform convergence on [0, T].

PROOF. Take any u in (L ,t))m. For every t in [0, T),
E(, O, 1,t) E
cp(t 0, u) = g (Fru u) by (6.6). Then g (rg cu) converges to
g (no-t u) as t -to, since g is continuous and 729'u -Ti o0 u by
Proposition (4.3). Hence g(u) is a continuous function.
Now let u -- 0 in (L2 )m and p be a continuous seminorm
E n [O,T) 2
on X. Since g is continuous on each (L -T,O]) there exists
such tpg()) < m
Ca > 0 such that p(g Y)) CC 1[_T,0] for every o in (L[T,O])
Then we obtain


pw((t, 0, un)) p(g ( tn)),

C< call~ruln [-T,0]'

< Ca l7Tunll [-T,01]'

C auni [,T for all t E [0, T].

Hence p(cp(t, 0, un)) 0 uniformly on [0, T] as n ->~. Thus
is continuous. 0

Now let X := (cp(, 0, u) E C(0, T; X): u (L0,T)m), i.e.,
S:= ((L[0,T) ). We now prove the following

(10.4) PROPOSITION. Let E = (X, p, H) be a realization of a linear
input/output map f. For each T > 0 there exists a continuous linear
h: X4 (L[OT]) such that

(10.5) h(cp(., 0, u)) = (u) for all u in (L[0,T)m







[o2 ,T]
FROOF. Define h: g((C[O,T]))m -> (L[O,T]) by (p(, 0, u))(t) :
Hp(t, O, u). By Proposition (10.1), h(p(., O, u)) = f(u) for all u in
(Co[O, T])m. Since g((C[0, T])m) is clearly dense in X and (L2,T )
is complete, there exists a unique continuous linear extension
h: X -(L2O T)P. Since (CG[O, T])m is dense in (LoT)m and
hg =ihg = f on (Co[O, T])m, it follows that hg = f on (L2,T))m, i.e.,
h(p(., 0, u)) f(u). 0

In other words, even while an L2-input is applied, the system Y
keeps producing an output and this output is equal to that given by the
extended input/output map T.

(10.6) REMARK. Clearly h satisfies h(p(, 0, u))[Ot) =
h(q(., 0, u)I[O,T))[Ot).


11. Compatible Systems and Differential Equations.

In this section we shall prove one of our main results: If a linear
input/output map f is sufficiently smooth, then its canonical realization
f admits a differential equation description. In order to avoid cumber-
some notation we assume that systems under consideration are single-input/
single-output systems. The general case can be treated in a similar way.

Following T. KOMURA [1968], we say that a strongly continuous semigroup
(which we shall abbreviate as simply "semigroup" in the sequel) ([(t))t>O in
a locally convex space X is locally equicontinuous iff for every T > 0
the family {N(t)} maps, i.e., for every continuous seminorm p there exists a continuous
seminorm q such that

(11.1) p(O(t)x) < q(x) for all t in [0, T].

We quote from T. KOMURA [1968] the following facts:

(11.2) LFMMA. Let [((t))t>0 be a semigroup in a complete locally
convex space X with the infinitesimal generator F. Then the following
statements are true:

(a) The domain of F, denoted by D(F), is dense in X.








(b) D(F) is i-invariant; O(t)F = FO(t) on D(F); and for
every x in D(F), I(t)x is differentiable with respect to the topology
of X. and


d -(t)x = F((t)x = D(t)Fx for all t > 0.
dt

(c) If X is a barreled space (in particular, if X is a Fr4chet
space), then ([(t))t>0 is locally equicontinuous.
-.------- tX)u~- --"--~'"---
(d) If (O(t))t>o is locally equicontinuous, then F is a
closed operator.

PROOF. See T. KOMURA [1968]. O

Now let (I(t))t>O be a locally equicontinuous semigroup in X. For
every continuous seminorm p of X we define

(11.3) pF(x) := p(x) + p(Fx) for x in D(F).

Clearly pF is a seminorm on D(F). We define a locally convex Hausdorff
topology TF on D(F) by the collection of seminorms (PF: p is a con-
tinuous seminorm on X). Clearly TF is finer than that induced on D(F)
from X. Furthermore, when (D(t))t>0 is locally equicontinuous, we have

(11.4) PROPOSITION. Suppose that ([(t))t>O is a locally equicontinuous
semigroup. Then D(F) is complete with respect to TF. Moreover,
c((t))t>O is again a (strongly continuous) semigroup in D(F) with
respect to TF

SKETCH OF PROOF. The completeness of D(F) is an immediate conse-
quence of the completeness of X and the fact that F is closed (Lenma (11.2)).

For strong continuity, observe that

pF(-(t)x a(to)x) = p(O(t)x 0(to)x) + p(O(t)Fx G(to)Fx)

by virtue of 0(t)F = Fr(t). O


Now we can give the following







(11.5) DEFINITION. A linear system E = (X, 9, H) is compatible iff
(a) the semigroup {F(t))jt> associated with E (given by (6.6))
is locally equicontinuous;

(b) H is well-defined on D(F), and H is continuous with
respect to T', that is, there exists a continuous seminorm p of X
such that IHxl S PF(x) = p(x) + p(Fx).

