(Lecture notes in mathematics 1770) heide gluesing luerssen (auth ) linear delay differential systems with commensurate delays an algebraic approach springer verlag berlin heidelberg (2002

Heide

Gluesing-Luerssen,

Linear

Delay- Differential

Systems with
Commensurate

Delays:
An Algebraic Approach

4

11,11
4%

Springer


Author
Heide

Gluesing-Luerssen

Department of Mathematics
University of Oldenburg

26111 Oldenburg, Germany
e-mail:



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CIP-Einheitsaufnahme

Gltising-Ltierssen, Heide:
delay differential systernswith commensurate'delays : an algebraic
approach / Heide Gluesing-Lueerssen. Berlin; Heidelberg; New York;
Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002
(Lecture notes in mathematics ; 1770)
Linear

-

ISBN 3-540-42821-6

Mathematics

Subject Classification (2000): 93CO5, 93B25, 93C23, 13B99, 39B72

ISSN 0075-8434
ISBN 3-540-42821-6

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Preface

delay-differential equation was coined to comprise all types of differequations in which the unknown function and its derivatives occur with

The term
ential

various values of the argument. In these notes we concentrate on (implicit)
linear delay-differential equations with constant coefficients and commensurate

point delays. We present

an

investigation of dynamical delay-differential

sys-

tems with respect to their general system-theoretic properties. To this end, an

algebraic setting for the equations under consideration is developed. A thorough

purely algebraic study shows that this setting is well-suited for an examination
of delay-differential systems from the behavioral point of view in modern systems theory. The central object is a suitably defined operator algebra which
turns out to be an elementary divisor domain and thus provides the main tool
for handling matrix equations of delay-differential type. The presentation is introductory and mostly self-contained, no prior knowledge of delay-differential
equations or (behavioral) systems theory will be assumed.
people whom I am pleased to thank for making this
grateful to Jan C. Willems for suggesting the topic "delaydifferential systems in the behavioral approach" to me. Agreeing with him,
that algebraic methods and the behavioral approach sound like a promising
combination for these systems, I started working on the project and had no idea
of what I was heading for. Many interesting problems had to be settled (resulting
in Chapter 3 of this book) before the behavioral approach could be started.
Special thanks go to Wiland Schmale for the numerous fruitful discussions we
had in particular at the beginning of the project. They finally brought me on
the right track for finding the appropriate algebraic setting. But also later on, he
kept discussing the subject with me in a very stimulating fashion. His interest in
computer algebra made me think about symbolic computability of the Bezout
identity and Section 3.6 owes a lot to his insight on symbolic computation.
I wish to thank him for his helpful feedback and criticisms. These notes grew
out of my Habilitationsschrift at the University of Oldenburg, Germany. The
readers Uwe Helmke, Joachim Rosenthal, Wiland Schmale, and Jan C. Willems
deserve special mention for their generous collaboration. I also want, to thank
the Springer-Verlag for the pleasant cooperation. Finally, my greatest thanks go
There

work

are

a

number of

possible.

I

am


VI

Preface

only for many hours carefully proofreading all
making
helpful suggestions, but also, and even more, for
and
so
patient, supportive,
being
encouraging during the time I was occupied
with writing the "Schrift".
to my

partner, Uwe Nagel,

these pages and

Oldenburg, July

not

various

2001

Heide

Gluesing-Luerssen


Table of Contents

1

Introduction

2

The

Algebraic

3

The

3.1

Algebraic Structure
Divisibility Properties

3.2

Matrices

3.3

Systems over Rings: A Brief Survey
Nonfinitely Generated Ideals of Ho
The Ring H as a Convolution Algebra
Computing the Bezout Identity

3.4
3.5

3.6
4

5

................................................

Framework

over

Ho

..................................

of

Wo

,

............................

25
35

.........................

43

.....................

45

......................

51

Delay-Differential Systems

4.1

The Lattice of Behaviors

4.2

Input/Output Systems

4.3

Transfer Classes and Controllable

4.4

Subbehaviors and Interconnections

Assigning

4.6

Biduals of

..........

59

.....................

73

..................................

76

....................................

89

Systems

Nonfinitely Generated

...................

.........................

the Characteristic Function

First-Order

.......................

Ideals

5.1

Representations
Multi-Operator Systems

5.2

The Realization Procedure of Fuhrmann

5.3

First-Order Realizations

5.4

Some

.....................

9&

104
115
129

................................

135

...................................

138

.....................

148

...................................

157'

...................................

162

......................................................

169

...........................................................

175

References
Index

23

.........................................

..................

4.5

7

......................................

The

Behaviors of

I

Minimality Issues


Introduction

I

equations
Delay-differential
(DDEs, for short) arise when dynamical systems
modeled. Such lags might for instance
if some
are being
occur
time-lags
time is involved
in the system or if the system needs
transportation
nonnegligible

with

a

amount of time

certain

feature

of

a

to

system

sense

information
is that

time-lags

with

and react

on

dynamics

the

The characteristic

it.

at

a

certain

time

does

only depend on
The dependence on the past can
that of a constant
for instance the
a so-called
retardation,
point delay, describing
reaction
time of a system. More generally,
the reaction
time itself
might depend
on time
Modeling such systems leads to differentialdifference
(or other effects).

also called
equations,
equations with a deviating
differential
argument, in which
the instantaneous

not

of the system but also on some past
take various shapes. The simplest
type is

state

values.

the

unknown function

various

time

and its

instants

if the process under

time interval.
a certain

t--rk.

In this

theory

case

equations,
the term

a

for

distributed

with

their

values

respective

at

on

form of past dependence arises
the full history
of the system over

ma*matical formulation
instance
integro-differential
delay, as opposed to point

leads

general
In
equations.

delay,

to

has been

use the term
type of past dependence. Wewill consistently
delayfor differential
differential'equation
equations having any kind of delay involved.

coined

All

for

occur

different

depends

investigation

functional-differential
control

derivatives

A completely

the

this

delay-differential

infinite-dimensional

twofold

equations
systems.

way.

