Athorough purely algebraic studyshows that thissetting is well-suited foranexamination ofdelay-differential systems from the behavioral point of view in modern sys-tems theory.. Wewill
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Trang 5The term delay-differential equationwas coined tocompriseall types of entialequations in which the unknown function and its derivatives occurwithvarious values of the argument. In these notes we concentrate on (implicit)
differ-lineardelay-differential equationswithconstantcoefficients and commensurate
point delays. We present an investigation ofdynamical delay-differential tems withrespectto theirgeneral system-theoretic properties. To this end, an
sys-algebraic settingfor theequationsunder consideration isdeveloped. Athorough purely algebraic studyshows that thissetting is well-suited foranexamination
ofdelay-differential systems from the behavioral point of view in modern
sys-tems theory. The central object is a suitably defined operator algebra which
turns out to bean elementarydivisor domain and thus providesthe main toolfor handlingmatrixequationsofdelay-differential type. Thepresentation is in-
troductory and mostly self-contained, no prior knowledge ofdelay-differential equationsor (behavioral) systems theory will be assumed
There are a number ofpeople whom I am pleased to thank for making thisworkpossible. Iam gratefulto Jan C Willems forsuggestingthetopic "delay-
differential systems in the behavioral approach" to me. Agreeing with him,
that algebraic methods and the behavioral approach sound like a promising
combinationfor thesesystems,I startedworkingontheproject and hadnoidea
of what Iwasheadingfor.Many interesting problemshadtobe 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 discussionswehad in particular at thebeginning of theproject They finally brought me on
therighttrack forfindingtheappropriate algebraic setting.But also lateron,he
kept discussingthesubjectwithmeinaverystimulatingfashion His interest in
computer algebra made methink 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 grewout ofmy Habilitationsschrift at the University of Oldenburg, Germany. Thereaders UweHelmke,JoachimRosenthal,WilandSchmale, and Jan C Willemsdeserve special mention for their generous collaboration I also want, to thanktheSpringer-Verlagfor thepleasant cooperation Finally,mygreatestthanksgo
Trang 6to mypartner, Uwe Nagel, not onlyfor many hours carefully proofreading allthesepagesandmakingvarioushelpful suggestions,butalso,andevenmore,for
being so patient, supportive, and encouraging duringthe timeI was occupied
withwritingthe "Schrift"
Trang 7Table of Contents
1 Introduction I
2 The Algebraic Framework 7
3 The Algebraic Structure of Wo , 23
3.1 Divisibility Properties 25
3.2 Matrices overHo 35
3.3 SystemsoverRings: A BriefSurvey 43
3.4 The Nonfinitely Generated Ideals ofHo 45
3.5 The RingH as aConvolutionAlgebra 51
3.6 Computingthe Bezout Identity 59
4 Behaviors ofDelay-Differential Systems 73
4.1 The Lattice of Behaviors 76
4.2 Input/Output Systems 89
4.3 Transfer Classes and Controllable Systems 9& 4.4 Subbehaviors and Interconnections 104
4.5 Assigning the Characteristic Function 115
4.6 Biduals ofNonfinitely Generated Ideals 129
5 First-Order Representations 135
5.1 Multi-Operator Systems 138
5.2 The Realization Procedure of Fuhrmann 148
5.3 First-Order Realizations 157'
5.4 SomeMinimality Issues 162
References 169
Index 175
Trang 8I Introduction
Delay-differential equations (DDEs, for short) arise when dynamical systems
nonnegligible transportation time is involved in the system or if the system needs
acertain amount of time to sense information andreact on it The characteristicfeature of a system with time-lags is that the dynamics at acertain time does
not only depend on the instantaneous state of the system but also on somepast
values The dependence onthe past cantake various shapes. Thesimplest type isthat of aconstant retardation, aso-called point delay, describing for instance thereaction time of asystem More generally, the reaction time itself might depend
on time (or other effects) Modeling such systems leads to differential- difference
equations, also called differential equations with a deviating argument, in whichthe unknown function and its derivatives occur with their respective values at
various time instants t rk Acompletely different form of past dependence arises
if the process under investigation depends on the full history of the system over
acertain time interval In this case ama*matical formulation leads to generalfunctional-differential equations, for instance integro-differential equations. Incontrol theory the term distributed delay, as opposed to point delay, has beencoined for this type of past dependence. Wewill consistently use the term delay- differential'equation for differential equations having any kind of delay involved.All the delay-differential equations described above fall in the category ofinfinite-dimensional systems. The evolution of these systems can be described in
atwofold way. On the one hand, the equations can, in certain circumstances, beformulated as abstract differential equations on an infinite-dimensional space.The space consists basically of all initial conditions, which in this case are seg-ments of functions over a time interval of appropriate length. This descriptionleads to an pperator-theoretic framework, well suited for the investigation of the
func-tional analytic methods werefer to the books Hale and Verduyn Lunel [49] andDiekmann et al [22] for functional-differential equations and to the introduc-
tory book Curtain and Zwart [20] on general infinite-dimensional linear systems
in control theory. On the other hand, DDEs deal with one-variable functionsand can be treated to acertain extent with "analysis onW' and transform tech-
niques. For an investigation, of DDEsin this spirit werefer to the books Bellmanand Cooke [3], Driver [23], El'sgol'ts and Norkin [28], and Kolmanovskii and
H Gluesing-Luerssen: LNM 1770, pp 1 - 5, 2002
© Springer-Verlag Berlin Heidelberg 2002
Trang 9Nosov [65] and the references t4erein. All the monographs mentioned so far aim
at analyzing the qualitative behavior of their respective equations, most of thetime with an emphasis on stability theory.
Our interest in DDEs is of a different nature Our goal is an investigation of
systems governed by DDEs with respect to their general control-theoretic erties To this end, wewill adopt an approach which goes back to Willems (see
prop-for instance [118, 119]) and is nowadays called the behavioral approach to tems theory. In this framework, the key notion for specifying a system is the
sys-space -of all possible trajectories of that system This space, the behavior, can
be regarded as the most intrinsic part of the dynamical system In case thedynamics can be described by aset of equations, it is simply the correspondingsolution space. Behavioral theory nowintroduces all fundamental system prop-erties and constructions in terms of the behavior, that meansat the level of the
trajectories of the system and independent of achosen representation. In order
to develop amathematical theory, one must be able to deduce these properties
from the equations governing the system, maybe even find characterizations interms of the equations. For systems governed by linear time-invariant ordinarydifferential equations this has been worked out in great detail and has led to
a successful theory, see, e. g., the book Polderman and Willems [87] Similarlyfor multidimensional systems, described by partial differential or discrete-timedifference equations, much progress has been made in this direction, see forinstance Oberst [84], Wood'et al [123], and Wood[122]. The notion of a con-
troller, the most important tool of control theory, can also be incorporated in
this framework A controller forms asystem itself, thus afamily of trajectories,
and the interconnection of a to-be-controlled system with a controller simplyleads to the intersection of the two respective behaviors
The aim of this monograph is to develop, and then to apply, a theory whichshows that dynamical systems described by DDEscan be successfully studiedfrom the behavioral point of view In order to pursue this goal, it is unavoidable
to understand the relationship between behaviors and their -describing tions in full detail For instance, wewill need to know the (algebraic) relationbetween two sets of equations which share the 'same solution space. Restricting
equa-to areasonable class of systems, this can indeed be achieved and leads to an gebraic setting, well suited for further investigations. To.be precise, the class of
al-systems we are going to study consists of (implicit) linear DDEs with constant
coefficients and commensurate point delays. The solutions being considered are
in the space of C'-functions Formulating all this in algebraic terms, one tains a setting where a polynomial ring in two operators acts on a module offunctions However, it turns out that in order to answer the problem raised
ob-above, this setting will not suffice, but rather has to be enlarged. More
specif-ically, certain distributed delay operators (in other words, integro-differential
have avery specific feature; just like point-delay-differential operators they are
determined by finitely many data, in fact they correspond to certain rational
Trang 101 Introduction
functions in two variables In order to get an idea of this larger algebraic
set-ting, only afew basic analytic properties of scalar DDEsare needed Yet, some
careful algebraic investigations are necessary to see that this provides indeed the
appropriate framework In fact, it subsequently allows oneto draw far-reaching
consequences, even for systems of DDEs, so that finally the behavioral approach
can be initiated Asaconsequence, the monographcontains aconsiderable part