(11.6) REMARK. The notion of compatible systems is clearly invariant
under isomorphisms of systems.

First we show that every topologically observable system is compatible.

(11.7) PROPOSITION. Every topologically observable linear system
S= (X, cp, H) is compatible.

PROOF. By Proposition (8.10), we may identify X with h (X).
Take E := h (X) C[O, m). Clearly H is well-defined on E by
H(h (x)) = h (x)(O). The semigroup of this system is given by ( tt>o
and its infinitesimal generator is (dt). Since h (X) is a Frechet
space as a closed subspace of P, ([t.t>O is locally equicontinuous by
Lemma (11.2).

Now one can easily prove that

(11.8) D(L) := (7 E h(X): 7 E h(X)),
2 h)2
= ( h (X): 7 E Llc[O, m)},

ShZ(X) Hoc[O, ),

1 ) 7 L
where H lo[O, ) := {7 E Loc[O, ): 7 L c[O, m)) with generating
seminorms

(11.9) K 1,Ol,,TI] 1 Ib[O,T] + ,T]"

This topology clearly induces T on D(!) as defined by (11.5).

Since E contains D( ), H is well-defined on D(d). We
0 TtL cdt








must prove that H is continuous with respect to TF. For every 7 in
Hoc [O, m), we have

t
(11.100) 7(0) = 7(t) f 7(T)dr.
0

It follows that

1
(11.11) IH71 = 17(0)1 = I 7(0O)dt,
1 It
< f [r(t)|dt + f f [7(T)ld-dt,
0 00

ll7|l[0,1] + 1 1[0o ] = 1,[0,1]'


by Schwarz's inequality. Thus E is compatible. 0

Before starting the discussion on differential equation descriptions
of systems, we need to prove the following lemma.

(11.12) LFEMMA. Let {((t)}t>0 be a locally equicontinuous semigroup in
a complete locally convex space X and F its infinitesimal generator.
Let G be an element of D(F). Then the functional differential equation


(11.13) dx(t) = Fx(t) + Gu(t)
Tu-t

with the initial condition

(11.14) x(0) = xo D(F)


admits a unique solution given by

t
(11.15) x(t) = D(t)x + / 0(t T)Gu(i)dr,
0 0

at least for uniformly continuous u.

This lemma is a standard well-known fact when X is a Banach space
(see, for example, YOSHIDA [19711). The proof for the present case is
essentially no different from that given for Banach spaces, in view of








Lemma (11.2) (especially (b), (d)). So we only give
t
SKETCH OF PROOF. The Riemann integral f 0(t T)Gu(T)dT exists
0
by the uniform continuity of u and local equicontinuity of ({(t))t>O. In
order to see that (11.14) gives a solution, we observe that the equality

t tt
(11.15) a 0 (t r)Gu(r)dr = f -(t T)Gu(T)dr + Gu(r),
t
= F f (t T)Gu(T)dT + Gu(t)
o

can be easily justified using Lemma (11.2) (b), and the facts that F is
closed, and the functions t >Gu(t), t ->FGu(t) are uniformly continuous
with respect to t.

In order to see the uniqueness of solutions, observe that

(11.16) (O(t T)x(T)) = 0(t T)Gu(r)


is valid for every solution x of (11.15). Integrating both sides from
0 to t, we obtain the desired uniqueness. O

We are now ready to give the definition of smooth systems. Roughly
speaking, a smooth system is a compatible system whose state-transition
is governed by a differential equation. To be more precise, we give
the following

(11.17) DEFINITION. A linear system Z = (X, cp, H) is smooth iff

(i) Y is compatible;

(ii) there exists G E D(F) such that the solution x(t) of
(11.13) and (11.14) is equal to p(t, xo, u) whenever x belongs to
D(F) and u is uniformly continuous.

(11.18) REMARK. Note that if the system Y is smooth then we may regard
H as defined only on D(F) and drop Conditions (f) (i) of Definition
(6.1). Indeed, D(F) is D-invariant, and for every u C Co[O, T],
I 0(s T)Gu(T)dT belongs to D(F) for every s E [0, T]. Further,
the correspondence: u t- 0 (t T)Gu(T)dT (E D(F)) is continuous








and linear as a map from C [O, t] to D(F) (endowed with TF). Since
H is continuous with respect to T the requirement (i) can be waived.

REMARK. BARAS and BROCKETT [1975] called smooth systems
"balanced" when X is a Hilbert space.

(11.19) REPARK. Observe that if a system Z is smooth, then for every
uniformly continuous input u

t
(11.20) f(u)(t) = f Hr(t )Gu(-r)d- (f = the extended input/output map
o
of Y).

In view of (5.5), this means that the weighting pattern of f is indeed
a (locally L2) function A(t) = HM(t)G = h (G)(t).

Moreover, we have

(11.21) PROPOSITION. If a system Z is smooth, then its weighting
pattern A(t) belongs to Ho[O, co).