On the

as

abstract

described

The evolution

hand,

above

fall

of these systems
the equations
can, in certain

in

the

can

category

be described

of
in

be
circumstances,
equations on an infinite-dimensional
space.
of all initial
The space consists
which in this case are segbasically
conditions,
interval
of appropriate
ments of functions
over
a time
length. This description
leads to an pperator-theoretic
of the
framework, well suited for the investigation
of these systems.
For, a treatment
of DDEs based on funcqualitativeIbehavior
tional
methods we refer to the books Hale and Verduyn Lunel [49] and

analytic
Diekmann et al. [22] for functional-differential
and to the introducequations
linear
tory book Curtain and Zwart [20] on general infinite-dimensional
systems
in control
functions
theory. On the other hand, DDEs deal with one-variable
and can be treated
with "analysis
to a certain
extent
techon W' and transform
of DDEs in this spirit
to the books Bellman
we refer
niques. For an investigation,
and Cooke [3], Driver
and Norkin
and
[23], El'sgol'ts
[28], and Kolmanovskii
a

formulated

one

differential

H. Gluesing-Luerssen: LNM 1770, pp. 1 - 5, 2002
© Springer-Verlag Berlin Heidelberg 2002


Introduction

2

1

Nosov

[65]

at

and the references

analyzing
with

time

All

t4erein.

the

the
an

Our interest

behavior
of their
qualitative
emphasis on stability
theory.

monographs mentioned so far aim
most of the
respective
equations,

DDEs is of

Our goal is an investigation
nature.
a different
of
by DDEs with respect to their general control-theoretic
propTo this end, we will adopt an approach which goes back to Willems
erties.
(see
for instance
[118, 119]) and is nowadays called the behavioral approach to sysIn this framework,
tems theory.
the key notion

for specifying
is the
a system
of that system. This space, the behavior,
can
trajectories
space -of all possible
be regarded
most intrinsic
as the
part of the dynamical
system. In case the
it is simply the corresponding
dynamics can be described by a set of equations,
solution
all fundamental
theory now introduces
space. Behavioral
system properties
and constructions
in terms of the behavior,
that means at the level of the
of the system and independent
of a chosen representation.
In order
trajectories
to develop a mathematical
be able to deduce these properties
must
theory, one

from the equations
in
governing the system, maybe even find characterizations
terms of the equations.
For systems governed by linear
time-invariant
ordinary
differential
this has been worked out in great detail
and has led to
equations
the
book
Polderman
a successful
and
Willems
e.
theory, see,
g.,
[87]. Similarly
for multidimensional
differential
or discrete-time
systems, described by partial
difference
much progress
has been made in this
equations,
see for

direction,

systems

instance

troller,
this

in

governed

most

framework.

Wood'et

[84],

Oberst
the

important
A controller

[123],

al.

tool

and

theory,
system itself,

forms

a

and the

interconnection

of

leads

the intersection

of the two respective

to

a

Wood

of control

to-be-controlled

[122].
can

thus

The notion

of

a con-

also
a

be incorporated
family of trajectories,

system with

a

in

simply

controller

behaviors.

The aim of this

monograph is to develop, and then to apply, a theory which
studied
dynamical systems described by DDEs can be successfully
from the behavioral
of
order
In
view.
to
this
it
is
unavoidable
point
goal,
pursue
the relationship
to understand
between behaviors
and their
-describing
equain full
tions
detail.
For instance,

we will
relation
need to know the (algebraic)
between two sets of equations
which share the 'same solution
space. Restricting
shows that

to

a

reasonable

gebraic
systems
coefficients

setting,
we are

class

well

going

of systems, this
for further

suited
to

study

and commensurate

consists

can

indeed

be achieved

investigations.
of

(implicit)

and leads

to

an

al-

To.be precise,
the class of

linear
DDEs with constant

delays. The solutions
being considered are
in the space of C'-functions.
all this in algebraic
Formulating
terms, one obtains
where a polynomial
a setting
acts on a module of
ring in two operators
functions.
However, it turns out that in order to answer the problem raised
but rather
has to be enlarged.
above, this setting will not suffice,
More specifcertain
distributed
ically,
delay operators
(in other words, integro-differential
in our framework.
These distributed
equations) have to be incorporated
delays
have a very specific
feature; just like point-delay-differential
operators

they are
determined
in fact they correspond
to certain
by finitely
rational
many data,
point


1

Introduction

setIn order to get an idea of this larger
in two variables.
algebraic
of scalar DDEs are needed. Yet, some
properties
only a few basic analytic
indeed the
careful
to see that this provides
are necessary
algebraic investigations
allows
draw
it
In fact,
framework.

one to
far-reaching
subsequently
appropriate
the behavioral
even for
approach
systems of DDEs, so that finally
consequences,

functions

ting,

can be initiated.

of

As

which in

algebra

a

consequence,

the

is

fairly

our,

opinion

monographcontains
interesting

by

a

considerable

part

itself.

remark that
delay-differential
systems have already been studpoint of view in the seventies,
algebraic
see, e. g., Kamen [61],
These
have
initiated
the theory of SysMorse [79],

and Sontag [105].
papers
of dynamical
towards
which developed
tems over rings,
an investigation
systhe
itself.
in
evolve
the
tems where
Although this point of view
trajectories
ring
whenever
leads away from the actual system, it has been (and still
is) fruitful
the
Furof
are
investigated.
ring
concerning solely
operators
system properties
and difficult
thermore it has led to interesting
problems.

purely ring-theoretic
of sysit is not in the spirit
Even though our approach is ring-theoretic
as well,
for simply the trajectories
live in a function
tems over rings,
space., Yet, there
between the theory of systems over rings. and our apexist
a few connections
proach; we will therefore
present some more detailed
aspects of systems over

We want
ied

rings

to

from

an

later

in the

book.

of the book. Chapgive a brief overview of the organization
the
class
of
DDEs
consideration
under
along with
introducing
A
above.
and
relation
bementioned
the algebraic
simple
setting
very specific
differential
and
to
tween linear
a
study
equations
DDEs'suggests
ordinary
ring
of operators

of point-delay-differential
as certain
as well
operators
consisting
distributed
delays; it will be denoted by H. In Chapter 3 we disregard the inthe ring 'H from a
and investigate
as delay-differential
operators
terpretation
purely algebraic point of view. The main result of this chapter will be that the
ring'H forms a so-called elementary divisor domain. Roughly speaking, this says
transformain that
that matrices
with entries
ring behave under unimodular
Wenow
ter

proceed

2 starts

Euclidean

domains.

The fact

that all operators
in H
question whether these data
(that is to say, a desired operator) can be determined exactly. Wewill address
this problem by discussing
of the relevant
constructions
symbolic computability
in that ring.
of H as a convolution
we will
Furthermore,
present a description
of distributions
with compact support.
In Chapter 4 we fialgebra consisting
nally turn to systems of DDEs. We'Start with deriving a Galois-correspondence
between behaviors
and the modules of annihilating
on the one side
operators on
the other.
of
Among other things, this comprises an algebraic characterization
systems of DDEs sharing the same solution
space. The correspondence
emerges
from a combination
of the algebraic
of 'H with the basic analytic

structure
of scalar
DDEs derived
in Chapter
properties
2; no further
analytic
study of
tions
are

like

to

with

matrices

determined

over

by finitely

many data

raises

the

1 Introduction

systems of DDEs is needed.*

machinery

for

addressing
Therein,

sections.