of algebra which in our, opinion is fairly interesting by itself
Wewant to remark that delay-differential systems have already been ied from an algebraic point of view in the seventies, see, e. g., Kamen [61],
stud-Morse [79], and Sontag [105]. These papers have initiated the theory of
Sys-tems over rings, which developed towards an investigation of dynamical tems where the trajectories evolve in the ring itself Although this point of viewleads away from the actual system, it has been (and still is) fruitful whenever
sys-system properties concerning solely the ring of operators are investigated. thermore it has led to interesting and difficult purely ring-theoretic problems.Even though our approach is ring-theoretic as well, it is not in the spirit of sys-
Fur-tems over rings, for simply the trajectories live in a function space., Yet, thereexist a few connections between the theory of systems over rings. and our ap-
proach; wewill therefore present some more detailed aspects of systems over
rings later in the book
Wenowproceed to give abrief overview of the organization of the book
Chap-ter 2 starts with introducing the class of DDEs under consideration along withthe algebraic setting mentioned above Avery specific and simple relation be-
tween linear ordinary differential equations and DDEs'suggests to study aring
of operators consisting of point-delay-differential operators as well as certaindistributed delays; it will be denoted by H In Chapter 3 wedisregard the in-
terpretation as delay-differential operators and investigate the ring 'H from a
purely algebraic point of view The main result of this chapter will be that thering'H forms aso-called elementary divisor domain Roughly speaking, this saysthat matrices with entries in that ring behave under unimodular transforma-
tions like matrices over Euclidean domains The fact that all operators in H
are determined by finitely many data raises the question whether these data
this problem by discussing symbolic computability of the relevant constructions
in that ring Furthermore, wewill present a description of Has a convolution
algebra consisting of distributions with compact support In Chapter 4 we
fi-nally turn to systems of DDEs We'Start with deriving aGalois-correspondencebetween behaviors onthe oneside and the modules of annihilating operators on
the other Amongother things, this comprises an algebraic characterization of
systems of DDEssharing the samesolution space. The correspondence emergesfrom a combination of the algebraic structure of 'H with the basic analytic
properties of scalar DDEs derived in Chapter 2; no further analytic study of
Trang 111 Introduction
systems of DDEsis needed.* The Galois-correspondence constitutes an efficientmachinery for addressing the system-theoretic problems studied in the subse-
quent sections Therein, someof the basic concepts of systems theory, defined
purely in terms of trajectories, will be characterized by algebraic properties ofthe associated equations. Wewill mainly be concerned with the notions of con-
trollability, input/output partitions (including causality) and the investigation
of interconnection of systems The latter touches upon the central concept ofcontrol theory, feedback control The algebraic characterizations generalize thewell-known results for systems described by linear -time-invariant ordinary dif-ferential equations. Anewversion of the finite-spectrum assignment problem,-well-studied in the analytic framework of time-delay systems, will begiven in thealgebraic setting. In the final Chapter 5 westudy aproblem which is known as
state-space realization in case of systems of ordinary differential equations. If we
cast this concept in the behavioral context for DDEs, the problem amounts to
finding system descriptions, which, upon introducing auxiliary variables, form
explicit DDEsof first -order (with respect to differentiation) and of retarded
type Hence, amongother things, weaim at transforming implicit system
de-scriptions into explicit ones. Explicit first order DDEs of retarded type form the
simplest kind of systems within our framework -Of the various classes of DDEsinvestigated in the literature, they are the best studied and, with respect to
applications, the most important ones. The construction of such adescription
no distributed delays arise Therefore, the methods of this chapter are differentfrom what has been used previously. Asaconsequence and by-product, the con-
struction even works for amuchbroader class of systems including for instance
certain partial differential equations. Acomplete characterization, however, of
systems allowing such anexplicit first order description, will be derived only forDDEs
Amore detailed description of the contents of each chapter is given in its
re-spective introduction
We close the introduction with some remarks on applications of DDEs One
of the first applications occurred in population dynamics, beginning with thepredator-prey models of Volterra in the 1920s Since population models are in
general nonlinear, wewill not discuss this area and refer to the books Kuang [66],
MacDonald [70], and Diekmann et al [22] and the references therein. The work
of Volterra remained basically unnoticed for almost two decades and only in
the early forties DDEsgot muchat"tention when Minorsky [77] began to study
ship stabilization and automatic steering. He pointed out that for these
sys-tems the existing delays in the feedback mechanism can by no means be
ne-glected. Because of the great interest in control theory during that time and
At this point the reader familiar with the paper [84] of Oberst will notice thestructural similarity of systems of DDEsto multidimensional systems Wewill point
out the similarities and differences between these two types of systems classes on
several occasions later on.
Trang 12the decades to follow the work of Minorsky led to other applications and a
rapid development of the theory of DDEs; for more details about that period
see for instance the preface of Kolmanovskii and Nosov [65] and the list of plications in Driver [23, pp 239]. It was Myschkis [81] who first introduced a
ap-class of functional-differential equations and laid 'the foundations of a general
theory of these systems Monographs and textbooks that appeared ever sinceinclude Bellman and Cooke [3], El'sgol'ts and Norkin [281, Hale [481, Driver [23],
Kolmanovskii and Nosov [65], Hale and Verduyn Lunel [49], and Diekmann et
be found in the book Kolmanovskii and Nosov [65], from which we extract
the following list In chemical engineering, reactors and mixing processes are
standard examples of systems with delay, because anatural time-lag arises due
to the time the process needs to complete its job; see also Ray [89, Sec 4.5]
for an explicit example given in transfer function form Furthermore, any kind
of system where substances, information, or energy (wave propagation in deep
space communication) is being transmitted to certain distances, experiences a
time-lag due to transportation time An additional time-lag might arise due to
the time needed for certain measurements to be taken (ship stabilization) or forthe system to sense information and react on it (biological models). A model
of a turbojet engine, given by a linear system of five first-order delay
equa-tions with three inputs and five to-be-controlled variables can be found in [65,
Sec 1.5]. Moreover asystem of fifth-order DDEsof neutral type arises as alinearmodel of a grinding process in [65, Sec 1.7] Finally wewould like to mention
a linearized model of the Mach number control in awind tunnel presented in
Manitius [75]. The system consists of three explicit equations of first order with
a time-delay occurring only in one of the state variables but not in the input
channel In that paper the problem of feedback control for the regulation of theMach number is studied and various different feedback controllers are derived
by transfer function methods This problem can be regarded as aspecial case ofthe finite-spectrum assignment problem and can therefore also be solved within
our algebraic approach developed in Section 4.5 Our procedure leads to one ofthe feedback controllers (in fact, the simplest and most practical one) derived
in [75].
Trang 132 The Algebraic Framework for
linear delay-differential equations with constant coefficients and commensurate
point-delays on the space C' (R, C). Weare not aiming at solving these tions and expressing the solutions -in terms of (appropriate) initial data For
equa-our purposes it will suffice to know that the solution space of a DDE(without
initial conditions), L e. the kernel of the associated delay-differential operator,.
is "sufficiently rich" In essence, weneed someknowledge about the exponential
polynomials in the solution space; hence about the zeros of a suitably defined
characteristic function in the complex plane.
Yet, in order to pursue by algebraic means, the appropriate setting has to befound first The driving force in this direction is our goal to handle also systems
of DDEs, in other words, matrix equations. In this chapter wewill develop thealgebraic context for these considerations Precisely, aring of delay-differential
operators acting on C1(R, C) will be defined, comprising not only the
point-delay differential operators induced by the above-mentioned equations but alsocertain distributed delays which arise from a simple comparison of ordinarydifferential equations and DDEs It is by no means clear that the so-defined
operator ring will be suitable for studying systems of DDEs That this is indeedthe case will turn out only after athorough algebraic study in Chapter 3 In the
present chapter weconfine ourselves with introducing that ring and providing
some standard results about DDEsnecessary for later exposition. In lar, wewill show that the delay-differential operators under consideration are
particu-surjections on C1(R, C).