PROOF. We have already seen that A(t) = HI(t)G belongs to
L2 oc[, "). In view of (11.8), it suffices to prove that h (G) (= A)
c d
belongs to D(t). We have


r () li ah h(G) h (G)
1-m n (G) = lim
t-4o 0 t>o t

= lim h (O(t)G G) (h commutes with shifts),
t->o t

= h7(lim Dt) IG) (continuity of h),
t-4o t

= h (FG).

This means that h (G) (= A) is locally absolutely continuous and its
derivative is given by h (FG) C Llo [O, ). O

We now consider the converse: Given a.weighting pattern
A(t) H4c[O, c), does there exist a smooth realization? We give an
affirmative answer in the following form.








(11.22) THEOREM. Suppose that f is a linear input/output map with the
weighting pattern A E H ocO, m). Then its canonical realization
f = (im f, tPf, Hf) is smooth.

PROOF. We have already seen, in Proposition (11.7), that Ef is
compatible.

By the proof of Proposition (11.7), we have D(F) = im f Hoc[O, m),
fbc
c(t) = at, and F = ( ). Now let G := f(5) = A(t) E D(F),
where f is the extension of f given in Proposition (3.14). As can be
clearly seen, HO(t)G = A(t). By Lemma (11.12), the functional differential
equation (11.13) together with the initial condition (11.14) admits a
unique solution

t
x(t) = '(t)x + f 0(t T)Gu(T)dT.
0 o

In view of the linearity cPf(t, x u) = D(t)xo + pf(t, O, u), we need
only to prove Y 0(t T)Gu(T)dT = P (t, 0, u).
o f
Recall that each state of f is a function of il > 0. By
definition (see (7.20))


cpf(t, 0, u)(t) = f(v. u)(n) = ft A(T t)u(T + t)dt
-7t
t
= / A( T + t)u(r)dr.
0

On the other hand,

t t
(f 0(t T)Gu(T)dT)(T) = HO(%) f T(t l)Gu(T)dT,
o o
t
= H f O(In)(t T)Gu(T)dT,
0
t
= f H(Tj T + t)Gu(T)dT.

Since H + t) = A + t), our assertion is proved
Since HO(T T + t)G = A(n T + t), our assertion is proved. O










CHAPTER IV. CANONICAL REALIZATIONS WITH
HILBERT STATE SPACES

We have already seen that for every input/output map f the state
space of its canonical realizations (which we shall abbreviate as the
canonical state space in the sequel) is at least a reflexive Frechet
space (Remark (9.8)). Our main concern in this chapter is to investigate
conditions under which a canonical realization has a Banach (or Hilbert)
space as a state space.

We start by introducing a particular notion of topological observa-
bility, which we call topological observability in bounded time. This
notion then leads to a necessary and sufficient condition that an input/
output map admit a canonical realization with a Banach (Hilbert) state
space.

Section 14 is devoted to the study of an example. The example is
described by the wave equation of one space-dimension. We convert this
description to our framework and prove that the system is indeed topo-
logically observable in bounded time.


12. Topological Observability in Bounded Time.

Topological observability guarantees the property that initial
states of the system can be continuously determined from observation data.
A new notion, topological observability in bounded time, requires further
that the initial state determination be done continuously based on
(uniformly) bounded time observation data. To be more precise, we give

(12.1) DEFINITION. A linear system Z = (X, H) is topologically
observable in bounded time iff there exists T > 0 such that the
following statement is true:

(12.2) For every continuous seminorm p of X, there exists Ca > 0
such that p (x) < C lhE(x) I0,T] for all x in X.


In view of Proposition (8.10), topological observability in bounded
time clearly implies topological observability. It is also obvious that

62









(12.2) is equivalent to the following statement:

(12.5) For every continuous seminorm p. of X, there exists C such
that p_(x) < CJHI(-)x[OT] for all x in D (H).

(cf. the proof of Proposition (8.10))

(12.4) REMARK. We may define observability in bounded time as the property
that there exists T > 0 such that .h(x) |, = 0 implies x = 0.
Clearly topological observability in bounded time implies observability
in bounded time, but not vice versa. An easy counterexample is supplied
by Example (8.9) (and Lemma (8.2)).

Our main objective in this section is to prove the following

(12.5) THEOREM. Let Z = (X, p, H) be a topologically observable system.
Then X is a Banach space iff E is topologically observable in bounded
time.

PROOF. Sufficiency. The continuous linear map h : X -L (L 2
E [O,T]
x h (x) I[O,T] is one-to-one if Y is topologically observable in
bounded time T. Furthermore, (h )-l: h (X)I o T] X is continuous
with respect to the topology induced from (L2 ,) by (12.2). Hence
2 2 [o,T]
X h (X) I[,T]. Since h(X) I[O,T] is a normed linear space, so is X.
But X is complete, so X must be a Banach space.

Necessity. Let 1I'] denote the norm cf X. Since E is topolo-
gically observable, there exist T > 0 and C > 0 such that
lixl| Cllh (x) I[O,T] for all x in X by Proposition (8.10). Since any
other continuous seminorm pC of X satisfies p,(x) < M~ix|II for all
x in X, 2 is topologically observable in bounded time. D

(12.6) COROLLARY. Let Z = (X, p, H) be a topologically observable
system. If X is a Banach space, then it is isomorphic to a Hilbert space.