quent

purely

Galois-correspondence
system-theoretic
problems

The

the

of the basic

some

constitutes

of systems

concepts

an

studied

in

the

theory,

efficient
subse-

defined

of trajectories,
will be characterized
of
by algebraic properties
will
We
equations.
mainly be concerned with the notions of conand the investigation

partitions
input/output
(including
causality)

in

terms

the associated

trollability,

of interconnection

theory,

control

well-known

of systems.
touches upon the central
The latter
concept of
control.
The algebraic
characterizations
the
generalize

feedback

results

for

described

by linear time-invariant
ordinary difequations.
finite-spectrum
assignment problem,well-studied
in the analytic
framework of time-delay
systems, will be given in the
In the final
algebraic setting.
Chapter 5 we study a problem which is known as
in case of systems of ordinary
realization
differential
If we
state-space
equations.
cast this
for DDEs, the problem amounts to
context
concept in the behavioral
form

finding system descriptions,
which, upon introducing
auxiliary
variables,
DDEs of first
-order
and of retarded
explicit
(with respect to differentiation)
we aim at transforming
type. Hence, among other things,
implicit
system deinto explicit
first
order DDEs of retarded
ones.
scriptions
Explicit
type form the
simplest kind of systems within our framework. -Of the various classes of DDEs
in the literature,
investigated
they are the best studied and, with respect to
the most important
The construction
of such a description
ones.
applications,
in other
(if it, exists) takes place in a completely polynomial setting,

words,
the methods of this chapter are different
no distributed
delays arise. Therefore,
from what has been used previously.
As a consequence and by-product,
the construction
class of systems including
even works for
a much broader
for instance
certain
differential
A complete characterization,
partial
equations.
however, of
first
order description,
will be derived only for
systems allowing such an explicit
systems

A

ferential

new

-

of the

version

DDEs.

A

more

detailed

We close
of the

description

of the

the

first

introduction

applications

with

occurred

MacDonald
of Volterra

early

ship

forties

the

glected.
At

in

is

given

in

its

re-

basically

the

several

and automatic

point the
similarity
similarities
occasions

[22]

unnoticed

DDEs got much at

existing

this

remarks

delays

the

in

great

reader

"tention
steering.

feedback

interest

familiar

on

population

and Diekmann et al.

Because of the

structural
out

[70],

remained

stabilization

tems

chapter

of DDEs. One
applications
dynamics, beginning with the
the 1920s. Since population
models are in
this area and refer to the books Kuang [66],

some

models of Volterra
in
predator-prey
not discuss
we will
general nonlinear,

the

of each

contents

introduction.

spective

with

in

and the references
for

almost

therein.
decades

The work

only in
Minorsky [77] began to study
He pointed
for these sysout that
mechanism can by no means be necontrol
theory during that time and
two

and

when

the

paper

[84]

of

Oberst

will

notice

the

of systems of DDEs to multidimensional
systems. Wewill point
and differences
between these two types of systems classes on
later

on.


1 Introduction

the

decades

rapid

work

of

Minorsky
DDEs; for

of Kolmanovskii

239].

led

to

other

more

details

and Nosov

[65]

applications
about

and

that
list

and the

a

period
of ap-

was Myschkis
a
[81]
of a general
equations and laid 'the foundations
that appeared ever since
theory of these systems. Monographs and textbooks
include
Bellman and Cooke [3], El'sgol'ts
and Norkin [281, Hale [481, Driver
[23],
and Nosov [65], Hale and Verduyn Lunel [49], and Diekmann et
Kolmanovskii
A nice and brief overview of applications
of DDEs in engineering
al. [22].
can

plications

[23,

of

theory

preface

the

instance

the

of the

development

for

see

follow

to

Driver

in

pp.

introduced

of functional-differential

class

be found

the

in

book

Kolmanovskii

following
examples of systems

the

who first

It

[65],

and Nosov

from

which

we

extract

and mixing processes
natural
time-lag arises

are
engineering,
because
due
a
delay,
needs to complete its job; see also Ray [89, Sec. 4.5]
to the time the process
function
for an explicit
form. Furthermore,
example given in transfer
any kind
of system where substances,
or energy
information,
(wave propagation in deep
transmitted
is

certain
to
a
being
distances,
experiences
space communication)
An additional
time.
time-lag might arise due to
time-lag due to transportation

list.

In

chemical

standard

the time
the

needed for

system

to

sense

certain

reactors

with

measurements

information

(ship stabilization)
(biological
models).

for

to be taken

or

it

A model

and react

on

delay equaengine, given by a linear system of five first-order

in [65,
variables
can be found
inputs and five to-be-controlled
DDEsof neutral
Sec. 1.5]. Moreover a system of fifth-order
type arises as a linear
model of a grinding
Finally we would like to mention
process in [65, Sec. 1.7].
model of the Mach number control
in a wind tunnel
a linearized
presented in
Manitius
equations of first order with
[75]. The system consists of three explicit
but not in the input
a time-delay
only in one of the state variables
occurring

of

a

turbojet

tions

with

channel.

three

In that

paper

Mach number is studied

the

problem of feedback
and various

different

control
feedback

for the

regulation

controllers

are

of the
derived

by transfer function methods. This problem can be regarded as a special case of
the finite-spectrum
also be solved within
assignment problem and can therefore
our algebraic
approach developed in Section 4.5. Our procedure leads to one of
the feedback controllers
(in fact, the simplest and most practical
one) derived
in

[75].


Algebraic
Delay-Differential
2 The

Framework

for

Equations

specific class of delay-differential
equations we
In this way

are
some basic,
yet important,
properties.
make clear that,
and how, the algebraic
we hope to
approach we are heading
for depends only on a few elementary
of the equations
under
analytic properties
consideration.
The fact that we can indeed proceed by mainly algebraic
argufrom the structure
ments results
of the equations
under consideration
together
with'the
in. To be precise,
restrict
to
we will
type of problems we are interested
linear
with
coefficients
and
constant

commensurate
delay-differential
equations
We are not aiming at solving
these equaon the space C' (R, C).
point-delays
chapter

In this

we

interested

tions
our

in

introduce

the

and derive

and expressing
the solutions
-in terms of (appropriate)
initial
data.

For
will
suffice
it
know
that
the
solution
to
DDE
of
a
purposes
(without
space

conditions),

initial

"sufficiently
polynomials

the

e.

kernel

of the

delay-differential

associated

operator,.

In essence, we need some knowledge about the exponential
solution
defined
space; hence about the zeros of a suitably

function

order

in

the

in

characteristic

Yet,

L

rich".

is

in the

complex plane.

by algebraic

the

has to be
appropriate
setting
handle
also
to
driving
goal
systems
of DDEs, in other words, matrix equations.
In this chapter we will
develop the
of delay-differential
a ring
algebraic context for these considerations.
Precisely,
not
operators
acting on C1 (R, C) will be defined,
only the pointcomprising

induced by the above-mentioned
delay differential
operators
equations but also
distributed
certain
delays which arise from a simple comparison of ordinary
differential
and DDEs. It is by no means clear that
the so-defined
equations
for studying
operator ring will be suitable
systems of DDEs. That this is indeed
the case will turn out only after a thorough algebraic
study in Chapter 3. In the
with introducing
that ring and providing
present chapter we confine ourselves
results
about DDEs necessary
for later
some standard
In particuexposition.
under consideration
lar, we will show that the delay-differential
are
operators
on C1 (R, C).
surjections

found

first.