As the starting point of our investigation, let us consider a homogeneous,
lin-ear DDEwith constant coefficients and commensurate point delays, that is an
equation of the type
H Gluesing-Luerssen: LNM 1770, pp 7 - 21, 2002
© Springer-Verlag Berlin Heidelberg 2002
Trang 142 The Algebraic Framework
i=0 j=0
where N, ME No, pij c R, and h > 0 is the smallest length of the point
delays involved Hence all delays are integer multiples of the constant h, thus
commensurate For our purposes it suffices to assume the smallest delay to be ofunit length, which can easily be achieved by rescaling the time axis Therefore,
from now on wewill only be concerned with the case h = I and the equation
above reads as
i=0 j=0
It will be important for our setting that the equation- is considered on the full
time axis R Moreover, we are not imposing any kind of initial conditions butrather focus on the solution space in C C' (R, C), hence on
B: = f f EL 1 (2 1) is satisfiedl.
The choice C = C' A C) is algebraically very convenient, for 'C is invariantunder differentiation and shift, hence a module over the corresponding ring ofdelay-differential operators In acertain way, however, larger classes of functions
can beincorporated in the algebraic approach; this will bediscussed occasionally
throughout the book
Observe that'equations of the type (2.1) cover in particular linear time-invariantordinary differential equations (ODEs, for short) as well as pure delay equations (N =0).
Let us briefly think about initial conditions for Equation (2.1) Disregardingthe requirement that solutions be smooth, it should be intuitively clear whatthe minimum amount 'of initial data should be in order for (2.1) to single out
a unique solution (if any). It is natural to require that f satisfy f (t) = fo(t)
and Mis the largest delay appearing in (2.1). Then finding asolution on thefull time axis Ramounts to solving the initial value problem in both forward
and backward direction This, is, of course, not always possible. It also fails if
one starts with an arbitrary smooth initial condition, i e. fo C- C' Q0, M], C),
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 exists; this will be shown in Proposition 2.14 The solvability ofthis restricted initial value problem for the quite general equation (2.1) rests on
the fact that weconsider C'-functions, so that wehave asufficient amount ofdifferentiability of the initial condition fo, necessary for solving the equation on
the whole of R
Remark 2.1
It is crucial for essentially all parts of our work to restrict to DDEs with
com-mensurate delays. As it turns out, the occurrence of noncommensurate delays
Trang 152 The Algebraic Framework
(like e. g delays of length 1and V2_or -7r) leads to serious obstacles preventing an
algebraic approach similar to the one to bepresented here; see [47, 109, 111, 26].
At this point we only want to remark that in the general case the according
operator ring lacks the advantageous algebraic properties which will be derivedfor our case in the next chapter. These differences will be pointed out in some more detail in later chapters (see 3.1-8, 4.1.15, 4.3.13).
Remark 2.2
In the theory of DDEsone distinguishes equations of retarded, neutral, andadvanced type. These notions describe whether or not the highest derivative in,
retarded if PNO: 0 and PNj = 0 for j = 1, ,M; it is said to be neutral if
PNO0 0 and PNj 0 0 for somej > 0, and advanced in all other cases, see [28,
p 4]. This classification is relevant whensolving initial value problems in forwarddirection Roughly speaking, it reflects how muchdifferentiability of the initialcondition on [0, M] is required for (2.1) being solvable in forward direction; see
for instance the results [3, Thms 6.1, 6.2, and the transformation on p 192].
Since we are dealing with infinitely differentiable functions and, additionally,
requite forward and backward solvability, these notions are not really relevantfor our purposes
Let us now rewrite Equation (2.1) in terms of the corresponding operators
Introducing the forward shift aof unit length
af (t) := f (t - 1), where f is a1unction defined on R,
and the ordinary differential operator D = dtd, Equation (2.1) reads as
For notational reasons, which will become clear in a moment, it will be
conve-nient to have an abstract polynomial ring R[s, z] with algebraically independentelements s and z at our disposal (The names chosen for the indeterminatesshould remind of the Laplace transform s of the differential operator Dand thez-transform of the shift-operator in discrete7time systems.) Since the shift Uis a
bijection on L, it will beadvantageous to introduce even the (partially) Laurentpolynomial ring
i=O j=m
Trang 1610 2 The Algebraic Framework
Associating with each Laurent polynomial the delay-differential operator
(in-cluding possibly backward shifts) weobtain the ring embedding
(of course, if p is a nonzero polynomial, then the operator p(D, 0') is not the
zero operator on C). In other words, the operators Dand a are algebraically
independent over R in the ring Endc(,C). Put yet another way, C is afaithfulmodule over the commutative operator ring R[D, a, o-1].
Let us now look for exponential functions eA* in the solution space (2.3). Justlike for ODEsone has for AEC
Before providing some more details on exponential polynomials, wewant to fix
somenotation
Definition 2.3
(1) Denote by H(C) (resp M(C)) the ring of entire (resp meromorphic)
func-tions on the full complex plane.
(2) For a subset SC H(C) define the variety V(S) as the set of all common
zeros of S, thus
V(S) := JAECI f (A) = 0 for all f ES}.
In case S fl, ,fj I is finite, wesimply write V(fl, ,fl) for V(S) (3) For q= P R(s) [z, z-'], where p = EN0EMmpijs'.zj ER[s, z, z-'] and
In case q* is entire, wecall the set V(q*) the characteristic variety and its
elements the characteristic zeros of q.
(4) For f EH(C) and AEC let
I f(k) (A) ord.\ (f minf k ENo 76 0}
denote the multiplicity of Aas a zero of f. If f =- 0, weput ord.\(f) = oo
for Ae C ' ,
Trang 172 The Algebraic Framework 11
Part (1) of the next proposition is standard in the theory of DDEs Just likefor ODEs, the multiplicities of the characteristic zeros correspond to exponen
tial monomials in the solution space. As a simple consequence we include thefact that delay-differential operators are surjective on the space of exponential polynomials.
(2) The operator p(D, o) is a surjective endomorphism. on the space of
pre-cisely, let a := ord,\(p*) ! 0 Then, for all el,,\ E B there exist constants
ao, al+a EC with al+a : 0 such that
+a
PROOF:(1) Let p = I:i,j pijs'zi E R[s, z, z-1]. The asserted identity is easilyverified in the following way:
The rest of (1) is clear
(2) It suffices to establish (2.6). Weproceed by induction on 1 Put c(p*) (a) (A). Then c : - 0 by assumption.
For I =' 0it follows from (1) that p(D, o) (c- 1
ea,,\) =eo,,\, as desired
Trang 1812 2 The Algebraic Framework
for some constants bj G C By induction the functions bjej,,\ have preimages
involving solely exponential monomials ei,.\ with i < 1+a - 1 Combining them
Theforegoing considerations show that characteristic functions play exactly the
samerole as for ODEs, in the sense that their zeros correspond to the tialmonomials in the solution space. The main difference to OI?Es is that thecharacteristic function has infinitely manyzeros in the complex plane unless itdegenerates to a polynomial. Since this property will be of central importance
exponen-for the algebraic setting (in fact, this will be the only information about thesolution spaces of DDEswe are going to need), weinclude ashort proof showinghowit can be deduced from Hadamard's Factorization Theorem The estimate
in part (1) below will be useful in a later section to embed R[s, z, z-'] in a
Paley-Wiener algebra.
Let p ER[s, z, z-1] 'Then
(1) there exist constants C, a > 0 and NGNo such that
jp*(S)1: C(I + ISI)N ealResi for all S C(C'
(2) the characteristic variety satisfies
#V(P*) < 00 4==> P=Zko for somek EZand 0 ER[s]\f 01.
In the classical paper [88] much more details about the location of the zeros
of p* can be found, see also [3, Ch 13]. As we are not dealing with stability
issues, the above information (2) suffices for our purposes
PROOF:(1) Letting p= 1,N.,i= 0EMj= MPij 5y, we can estimate straightforwardly
where C> 0 is asuitable constant and a = maxflml, IMIJ.