PROOF. From the proof of the previous theorem, we see that if X
is a Banach space then it is isomorphic to a closed subspace of (L ,T])
for some T > 0. Since (L2 ,T]) is a Hilbert space, X is isomorphic
to a Hilbert space. 0








15. Necessary and Sufficient Conditions for a Hilbert Space Canonical
Realization.

Let f: -4 r be a linear input/output map. We ask the question:
What is a condition for f to have a canonical realization with a Hilbert
state space? In view of Corollary (12.6), this question is equivalent
to asking whether the canonical realization possesses a Banach state space.
Since we already know that a canonical state space is isomorphic to im f,
we only have to find conditions under which im f is a Banach space. Let
us start our discussion with the following

(13.1) THEOREM. The canonical realization of an input/output map f
has a Hilbert state space iff there exist T, M, 0 > 0 such that

(15.2) If(%t) [O,T] Me tllf(m) IO,T]

for all (0 E 0 and t > 0.

PROOF. Necessity. Let S = (X, cp, H) be the canonical reali-
zation of f with X being a Hilbert space. 'Let (0(t)}t>O be the
semigroup given by (6.6). By Theorem (12.5) there exist T, C1 > O such
that lxii CljhZ(x) [OT]. It follows that llh(x) l[O,T] < C21XII x_
CIC2lh(x) I[OT]. We now note from YOSHIDA [1971, IX.1, Proposition 1]
that there exist K, 0 > 0 such that

(15.3) IIQ(t)x|| < Ke tllxll,

since X is a Banach space. Then we obtain

llhz(l(t)x) l[O,T l C21q(t)xil,

S CeKet Hxl ((15.5)),

5 ClC2Ket llhS (x) II[, T]

For every m in n, this yields


I|f(V) I|[o,T] = llhgE(4) j[OT]'








= [IhZ((t)gZ ()) [O,T] (Proposition (6.12)),

C1C2Ket IlhZg(m) I[O,T]'

= Me tlf(') O,T]'

where M := CIC2K.
Sufficiency. Take any 7 in im f. Clearly,


I tT [O,T] < Me Btl7I[O,T].

We want to prove that the canonical realization Zf = (im f, if, Hf) is
topologically observable in bounded time T.
Take any a > 0. If Q < T, then !Hl7[o] <_ I71[[OT] for all 7
in im f. Suppose a > T. Let i be the integer that satisfies
T < a< (2 + l)T. Then we obtain inductively

1171 [o,0,] ,T]+ 1 [T,a]

= IlI[O,T] + T TI I [O,a -T]'

S1YII[O,T] + II&711[o,T] + IITT7l[T,-TJ]'
< (1 + MeHT) 117ll[o,T] + II& [T,la-T]'


<(1 + MeT + ... + MeT )TIII [OT]

for all 7 in im f. This shows that the canonical realization f is
topologically observable in bounded time T. By Corollary (12.6), im f
is indeed isomorphic to a Hilbert space. O

We can yet relax the condition (15.2) as follows:

(13.4) THEOREM. The canonical realization of an input/output map f
has a Hilbert state space iff there exists T > 0 such that the
following statement is true:








(13.5) For each t > 0 there exists Ct > 0 such that
IPtll[,T1 Ct subspace of im f.

PROOF. Necessity is obvious from Theorem (15.1) (take Ct := Me t).
Sufficiency. Note that (13.5) holds for every 7 in in f by
continuity. Take any a> 0. If a < T, then I)ll[0,a] < IT1[O,T].
Let a > T, and X the integer that satisfies AT < a < (, + l)T. It
follows that

11711[0,a] < I111[0,T] + iz (I[T,a],

= IYl[O,T] + ITYl[O,a-T],

< 11/11[0,T] T+ II'T/ O,T] + I+ I [T'II[T,a-T]'
< (1 + CT)1711[O,T] + lT7II[T,a-T]


S(1+ CT + ... + CT) [,T] for all 7 in im f.

This shows topological observability in bounded time T. Hence im f
is isomorphic to a Hilbert space. O

Let Mc(- ( 0] := EC_-,], i.e., the space of Radon measures with
compact support contained in (- m, 0]. If a weighting pattern A(t) is
continuous, then its associated input/output map f has an extension
f: (Mc(- m, O])m -> r, as was proved in Proposition (3.14). By
Proposition (9.6), it follows that A = f(B5) C im f. Then in order that
in f be isomorphic to a Hilbert space it is necessary that A satisfy

(15.6) KIAll[O,T] < Me tlI 0,T]

for some T, M, 4 > 0 by Theorem (15.1). Thus we obtain an example of a
weighting pattern whose canonical realization cannot have a Hilbert (Banach)
state space as follows:









(15.7) COUNTEREXAMPLE. Let the weighting pattern be given by
A(t) := exp(exp(t)). Clearly A does not satisfy (15.6) for any T, M,
P > 0. Hence its canonical realization cannot have a Hilbert (Banach)
state space.