As the starting
ear DDEwith

equation

to

pursue

force

The

point
constant

of

our

in this

means,

direction

investigation,

coefficients

is

let

our

us

consider

and commensurate

of the type

H. Gluesing-Luerssen: LNM 1770, pp. 7 - 21, 2002
© Springer-Verlag Berlin Heidelberg 2002

a

point

homogeneous,
delays, that

linis

an


2 The

Algebraic

Framework
N

M

EEpijf( )(t-jh)=O,
i=0

where

N,

delays

involved.

ME

length,

from

No, pij

For

now on we

purposes

our

which

above reads

R, and h > 0 is the smallest
of
delays are integer multiples

c

Hence all

commensurate.

unit

tER,

j=0

it

suffices

to

easily be achieved
only be concerned

will

assume

the

with

be important
for
R. Moreover,

focus

the solution

on

The choice

C

=

C'

:

=

is

ff

C

hence

a

of initial

(R, C),

is

satisfiedl.

1 (2. 1)

L

is considered

any kind

hence

that'equations

(N

=

differential

briefly

about

think

Let

us

requirement

the

minimum amount

for

E

that

solution

unique
t

cover

particular

in

short)

for

for

as

'C is

invariant

corresponding
ring of

larger classes of functions
be discussed occasionally

well

linear
as

pure

time-invariant

delay equations

0).

the

a

(2.1)
(ODEs,

of the type

equations

full
but

the

over

In a certain
delay-differential
operators.
way, however,
be incorporated
in the algebraic
approach; this will
the book.
throughout

Observe

the

on

conditions

on

convenient,

very

module

can

ordinary

equation

(2.1)

equation-

C'

algebraically

shift,

and

E

the

imposing

space in

A C)

differentiation

not

we are

B

under

and the

I

tGR.

that

setting

our

axis

rather

=

j=0

i=0

will

h

delay to be of
Therefore,

axis.

M

EEpjjf(')(t-j)=0,
time

the time

case

point
h, thus

as

N

It

of the

constant

the smallest

by rescaling

can

length
the

[0, M],

and M is the

initial

solutions

'of

initial

(if any).

It

conditions
be

data

is

for

smooth,

it

should

natural

to

be

be in order

require

fo is some prespecified
delay appearing in (2.1).
amounts to solving
the initial

where

largest

(2.1).

Equation

should

that

function

Disregarding

intuitively

(2.1)

for

f satisfy
on

the

clear
to

what

single

f (t)

=

interval

out

fo(t)
[0, M]

on the
finding a solution
full
time axis R
value problem in both forward
and backward direction.
It also fails
This, is, of course, not always possible.
if
with an arbitrary
smooth initial
one starts
i. e. fo C- C' Q0, M], C),
condition,
and seeks solutions
in L. But,
if fo is chosen correctly
(that is, with correct
data at the endpoints
of the interval

[0, M]), a unique forward and backward
&-solution
this will
be shown in Proposition
2.14.
The solvability
of
exists;
this restricted
initial
value problem for the quite general equation
(2.1) rests on
the fact that we consider
so that
we have a sufficient
amount of
C'-functions,
of
the
initial
condition
differentiability
fo, necessary for solving the equation on

Then

the whole of R.
Remark 2.1
It

is crucial

mensurate

for essentially
delays. As it

all

parts

of

turns

out,

the

our

work to restrict

occurrence

to

DDEs with

of noncommensurate

com-

delays


Algebraic

The

2

Framework

delays of length 1 and V2_ or -7r) leads to serious obstacles preventing an
approach similar to the one to be presented here; see [47, 109, 111, 26].
At this point
in the general
want to remark that
we only
case the
according
which will be derived
properties
operator ring lacks the advantageous algebraic
These differences
will be pointed
for our case in the next chapter.
out in some
in later

more detail
chapters (see 3.1-8, 4.1.15, 4.3.13).

(like

e.

g.

algebraic

Remark 2.2
advanced

retarded
PNO 0

4].

distinguishes

one

These notions

type.

(2.1),

say

of DDEs

theory

the

In

with

occurs

if PNO :
0 and PNj

a

describe

delayed

0 and PNj
0 0 for some

argument.
0 for

=

This classification

j

>

of

equations
whether

j

=

1,

Precisely,
M;
.

.

.

,

0, and advanced

and

retarded,
neutral,
in,
highest derivative
Equation (2.1) is called

the

not

or

said

it

is

in

all

other

if

be neutral

to

cases,

[28,

see

problems in forward
how much differentiability
direction.
of the initial
Roughly speaking, it reflects
for (2.1) being solvable
condition
in forward
on [0, M] is required
see
direction;
the results
for instance
on p.
[3, Thms. 6.1, 6.2, and the transformation
192].
with infinitely
differentiable
Since we are dealing
functions
and, additionally,
these notions
are not
requite forward and backward solvability,

really relevant
p.

for

our

Let

us

the

(2.1)

Equation

rewrite

now

shift

the forward

af (t)
and

when

:=

f (t

1),

-

in

f

where

differential
ordinary
0, where

is a1unction
D

operator

in

the

two

commuting

B

For notational

reasons,

which

ker

will

corresponding

operators.

defined

on

d, Equation

=

dt

R,

(2.1)

reads

as

M

1: 1:
i=0

a polynomial
simply

of the

terms

N

is

value

length

of unit

a

p(D, a)
is

initial

solving

purposes.

Introducing

p(D, u)f

is relevant

pij D'ai

(2.2)

j=0

D and

operators

p(D, a)

C

become clear

a.

The solution

(2.3)

L.
in

space

a

moment, it will

be

conve-

polynomial ring R[s, z]
algebraically
independent
elements
s and z at
our
disposal.
(The names chosen for the indeterminates
should remind of the Laplace transform
s of the differential
operator D and the
z-transform

of the shift-operator
in discrete7time
Since
the shift
U is a
systems.)
be advantageous to introduce
Laurent
on L, it will
even the (partially)
bijection
polynomial
ring
nient

to have

R[s,

an

z

abstract

Z-1

with

N

i=O

pijSY
j=m

Tn, MEZ N E=

No, pij

E R


10

Algebraic

2 The

Associating
cluding

with

Framework

each Laurent

possibly

R[s,

(of

polynomial

shifts)

backward

we

z-1]

z,

delay-differential
embedding

the

EndC (,C),

)

(in-

operator

the ring

obtain

p

p(D, o,)

)

i

(2.4)

then the operator
polynomial,
p(D, 0') is not the
the
D
and
a are
words,
operator
operators
C).
algebraically
over R in the ring
independent
Endc(,C). Put yet another way, C is a faithful
module over the commutative
operator
ring R[D, a, o-1].
if p is

course,

zero

a nonzero

Let

us

like

for

look

now

other

In

on

for

exponential

ODEs one has for

eA*

functions

in the

(NE E pjjDY)
M

p(D, o,) (e A.)

i=O

solution

(2.3).

space

Just

A E C
N

M

E.Y pjjA e-

(e\')

)

\j

,

j=M

i=O

e

A-

(2-5)

j=M

p(A, e--\)e"'
Hence the

exponential

p(s, e-')

function

Before

providing

details

some more

only if

A is

is

it

function

we

of

function,

entire

an

of the

a zero

characteristic

polynomials,

exponential

on

by H(C) (resp. M(C))
on the full
complex plane.