(2) It suffices to show Let p be as in the proof of (1) and assume
Factoriza-tion Theorem, one simply has to makesure that the order (of growth) of p*,
defined as
log log M(r; P*)
deduced either from (1) from simple properties of the order concerning
Trang 192 The Algebraic Ramework 13
and products of entire functions, see [54, Sec 4.2]. NowHadamard's tion Theorem [54, 4.9] implies that p* is of the form p*(s) = 0(s)e"+O, where
Factoriza-0 GC[s] collects the finitely manyzeros of p* anda, 0 are constants in C
Com-paring with p* (s) =j:N 0EM,,, pij s'e-j" and using the linear independence of
Let us nowexpress the results obtained so far in terms of solution spaces Forthe first, we have that a delay-differential operator has a finite-dimensionalsolution space if and only if it is a (shifted) differential operator Secondly,
Proposition 2.4(l) leads to the simple but important characterization (b) belowfor the inclusion of kernels in case ODEsare involved
dim ker p(D, o,) < oo 4= p= zko for somek EZand 0 ER[s]\101.
0
Part (b) can also be interpreted as follows Each pair (p, 0) which satisfies theequivalent conditions in (b) gives rise to an operator on L Precisely, using theinclusion ker O(D) Ckerp(D, o,) and the surjectivity of the differential operator
O(D) one obtains a unique well-defined map4:,C ->,C making the diagram
p(D,1)\
commutative Thecollection of all these operators 4will constitute the algebraic
setting for our approach to DDEs Let us first give an example.
In infinite-dimensional control theory, this operator is called adistributed delay,
since the value of qf at time t depends on the past of f on the full time segment [t L, t].
Trang 2014 2 The Algebraic Framework
to show that p(D, u)g =P(D, u). To doso, wepick h GL such that O(D) h= g
Then, using p = P0, weobtain p(D, u)g - P(D, u) = P(D, o) (O(D)h - ),
which is indeed zero, since (D) (O(D)h - ) = O(D)g - (D) = f - f = 0and ker (D) 9 ker P(D, o,). As aconsequence, the map4 depends only on the
quotient P-0 and not onthe particular representation.
Nowwe are ready to introduce the ring of operators 4 as they occur in (2.7).
Wealso define the analogue where the backward shift a-' is omitted This will
be quite convenient for causality considerations later on and, occasionally, fornormalization purposes.
Define 4as the operator
0
Just like p(D, o,),, the map4 is simply called adelay-differential operator
Henceforth the term DDErefers to any equation of the form df = h
Obviously, 'H and Hoare subrings with unity of R(s) [z, z- 1] inducing the
Furthermore, the operators 4are C-linear and wehave the injection
Trang 212 The Algebraic Frarnework 15
Using commutativity of R[D, a, a-'] CL, it is easily seen that (2.9). is aring
each other Notice that the embedding extends (2.4), turning L into a faithful
H-module
In Section 3.5 wewill describe the ring Hin terms of distributions, showingthat the mappings 4 are convolution operators on L
Part (b) of Corollary 2.6 can nowbe translated into
for all R[s] and p E R[s, z, z-']. Recall from the introduction, that it will
be one of our objectives to describe the algebraic relation between systems ofdelay-differential equations which share the samesolution space. Characterizingthe inclusion of solution spaces is only a slightly more general task for which
now aspecial, and simple, case has been settled by simply defining the operator
ring suitably. Theequivalence (2.10) suggests that the operators in H should betaken into consideration for the algebraic investigation of DDEs This extensionwill turn out to bejust right in Section 4.1. where wewill see that (2 10) holds
true for arbitrary delay-differential operators, even in matrix form
Remark 2.10
Thering Has given in Definition 2.9 has been introduced first in the paper [42].
It has appeared in different shapes in the control-theoretic literature before In a
very different context, the ring of Laplace transforms of H has been introduced
in the paper [85] to show the coincidence of null controllability and spectral
controllability for acertain class of systems under consideration In acompletelydifferent way, the ring Howasalso considered in [63] Therein, aring egenerated
by the entire functions 0,\(s) = and their derivatives is introduced in
order to achieve Bezout identities sl - A(e-'))M(s) +B(e-')N(s) = I with
coefficient matrices over the extension (9[s, e ]. Onecan showby somelengthy
computations that 'Ho is isomorphic to this ring (9[s, e-']. Notice for instance
approach of [63] has been resumed
At this point wewish to take a brief excursion and compare the situation forDDEs with that for partial differential equations.
Remark 2.11
In the paper [84] a very comprehensive algebraic study of multidimensional
systems has been performed. The commonfeature of the various kinds of
sys-tems covered in [84] is apolynomial ring K[si, ' s,,,] of operators acting on
afunction space A This model covers linear partial differential operators with
constant coefficients acting on C' (RI, C) or on D'(Rm) as well as their real
counterparts and discrete-time versions of partial shift-operators on sequence
Trang 2216 2 The AlgebraicFramework
spaces It has been shown in [84, (54), p 33] that in all these cases the
cor-responding module Aconstitutes a large injective cogenerator within the egory of K[sj, ,s,,,]-modules. From this a duality between solution spacesand finitely generated submodules of K[sl, , s,,,] (the sets of annihilating
commutative algebra to problems in multidimensional systems theory (see
Ex-ample 5.1.3 for abrief overview of the structural properties of multidimensional
"suffices" to stay in the setting of apolynomial operator ring in order to achieve
atranslation of relations between solution spaces into algebraic terms At [84,
p 171 Oberst has observed that his approach does not cover delay-differential
equations. We wish to illustrate this fact by giving a simple example whichshows that L is not injective in the category of R[s, z]-modules. -
Recall that an R[s, z]-module M is said to be injective, if the functor
HomR[,,,] (-, M) is exact on the category of R[s, z]-modules [67, 111, 8]. For
our purposes it suffices to note that HomR[,,,,] (R[s, Z]n ,L) -C Lnwhere the morphism is given by f -4 (f (el), ,f (en) T. The inverse associates with each
element Enj= Ipi (D, a)ai EL As a consequence, for a matrix PER[s, z] nXm,considered as amapfrom R[s, z]m to R[s, Z]n, its dual with respect to the above-mentioned functor is given by P(D, a : Ln
> Lm Nowwe can present theexample. Consider the matrices
P= [Z S 'I, Q= [8, 1 - Z1.
Then ker]5r = im(jr in R[s, Z]2, while for the dual maps one only has
W = (0, IT CC2. Hence L is not injective. It can be seen straightforwardly fromthe very definition of 4in Definition 2.9 that imP(D, o,) =ker[l, TS4 indicat-
ing again that it is natural to enlarge the operator ring from R[s, z] to Ho. Weremark that the fact, that multidimensional systems theory "takes place in a
polynomial setting", by no meansimplies that it is simpler than our setting forDDEs Quite the contrary, wewill see that every finitely generated submodule
of a free R-module is free, which simplifies matters enormously when dealingwith matrices
Despite the complete different algebraic setting there will arise astructural ilarity of systems of DDEsto multidimensional systems, which will be pointed
sim-out on several occasions in Chapter 4 In Chapter 5, multidimensional systems
will be part of our investigations on multi-operator systems
For completeness and later use wewant to present the generalization of sition 2.4 about exponential monomials
Trang 23Propo-2 The Algebraic Framework 17
(b) If m > 1, then qf =E'-'V=0 b,e,,,\ for someb, e C and b,,,-, 7 0
Asaconsequence, f E. ker 4if and only if ord,\ (q*) > m. Thefunction q* EH(C)
is said to be the characteristic function of the operator 4.