We now give easy examples of weighting patterns whose canonical reali-
zation has a Hilbert state space.

(13.8) EXAMPLE. Let pi be a weighting pattern with support contained
in [0, T]. Then its canonical realization has a Hilbert state space.
Indeed, for every c in 0,

ST
f(c)(t) f= c(t r)dj. (T) f C(t )dp()
t t

Hence f(u)(t) = 0 if t > T. Thus im f can be identified with a
subspace of (L2 T)P. Hence im f is isomorphic to a Hilbert space.
e 0,T]
2
(15.9) EXAMPLE. Let A(t) be a locally L -weighting pattern given by


A(t)= 2 a exp(inrt/L),
n=-o n

with Ia n2 < w. Then A(t) is a function of period 2L. One can
easily check that (15.2) is satisfied in this case.


14. An Example of a Topologically Observable System.

Consider the wave equation of one space-dimension:


(14.1) (t2 /t tv(t, 2)= (2a 2)v(t, ), t > 0 < < 1,

with the initial condition

(14.2) v(O, ) = Vo(S), (Ov/t)(0, ) = v 1()


subject to the boundary condition


(14.5) (v/l)(t, O) E (v/Wi)(t, 1) 0.







This initial-boundary value problem occurs when we describe the vibrating
string of length 1 with both ends at 0 = 0 and 5 = 1 free to slide
along fixed parallel rails.
Now assume that

(14.4) yo(t) := v(t, 0), yl(t) := (Qv/at)(t, 0)

can be observed as outputs. We pose the question: Is it possible
to determine unknown initial states vo(0), vl() by observing Yo(t),
yl(t) for a sufficiently long time? If it is possible, can it be done
continuously? In order to answer these questions we must formulate
the problem more precisely.
We convert the equation (14.1) to a first-order differential equation
1 2
given in the space H[O,1 X L ,1 as follows:
0 [0,1]

(t) 0 l\/x (t)\ /X (t)\
(14.5) ( )) ((2a 2) )o( )) =:


where xo(t)(S) and xl(t)(a) are functions of (0 < <1) for
each t > 0, and xo(t)(.) E H0, ], xl(t)(.) E Lo2

REMARK. We drop the input term, since it is not relevant to the
observability question. The space H1 is the first order Sobolev
[0,1]
space, i.e.,


H0,1]= ( () L[Oi]: (d/d).z E L2

with the norm

jIzII1 := (IIzIl2 + daz/d~if2)1/2,


where 11I1 denotes the L2-norm.
Define the domain of F by

x\ H-9 1 2 2 01 E 2
(14.6) D(F) := x X L[0,1]: x1 C H[0,1] (d2 d )x L O

(dxo/d) = = (dxo/d) =1= 03.









REMARK. If (d2/2)x belongs to L,1], then (dxo/d) is
a continuous function of . Hence (dxo/d) g=o is well-defined.
1 2
Clearly D(F) is dense in H[,1] x L[0,l]. However, it is a non-
trivial problem to check whether F generates a strongly continuous semi-
group using the Hille-Yoshida theorem; see YOSHIDA [1971, XIV.5] for this
type of treatment in case of the Dirichlet boundary condition.

Fortunately we can bypass this problem by explicitly giving the formula
for the strongly continuous semigroup ([(t))t>O generated by F, with
the aid of the Fourier series expansion of the solution of the initial-
boundary value problem (14.1) (14.3). But before doing this, we remark
that the initial condition (14.2) must be rewritten as

(14.7) xo(0)(0) = vo(), xl(O)() = vl().

Now for each t > O, define a linear operator O(t) from
X := H,1 x 0,1] into itself by
[0,11 [0,1]


z(14.) cos new[A cos nrt + Bn sin nrt] + Ao + B t
(14.8) O(t) =
z \n=l cos nwr[- nnrA sin nTt + n7rB cos nTt] + Bo/

where

1 1
(14.9) An:= 2 f z o() cos inrdq, B := (2/mn) f z (I) cos rnrndr, n > 1;
o O

1 1
(14.10) Ao := z o(q)di, Bo := I zl()dn.
o o

Note that z and z are indeed expanded in Fourier series as
o

(14.11) Zo() = 1 An cos n7S + Ao'


(14.12) z (Y ) = l nTrBn cos nm7 + Bo, 0 < < 1,









since we can always extend z and z1 as even functions on (- c, m)
o 1
of period 2. Furthermore, since zo belongs to H[O,1] and z1
belongs to L O,1],


(14.13) Y InmAn 2 < c, InwB 2 n=l *An n=1 n
c 1
(14.14) REMARK. If z = 2 A cos nrt + A belongs to H[0,, then
c o n=1 n o [0,11 t
(d/dS)zo() = 1n R~EA sin nmr in the sense of L Indeed, (d/da)z
equals the right-hand series in the sense of distributions. But since
(d/dF)zO belongs to L2, the series n=l AnrA sin rT is indeed a
function and belongs to L2 0,]; see MIZOHATA [1975, Theorem 2.7]. But
n~ nAn sin nrf belongs to LO,1] iff n xA I < by
n-L n L0,1] n=1 0
Parseval's identity.