For

a

S C

subset

want to fix

H(C)

In

fl,

S

case

For q

=

0

.

q*

case

elements

f

.

.

01,

denote

,

:=

fj I

JA

denote

V(S)

variety

G

meromorphic)
of all

the set

as

M(C)

entire,

we

call

the set
zeros

f

all

E

func-

common

S}.

V (fl,

write

.

EN0 EM
pijs'.zj
j=
j=

the

p(s, e_S)
0(s)

V(q*)

'

for

.

.

E

m

meromorphic

O(S)

is

the

=

0 for

simply

we

where p

by q*

=

M

H(C)

A e C.

I f (A)

finite,

is

the characteristic

E

C

E

fl)

,

R[s,

function

S

G

V (S).

for
z,

z-']

given

and

by

C\V(O).

the characteristic

variety

and its

of q.

and A E C let

ord.\ (f

for

the

(resp.

of entire

EN0 EM
Pijsie-i'
j=

q*(s)

For

define

R(s) [z, z-'],

P

R[s]\f

In

the ring

S, thus

of

V(S)

(4)

the

2.3

zeros

(3)

if and

be called
=

Denote
tions

(2)

solution

a

will

notation.

Definition

(1)

is

therefore

0. Obviously,
equation p(D, o,)f
polynomial
(or quasi polynomial).

delay-differential
known as exponential
the

some

e,\*

function

which

multiplicity
'

,

of A

minf

k E

as a zero

No I
of

f.

f(k) (A)
If

f

=-

76 0}
0,

we

put

ord.\(f)

=

oo


2

(1) of the next proposition
of
ODEs, the multiplicities

The

Algebraic

Framework

11

standard
of DDEs. Just like
in the theory
zeros
correspond to exponen
characteristic
monomials in the solution

the
tial
space. As a simple consequence we include
fact that delay-differential
are surjective
on the space of exponential
operators
polynomials.
Part

for

Proposition

(1)

2.4

R[s,

Let p e
For

k

ek,A

(t)

is

the

z-'] \10}.

z,

by ek,A

and A E C denote

No
tke,\'.

p(D, u)ek,,%
,=o

In

(2)

particular,

ao,

is

polynomials
a

:=

al+a

E

L the

exponential

monomial

(p*)()(A)ek-K,A.
only if ord,\ (p*)

if and

surjective

a

B

ord,\(p*)

:

!

C with

(k)'

characteristic

the

E

ponential
cisely, let

p(D, o)

C ker

ek,X

H(C) is called
operator p(D, o).
The operator
p(D, o)

p*

E

Then

al+a

function

endomorphism.
f ek,A I k E No,

=

span(C

0.

Then, for

:

0 such that

( 1=0

all

E B

el,,\

k.

>

of the

on

The function

delay-differential
the

A E
there

space

of

ex-

C}.

More pre-

exist

constants

+a

p(D, a)

E a,,

(2.6)

el,,\.

e,,,,\

r.

PROOF:
verified

(1)

Let p

=

following

in the

(p(D, u)ek,.\)

pijs'zi

I:i,j

E

R[s,

z,

z-1].

The asserted

identity

is

easily

way:

di

(t)

Pij

[(t

Tt

i

_

j)k

e)(t-j)]

EP'j

10

di

dk

Tti dAk

(eA(t-j)

1,3

dk

(E pjjA'e'X(t-j)

dAk

)

dk
dAk

(p*(A)e\t)

1,3

k
=

E
K=o

The rest

of

(1)

(k)

(p*)( ')(A)ek-r.,A(t)-

K

is clear.

(2)
(p*)

It suffices
to establish
on 1.

(2.6). We proceed by induction
(a)
Then
0
c
:by assumption.
(A).
For I =' 0 it follows
from (1) that p(D, o) (c- 1 ea,,\)
as desired.
eo,,\,

Put

=

For 1 > 0 put

al+a

1+a)a

1
c

-1

1+a

p(D, o,)(al+ael+a,,\)

=

al+a

E
r.=a

.

Then, by

(1 a)

virtue

of

(1),

+

K

el+a-r.,,X

=

el,,x

+

1:'bjej,,\
j=o

c


Algebraic

2 The

12

for

The

solely
with

role

C.

G

exponential
the equation

foregoing

same

bj

constants

some

involving
suitably

Framework

By

ODEs,
in the

that

sense

solution

bjej,,\

their

have preimages
them

Combining

1.

-

El

play exactly

functions

characteristic

show that

in the

functions

the

ei,.\ with i < 1 + a
the desired result.

yields

above

considerations
for

as

induction

monomials

to the

correspond

zeros

the

exponen-

the
to OI?Es is that
complex plane unless it
Since this property
will be of central
degenerates to a polynomial.
importance
for the algebraic
about the

setting
(in fact, this will be the only information
solution
a short
proof showing
spaces of DDEs we are going to need), we include

tialmonomials

function

characteristic

how it
in

(1)

(1)

z-1]

z,

exist

the characteristic

the

of

p*
issues,

classical
be

can

C,

C(I

< 00 4==> P

[88]

found,

see

(1) Letting

p

=

tion

suffices
oo.

Theorem,

defined

a

much

[3,
(2)

also

In

i=O

j =M

C(l

+

order

R[s,

z,

z-']

in

a

all

S

C

k E Z and

0

(C'

details

13].

about

As
for

our

5y,

Pij

M

we are

the

R[s]\f

E

of the

location

dealing

not

01.
zeros

stability

with

purposes.

we can

straightforwardly

estimate
M

<

C(l

+

1:

ISI)N

e-jRe'

j=M

ISI)Neaftesj,
and

constant

get

to

simply

one

for

some

suffices

.,i=

show

a

Let

p be

the

desired

has to make

maxflml,

=

as

the

in

result

sure

IMIJ.
proof

from

the

that

of

(1)

and

Hadamard's
order

assume

Factoriza-

(of growth)

of

p*,

as

ri-M,
(see [54,

for

more

Ch.

1,N 0 EM
j=

suitable

to

Zko

=

1: 1: jp,jjjSjie-jRes

:5

<

where C > 0 is

ISI)N ealResi

+

M

N

<

to

The estimate

satisfies

variety

paper

1P* (S) 1

(2) It
#V(p*)

Theorem.
embed

0 and N G No such that

>

a

the above information

PROOF:

a

section

.'Then

constants

#V(P*)
In

Factorization

later

2.5

R[s,

jp*(S)1:
(2)

in

in the

zeros

algebra.

Proposition
there

be useful

will

below

Paley-Wiener

Let p E

many

be deduced from Hadamard's

can

part

The main difference

space.

infinitely

has

Def.

deduced either

log log M(r; P*)

log

1.11.1])
from

is

(1)

r

bounded
or

,

where M(r;
from

from simple

above

properties

p*)

by

one.

max

lsl=r
But

of the order

jp* (s) 1,
be
can easily
concerning sums

this


Algebraic

2 The

functions,

of entire

products

and

see

[54,

4.2].