PROOF: Let ord,\(0) =k, thus ord,\(p*) =1 +k Proposition 2.4(2) (applied to
the ordinary differential operator O(D)) guarantees the existence of a function
E,rn+k g,,,",
9 V=0 where gv G (C) gm+k =7 0, satisfying O(D)g f Using
Em+kg,'Ev=0(v)
Proposition 2.4(l), weobtain qf = Pg _.,v=O K K (P (A)e,-,.,,\
and the desired result follows since (p*) W(A) =0 for n < 1 + k EJ
Remark 2.13
Notice that we did not consider any expansions of solutions as infinite series
of exponential polynomials. Such expansions do exist, see [102] and [3, Ch 6],
the latter for solutions of retarded equations on R+. Wewill not utilize thesefacts since the only case, where the full information about the solution space isneeded, is that of ODEs, see also (2.10). For the general case it will be sufficientfor us to knowwhich 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 results from the literature
Weconclude our considerations on scalar DDEs with the surjectivity of differential operators on L This fact is well-known and can be found in [25,
rather elaborate methods However, wewould like to prove aversion which alsoshows what kind of initial conditions can be imposed for the DDE(2.1). Thisalso gives us the opportunity to present the method of steps, the standard pro-cedure for-solving initial value problems of DDEs
Earlier in this chapter webriefly addressed what kind of initial data should
be specified in order for (2.1) to single out a unique solution f Apart fromsmoothness requirements, wesuggested that f has to be specified on an interval
of length M, the largest delay occurring in (2.1). For instance, a solution ofthe pure delay equation of - f = 0 is determined completely by the restriction
fo := f 1[0, 1) But in order that f be smooth, it is certainly necessary that theinitial condition'fo can be extended to asmooth function on [0, 1] having equal
derivatives f0(v) (0) =f0(v) (1) of all orders v ENoat the endpoints of the interval
In other words, fo and all its derivatives have to satisfy the delay equation for
t = 1 This idea generalizes to arbitrary DDEs and leads to the restriction given
Trang 2418 2 The Algebraic Framework
compati-ble with the given DDE As our approach comprises retarded, neutral, and alsoadvanced equations of arbitrary order, and, additionally, requires smoothness,
wecould not find areference for the result as stated below However, the cedure is standard and one should notice the similarity of the proof given belowfor part (1) with, e. g., those presented in the book [3, Thms 3.1 and 5.21. Inthe sequel the notation C' [a, b] := C' ([a, b], (C) as well as f M for f C- C' [a, b]
pro-refers, of course, to one-sided derivatives when taken at the endpoints a or b
Let q = po' G 7io 01 where p = Ej' o pj zi, pj, 0 G R[s], po 0 : pm,
0 : 0, and M> 1 Furthermore, let g EL
(1) For every fo EC' [0, M] satisf ying
(p(D, a)f 0()) (M) g() (M) for all v ENo (2.11)
there exists a unique function f E L such that p(D, o,)f g and
f I[0,M] = fo. Asa consequence, the map4is surjective on L
(2) If f (=- ker 49 L satisfies f I[k,k+M] =_0 for somek c R, then f -= 0
PROOF:(1) To prove the existence of f, we show: every fo CC' [a, b] defined
on an interval of length b- a > Mwhich satisfies the condition
M
(p(D, u)f(v))0 (t) Epj (D)oif 0(v)) (t) - g(') (t) for all v ENo (2.12)
j=o
for all t G [a + M, b] can be extended in a unique way to a solution
C' [a - 1, b + 1] which satisfies (2.12) on [a - 1 + M,b +1] (Notice that theinitial condition given in the proposition is included as an extreme case wherea=O and b=:M.)
To this end, write po(s) ai sz+Sr and consider the inhomogeneous ODE
aunique solution c C' [b, b +1] to (2.13), (2.14) and j satisfies
M(b) =g(b)
-pj (D) ai fo) (b) - Eai j(') (b) = f0() (b).
Trang 252 The Algebraic Framework 19
Differentiating (2.13) and using (2.12) shows successively j(') (b) = f0(") (b) for
all v ENo Therefore, the function f, defined by f, (t) = fo (t) for t E[a, b] and
f, (t) =j (t) for t E (b, b+1] is in C'[a, b+1] and, by construction, satisfies (2.12)
on [a +M,b +1]. In the same manner one can extend f, to asmooth solution
on [a - 1, b+1]; one takes the unique solution of the ODE
Then f2 EC' [a - 1, b +1] and satisfies (2.12) on [a + M- 1, b + 1].
Repeating this extension leads to asolution in L It is clear from the procedure
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 = p(D, a). The latter can be accomplished by providing a function fo E
C' [0, M] satisfying (2 11). Asimple choice for fo is as follows: pick asolution
C' [0, M] be such that h2 I[0,M-0.75) =_0 and h2 I[M-0.5,M] =_ 1 Then one checksthat fo := h1h2 is adesired initial condition
(2) is aconsequence of the uniqueness in (1). El
Remark 2.15
It is immediate to seethat for all f C- L and q cz H the inclusion f C- ker 4 implies
Ref, Im f Eker 4, too As a consequence, Corollary 2.6, Equation (2 10), andProposition 2.14 remain valid when L is replaced by its real-valued analogueC' (R, R).
Wenowclose the considerations about scalar DDEs and want to spend the rest
of this chapter on discussing a (somewhat extreme) example illustrating some
features of systems of delay-differential equations. The general theory has to be
postponed until Chapter 4, when the algebraic results concerning matrices withentries in 'H are available
This example is taken from [23, pp 249] (see also the references given therein),
where it is presented in theform.
Trang 2620 2 The Algebraic Framework
solve (2.15) uniquely in forward direction by using the (corresponding version of
the) method of steps The solution is continuously differentiable on (to + 1, 00)
and satisfies (2.15) on that interval, see [3, Thm 6.2] (In fact, the solution is ofclass Ckon (to +k, oo) for each k GNo.) If the initial condition xo is of class C'
on [to, to +1] and satisfies (2.15) for t =to + 1 in all derivatives, the solution is
of class C' on [to, oo) and fulfills (2.15) for all t > to + 1 In order to discussbackward solvability, it is shown by someelementary ealculations in [23] that
every differentiable function on [to, oo) which satisfies (2.15) for t > to +1 is on
the interval [to +2, oo) apolynomial of the form
As a consequence, even an initial condition imposed at just one single point
might not allow a backward solution For instance, there is no differentiablesolution x = (Xl) X2) X3 on (-oo, to] satisfying xi (to) - 2X2(to) - X3(to) : 0-
Of course, this is due to the singular coefficient matrix of x(t - 1) in (2.15).
solutions on the whole of R are being considered In this particular example
it is possible to achieve a triangular form for Rby applying elementary row
operations over R[s, z]; indeed wehave
0-Since V-1 CR[s, zJ"', too, the operator V(D, u) is bijective on L3. As a
con-sequence, the kernel of R(D, a) remains unchanged under this transformationand can be read off explicitly from the triangular form In fact, one obtainsker R(D, a) =ker R, (D, or) = fp E(C[t] 3I'p is of the form (2.16)1 (2.17)
Wesee that the triangular form does not only have the advantage of providingthe solution space of (2.15), it also exhibits via its diagonal elements, where andhow initial data can be imposed so that, by virtue of Proposition 2.14, forward
Trang 272 The Algebraic Framework
and backward solutions are guaranteed. Wecan go even one step further Thekernel (2.17), being afinite-dimensional space of polynomials, is in fact asolu-tion space of anordinary differential operator This operator can be determinedexplicitly by exploiting the linear equations governing the coefficients in (2.16).
Elementary calculations show
53 0 0
8-1 0 1
see also [44, pp 227] for ageneral method Let us determine the row
transforma-tions relating the matrices R and R2. To this end, wesimply have to calculate
This matrix is easily seen to have determinant 2 and entries in Ro. Put another
the operator algebra Ro. In Section 4.1 wewill see that operator matrices
shar-ing the same kernel are always related like this
Let us briefly draw a link to the forward solutions discussed above Since thetransformation matrices V and U contain the shift operator z, they don't pre-
serve the solution space of R(D, u)x = 0 in C1 ([to, 00), (C3) , but only respect the
"tails of the solutions" A few time units have to elapse (the number depending
differentiability required for applying the transformations), before the solutions
weobserved in (2.16) and (2.17).
Finally weshould mention that in this example a first information about thekernel of Rin C3 could have been obtained by noticing'that det R= 93. The
equation (adjR) (D, a) oR(D, a) = D3J3 shows immediately that the solutions
space is finite-dimensional In this case one could goeven further and determinethe full solution space by substituting the general polynomial of degree three
in the original equation R(D, c7)f = 0 This idea, of course, applies wheneverthe determinant of the system matrix is in R[s] but fails in the general case
det RGR[s, z].
Eventhough the example is extreme in the sense that it describes actually atem of ODEs, it should demonstrate that atriangular form is of advantage for
sys-getting some more information about the solution space of a delay-differential
operator But not every polynomial matrix in two variables can berow reduced
to atriangular form Fortunately, this can always be achieved with tion matrices having entries in R, as wewill see in the next chapter.
