key problems in the extension of module-behaviour duality · key problems in the extension of...

Linear Algebra and its Applications 351–352 (2002) 761–798www.elsevier.com/locate/laa

Key problems in the extension ofmodule-behaviour duality

Jeffrey Wood∗,1

Department of Electronics and Computer Science, University of Southampton, ISIS Group,Southampton SO17 1BJ, UK

Received 20 February 2001; accepted 9 October 2001

Submitted by J. Rosenthal

Abstract

The duality for linear constant coefficient partial differential equations between behavioursand finitely generated modules over the operator ring is a very powerful tool linking equationstructure to dynamic behaviour. This duality is critically dependent on the choice of signalspace. In this paper we discuss two key algebraic problems which form an obstacle to theextension of this theory to general signal spaces. The first of these is the so-called Willemsclosure problem, which limits the ability of system equations to directly describe the sys-tem. The second is the elimination problem, the general solution of which depends upon analgebraic property (injectivity) of the signal space. We demonstrate the importance of theseproblems in the module-behaviour framework, and some of the useful consequences of a fullor partial solution. The issues here are of particular relevance to the extension of the currentduality theory for behaviours defined by linear partial differential equations from the case ofconstant to non-constant coefficients. © 2002 Elsevier Science Inc. All rights reserved.

Keywords: Linear systems; Multidimensional systems; Behaviors; Module theory; Willems closure; Elim-ination problem; Image representations

1. Introduction

In most branches of systems theory, we inevitably describe a system by someset of equations. The question as to what constitutes a solution of those equationsthen becomes paramount for any detailed analysis of the system and its dynamical

∗ Corresponding author. Tel.: +44-23-8059-4984; fax: +44-23-8059-4498.E-mail address:[email protected] (J. Wood).

1 Royal Society University Research Fellow.

0024-3795/02/$ - see front matter� 2002 Elsevier Science Inc. All rights reserved.PII: S0024-3795(01)00532-8

762 J. Wood / Linear Algebra and its Applications 351–352 (2002) 761–798

properties. This question becomes particularly sharp when the notion of a solution isformalized and used as a central object of study, as in the behavioural approach dueto Willems [39,40].

The behaviour of a system given by some (say, differential) equations is the solu-tion set of those equations with respect to some given signal space (which is wherethe individual components of the solution are taken to lie). This is a natural model ofthe physical or dynamical phenomena which can be exhibited by the system. Basicsystems theory then proceeds by elucidating the relationship between the propertiesof the equations and the properties of the behaviour, which cannot be representeddirectly. This relationship can be formalized as a correspondence (not necessarilyone-to-one) between behaviours and finitely generated modules over the ring of op-erators. The finitely generated module corresponding to a given behaviour is thesame as the module used as a model of the system by Fliess and co-authors [1,6] andby Pommaret and Quadrat [29,31], and is interpretable as the set of distinct single-valued differential operators on the behaviour [41]. Conversely, given a module, thecorresponding behaviour is the solution set of any set of equations (re)presentingthe module in a certain sense. Thusthe module-behaviour correspondence is not anartificial construct but a natural relationship.

However, the extent to which the structure of the module reflects the structure ofthe corresponding behaviour, and vice versa, depends crucially upon the choice ofsignal space. In the case of systems defined by linear ordinary or partial differentialequations (PDEs) with constant coefficients, we find that the signal spaces of smoothfunctions and of distributions are “injective cogenerators” (these terms are explainedlater). This result is due to Oberst [24], and essentially entails that the module-behaviour correspondence becomes a very powerful categorical duality. Using thisduality, it has been shown for linear constant coefficient PDEs that controllabilitycorresponds to torsionfreeness, autonomy to torsion, exponential modes to the pointsof the characteristic variety of the module, and much more (e.g. [24,27,41,44]).Moreover, from the duality we obtain constructive techniques in the form of alge-braic computations, generally using Gröbner bases.

For other choices of signal space, the module-behaviour relationship is not sostrong or is unknown. For a given signal space, a weak relationship between modulesand behaviours means that the ability of system equations and symbolic manipulationto capture the system dynamics is considerably limited. Thus the problems discus-sed here, though formulated in the behavioural framework, are not confined to thatframework but are intrinsic to the use of algebraic/symbolic tools in systems theory.

One main arena where an extension of the module-behaviour duality would bevery useful is that of linear differential equations with variable coefficients. In thecase of ordinary differential equations, Fröhler and Oberst [10] have shown that asuitable choice of signal space is given by the hyperfunctions. For PDEs, the situationis unknown, and consequently results linking the structure of an arbitrary system ofsuch equations to the structure of the behaviour they define are to the best of ourknowledge entirely lacking to date.

J. Wood / Linear Algebra and its Applications 351–352 (2002) 761–798 763

In this paper, we aim firstly to identify the key problems which currently forman obstacle to the extension of this duality theory, and secondly to fully or partiallysolve these problems in terms of algebraic conditions on the signal space. There aretwo “key problems”. The first is the so-called Willems closure problem, which isconcerned with whether a given system of equations defining a behaviour generatesall the equations satisfied by that behaviour. When this is not the case, the ability ofthe equations (or the module they generate) to describe the system is very limited.The Willems closure problem is related to the cogenerator property of the signalspace. The second of the problems is the elimination problem, i.e. given a system ofequations involving two sets of variables,w, l, can we find some equations whichgive the conditions onw for a solution to exist? This problem is clearly of greatimportance in systems modelling, and is related to the injectivity property of thesignal space.