Again by Parseval's identity one obtains


(14.15) izoll = [I oll 2 + lId o/d112}1/2,

= Io12 + (1/2)n IAn12 + (1/2) n1 nAn 2)1/2;


(14.16) 11z1 = (IBo 2 + (1/2) ,1 I[nBn 21/2.


We are now ready to prove

(14.17) PROPOSITION. The family of linear operators (t(t)t>0 given
t>0
by (14.8) (14.10) forms a strongly continuous semigroup in
H[,1] x L20,1], and its infinitesimal generator F is precisely F
given by (14.5) and (14.6).

PROOF. Clearly s(0) = I by (14.11) and (14.12). The semigroup
property i(t + s) = 0(t)D(s) follows easily from direct calculation.

We prove strong continuity. It suffices to prove strong conti-
1 2
nuity at t = 0 since HI, X L[1] is a Banach space (see YOSHIDA
[0,1] [0,1]
[1971, IX.1]). For the first coordinate of D(t)(zo, zl) (we shall use
the row vector notation as well as the column vector notation), we have








(14.18) (t)(zo, z) o z = ~cos r[An(cos nT( t 1) + B sin n7t] + B t.

By Parseval's identity



22
(14.19) I||(t)(zo, z) o zo| =

IB I2t2 + (1/2) IA (cos nrt 1) + B sin nrt 2

+ (1/2) n2 7A (cos n7t 1) + B sin mrt .

The right hand side of (14.19) is uniformly convergent because we have
the following estimate for each n:

(14.20) IAn(cos rmt 1) + B sin nrtl2 < 2(4 An + IBn 2)

and because Y 2(4 1A 12 + EBn 2), 2 2n2 (4 An 2 + IBn12) are finite.
Hence we can take the termwise limit as t -> 0 in (14.19), yielding
II|(t)(zo, zl) o1 zol1 0 as t 0.
For the second coordinate of o(t)(zo, zl), we have

(14.21) O(t)(zo, z) I zl =

n cos n7rS[- nrAn sin nrt + nrBn(cos nrt 1)].
n=l n n

The L2-norm |l|(t)(zo, z II zljI can be estimated similarly as is
done for llq(t)(zo, Zl) o zol 1, and it approaches 0 as t 0. Hence
[I(t)}t>0 is strongly continuous.
We now calculate the infinitesimal generator of (m(t)}t>O. First
suppose that (z, zl) belongs to D(F), that is, (d2/dg2)z belongs
to L20,1] and z1 belongs to HI0 and (dzo/d )l = (dzo/dg),=1
t Cl1 a [0,1] (os ( /a)I=
= 0. These conditions imply that


(14.22) n1 n An 2 < n=ln 4Bn
(Note that (zo, z ) belongs to D(F) iff (14.22) holds.) It follows that








(14.25) (l/t){$(t)(zo, z 1o) zo =

l cos nrri[An(cos n7t l)/t + B(sin nrt)/t] + Bo

By the same reasoning as was given for strong continuity of (O(t))t>O, the
right hand side of (14.25) converges to z = E n-rBn cos nir + Bo in
[ n1] as t 0 (we can take the termwise limit by virtue of the fact that
n4[ A I2 < and n 1B 12 For the second coordinate, we obtain

(14.24) (l/t)({(t)(zo, z) I1 Zl =

Zn cos anr[- nrA (sin nTt)/t + n7rB (cos nTt 1)/t].

As before, it is easily seen that the right side converges to
n nr2 An cos nurT in L O,~ which is equal to (d2/d 2)zo.

Thus we see that the infinitesimal generator F of (e(t))t>o
coincides with F on D(F), i.e., F is an extension of F. We
postpone the proof of the fact D(F) = D(F) until the end of this section. O

(14.25) REMARK. It is known that the first coordinate of D(t)(vo, Vl)
indeed gives the genuine solution of the initial-boundary value problem
(14.1) (14.5) for sufficiently smooth initial data v vl; see,
for example, L. SCHWARTZ [1961, VII.1].

Now define the readout map H: D(F) -4 k2 by


(14.26) H x : (


Also define a linear map h : D(F) -r F by
o

(14.27) h ( (t) := H(t) j = I [A cos nart + B sin n-t] + A + B t
x l [ n= An n nn cos n o

[- nrA sin nt + n-rB cos n-t] + B
n=l n n o








where x() = A cos nr + A, x = n nrBn cos n7 + B .
o n=l n' I ni1 n 0
We claim

(14.28) LEUMA. Iho(X, xl)O,2c= q( (n2v + 1)( A 2 + IB2) +

2 1Ao12 + 2(A 0B + ATB) + (14/5) IBo12

where q is a positive integer and A is the complex conjugate of A.

PROOF. Immediate from Parseval's identity and direct calculation. O

In view of (14.15) and (14.16), Lemma (14.28) clearly implies the conti-
nuity of h (with respect to the topology of X = H,1] x L2, )
Hence there exists a unique continuous extension h : X -> r, the obser-
vability map. Therefore our system is well defined. Then we claim

(14.29) THEOREM. The map h : X P is an isomorphism into P, i.e.,
the system is topologically observable (in bounded time).