Trang 28transforma-3 The Algebraic Structure of Wo
In this chapter. wewill concentrate on the purely algebraic part of the theoryand analyze the ring structure of Ho. As the following sections will show, the
operator ring Hocarries arich algebraic structure, interesting by itself, but alsonicely suited for an algebraic study of delay-differential equations and systems
thereof later on. The combination of the two embeddings Ho 9 R(s)[z] and
7io g H(C) proves to be apowerful tool for the upcoming investigations; the
ring R(s)[z] is a principal ideal domain, while H(C) is a Bezout domain (it is
even known to be anelementary divisor domain, but for us the Bezout property
once in Proposition 2.5(2), where weestablished that aproper delay-differential
operator has infinitely manycharacteristic zeros. This fact in combination with
an easy handling of the finitely many zeros of each possible denominator in
In that section wederive the main results about 'Ho (The corresponding factsabout the ring R, which is simply the localization (Ho), are readily derived.)
On the one hand, it will be shown that Ho is a Bezout domain, that is, eachfinitely generated ideal is principal. Put another way, each two nonzero elements
have agreatest commondivisor which, additionally, can beexpressed as alinearcombination of the given elements with coefficients in Ro. On the other hand,
wewill also see that Hois aso-called adequate ring, meaning that each element
can be factored in a certain desired manner concerning its characteristic zeros.
In Section 3.2, general ring theory is applied to deduce that lio is aso-called
el-ementary divisor domain This says essentially, that matrices with entries in Ho
admit triangular and diagonal forms in a way quite similar to matrices over
Euclidean domains This fact will be of fundamental importance in the next
chapter when westudy systems of delay-differential equations Furthermore,the Bezout property of 'Ho will be utilized to generalize the notions of greatest
commondivisors and least commonmultiples to matrices over Ro. As has just
been indicated, the reason for focusing on matrix theory rather than general
module theory over Hois the fact that in the next chapter our main objects ofstudy -will be systems of delay-differential equations, hence operators which are
matrices over 'Ho or n However, atranslation and interpretation of the theoretic results into the language of general module theory will begiven at theend of Section 3.2
matrix-H Gluesing-Luerssen: LNM 1770, pp 23 - 72, 2002
© Springer-Verlag Berlin Heidelberg 2002
Trang 293 The Algebraic Structure of Ho
In Section 3.3 wewill take abrief excursion into the theory of systems over rings.
Even though this area of control theory is not directly related to our approach
to delay-differential systems, it has been ahelpful source for getting acquaintedwith various ring structures and their matrix-theoretic implications.
In order to further study the algebraic structure of HO, a description of thenonfinitely generated ideals of HOfollows in Section 3.4 Amongother things,
it will be shown that HOhas Krull-dimension one, that is, all nonzero prime
ideals are maximal
In Section 3.5 werecall the original meaning of Has an operator algebra acting
dis-tributions with compact support. This can also be understood as anembedding
of H into asuitable Paley-Wiener algebra of entire functions The structure ofthe distributions associated with the elements of H will be given explicitly Finally, in Section 3.6 the question is posed whether the ingredients required forthe calculation of the various objects introduced earlier can actually be com-
puted. Wewill concentrate on symbolic computability; numerical questions will
not be addressed Although thevarious calculations of the previous sections are
in acertain sense constructive, someadditional work is necessary with respect
to their symbolic computability. This is mainly due to the central role played bythe zeros of the (transcendental) functions in R[s, e-'] for the construction ofsuitable ring elements Starting with functions in, say Q[s, e-'], one is necessar-
ily led to certain transcendental field extensions of Q Interestingly enough, it
is exactly Schanuel's (still open) conjecture about the transcendence degree ofsuch extensions, which, in case it is valid, enables symbolic computability of Be-
zout equations in HO and, consequently, of all relevant constructions presented
in this chapter. In this context it is also interesting to note that generically thecomputational difficulties for the Bezout identity of three or more polynomials
Wewill makeuse of the following
Notation 3.1
Let R be any commutative domain with unity.
(a) By R' we denote the group of units of R We will write pJ." q if p vides q in R Any greatest commondivisor of P1, ,pl c R(if it exists)
di-will be denoted by gcd,.(p1, 'p1) Consequently, each expression
con-taining gcd., (pi I )pl) has to be understood as up to units in the ring.
In the same way, a least commonmultiple (if it exists) will be denoted by lCMR (P1I Ipi). In case R=R[s], weomit the index R This will not cause
any confusion due to the specilic rings under consideration
(b) A matrix UER"' is called unimodular if det UER' and U is said to benonsingular if det UER\10}. Thegroup of unimodular matrices over Ris
denoted by Gl,,(R), while E,,(R) is the subgroup
Trang 30Divisibility Properties
where -elementary matrices are understood in the usual sense, see, e.g., [55,
p 338].
(C) The rank of amatrix MERPx qis defined to be the rank of Mregarded as
amatrix over the quotient field of R Hence it denotes the maximal number
of linearly independent rows or columns in the matrix M
(d) Wecall two matrices P, QERPx q left equivalent (resp equivalent) over R
if there exists UCGlp(R) (resp. UE Glp(R) and V(=- Glq(R)) such that
(e) For a polynomial p GR[xl, , x,,] let deg,,,, p denote the degree of p as
define degs q= degsp- deg
(f) For an indeterminate x over R the notation R((x)) stands for the ring offormal Laurent series in x with coefficients in R, that is,
In this section wewant to lay down the foundations for our algebraic
considera-tions Tobegin with, wepresent somebasic, yet important, rules for calculating
can be expressed as a linear combination of the given elements On the other
hand, it is asimple fact that Hois not aprincipal ideal domain A combination
of the embeddings 'Ho g R(s) [z] and Ho 9 H(C) will finally show that Ho
admits adequate factorizations in a sense to be madeprecise below
Remark 3.1.1
In the sequel wewill makefrequent use of the fact that the ring H(C) of entire
functions is aBezout domain [82, p 136] Precisely, each two nonzero functions
f, g c H(C) have a-greatest commondivisor dCH(C) which satifies
Trang 3126 3 The Algebraic Structure HO
ord,\ (d) =minf ord.\ (f ), ord,\ (g) I for all Ac Cand can be expressed as linear combination
d= af +bg with suitable functions a, bEH(C).
Let us start with the following list of properties. Comments will be given below.Recall the notation from Definition 2.3
Proposition 3.1.2
(a) For all p EHo and AGCwehave p* (X) =p* (A), where denotes complex
conjugation.
(b) The units of Hoare given by HO R\ f0
Moreover, fp E Ho I p* c H(C) I} fazkI a ER\fOJ, k c Nol.
(C) Let p, q GHo and let zt,. p Then P* IH(,) q* p1,0 q
(d) For all p E Ho the following assertions are equivalent: i) p is irreducible, ii) either p = 0 for someirreducible R[s] or p = az for some nonzero
a cz R, iii) p is prime.
(e) Thering Hois not factorial and not Noetherian
(f) For all k c No and R[s] \f 0} there exists a polynomial J ER[s] suchthat z'0 7io
(g) For all p, q GHo there exist elements GHo Such that p= fq +P anddegz P< degz q
(h) If 0 (E R[s] and a, b G Ho are elements such that 0 1,0 (ab), then thereexists afactorization 0= 0102 in such a waythat A-,01 - -02 EHo Moreover,
one can arrange the factorization such that V(01, (b02 T) 0'
(i) Each two elements p, q G Ho, not both zero, have a greatest common
divisor d = gcdo (p, q) G Ho\101 Moreover, d* = gcd,,(c) (p*, q*) andhence V(d*) = V(p*, q*). In particular,
gcdH(C) (p*, q*) = 1 < ==> gcdo (p, q) = z' for some1 ENo
4==>- V(p*, q*) = 0.
(j) Let p, a, b c Ho be such that pI, (ab) and gcd, (p, a) = 1 Then pI,. b
(k) Each pair of elements p, q E 'Ho\f 01 has a least common multiple
lcm,,o (p, q) E ho, given by ged pq
(p,q) *In other words, the intersection:no .
of finitely manyprincipal ideals is principal.