The paper is arranged as follows. In Section 2 we set up the formalism requiredthroughout the paper, i.e. the language of behaviours and modules. We also pro-vide a collection of background definitions and results from module theory. Thenin Section 3 we look at the Willems closure problem for a given signal space. Wepresent the problem and discuss its importance, giving some useful consequencesof using a Willems closed module. We also demonstrate that this problem disap-pears when the signal space is a cogenerator. Finally in this section we provide somepartial results for general signal spaces satisfying certain weak conditions, and forcompleteness a general solution for the constant coefficients case which holds undercertain algebraic assumptions on the signal space.

Section 4 studies the elimination problem. We show that, under very minor as-sumptions, the elimination problem is solvable in full generality precisely when thesignal space is injective, and moreover we can then apply the algorithm already givenby Oberst [24] and Komorník et al. [17]. By use of the Ext functor, which moreaccurately describes the ability to eliminate, we are able to give a variety of weakerconditions on the signal space under which various elimination problems can besolved. We conclude in Section 5 with some further consequences of having a Wil-lems closed equation module and/or an injective signal space. These consequencesinclude some characterizations of when a given behaviour has an image representa-tion.

2. The formal setting

A systemis a triple (A, q,B), whereA is a function space called thesignalspace, q is the number of components (dependent system variables), andB, thebe-haviour, is a subset ofAq . The elements ofB, or more generally ofAq , are calledtrajectories.

In this paper our interest centres on systems described by linear partial differentialequations, not necessarily with constant coefficients. However, our results also apply


when the ring of linear partial differential operators is replaced by some arbitraryNoetherian domain. Thus letD be a left and right Noetherian domain,2 which isnot necessarily commutative; this covers the majority of operator rings of interest insystems theory. The examples in which we are particularly interested are those of thefollowing form:• D = k[z1, . . . , zn], the (commutative) ring of polynomials inn indeterminates

over a fieldk. By identifyingzi with the partial derivative operator�/�ti , we canregard this as the ring of linear partial derivative operatorsk[�/�t1, . . . , �/�tn]with coefficients ink (normallyk = R, or perhapsC).Of particular interest in this context are the classical spaces from the theory ofPDEs: the spaces of smooth functionsC∞(Rn,R), of distributionsD′(Rn,R),of compactly supported smooth functionsC∞

0 (Rn,R), of compactly supported

distributionsE′(Rn,R), of rapidly decreasing functionsS(Rn,R) and of tem-pered distributionsS′(Rn,R). The problems described in this paper have beenfully resolved for these spaces over the ring of linear constant coefficient partialdifferential operators [20,24,26,27,35,36], as we will indicate throughout.

• D = k[z1, . . . , zn], where this timezi is interpreted as a unit shift operator actingon some functionw defined onRn or a sublattice (oftenNn or Zn)

(ziw)(t1, . . . , tn) := w(t1, . . . , ti−1, ti + 1, ti+1, . . . , tn). (1)

Thus we can treat multidimensional difference equations in an analogous way toPDEs.

• D = k[s, z], wheres is the derivative operator d/dt , andz is a shiftt �→ (t + 1)as in (1). This enables us to formally model delay-differential systems, and canbe generalized to multiple non-commensurate delays in the obvious way.

• D = K[z1, . . . , zn], whereK is a differential field withn derivative operators, i.e.for anyf ∈ K, �f/�t1, . . . , �f/�tn are defined as elements ofK. For example,K might be the fieldR(t1, . . . , tn) of rational functions, or the ringR[t1, . . . , tn].Another example, used in [10] and perhaps the most appropriate operator ringfor linear variable coefficient differential equations is the ringR(t1, . . . , tn) ∩C�(U,R), whereC�(U,R) denotes analytic functions on some open setU ⊆Rn, i.e. R(t1, . . . , tn) ∩ C�(U,R) is the ring of rational functions with no polesin U. The indeterminateszi again represent partial derivative operators, and themultiplication for elements ofD is defined by the commutator law for an elementof K and a derivative operator

∀f ∈ K, i = 1, . . . , n, zif := f zi + �f�ti. (2)

Except in the case where�f/�ti = 0 for all f ∈ K, we see that these rings arenon-commutative. However, the Weyl algebraR[t1, . . . , tn, z1, . . . , zn] is wellknown to be left and right Noetherian, and since the ringsK[z1, . . . , zn] for the

2 It is possible to generalize most if not all of our results from domains to semiprime rings.


fieldsK specified above are rings of fractions of the Weyl algebra, they are all leftand right Noetherian as well (see, e.g. [14, Corollary 9.18]), and also domains.Note that we can describe linear multidimensional difference equations with vari-able coefficients in an analogous way.We assume that our signal spaceA is a leftD-module, i.e. that for anyp ∈ D

andw ∈ A, the operatorp can be applied tow to give some other elementpw ∈ A.We will be particularly interested in behavioursB which are defined by a systemof equations over the ringD. In other words, we have a matrixR ∈ Dg,q , andB isdefined by

B :={w ∈ Aq |Rw = 0

}=

w ∈ Aq

∣∣∣∣∣∣q∑j=1

Ri,jwj = 0 for i = 1, . . . , g

. (3)

ThusB is the set of solutions of the given operatorRwithin the signal spaceA. Wesay thatR is akernel representationof B, and we writeB = kerAR.

Example 2.1.1. The real solutions of the 2D difference equations

w1(t1 + 1, t2 + 1)− w1(t1, t2 + 1)− w2(t1 + 1, t2) = 0,2w1(t1, t2)− w2(t1 + 2, t2) = 0

on N2 can be encoded by the kernel representation

B = kerAR, R =(z1z2 − z2 −z1

2 −z21

),

whereA = RN2.

2. The time-varying ordinary differential equation

d2w1

dt2(t)+ (t2 − 1)

dw1

dt(t)− tw2(t) = 0

on the open interval(0, 1) can be represented by theD-matrix

R = (z2 + (t2 − 1)z −t),whereD = (R(t) ∩ C�((0, 1),R))[z], and the smooth solutions are given byB =kerAR with A = C∞((0, 1),R).