Sec.

Ramework

Now Hadamard's

13

Factoriza-

0(s)e"+O, where
the form p*(s)
in C. Com0 G C[s] collects the finitely
many zeros of p* and a, 0 are constants
s'e-j"
of
the
linear
and
j:Nj= 0 EM,,,
independence
using
paring with p* (s)
pij
j=
0 for
monomials over C yields
the exponential
a E I-M,...,
-ml and pij,
El
j =34 -a, which is what we wanted.
[54, 4.9] implies

Theorem

tion

p*

that

is of

=

=

=

Let

us

the

first,

now express

the

we

have

that

space

if

solution
for

leads

Corollary

dim ker

(b)

0

For

(b)

Cz

can

equivalent

z-']

far

is

a

simple

(shifted)
but

differential

R[s]

< oo 4= .

p

R[s,

z-1]

and p E

also

ker

be

one

O(D)

C

obtains

a

z,

O(D)

=

z

ko

we

for

0

t

H(C).

E

In

kerp(D,

for

some

o,)

follows.

rise

eA('-)f

In infinite-dimensional

[t

-

the value

L, t].

to

of

=

,

E

R[s]\101.

0

H(C).

G

Each

us

(2.7)

4 will constitute
give an example.

first

s

-

qf
L,
f
satisfying
(D
A)g
ODE, we then obtain
for

c

A. Since

p*(A)

where

is

t

_

1)g)(t)

theory,
depends

this
on

=

_

I

operator
the past

4

Using

-

((e,\LOL

diagram

the

these operators

e,\LZL

=

making

4:,C --->,C

1)\

calculate

at time

0

'C

of all

control

qf

k E Z and

P*

-.#

map

g c L

of this

(,r)d-r

(4f)(t)
since

below

to

DDEs. Let

to

Z and p

order

need to find

first

(b)

the
pair (p, 0) which satisfies

L.
an operator
on
Precisely,
using the
differential
and
the
of
the
operator
surjectivity
o,)
as

unique well-defined

approach

our

Example 2.7
Let A E R, L
we

Secondly,

operator.

characterization

have

kerp(D,

C

interpreted
in (b) gives

The collection

commutative.

E

For

spaces..

finite-dimensional

a

ODEs are involved.

case

p(D

P*

has

important

Ic

setting

of solution

in terms

operator

Then

p(D, o,)

conditions

inclusion

O(D)

it

in

ker

Part

so

2.6

Let p E R[s, z,

(a)

if

to the

of kernels

the inclusion

obtained

delay-differential

a

only

and

2.4(l)

Proposition

results

the
the

the

=

0,

algebraic

we

map in

solution

have

(2.7),
g(t)

L

eAf (t
is called

of

f

on

-

-r)d-r.
a

distributed

the full

time

delay,
segment


Algebraic

2 The

14

Framework

Remark 2.8
Let

us

tion

P

verify

that

as a

quotient

such that

R(s, z).

f

Then,

using

which

is

p

(D)
P-

quotient

0

Now we

and not

ready

are

define

P0,

be

z, z-

R[s] \f 01

as

O(D)g

satisfying

L

c-

of the

P(D, u).

we

obtain

p(D, u)g

P(D, u)
O(D)g
)

(D) (O(D)h

since

P(D, o,).

As

introduce

pick
=

-

in

particular
Corollary

representa-

2.6(b)

O(D)

=

(D)

-

the map 4

and
Py

=

in

f Wewish

=

P(D, o)

=

P-

such that
.

O(D)

h G L such that

-

consequence,

a

particular

the

on

independent
end, let p,

To do so,

we

to

the

convenient

quite

=

zero,

9 ker

normalization

h

(O(D)h

f

=

=

-

f

-

depends only

g.

),
=

on

representation.

the

4 as they occur in (2.7).
a-' is omitted.
This will

of operators

ring

analogue where the backward shift
later
considerations
for causality

for

and, occasionally,

on

2.9

Define

7j:=

p0 Ip
q

Ho

E

R(s)[z, z-1]
in R(s).
Letp c R[s, z, z-1]

R[s,

z-1],

z,

R[s]\101,

0

E

E

H(C)

P*

0

-

I q*

R(s) [z, z-1]

R(s) [z]

Hn

where

G

q E

=

ring of

denotes

the

and

R[s]\f0J

I

H(C)

1,

Iq*

R(s)[z]

E

E

H(C)

polynomials

Laurent

in

z

with

coef-

ficients

(2)

Define

4

4: L
Just

like

Henceforth

be

polynomials

such that

q:=

P-

0

the operator

as

)

L,

p(D, o,),,

f

)

1

p(D, a)g,

the map 4 is

where g E L is such that

simply

the term DDErefers

'H and Ho are subrings
Obviously,
ring homomorphism

called

to any

with

delay-differential

equation
of

unity

a

O(D)g

R(s) [z,

of the form

z-

=

(=-'H.

f.

operator.

df

1] inducing

=

h.

the

injec-

tive

H(C),

H

Furthermore,

the operators

H

4

are

)

0

the

purposes.

Definition

(1)

L and choose g,

=

indeed

and ker

be

E

R[s,

and let

p(D, u)g

to show that

Wealso

H(C)

E

Pick

the map 4 in (2.7) is
in R(s, z). To this

C-linear

Endr_(L),,

q*.

q

and

we

ql

(2.8)

have the injection

)4.

(2.9)


Algebraic

2 The

Frarnework

15

C L, it is easily
of R[D, a, a-']
seen that

Using commutativity
(2.9). is a ring
that the operators
homomorphism, which means in particular
4 commute with
each other.
Notice that the embedding extends (2.4),
turning L into a faithful

H-module.
In

Section

3.5

we

that

the mappings

Part

(b)

for

R[s]

all

of

one

4

describe

our

2.6

O(D)

can

now

ring

R[s,

and p E

to

on

0

divides

Recall

from

- =*

describe

the

showing

of distributions,

L.

into

the

p in

the

algebraic

which share the

equations

terms

be translated

z-'].

z,

H in

operators

o-)

kerp(D,

C

objectives

delay-differential

the

convolution

are

Corollary

of

ker

be

will

introduction,

relation
solution

same

(2.10)

ring.H
that

between

it

will

systems

of

Characterizing

space.

of solution

task for which
more general
spaces is only a slightly
simple, case has been settled by simply defining the operator
The equivalence
(2.10) suggests that the operators in H should be
taken into consideration
for the algebraic
of DDEs. This extension
investigation
will turn out to be just right
in Section
4. 1. where we will
see that
(2. 10) holds
true for arbitrary
form.
even in matrix
delay-differential
operators,

the

inclusion

special,
ring suitably.

and

now a

Remark 2.10
2.9 has been introduced
in the paper [42].
first
ring H as given in Definition
literature
In a
before.
appeared in different
shapes in the control-theoretic
the ring of Laplace transforms
of H has been introduced
context,
very different
in the paper [85] to show the coincidence
of null controllability
and spectral
for a certain
class of systems under consideration.