Before turning to the proof wewant to briefly commenton someof the assertionsabove
Remark 3.1.3
(i) Part (c) is avery simple instance of aresult given in [4, p 270], where this
property is called stability and proven for an analogous situation
includ-ing polynomials and exponential functions of several complex variables In
Trang 32Divisibility Properties
that generality the proof requires muchmore sophisticated methods fromharmonic analysis, whereas in the case of interest to us, namely complexfunctions in one variable, the result can easily be obtained from the Theo-
rem of Bezout for algebraic plane curves as will be shown below
(ii) The sole reason for 'Ho not being factorial, as stated in (e), is that elementsusually have infinitely many characteristic zeros, hence, due to the very
definition of 7io, infinitely many irreducible (linear) factors In fact, we
will, make frequent use of this property. With a suitable adaptation
guar-anteeing convergence, these linear factors can be arranged in an infinite
product according to WelerstraB's factorization theorem However, wewill
not utilize this nice result from complex analysis.
the polynomial part R[s, z] of 7io Regard q ER[s, z] as a polynomial in z
with coefficients in R[s]. It has in general a highest coefficient, say 0, that
is not a unit in R[s] and thereffire prevents division with remainder by q
in R[s, z]. While simply dividing by 0 would in general not be admissible
in Ho, we can "normalize" q via multiplication with a suitable function
(z - 6)0-1 G7io. With an appropriate handling of possible denominatorsthis basic idea is easily exploited to establish the division with remainder
as given in (g). Dueto the increase of the degree (with respect to z) in the
normalization, one cannot achieve, in general, astrictly smaller remainder
(iv) Part (i) will be of central importance for the algebraic structure of 7io. InTheorem 3.1.6 wewill encounter an alternative proof for the existence of
greatest commondivisors Yet wethink it is worth presenting the versionabove as it shows more directly the connection to the greatest common
divisor for entire functions
PROOFOFPROPOSITION 3 1.2: (a) is obvious
(b) The first part is clear, while the second one follows from Proposition 2.5(2) (c) Thedirection is true since p -4 p* is aring homomorphism. Asfor "=*",let q* (p*)-l E H(C). In the field R(s, z) we can write qp-' = ab-1 for some
coprime polynomials a, b E R[s,z]. Then a*(b*)-' = q*(p*)-l C H(C), thus
plane curves and obtain #1 (A, /-t) E (C2 I a(A, p) = 0 = b(A, p) I.< oo, hence
of a and b provides k =0 and thus wearrive at qp-1 = a0-1 c Ho.
(d) "i) = > ii) " Let p E'Ho be irreducible andp=7 az for all a CR By (b) there
exists A EC such that p* (A) = 0 If A(=- R, then p=
Trang 3328 3 The Algebraic Structure of Ro
(e) Consider Z - I E Ro. The polynomials p, = (s - 27riv)(s +2,7riv) GR[s]
for v c- Nsatisfy M-1w'-' P., Glio for all n c N Hence z - 1 has infinitely many
irreducible factors in ? to and
) C
is an infinite properly ascending chain of ideals in 'Ho.
(f) is a simple interpolation: one needs J(') (A) = (-k)ve-k,\ for each root A E
V(0) and 0 :!5 v :5 ord,\ (0) - 1 Using (a), such apolynomial 5exists, even withcoefficients in R, cf [21, p 371.
qM
Pf:=P -PLZL-M-lz-'5qqM GHo and deg, p' < deg, p This way, we can proceed
until the degree of the remainder is reduced to M
(h) Simply distribute the zeros of 0in an appropriate way, see also (c).
(i) Only the case p 7 0 : q needs consideration Let p and q = - where
divisor of a, b in R[s, z] is extracted Thereafter only finitely many common
characteristic zeros are.left producing a polynomial gcd in H(C). The details
are as follows Define g = gcd,,,,,,,,;, (a, b) c R[s, z] and let gal = a, gbl = b.Moreover, write 0= 0102 in R[s] such that
pand q'has been movedinto the factor 9, this commondivisor can be cancelled
and, upon using (b) and (c), weobtain that 'P9 is a greatest commondivisor
of p and q in Ho, which is what wewanted 02
(j) is a consequence of (c) and (i).
(k) can be shown by standard calculations in H(C) Alternatively, aproof will
beprovided for amatrix version of the assertion in Theorem 3.2.8 It will make
use of the Bezout property (proven for Hoin Theorem 3.1.6). 0
Trang 34Divisibility PropertiesRemark 3.1.4
Aglance at the proof of (i) shows that the greatest commondivisor in Woof
polynomials pand q is apolynomial, too
Remark 3.1.5
For the ring R the situation becomes even smoother Since the units of Rare
given by the set
H' = faz k
and because of the relationship
p, ER===: - 3 k EZ such that zkp EHoand zj,. zkA
the results above translate easily into according properties for H Onesimplyhas to adapt the formulations, whenever the element zis involved In particular,
p and q are coprime in H if and only if p* and q* are coprime in H(C).
Note also that Proposition 3.1.2(c) can be rephrased as saying that R is thelargest ring extension of R[s, z] within R(s, z) to which the embedding (2.8) can
be extended Put another way, the ring Rcan be written as
The proof of the existence of the greatest commondivisor given above is
con-structive in the sense that it shows exactly which steps lead to the desired result
However, the practical computations involve serious difficulties as one needs to
compute the common zeros of exponential polynomials. Before presenting some
examples, wewant to establish the main results of this section Its proof
demon-strates an alternative way for the computation of a greatest commondivisor.But even more will be obtained The procedure generates alinear combinationfor the greatest commondivisor, showing that Ho is a Bezout domain As a
by-product - and as aconsequence of the sort of division with remainder given
in Proposition 3.1.2(g) - one observes that each unimodular matrix is a finiteproduct of elementary matrices Weremark that this is true, for the same rea-
son, over every Euclidean domain, but not, in general, for the ring R[S, z]. Acounterexample in form of a2x 2-matrix over R[s, z] has been found in [16]. We
present this'matrix along with afactorization into elementary matrices over 'Ho
in Example 3.2.3(2) in the next section It is worth mentioning that for n > 2unimodular n x n-matrices over R[s, z] are always finite products of elementarymatrices This is aspecial case of Suslin's stability theorem [106] Interestingly enough, the unimodular matrices over the ring H(C) of entire functions are alsofinite products of elementary matrices, see [82, p 141]. In this case the argu-
ment is completely different from that for 'H and will be addressed briefly in
Remark 3.1.10 below
Part (c) below is a technical fact which will be needed in the next section in
Trang 3530 3 The Algebraic Structure of ?io
order to prove that His an elementary divisor domain If one translates the
adequate factorization stated in (c) into entire functions one observes that thefactor b* is madeup of exactly all common zeros of p* and q* with the mul-tiplicity they have in p*. This formulation shows that the ring H(C) itself isadequate, too In our approach, the adequate factorization will be mainly used
to prove that His an elementary divisor domain; see the next section Recallthe notation given in 3.1(b).
Theorem 3.1.6
Let K be any of the rings H and 'Ho
(a) IC is a Bezout domain, that is, each finitely generated ideal is pal. In other words, for all pl, -,p,, c IC (not all zero) and every
princi-d= gcdr- (P1i Pn) there exist a,, , an EIC such that
and gcd,, (6, q) V)Cx for every divisor E/C\)C x of b
In Section 3.4, where the nonfinitely generated ideals are described, an
alterna-tive argument for Hbeing adequate will comealong as a by-product.
PROOF: It is easy to see that we can restrict to the ring H0, cf (3-1.1).
(a) Using the sbrt of division with remainder given in Proposition 3.1.2(g) one
mayassume pi = 6 0 for i = 1, , n. Write
Emi _,3.
where pij, R[s], pimi :7 0
0
Without restriction let M, :! Mk for k 1, , n. Wewill show that by
ele-mentary row transformations applied to the vector (Pi Pn the degrees ofthe elements Pk with respect to z can be reduced In order to do so consider thefollowing two cases.
Trang 363.1 Divisibility Properties 31
i) If Mk > M, for some k, we use Proposition 3.1.2(g) to accomplish Pk ::: :
Pk - fP1 for some f E7io and deg, Pk :5 deg, pl Proceeding this way, we can
achieve via elementary operations that the degrees OfP27 p, are at most M1.