We will also say that a behaviourB ⊆ Aq has animage representation, and writeB = imAM, if there exists a matrixM ∈ Dq,c for somec with

B = {w ∈ Aq | ∃ l ∈ Ac with w = Ml}. (4)

Returning to the case of a kernel representationB = kerAR, consider the rowspan overD of the matrixR ∈ Dg,q , i.e. the setD1,gR, which we will sometimes


write this as ImDR. This is a submodule of the leftD-moduleD1,q , and we will referto it as anequation module; however, it must be distinguished from themodule of allsystem equationsdefined in Section 3.1.

The factorD1,q/D1,gR, which we denote by CokerDR, is also a finitely generatedleft D-module. The elements of CokerDR can be interpreted as mappings fromB toA; for anyx ∈ D1,q and anyw ∈ B, we have

(x + ImDR)w := xw ∈ A, (5)

wherexw denotes the operatorx applied to the trajectoryw in the obvious way.Notice that, since the elements of ImDR are all system equations, this is indeedwell defined, i.e. ifx + ImDR = x′ + ImDR then(x + ImDR)w = (x′ + ImDR)w

for anyw ∈ B. We therefore introduce the obvious notationxB or xB (wherex =x + ImDR) for the set of all elements of form (5), wherew ranges overB.

The module CokerDR is used by some authors as a model for the system itself;Fliess refers to it as the “system dynamics” [6] (note however that it is not alwaysuniquely defined given the system behaviour, as we will see in Section 3). See[1,6,7,29] for examples of work which use this module as the main tool in describingthe system.

Malgrange [21] observed that CokerDR andB = kerAR are formally related

kerAR = HomD(CokerDR,A). (6)

Thus the behaviour is (as an additive Abelian group) equal to the set of allD-linearmaps from the leftD-module CokerDR to the leftD-moduleA. This is seen asfollows: for anyw ∈ kerAR andx ∈ D1,q , define

w(x + ImDR) := xw.Conversely, anyD-linear map from CokerDR to A can be identified with a systemtrajectory, the components of which are the values of the map on the generatorse1 + ImDR, . . . , eq + ImDR, wheree1, . . . , eq are the natural basis vectors ofD1,q .

Notice however thatB = HomD(CokerDR,A) is not in general aD-module.Consider for example the behaviour defined inC∞(R,R) by

dw

dt− 2tw = 0.

A solution of this isw = et2; however the derivative of thisw is not a solution. This

can also be viewed as a consequence of the fact that ImDR is a leftD-module butgenerally not a rightD-module. Of course, in the case whereD is commutative (i.e.constant coefficient equations),B does have aD-module structure.

2.1. Modules over non-commutative Noetherian rings

As elsewhere in the paper, we takeD here to be a left and right Noetherian do-main. Where specific references or proofs are not given for a definition or a result,the material is completely standard (a good reference is [18]).


Definition 2.2. We say that a leftD-moduleM is:• free if it has a basis overD.• projectiveif for any surjection of leftD-modulesφ : L �→ N, and any mapψ :M �→ N, there is a mapτ : M �→ L with ψ = φ ◦ τ . Equivalently,M is a di-rect summand of some free module.

• injective if for any injection of leftD-modulesφ : L �→ N, and any mapψ :L �→ M, there is an extension ofψ to a mapτ : N �→ M, i.e. withψ = τ ◦ φ.

• torsion if for eachx ∈ M there is some non-zero elementp ∈ D with px = 0.• torsionfreeif for eachx ∈ M, x /= 0, there is no non-zero elementp ∈ D withpx = 0.

Any finitely generated free module is isomorphic toD1,q for someq. Free mod-ules are projective, and projective modules are torsionfree. The following charac-terization of torsionfree modules is due to Gentile/Levy; see, e.g. [14, Proposition6.19].

Lemma 2.3. Any finitely generated torsionfree leftD-module can be embedded ina finitely generated freeD-module.

Given a leftD-moduleM, we can define thetorsion submodule, denotedt (M),which is the left submodule of all elementsx with px = 0 for some non-zerop ∈ D.Furthermore,M/t (M) is torsionfree.

Definition 2.4. Given a matrixM ∈ Dg,h, another matrixC ∈ Dc,g is called auni-versal left annihilatorof M if the rows ofC generate all the relations (syzygies) onthe rows ofM, i.e. if the sequence

D1,c C−→D1,g M−→D1,h

is exact. Similarly, if the columns ofM generate the syzygies on the columns ofC,i.e. if the sequence

DhM−→Dg

C−→Dc

is exact (where now the maps denote natural action on column vectors),M is said tobe auniversal right annihilatorof C.

Universal left and right annihilators have formally been calledminimal left andright annihilators. It follows from the Noetherian property ofD that everyD-matrixadmits a universal left annihilator and a universal right annihilator. The followingresult is elementary:

Corollary 2.5. LetM = CokerDR be a finitely generated leftD-module. ThenMis torsionfree if and only if R is a universal left annihilator(of some matrix).


Proof. Suppose thatR ∈ Dg,q , so that there is a projection fromD1,q to M. ThenM is torsionfree if and only ifM can be embedded inD1,c for somec (by Lem-ma 2.3), i.e. if and only if there is a mapD1,q �→ D1,c (which can be represented byaD-matrix) with kernel ImDR. Equivalently,R is a universal left annihilator. �

We now define the well-known Ext and Tor modules.

Definition 2.6. Let M be a finitely generated leftD-module withM = CokerDMfor some matrixM ∈ Dg,h, and letA be an arbitrary leftD-module.

Further, letC ∈ Dc,g be a universal left annihilator ofM so that

D1,c C−→D1,g M−→D1,h −→M −→ 0

is an exact sequence. Then we have a complex

0−→Ah M−→Ag C−→Ac.