PROOF. By Lemma (14.28) we obtain


(14.50) Jhzx, l) 11,2] n= (n22 + 1)(|An12 + B2) +

21 1A2 +2(AoB + ABo) + (14/5) IB 2,

> n 2 (r 2 + 1)(IAn 2 + iBn2) + 21Ao 2 + 21Bo 2 2IAo IBoj'

> (n2 r + 1)(IAn2 + IBn2) + 1A2 + B12

It now follows that

(14.51) llxo l2 + IxlII2 < lhz(xo, x)ll 02]

by (14.15) and (14.16). Hence h is an isomorphism of X into r,
and the system is topologically observable in bounded time 2. O

Let us now finish the proof of Proposition (14.17). We must prove







(14.32) LEMMA. D(F) = D(F).

PROOF. The inclusion D(F) C D(F) has already been proved.
Assume that z = (zo, 1) belongs to D(F). Consider the function

(14.35) a: [0, 1] --R: t fl|(((t) I)/t)z|Jx if t > 0,

I|Fzi|X if t = 0,

where

(14.34) Il((e(t) I)/t)z =

Z- (n2w2 + 1) A (cos nt l)/t + B (sin n7t)/t2 +

cl n2w2 An(sin n7t)/t + Bn(cos nTt l)/t2 + IBo 2

Since z belongs to D(F), a is a continuous function of t on [0, 1].
Furthermore, the q-partial sum of the right side of (14.54) converges
monotonically to a(t) for each t as q -'=. Hence by Dini's theorem
(RUDIN [1964, Theorem 7.113), the q-partial sum of the right side of
(14.34) converges uniformly on [0, 1] as q ->m. Hence we can take the
termwise limit of the right side of (14.34) as t -~m, and obtain


(14.35) =lnA2 i nlnJBn2 < 0.


These inequalities readily imply that z C D(F).. 0









APPENDIX


In this appendix we review some basic facts on inductive limits and
projective limits of sequences of locally convex Hausdorff spaces. We
remark that an inductive limit projectivee limit) is also known as a
colimit (limit) in categorical terms. In most cases we shall omit proofs
since they are available in the following standard references: BOURBAKI
[1966]; KOTHE [1969]; SCHAEFER [1971]; TREVES [1967].

Suppose that we are given a sequence of locally convex Hausdorff spaces
[(En such that for every n there exists an inclusion j +: E E
n n=l n,n+l n n+1
and each n,n+l is an isomorphism of En into En+ i.e., the topo-
logy of E is identical to that induced on jn,n+l(En) from En+F
Clearly for every pair n, m such that n < m, there exists an isomor-
phic inclusion j : En E thereby enabling us to identify an element
of E with an element of E whenever n < m. let E := n= E ; E is
clearly a vector space (over k) by the preceding identification. We
try to introduce a natural topology on E.

Let j : E > E be the inclusion. The inductive limit topology on E
is the finest locally convex topology on E such that all jn are con-
tinuous; E is called the (strict) inductive limit of a sequence (E n
and is denoted as E = lim E A strict inductive limit of a sequence has
-4 n
the following remarkable property.

(A.1) PROPOSITION. Let E = lim E be the (strict) inductive limit
of a sequence (En). Then the topology of E is locally convex Hausdorff
and j : E -n E is an isomorphism for every n.

PROOF. See SCHAEFER [1971, 11.6.4]. D

In order to check the continuity of a linear mapping f: lim E ->F,
n
the following Proposition (A.2) is extremely useful.

(A.2) PROPOSITION. Let E = lim E be the (strict) inductive limit
n
of (E }]n and F a locally convex space. A linear map f: E -,F is
n n=
continuous iff each fj n: En -> F is continuous for every n, where
j : E -4 E is the inclusion.
n n









PROOF. See SCHAEFER [1971, 11.6.1]. n

If F is also a (strict) inductive limit of a sequence (Fn with
inclusions i : F -4 F, and if a linear map f: E -4 F satisfies

(A.3) for every n there exists m such that f(En) C Fm'


we have the following

(A.4) COROLIARY. Let f: E F be a linear map that satisfies (A.3),
where E := lim E and F := lim F Then f is continuous iff each
n n-
fj is continuous as a map from En to F.

PROOF. Necessity is obvious. Conversely, if fojn is continuous
as a map from En to Fm, it is also continuous as a map from En to
F because i : F -4 F is an isomorphism by Proposition (A.1). Thus
m m
by Proposition (A.2) f is continuous. O

We now turn our attention to the dual notion of inductive limits,
namely projective limits of locally convex spaces. Let E be a vector
space and (E I= a sequence of locally convex Hausdorff spaces. Suppose
that for each n there is given a linear map 7n: E ->En, and to each
pair m, n with m> n there is associated a continuous linear map
7 m: E E such that
mn m n
(i) rnn = En for every n;

(ii) VJvmn = v whenever m> n > Z;

(iii) iTmnm = Vn whenever m > n.