In a completely
controllability
different
in [63]. Therein,
e generated
a ring
way, the ring Ho was also considered

The

It has

by the

0,\(s)

functions

entire

and their

=

derivatives

is

introduced

in

+ B(e-')N(s)
A(e-'))M(s)
(9[s, e--]. One can show by some lengthy
that 'Ho is isomorphic
to this
Notice for instance
computations
ring (9[s, e-'].
that 0,\ (s)
1. In [9] and [81 the
(s) for p and 0 in Example 2.7 and L
approach of [63] has been resumed.

order

to

achieve

coefficient

identities

Bezout

matrices

over

sl

=

-

I with

the extension

=

At this

point

DDEs with

wish

we

that

for

to

partial

take

a

brief

differential

excursion

and compare the

situation

for

equations.

Remark 2.11
In

the

[84]

a very
comprehensive
algebraic
study of multidimensional

The common feature
of the various
kinds of sysperformed.
tems covered in [84] is a polynomial
of
ring K[si,...'
operators
acting on
s,,,]
a function
differential
with
partial
space A. This model covers linear
operators
coefficients
constant
acting on C' (RI, C) or on D'(Rm) as well as their real
and discrete-time
of partial
versions
on sequence
counterparts
shift-operators

systems

paper

has been

16

2 The

spaces.

It

module A constitutes

finitely.

and

equations)

generated
derived,
algebra

is

commutative

ample

for

5.1.3

a

systems).

From

"suffices"

to

33]

p.

large

a

K[sj,...,s,,,]-modules.

of

(54),

[84,

has been shown in

responding
egory

AlgebraicFramework

From this

submodules

that

K[sl,...,

of

brief

of the structural

overview

of view this

point
stay in the setting

of

between

s,,,]

to apply
making it feasible
in multidimensional
to problems

our

(the

for

polynomial

the

cor-

the

cat-

solution

spaces

of

annihilating

machinery of
systems theory (see Exsets

powerful

the

of multidimensional

properties

says that

cases

within

cogenerator

duality

a

these

all

in

injective

multidimensional

systems it

ring in order to achieve
of relations
between solution
into
a translation
At [84,
terms.
algebraic
spaces
Oberst
has
observed
that
his
does
not
cover
approach
p. 171
delay-differential
We wish to illustrate
fact by giving
this
a simple
equations.

example which
shows that L is not injective
in the category
of R[s, z]-modules.
Recall
that
M is said
be injective,
to
if the functor
an
R[s, z]-module
is
of
the
exact
on
R[s, z]-modules [67, 111, 8]. For
category
HomR[,,,] (-, M)
a

operator

-

morphism

Ln the

E
I

HomR[,,,,]

that

note

(f (el),

-4

.

.

,

.

E

ai

L As

a

mentioned

functor

is

Consider

-C--

The inverse

associates

,

that

takes

consequence,

R[s, z]m to R[s, Z]n,
given by P(D, a : L

map from

as a

(R[s, Z]n L)

f (en) T.

homomorphism

(D, a)

pi

considered

example.

to.

given by f

is

anT
Enj=

(aj,...,
element

suffices

it

purposes

our

n

(pj,...,pj

G

P E R[s,

matrix

a

R[s, Z]n
to the

respect

Lm. Now we

-->

with

for

dual with

its

Lnwhere the isoeach
to

z] nXm,
above-

present

can

the

the

the matrices

P

[Z 'I,

=

Q

S

=

[8,

1

Z1.

-

ker]5r
in R[s, Z]2,
while
for
the dual maps one only
has
im(jr
P(D, a) C ker Q(D, a) in C2 as can readily be seen by the constant function
It can be seen straightforwardly
W
from
(0, IT CC2 Hence L is not injective.
Then

=

im

,

=

.

definition

the very

ing again

that

it

of 4 in Definition
2.9 that im P(D, o,)
ker[l,
is natural
the operator
to enlarge
ring from R[s,
=

T

S4 indicat-

z]

to

Ho. We

"takes place in a
fact, that multidimensional
systems theory
polynomial setting",
by no means implies that it is simpler than our setting for
DDEs. Quite the contrary,
we will
see that
generated submodule
every finitely
of a free R-module is free,
which simplifies
matters
enormously when dealing
remark

with

that

the

matrices.

Despite
ilarity

the

complete

different

out

on

will

be part

For

completeness

and later

2.4 about

exponential

sition

algebraic

setting

there

will

arise

a

structural

of systems of DDEs to multidimensional
systems, which will
several occasions
in Chapter 4. In Chapter 5, multidimensional

of

our

investigations

on

use we

multi-operator

want to

monomials.

present

sim-

be pointed

systems

systems.
the

generalization

of

Propo-


Algebraic

2 The

Framework

17

Lemma2.12
Let

R[s,

p G

z-']

z,

R[s]\f

and

A E C and put 1
ord,%
coefficients
f, E C and

(q*).

(a)

If

m<

1,

then

qf

=

(b)

If

m>

1,

then

qf

=

Consider

:

f..

01

that

such

the finite

q

2

:=

sum

Moreover,

R.

E

0

E Lo

f

f, e,,,\

V=

is

f

a

0.

E'-'V=0

b,e,,,\

for

some

b,

C and b,,,-,

e

7

0.

ordinary

=

k, thus ord,\(p*)

differential

E,rn+k
V=0

9

E. ker

ord,\(0)

PROOF: Let
the

2.4(l),

Proposition

and the desired

we

gv

follows

result

(C)

G

qf

obtain

=

1 + k.

Em+kg,'Ev=0(v)

Pg

_.,v=O

(p*) W(A)

since

the existence

=7 0, satisfying

gm+k

=

=

K

0 for

n

q*

of

to

function

a

O(D)g
(P

f.

Using

(A)e,-,.,,\

K

< 1 +

H(C)

E

(applied

2.4(2)

Proposition

O(D)) guarantees

operator
where

g,,,",

let

L with

0.

4 if and only if ord,\ (q*) > m. The function
consequence,
said to be the characteristic
function
of the operator
4.

As

G

k.

EJ

Remark 2.13

Notice
of

that

the

latter

facts

did

we

consider

not

any

for

Such expansions
of retarded
equations

solutions

of solutions

expansions

polynomials.

exponential

do exist,

R+.

on

infinite

as

[102]

see

Wewill

and
not

series

[3,

6],

Ch.

these

utilize

about the solution
only case, where the full information
space is
of ODEs, see also (2.10).
For the general case it will be sufficient
for us to know which exponential
monomials are contained
in the solution
space.
Series expansions
of the type above are important
when dealing
with stability
of DDEs. Wewill briefly
discuss the issue of stability
in Section
4.5, where we
will simply quote the relevant
from the literature.

results
since

needed,

the

is that

Weconclude

differential

697],

p.

our

where

it

elaborate

rather

shows what kind
also

on

L.

stated

is

on

in

a

scalar

fact

This

much

methods.