The homology of this complex depends only uponM and A. The homolo-gy at Ah is denoted by Ext0

D(M,A), and the homology atAg is denoted byExt1D(M,A). That is,

Ext0D(M,A) :=kerAM = HomD(M,A), (7)

Ext1D(M,A) :=kerAC

imAM. (8)

Ext is defined analogously for rightD-modules. Now letL be a finitely generatedright D-module of the formL = Dg/RDq , whereR ∈ Dg,q . Further, letC ∈ Dq,c

be a universal right annihilator ofRso that

DcC−→Dq

R−→Dg −→L −→ 0

is an exact sequence of rightD-modules, where the matrices act on column vectors.Then we have a complex

Ac C−→Aq R−→Ag −→ 0,

given formally by right tensoring withA. The homology of this complex atAq

depends only onL andA, and is denoted by TorD1 (L,A). In other words,

TorD1 (L,A) := kerAR

imAC. (9)

Lemma 2.7. A leftD-moduleA is injective if and only ifExt1D(M,A) vanishes forall finitely generatedM. A leftD-moduleM is projective if and only ifExt1D(M,A)vanishes for allA.


Injectivity of A is also equivalent to exactness of the contravariant functorHomD(−,A). The following result proved in [31] (see also [45]) will be usefullater:

Lemma 2.8. LetR ∈ Dg,q be arbitrary, and setM = D1,q/D1,gR,L = Dg/RDq .Then

t (M) ∼= Ext1D(L,D), (10)

where∼= denotes isomorphism of leftD-modules.

Definition 2.9. A left D-moduleA is calledflat if the functor− ⊗ A is exact, orequivalently if TorD1 (L,A) vanishes for all finitely generated rightD-modulesL.

Note that flat modules have nothing to do with the class of “flat systems” as stud-ied for example in [9]. A projective module is flat, and a flat module is torsionfree.

Definition 2.10. The injective dimensionof a leftD-moduleA, written as id(A),is the minimum lengthl of an exact sequence of the form

0−→A−→ I0 −→ I1 −→ I2 −→ · · · −→ Il −→ 0

with I0, I1, . . . , Il injective leftD-modules.

We will not need the full definition of projective dimension, but only the follow-ing:

Definition 2.11. A finitely generated left moduleM over a left Noetherian ringDhasprojective dimension� 1 if there exists a matrixM ∈ Dg,h for someg, h suchthat

0−→D1,g M−→D1,h −→ M −→ 0

is an exact sequence.

In the case where the ringD is a commutative domain, the matrixM in Defini-tion 2.11 will have full row rank (over the quotient field ofD).

We now introduce the concept of torsionfree degree, which has not previouslybeen formalized although it is implicit in papers by Pommaret and Quadrat [29,31].

Definition 2.12. Let D be a left Noetherian domain with left global dimension3

n, and letM be a finitely generated leftD-module. Then thetorsionfree degreeofM, denoted tf(M), is the maximum value ofd ∈ 0, 1, . . . , n such that there exists asequence ofD-matricesC1, C2, . . . , Cd−1, and a sequence of finitely generated freeleft D-modulesD1,h1,D1,h2, . . . ,D1,hd giving us an exact sequence

3 In a polynomial ring, this is the number of indeterminates.


0−→M−→D1,h1C1−→D1,h2

C2−→· · · Cd−1−→D1,hd .

By Lemma 2.3, a finitely generated leftD-moduleM is torsionfree if and onlyif tf (M) � 1. As shown in [29], the condition tf(M) � 2 is captured by the con-cept of reflexivity. It follows in fact from Lemma 2.7 thatM is projective preciselywhen it has torsionfree degreen. Thus the torsionfree degree captures some generallyinteresting algebraic properties.

Next, given a subsetSof a leftD-moduleA, we define theannihilatorof Sby

annS := {p ∈ D |px = 0 for all x ∈ S}. (11)

This is a left ideal or a two-sided ideal whenS is a left submodule ofA. WhenS isa singleton set,S = {x}, we just write annx for the annihilator.

For any idealJ we use the notation

(0 : J )A := {w ∈ A |pw = 0 for all p ∈ J}

,

(0 : J∞)A :=∞⋃i=1

(0 : J i)A.

Finally, in the case whereD is a commutative domain, recall that aprime ideal Pis one for whichD\P is multiplicatively closed. Given a moduleM, theassociatedprimesof M are the prime idealsP such thatP = annx for somex ∈ M.

3. The Willems closure problem

Systems theory rests on the principle that the structure of a system (i.e. of itstrajectories) is reflected in the structure of the defining equations. However this re-lationship may be very limited, since it is possible that the system satisfies moreequations than those generated by the defining set! Consequently, even thealgebraicstructure of the system behaviour may not be adequately described by the algebraicstructure of the equations and the module they generate. The problem of finding agenerating set for the module of all equations satisfied by the system is the Willemsclosure problem.

In systems modelling, the Willems closure problem arises in two main contexts:identification of a system from trajectories, and interconnection of two known sys-tems. In each of these contexts, it is important that the resulting model of the system,i.e. the resulting set of system equations, satisfies the property of Willems closure;otherwise, little information on the dynamical behaviour can be obtained from theequational model. However here we are concerned not with modelling issues butwith the more fundamental problems of computing the Willems closure and of char-acterizing Willems closed submodules.


This section is divided as follows. We begin in Section 3.1 by formalizing theproblem and presenting some examples to demonstrate that it is non-trivial. In Sec-tion 3.2 we show that Willems closure is captured by the standard algebraic propertyof A-torsionlessness, and also that all submodules ofD1,q are Willems closed whenthe signal space is a cogenerator. In Section 3.3 we present some results characteriz-ing Willems closed modules.