= deg, p,, = MI, we can handle the vector of highestcoefficients (PlM,) P2M1i IpnM1T via elementary transformations in the Eu-clidean domain R[s]. Let 5 := gcdR[.] (P'Ml I - -,, PnMi) GR[s]. Then there exists
a transformation matrix V GEn (R [s]) such that V(PIMI 7P2MIi PnMi
C7in
and
deg, Pj < M, =deg, P, for j = 2, . '., n.
Combining these two methods wearrive after finitely manysteps at
(c) The idea of the proof is as follows: factor p= ab such that V(b*) =V(p*, q*)
ring 7-10 if #V(p*, q*) < oo. In the other case an iterative procedure is needed.First of all, it is easy to see that wemayrestrict to the case where zt,0 q, whichwill simplify the use of Proposition 3.1.2(i) later in the proof.
As for the iteration, start with bi := gcd,0 (p, q) G7io and put a, bi Next,
define successively for p= aibi, i EN, the following elements:
ai
ci := gcd, (ai, bi), aj+j :=
Ci, bj+j := cibi. (3-1-3)
Hencep= ai bi =ai+ 1bi+ 1. This produces asequence of elements ai (E 'Howhere
aj+j IHO aj. But then aj+j divides ai also in the principal ideal ring R(S)[z] withthe consequence that for somek ENthe element Ck EHois a unit in R(s)[z],
hence Ck ER[s]\101. As a consequence, V(a*, b*)k k is finite, say
V(a*, b*)k k = f Al, Anf,
and we can define
n
f (s - Aj) R[s], where 1i =ord.\j (a*).k (3.1.4)
show that this factorization satisfies the requirements of the theorem
Trang 3732 3 TheAlgebraic Structure of HO
1) To establish the coprimeness of a and q, suppose V(a*, q*) 7 0 and let
JA1, A,, 1. But for A = Aj we have ord,\ (a*) = ordX, (a*)k - ord.\, (f 0.Hence V(a*, q*) 0 and from Proposition 3.1.2(i) weconclude the coprimeness
of a and q.
consequence there is some A G V(b*) such that b*(A) = 0 The construction
(3.1.3) of the sequences (ci) and (bi) leads to the following identities of varieties
(recall that wecount zeros in V without multiplicity)
= V(p*, q*).
Thus AGV(q*, 6*) and therefore and q are not coprime.
Note that in the case V(p*, q*) A, IA,, I is finite, the construction aboveleads directly to the factorization p = 2bb where b fl'j=1(s - Aj)'i and 1i
Theprocedure given in (b) for the Bezout identity is, although somehownatural,
not very practical as the examples will show Abetter procedure, requiring less
steps, can be found in [39, Rem 2.5]. But that one has someshortcomings, too,
for it needs apriori knowledge of agreatest commondivisor and does not imply
part (b) about unimodular matrices We will demonstrate that procedure in
Example 3.1.9(3).
Remark 3.1.7
The result as stated above has been proven first in [42]. The adequateness hasbeen obtained in discussion with Schmale [98]. In special cases, basically if theelements are coprime and one of them is monic in s, aBezout identity has beenearlier derived in afairly different setting, see [85, Sec 4], [63, (3.2),(4.14)] as
well as later [9]. In [5, Prop 7.8] a Bezout 'identity I = Ej'=, fjgj has beenobtained for exponential polynomials fj G C[s, e"] with coefficients gj in a
Paley-Wiener algebra.
Remark 3.1.8
In [47] the approach of our Chapter 2 is applied to delay-differential equations
with noncommensurate delays. In the language of Cl: apter 2, that case can bedescribed by polynomial operators P p(D, al, ,al), where ai f (t) = f (t - Tj)are shifts of lengths rl rl > 0, that are linearly independent over Q. Asshown in [47, Thms 5.4 and 5.9], the algebraic approach leads to thepperator
algebra
Trang 38consequence, serious obstacles arise for an algebraic approach to equations with
noncommensurate delays. Wewill touch upon these issues in Chapter 4
Weillustrate the determination of agreatest commondivisor along with a
Be-zout identity by some simple examples. Part (b) of the theorem above will beaddressed in the next section when considering matrices over H, see Exam-ple 3.2.3
For computational issues, which will be addressed in Section 3.6, wewill keep
track of the coefficients of the indeterminates s and z in the calculations below,
starting with coefficients in Q.
easily be found by rewriting it as
b s
Now the requirement b* EH(C) forces a GRoto be such that the function
simple choice a = -1 GRosuffices and leads to the Bezout identity
82
over HO. Notice that aand bare in Q (s) [z] nHO,that is, all the coefficients
of the indeterminates s and z in the equation above are in Q.
Bezout equation I = ap + bq one needs b* 1-a*e-' E: H(C), leading to
8+1
the sole condition a*(-l) = e-1. Hence
1 = e-1Z + 1
- e-1z (8+1)
S+1
is aBezout identity as desired In this case the coefficients of 8 and z are
in the field Q(e). It is easy to see that no Bezout equation with coefficients
in the field Aof algebraic numbers exists
Trang 39Algebraic Structure of 'HO
(3) Let p +1 q = s+z GRo. The elements are coprime since the equations
e eand A+e-\ =0 have no common zeros in C To obtain aBezoutidentity wefirst let a =I and b = - (s +1) and get
This is indeed the first step in the procedure given in the proof of
where 6 E R[s] satisfies J(-I) e, 6(-e) ee Instead of going this
way, which would require another step thereafter, weproceed as follows.Equation (3.1.5) implies
[q*(-e),p*(-e)] (b*(-e))a (' e 0,
thus, by coprimeness of p and q, it follows
a*(-e) b*(-e)) EimR [ q*(-e) P*(-e)
with coefficients in Q(e, ee).
Theexamples (2) and (3) should demonstrate how(successive) Bezout identitiesforce to extend step by step the field of coefficients, in this case from Q through
Q(e) to Q(e, ee). It seemsunknown whether the transcendence degree of (Q(e, ee)
is two, which is what one would expect This is avery specific of
Trang 403.2 Matrices over WO 35
general conjecture of Schanuel in transcendental number theory, which wewill
present in 3.6.5 However, very little is known about this conjecture (just to
give an example, it is only known that at least one of the numbers el or ee2
is transcendental, gee. [1, p 119]) Handling of the successive field extensionsforms animportant (and troublesome) issue in symbolic computations of Bezoutidentities in 7to. Wewill turn to these questions in Section 3.6
The results stated so far show a striking resemblance of 'H and H(C) with
respect to their algebraic structure But there are also differences, one of thembeing presented next, another one is the dimension of the rings and has to bepostponed until Section 3.4
Remark 3.1.10
For acommutative domain R with unity one says that 1 is in the stable range
of R if for all n > 1 and a,, ,an+1 ERsatisfying R= (a,, ,an+1) thereexist bl, , bn c R such that R= (a, + blan+l, , an +bnan+1), see e.g.,
[30, p 345]. It is easy to see that this is equivalent to the property that for all
al, , an+1 ERsatisfying R= (a,, , an+1) there exist C2, Cn+1 c R suchthat a, +En+1i=2 ciai is aunit in R While this is true for the ring H(C), see [82,
shows Let a, = z - 1 and a2 = (s - 1) (s - 2) EX Then a, and a2 are coprime
in 'H and aBezout equation 1 = cial +C2a2 implies for the coefficients
Cl
-clal
Considering the roots of the denominators it can be seen that neither of the
coefficlents cl and C2 can be aunit in X
In [82, p 139] it has been proven that for every Bezout domain with 1 in the
sta-ble range unimodular matrices are finite products of elementary matrices Thisresult applies in particular to the ring H(C) and wearrive at Theorem 3.1.6(b)
for IC =H(C).
3.2 Matrices over WO
In this section we turn our attention to matrices over 'Ho. First of all, it is an
easy consequence of the Bezout property that one can always achieve'left
equiv-alent triangular forms RomTheorem 3.1.6(b) weknow that this can even bedone by elementary row transformations But even more can be accomplished.
It is aclassical result that an adequate commutative Bezout domain allows
di-agonal reductions via left and right equivalence for its matrices In other words,
matrices admit aSmith-form, just like matrices with entries in aEuclidean
do-main This will be dealt with in the first theorem below and someconsequences