However,

of initial

conditions

Earlier

in

we

chapter

we

briefly

more

general

would like

the

of

surjectivity

and

can

context

to prove

a

be found

delayin

[25,

and proven with
version
which also

imposed for the DDE (2.1).
the method of steps, the standard

can

us

this

DDEs with
well-known

is

the opportunity
to present

initial
value problems
for-solving

gives

cedure

considerations

operators

be

This
pro-

of DDEs.

addressed

what

kind

of initial

data

should

single
(2.1)
unique
f. Apart from
that f has to be specified
we suggested
requirements,
on an interval
of length
in (2.1).
For instance,
M, the largest delay occurring
of
a solution
the pure delay equation
0 is determined
the
restriction
of
f
completely by
fo := f 1 [0, 1). But in order that f be smooth, it is certainly
that the
necessary
specified

be

in

order

for

to

out

solution

a

smoothness

-

initial

condition'fo

In
t

(v)

be extended to a smooth function
on [0,
(v)
of

all
orders
v E No at the endpoints
(1)
0

can

f 0 (0)
f
other words, fo and all
1. This idea generalizes

derivatives

=

=

=

its
to

derivatives

arbitrary

have to

satisfy

DDEs and leads

the

1] having equal
of the interval.

delay equation

to the restriction

for

given


2 The

18

Algebraic

Framework

(2.11)

has to be compaticondition
below, which simply says that the initial

and also
As
DDE.
ble with the given
our approach
neutral,
comprises retarded,
of
advanced equations
smoothness,
requires
order, and, additionally,
arbitrary
below. However, the profor the result
as stated
not find
a reference
we could
of the proof given below
cedure is standard and one should notice the similarity
the
book
in
those
for part (1) with,
e. g.,
[3, Thms. 3.1 and 5.21. In
presented
C' [a, b] := C' ([a, b], (C) as well as f M for f C- C' [a, b]
the sequel the notation

when taken at the endpoints
a or b.
refers, of course, to one-sided derivatives
in

Proposition

2.14

po'

Let

q

0 :

0, and M>

(1)

For every

=

E

C'

[0, M]

(p(D,
there

exists

f I [0,M]

(2)

f

If

an

(1)

interval

all

[a

f

g() (M)

for

all

0

po

:

pm,

L such

E

f I [k,k+M]

0 for

=_

some

(2.11)

No

E

v

p(D, o,)f

that

the map 4 is surjective

M

on

R, then f

k c

g

and

L.
-=

0.

[a, b]

defined

(v))

(t)

0

g(') (t)

-

for

all

v

E

(2.12)

No

j=o

[a

+

M, b]

1]

which

given

be extended

can

(2.12)

satisfies
in

the

in

unique way to a solution
that
1 + M, b + 1]. (Notice

a

[a

on

-

is included

proposition

as

an

extreme

case

the

where

b=:M.)

end,

To this

--

Epj (D)oif

+

a=O and

(M)

consequence,

(t)

G

condition

initial

G R

-

1, b

t
-

[s],

0

pj,

of f , we show: every fo C C'
To prove the existence
the condition
satisfies
of length b

a > Mwhich

0

for

())

0

function

4 9 L satisfies

(p(D, u)f(v))
C'

a

zi,

pj

g E L

satisf ying

a)f

unique

a

As

fo.

(=- ker

PROOF:
on

=

Ej' o

=

Furthermore,

1.

fo

0 1 where p
let

7io

G

write

po

(s)

=0

consider

ai sz + Sr and
M

po(D)f (t)

g(t)

=

1:

-

fo)

(D) ci

pj

inhomogeneous

the

ODE

(t)

(2.13)

1.

(2.14)

j=1

for

t G

[b,

b +

1]

with

condition

initial

j(v) (b)
(If
a

r

=

0, then

unique

po

solution

1 and
c

C'

=

no

[b,

(v)

f0

(b)

1]

to

(b)

=

g (b)

pj

-

j=1

=

(2.13),

M

M

v

condition

initial

b +

for

(D) ai

fo)

0,...,

r

-

imposed).
(2.14) and j

In any case,

is

satisfies

r-1

(b)

-

E ai j(')
i=O

(b)

=

()

f0

(b).

there

is


Algebraic

2 The

Framework

19

(")

j(') (b)
successively
f 0 (b) for
for
t
No. Therefore,
E [a, b] and
f,
fo (t)
by f, (t)
satisfies
f, (t) =j (t) for t E (b, b+1] is in C'[a, b+1] and, by construction,
(2.12)
In the same manner one can extend f, to a smooth solution
on [a + M, b + 1].
of the ODE
on [a
1, b + 1]; one takes the unique solution
(2.13)

Differentiating
all

and

(2.12)

using

shows

=

defined

the function

E

v

=

-

M-1

pm(D)f (t)

=

g(t)

E pj(D)fj(t

-

-

j)

[a

on

+ M

1,

-

a

+

M]

j=0

initial'data

with

(a

+

M)

fl(')

=

(a)

for

v

=

0,

.

.

.

,

deg pm

-

1 and

puts

f2(t):=f(t+M)f6ra-1
f2(t):=fj(t)fora
1, b + 1] and satisfies
f2 E C' [a
(2.12) on [a + M 1, b + 1].
this
leads to a solution
extension
in L. It is clear from the procedure
Repeating
that the solution
of the initial
value problem is unique.
As for the surjectivity
of 4, observe that it suffices
to show the surjectivity
of P
The
latter
be
can
a function
p(D, a).
accomplished by providing
fo E
C' [0, M] satisfying
A

choice
for
is
follows:
as
simple
(2. 11).
fo
pick a solution
Then

-

-

=

COO[0,

hi

c

C'

[0, M]

that

fo

(2)

M]

of the

ODEpo

:=

h1h2

is

a

desired

is aconsequence

(D) hi

g on the interval
0 and h2 I
[M-0.5,M]
condition.

h2 I [0,M-0.75)

be such that

initial

[0, M]

=

=_

of the uniqueness

in

=_

h2

and let

1. Then

one

E

checks

(1).

El

Remark 2.15
It is immediate

Re f,

f

Im.

E

Proposition
C'

to

see

that

f

for all

L and q cz H the inclusion

C-

ker 4, too. As a consequence,
2.14 remain valid
when L is

replaced

2.6,

by

its

f CEquation

ker

4 implies

(2. 10),

real-valued

and

analogue

(R, R).

Wenow close
of this

the considerations

about scalar DDEs and want to spend the rest
a (somewhat
discussing
some
extreme) example illustrating
The general theory has to be
systems of delay-differential
equations.
until
Chapter 4, when the algebraic results concerning matrices with

chapter

features

of

postponed

in 'H

entries

Example
Weconsider

are

on

available.

2.16

the

homogeneous system

R

s

-2

0-

s

I

0

example is taken from [23,
is presented
in theform.

where it

of DDEs R(D,

-z

-

This

Corollary

-2z

pp.

E

o,)w

=

0, where

R[s, Z]3X3

s_

249] (see

also

the references

given therein),