3.1. The Willems closure of a module

Given a behaviourB = kerAR, we define themodule of all system equationsB⊥,also called theorthogonal module, by

B⊥ := {v ∈ D1,q | vw = 0 for allw ∈ B

}, (12)

wherevw is interpreted in the usual way as the action of the operatorv on w. SinceD is Noetherian,B⊥ is a finitely generated leftD-submodule ofD1,q , and so thereexists some matrixRcl ∈ Dg

′,q for someg′ such thatB⊥ = ImDRcl.

Clearly ImDR ⊆ B⊥, i.e. anyD-linear combination of the original equations isactually a system equation. The problem is that the reverse inclusion may not hold,i.e. there may exist system equations which cannot be obtained algebraically fromthe defining set of equations. Such equations can only be derived throughanalysisofthe properties of the signal space and the operators acting on it.

Example 3.1.1. This example is well known in the delay-differential systems literature (e.g. [13,

Example 2.3] and [15]): letD = R[s, z], wheres denotes the derivative operatord/dt , andz the unit shift operator; letA = C∞(R,R). Consider the behaviourdefined by

B = kerAR, R = (s).Now B consists of all constant functions, and so the module of system equationsis

B⊥ = ImD

(s

z− 1

)/= ImDR.

This particular example can be explained by arguing that the operatorssandzarenot independent, and indeed the delay-differential case can be properly rectifiedby considering a different ring of operators (see, e.g. [13,15]); nevertheless, thisexample illustrates the dangers in the naïve approach.

2. The following example is given in [10, Example 6]. TakeA = D′(R,R) andD = R(t)[z] with z representing the derivative operator or any subring containingthe elementt3z+ 1, and consider

B = kerA(t3z+ 1).

It can be shown that the equationt3dw/dt + w = 0 has no non-zero distributionalsolution [34, Vol. 6:15] so thatB = 0 andB⊥ = D.


3. The following example demonstrates the possible existence of “hidden constraints”on derived quantitiesxB.TakeD = R[z1, . . . , zn] treated as the ring of linear partial differential opera-tors with real coefficients, andA = S′(Rn,R), the space of rapidly decreasingfunctions. Consider the behaviourB given by

B = kerAR, R =z1 z1 − 1z2 z1 − 10 z11 − z22

.

Let x = (0 1) ∈ D1,2, and consider the quantityxw = w2, w ∈ B. Computingthe annihilator of the formal elementx + ImDR, we obtain

ann(x + ImDR)=((z1 − 1)(z1 − z2), (z1 + z2)(z1 − z2)

)=((z1 − 1)(z1 − z2), (z2 + 1)(z1 − z2)

).

However these are not the only constraints onxB = {w2 ∈ A |w ∈ B}. We alsofind that if(z1 − 1)(z1 − z2)w = 0 and(z2 + 1)(z1 − z2)w = 0, then(z1 − z2)wis killed by both(z1 − 1) and(z2 − 1), so is therefore of the formα exp(t1 − t2).Since(z1 − z2)w is a rapidly decreasing function,α = 0 and so(z1 − z2)w mustalso vanish. Thus(z1 − z2) kills xB, although it does not kill the module elementx + ImDR, i.e.(z1 − z2)x �∈ ImDR.

In the third example above, we see that the annihilator of the formal quantityx + ImDR is distinct from the annihilator of the corresponding signal setxB (the setof values which this quantity can take). This is due to a phenomenon common to allthree examples above: the module of system equations is not generated by the givenequations.

We now need some further notation. For any leftD-submoduleN of D1,q (nec-essarily finitely generated), denote byN⊥ the behaviour which is the set of solutionsof the equations ofN (with respect to a given signal space)

N⊥ := {w ∈ Aq | vw = 0 for all v ∈ N}

. (13)

For an arbitrary behaviour (set of trajectories)B ⊆ Aq , we do not necessari-ly have (B⊥)⊥ = B. Indeed, this equation implies thatB has a kernel represen-tation, sinceB = N⊥ for N = B⊥. Conversely, ifB has a kernel representationB = kerAR, thenB = N⊥ for the moduleN = ImDR, and it is easy to show thatN⊥⊥⊥ = N⊥ for anyN, soB⊥⊥ = B. Thus the solution set of the module ofsystem equations is equal to the original behaviour if and only if that behaviour hasa kernel representation.

The following definition is due to Pillai and Shankar [27]:

Definition 3.2. Let N be a leftD-submodule ofD1,q . Then the submoduleN⊥⊥is called theWillems closure ofN (with respect toA). If N = N⊥⊥, thenN issaid to beWillems closed(with respect toA).


Thus the Willems closure ofN is the module of all system equationsB⊥ sat-isfied by the solution setB = N⊥ of N. ClearlyN ⊆ N⊥⊥. One also sees that(N⊥⊥)⊥⊥ = (N⊥⊥⊥)⊥ = N⊥⊥, so that the Willems closure of a submodule isalways itself Willems closed.

We can also introduce the moduleD1,q/B⊥, which as before can be interpreted asa set of mappings fromB toA. However, unlike in the case of the modulesD1,q/Nfor arbitrary submodulesN, we see that the mappings corresponding to elements ofD1,q/B⊥ must be non-zero. This will be formalized in Section 3.2. The modulesB⊥ andD1,q/B⊥ reflect the system structure far more tightly than do ImDR andCokerDR for an arbitrary kernel representationR.

We now demonstrate that, when we start with a Willems closed equation mod-ule, the constraints on elements of the moduleM (formal quantities) agree with theconstraints on the corresponding sets of signals. In the case whenN is not Willemsclosed, as in Example 3.1.3, without consideration of the analytic properties of thesignal space we cannot deduce the constraints on the system variables or derivedquantities.