The projective limit topology on E is the coarsest (locally convex)
topology on E such that all Tr are continuous; E is called the
projective limit of the sequence (E n= and denoted as E = lim vmn E
nn=l n- mn n
or simply E = lim En. The projective limit E = lim E is reduced
n n
when mn(E) is dense in En for every n.

A projective limit is not a priori a Hausdorff space. Hence we quote

(A.5) PROPOSITION. The projective limit E = lim En is a Hausdorff
'-- n








space iff for each nonzero x E E, there exist n and a 0-neighborhood
U in E such that T (x) V U.

PROOF. See SCHAEFER [1971, 11.5.1]. 0

The following Proposition (A.6) is very useful in checking the conti-
nuity of a linear map: F -> lim E.

(A.6) PROPOSITION. Let E = lm En be the protective limit and F
a locally convex space. A linear map f: F E is continuous iff each
n- of is continuous for every n.

PROOF. See SCHAEFER [1971, 11.5.2]. 0

We now prove that the space L c[O, m)) defined in Chapter II is
indeed the projective limit of ([LO,] n

(A.7) PROPOSITION. The space L2 [0O, j), with the seminorms [(IHI[[On
given in Section 2, is the projective limit of the sequence (Lon }.
Moreover, this topology is Hausdorff.

PROOF. Define -7 and v as follows:
n mn

(A.8) 7r(p) := l[O,n], P E Loc[O, );

(A.9) mn() := '[,n, Lo,m] m > n.

Clearly rnn = I, ne7mn = vm and i7rmn = Trn (m > n > ). For every
n > 1, we have


11hrTn(P) = ( 17 in(p)(t)i =2dt)}/2 [o,n]'


Hence vn is continuous for every n. Furthermore, it is necessary that
I1i[o,0-n] be a continuous seminorm for every n in order that each mn
be continuous. But since the topology of Loc[O, ") is generated by
the family of seminorms (1- lr[O,n]) this must be the coarsest (clearly
locally convex) topology making each 7n continuous. Hence









L2 [0, ) = lm L2 ,. The topology of L2 [O, c) is clearly Hausdorff.
loc 4- [0,n] loc
This completes the proof. O

We now want to prove that the space of Laurent functions A is indeed
the inductive limit of spaces (L c[- n, c)), where L oc[- n, m) is
topologized in the same way as L [0l m).
2 0 )
(A.10) PROPOSITION. A = li L oc[- n, ).
0o

PROOF. By definition, A = L2[- n, m). Take any n, and
2 Yi ?2 ncl bc 2 2
define pn: Lbc[- n, ) [-n,0] and qn: Lo[- n, m) 1m L20 by


pn(q) := (l[n,0]' LOi- ", m);
2
qn() :=([O,Bm),' Lle oc[- n,).

Clearly p and q are continuous for all n. Since
2 2 2 2
A = lm -n,] 1l L[0,n] lim l2-n,0] x 1m O,] by definition,
there exists a unique continuous linear map an: L oc[- n, m) A for
each n such that the diagram


n im L -m,0]

L oc[-n, m) n



2
a^'li^ [,Om]

commutes. It then follows that n is the inclusion of Lloc[- n,
into A. Hence the topology of A is coarser than that of lim L o[- n, )
in view of the definition of the inductive limit topology.

Conversely, define P: lim L -n, ] lim L oc[- n, c) and
Y: lnm L[,n] 1im Lloc[- n, m) by

2
P(p) := P, C lim L[_n, 0]

(q)) := cp, C lim L
[o,n]'









For any C L-nO we have P(y) C LnO] and II)ll_ no| = Il[_n,o]
Hence P is continuous by Corollary (A.4). Similarly, 7 is continuous.
2
Therefore there exists a unique continuous map b: A lim L oc[- n, m)
such that the diagram


1 [-n,O]


A lim Loc[- n, )


lim L2 ]
I- [o,n]

commutes. But b is clearly equal to the identity. Hence the topology
of A is finer than that of lim L oc[- n, '). D









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BIOGRAPHICAL SKETCH


Yutaka YAMAMOTO was born on March 29, 1950, in Kyoto, JAPAN, son of
Minoru YAMAMOTO and Sumiko YAMAMOTO. He obtained his Bachelor of Science
degree and Master of Science degree in Applied Mathematics and Physics
at the Kyoto University in 1972 and in 1974, respectively. In 1976
he also obtained his Master of Science degree in mathematics at the
University of Florida. He is a member of the American Mathematical
Society and the Japan Association of Automatic Control Engineers.













I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.





Dr. R. E. Kalman, Chairman
Graduate Research Professor of
Mathematics





I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.





Dr. V. M. Popov
Professor of Mathematics







I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.





Dr. R. R. Kallman
Professor of Mathematics












I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.



P->L P i{2 P t
Dr. M. P. Hale, Jr.
Associate Professor of Mathematics





I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




Dr. M. E. Warren,
Assistant Professor of
Electrical Engineering





This dissertation was submitted to the Graduate Faculty of the
Department of Mathematics in the College of Arts and Sciences and to
the Graduate Council, and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.

August, 1978


Dean, Graduate School




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