Lemma 3.3. Let N be an arbitrary leftD-submodule ofD1,q which defines thebehaviourB = N⊥. Let x1, . . . , xl ∈ D1,q be arbitrary operators which project,respectively, ontox1, . . . , xl ∈ D1,q/N.We have that

(p1 · · ·pl) ∈ D1,l

∣∣∣∣∣∣∣ (p1 · · ·pl)x1...

xl

= 0

⊆

(p1 · · ·pl) ∈ D1,l

∣∣∣∣∣∣∣ (p1 · · ·pl)x1w...

xlw

= 0 for all w ∈ B

. (14)

In particular, for anyx ∈ D1,q projecting ontox ∈ D1,q/N, we have

ann(x) ⊆ annxB (15)

and

annD1,q/N ⊆ annB. (16)

WhenN is Willems closed with respect toA, inclusions(14)–(16) become equali-ties.

Proof. It is sufficient to prove (14), since (15) follows by takingl = 1, and then (16)follows from (15) by intersecting over allx (or a generating set).

Observe that the left-hand side of (14) is given by{(p1 · · ·pl) ∈ D1,l

∣∣∣∣∣l∑i=1

pixi ∈ N

},


whereas the right-hand side is given by{(p1 · · ·pl) ∈ D1,l

∣∣∣∣∣l∑i=1

pixi ∈ B⊥}.

SinceN ⊆ N⊥⊥ = B⊥, we have the desired inclusions, with equality whenN isWillems closed with respect toA. �

In Section 5 we will see some further advantages of using a Willems closed equa-tion module. It turns out that the condition of Willems closure is precisely what isneeded in order to test for inclusion of the behaviour in other behaviours. Also, whenA satisfies the property of injectivity (and is not a torsion module) and ImDR is Wil-lems closed we obtain an algebraic test for the existence of an image representationfor kerAR.

3.2. Willems closure and cogenerators

In this section we demonstrate that the idea of Willems closure is in fact tied upwith some standard concepts in algebra. To begin with, we need the endomorphismring ofA:

E := HomD(A,A). (17)

A is a rightE-module and therefore a(D,E)-bimodule. It follows that a behaviourB given byB = HomD(M,A), for M a finitely generated leftD-module, becomesa rightE-module; for anyw ∈ B = HomD(M,A) ande ∈ E we definewe ∈ B bycomposition:we := e ◦ w ∈ HomD(M,A).

We now define a map, sometimes called theGel’fand map, which for a given leftD-moduleM is given by [18, Section 19D]:

θM : M −→ HomE(HomD(M,A),A),

θM : x −→ (w �→ w(x)).

This map sends a module elementx ∈ M to the ‘evaluation map’ which is the inter-pretation ofx as an (e.g. differential) operator sendingB toA.

The kernel ofθM is of particular interest. Notice that this consists of all elementsx ∈ M such thatw(x) = xw = 0 for allw ∈ B. These are the elements ofM whichare identified with the zero operator onB.

Definition 3.4 (e.g.[18, (19.5), (19.36)]). A left moduleM is said to beA-torsionlessif θM is injective. It is said to beA-torsion if the kernel ofθM is the whole ofM. A is said to be acogeneratorif all left modulesM areA-torsionless.

We soon conclude the following (in which the equivalence of conditions 2 and 3is standard, e.g. [18, (19.37)]):


Lemma 3.5. LetM = D1,q/N for some leftD-submoduleN of D1,q , and letAbe arbitrary. The following are equivalent:1. N is Willems closed with respect toA.2. M is A-torsionless.3. M can be embedded in a direct product of copies ofA.More generally,

ker θM = N⊥⊥/N, M/ker θM = D1,q/N⊥⊥. (18)

If A is a cogenerator, then all submodules ofD1,q are Willems closed with respecttoA.

Proof. From the preceding remarks, kerθM is the set of all elements ofM whichvanish as operators onB. Writing a given element asx + N, we see that it van-ishes onB precisely whenx ∈ N⊥⊥. This establishes Eq. (18), and so the equiv-alence of conditions 1 and 2. Now ifM is A-torsionless thenM can be embed-ded intoAHomD(M,A) via the Gel’fand map. Conversely, suppose there is an in-jection ι : M �→ AX for some setX. Then for anyy ∈ M there is someα ∈ Xwith (ι(y))(α) /= 0. Composingι with the projectionAX �→ A given by evaluat-ing atα, we have a mapw ∈ HomD(M,A) with w(y) /= 0. ThusθM(y) /= 0, andy �∈ ker θM. As this holds for ally ∈ M, M is A-torsionless. This establishes theequivalence of 1–3. The final claim is immediate.�

The ideal solution to the Willems closure problem is to use a signal space whichis a cogenerator or at least with respect to which all submodules ofD1,q (for all q)are Willems closed. This is the case for linear PDEs with constant coefficients (D =R[z1, . . . , zn]) whenA = C∞(Rn,R) orA = D′(Rn,R), as shown by Oberst [24].In the caseD = K[d/dt], whereK = R(t) ∩ C�(U,R), Fröhler and Oberst [10]have shown that the signal space of hyperfunctions is a cogenerator. Thus for theseoperator rings and signal spaces, the problem is fully solved.

In fact, in order for all submodules ofD1,q to be Willems closed, it is not nec-essary thatA be a cogenerator. The following demonstration of this is due to Ob-erst [23]: takeD to be a commutative Noetherian domain but not a field,Q(D) tobe its quotient field andA to be the direct product of all finite length modules.Then it can be shown that HomD(Q(D),M) = 0 for all finitely generatedM so thatHomD(Q,A) = 0 also. It follows thatA is not a cogenerator. However any finitelygeneratedM can be embedded in a direct product of modules of the formM/J kM,J a maximal ideal, and therefore in a power ofA.

For practical purposes it is not strictly necessary even that all submodules ofD1,q

be Willems closed, but only that we have an algorithm (preferably a reasonably ef-ficient and numerically stable one) for computing the Willems closure of a givensubmodule. Equivalently, we want to be able to compute the kernel of the Gel’fandmapθM for a given finitely generated moduleM. Furthermore, it may be sufficient,at least in principle, to have an explicit description of the Willems closed submodules


(or theA-torsionless modules), since the Willems closure of someN ⊆ D1,q isalways the smallest Willems closed module containingN. This follows since ifN ⊆ L ⊆ N⊥⊥ andL is Willems closed, then(N)⊥⊥ ⊆ (L)⊥⊥ = L so thatL = N⊥⊥.

The following result will prove useful later:

Corollary 3.6. Let A be a leftD-module, andM be a finitely generated leftD-module. Then for anyM1 ⊆ ker θM, we have

HomD(M/M1,A) = HomD(M,A).

In particular, the moduleM/kerθM defines the same behaviour asM, and is itselfA-torsionless. If L ⊆ M, then

kerθL ⊆ kerθM.

Proof. Without loss of generality, suppose thatM = D1,q/N for someq,N. IfM1 ⊆ kerθM, then we can writeM1 = N1/N for someN1 with N ⊆ N1, andby (18) we also haveN1 ⊆ N⊥⊥. HenceN⊥

1 = N⊥, and we have

HomD(M/M1,A)= HomD(D1,q/N1,A) = N⊥

1

= N⊥ = HomD(M,A).

Next, sinceM/kerθM = D1,q/N⊥⊥ andN⊥⊥ is Willems closed with respect toA, M/kerθM must beA-torsionless.

Finally, supposeL ⊆ M. There is a map� : HomD(M,A) �→ HomD(L,A),given by restricting a homomorphism fromM to L. Thus if x ∈ kerθL andw ∈HomD(M,A), then the value ofw at x equals the value of�(w) at x, which is 0,proving thatx ∈ kerθM. �

In the following section we present some results which characterize Willems clo-sure under certain conditions. First, notice that, if we have one cogeneratorA, thenany signal spaceA′ containingA must also be a cogenerator. More generally, if amoduleM is A-torsionless, then it is alsoA′-torsionless. These results follow fromcondition 3 in Lemma 3.5.

3.3. Some results on Willems closure

We begin with a new result which, under a minor assumption, identifies a largeclass ofA-torsionless modules.

Lemma 3.7. SupposeA is not torsion. ThenkerθM ⊆ t (M), and in particular anyfinitely generated torsionfree leftD-module isA-torsionless. Thus ifN′/N is thetorsion submodule ofD1,q/N, thenN⊥⊥ ⊆ N′, andN′ is Willems closed withrespect toA.


Proof. We need only prove the first claim, since the second is just a translation ofthis into terms of the modulesN andN⊥⊥. If A is not torsion, so contains a non-torsion elementw′, thenD can be embedded as a leftD-module intoA, byr �→ rw′.Hence any finitely generated free leftD-module, and therefore (by Lemma 2.3) anyfinitely generated torsionfree leftD-module, can be embedded in a direct product ofcopies ofA, i.e. such modules areA-torsionless. Now, HomD(−,A) is left exact,from which there is an embedding HomD(M/t (M),A) �→ HomD(M,A) givenfor anyw ∈ HomR(M/t (M),A) and anyx ∈ M by w �→ (x �→ w(x + t (M))).Hence we have a mapx �→ x + t (M) from kerθM to kerθM/t (M). In the case whereA is not torsion, kerθM/t (M) = 0 by the preceding argument, and sox ∈ t (M) foranyx ∈ kerθM. �

Note that few signal spaces used in practice are torsion modules (an exception isthe space of finitely supported signals onNn under the ring of shift operators).

Example 3.8. TakeA = D′(Rn,R), and consider systems defined by linear ordi-nary differential equations with coefficients inR[t], i.e.D = R[t, z], wherez is thederivative operator. We look at the behaviour

B = kerAR, R =(z −tzz t4z2 − 1

).

Then the element(z − 1) is in B⊥, since(z − 1)w satisfies the equation

(t3z+ 1)(z − 1)w = (t3z2 + z −t3z− 1)w = (t3z 1)Rw = 0, (19)

which implies that(z − 1)w ∈ kerA(t3z+ 1), so (z − 1)w = 0 as discussed inExample 3.1.2. So, denotingM := CokerDR, we see that(z − 1)+ ImDR ∈ M isin kerθM. Clearly the signal space is not a torsion module, so we can apply Lem-ma 3.7. Indeed,(z − 1) is a torsion element, since(t3z+ 1)(z − 1) ∈ ImDR asshown by Eq. (19). However it is not hard to show that(z − 1) is not in ImDR.

For the same signal space and operator ring, the module

M = CokerDR, R = (z −tz)can easily be shown to be torsionfree. By Lemma 3.7, this module is thereforeA-torsionless, and so the row span ofR is Willems closed.

In the remainder of this section, we are forced to restrict ourselves to the casewhereD is a commutative domain (so in the case of PDEs we assume constantcoefficients).

The following new result is inspired by the characterization of [35, Theorem 2.3]of the Willems submodules with respect to the classical spaceS′(Rn,R) over thering of linear constant coefficient partial differential operators. For a description ofprimary decomposition, see for example [4].


Theorem 3.9. Let D be a commutative Noetherian domain, A an arbitrary D-module, andN a submodule ofD1,q for some q. Suppose thatN = ⋂t

i=1Ni isan irredundant primary decomposition ofN in D1,q , whereNi is a Pi-prima-ry submodule ofD1,q for some prime idealsP1, . . . , Pt . Assume the componentsto be ordered so that for somer ∈ 0, . . . , t, (0 : P1)A /= 0, . . . , (0 : Pr)A /= 0 but(0 : Pr+1)A = · · · = (0 : Pt)A = 0. Then

r⋂i=1

Ni ⊆ N⊥⊥. (20)

In the case where(0 : P∞1 )A, . . . , (0 : P∞

r )A are injective modules, we moreoverhave

r⋂i=1

Ni = N⊥⊥. (21)

In particular, if N is Willems closed with respect toA then(0 : Pi)A /= 0 for allassociated primesPi of D1,q/N, and conversely when the injectivity conditionshold.

Proof. Note first that the last sentence follows from the earlier claims. LetN,

N1, . . . ,Nt be as specified, and writeM = D1,q/N,Mi = Ni/N, i = 1, . . . , t ,so thatD1,q/Mi is Pi-coprimary. Then 0= ⋂t

i=1Mi is an irredundant primarydecomposition of 0 inM, and by Eq. (18) we must show that

r⋂i=1

Mi ⊆ kerθM (22)

with equality when the given injectivity conditions hold.Thus letL := ⋂r

i=1Mi ⊆ M. Supposex ∈ L; we must show thatx ∈ kerθM.As M/Mr+1 is Pr+1-coprimary, there must exist a power ofPr+1 which takesx tosomex′ ∈ Mr+1 [4, Proposition 3.9], and similarly a power ofPr+2 which takesx′ into Mr+2, and so on. Putting this together, there is some productS of pow-ers ofPr+1, . . . , Pt with Sx = 0. Now for everyw ∈ HomD(M,A), writing vw =w(x) ∈ A we findSvw = 0. Since none of the primesPr+1, . . . , Pt annihilate anyelement ofA, we must havevw = 0 for all suchw. Hencex ∈ kerθM, which estab-lishes (20).

Now suppose that(0 : P∞1 )A, . . . , (0 : P∞

r )A are injective. Pickx ∈ M\L; wehave to show thatx �∈ kerθM, i.e. that there existsw ∈ HomD(M,A) with w(x) /=0. Now for somej ∈ 1, . . . , r, we havex �∈ Mj , sox + Mj ∈ M/Mj is non-zero.HenceD(x + Mj ) is a non-zero submodule of thePj -coprimary moduleM/Mj ,so that some submodule ofD(x + Mj )must be isomorphic toD/Pj . So there existsa ∈ D with ann(ax + Mj ) = Pj .

Since(0 : P∞j )A is injective and non-zero (because(0 : Pj )A is non-zero), it can

be written as a direct sum of indecomposable injectives of the formE(D/Q), where


E(·) denotes the injective hull andQ is prime [4, Example A3.2]. For each suchQ,D/Q is embedded in(0 : P∞

j )A, soPdj (D/Q) = 0 for somed, i.e.Pdj ⊆ Q. As bothPj andQ are prime, we havePj ⊆ Q. Hence there is a projectionρ : D/Pj �→ D/Qwith kernelQ/Pj . Now we have a mapw1 : D(ax + Mj ) �→ (0 : P∞

j )A, given bycomposition

D(ax + Mj )∼=�→ D/Pj

ρ�→ D/Q �→ E(D/Q) �→ (0 : P∞j )A.

Now if w1(ax + Mj ) = 0 thenρ(1 + Pj ) = 0 ∈ D/Q, which is impossible as itmeans that 1∈ Q. Hencew1(ax + Mj ) /= 0. By injectivity of (0 : P∞

j )A there isan extension ofw1 to a mapw2 : M/Mj �→ (0 : P∞

j )A. This map is non-zero asan extension of a non-zero map, and sincew2(ax + Mj ) /= 0 we also havew2(x +Mj ) /= 0. Next, define a map

w3 :t⊕i=1

M/Mi �→ (0 : P∞j )A

byw3|M/Mj= w2, w3|M/Mi

= 0 for i /= j . Clearlyw3 is a non-zero map.Finally,M is embedded in

⊕ti=1M/Mi under the mapx �→ ⊕(x + Mi ). Com-

posing this embedding withw3 and with the natural injection of(0 : P∞j )A into A,

we obtain a mapw : M �→ A which is non-zero atx. This completes the proof.�

In the case where(0 : P∞1 )A, . . . , (0 : P∞

r )A are known to be injective, Theo-rem 3.9 gives a constructive description of the Willems closure ofN as the inter-section of primary components, which can be calculated, e.g. using Gröbner bases[5,12].

The injectivity condition of Theorem 3.9 is restrictive, but will be met in particularwhenA itself is injective, since in this case(0 : J∞)A is injective for any idealJ(see [25, Theorem (1.14)(i)], following the work of Matlis [22]). In this case, the onlyanalysis we need to do to compute Willems closures is to identify once and for all theprimes for which(0 : P)A = 0. The condition of Theorem 3.9 can also be weakenedto: each primePj , j = 1, . . . , r, is contained in a primeQj with (0 : Q∞

j )A injectiveand non-zero.

Example 3.10. TakeD = R[z] andA to be the set of all piecewise polynomi-al functions fromR to R, treated as a subspace ofD′(R,R) with the ring actiongiven by differentiation. Then(0 : P∞)A = 0 for all non-zero primes exceptP =(z), and forP = (z) is equal to the subspace of polynomial functions, which is in-jective as it is equal to(0 : P∞)A1 for A1 the space of smooth functions. Thus(0 : P∞)A is injective for all non-zero primesP, and Eq. (21) holds; the Willemsclosure of anyN ⊆ D1,q is equal to the intersection of its 0-primary and(z)-primarycomponents.


We can also use Theorem 3.9 to strengthen Lemma 3.7 in the commutative case:

Corollary 3.11. LetD be a commutative Noetherian domain. ThenA is torsionfreeand non-zero if and only if the finitely generatedA-torsionless modules are preciselythe finitely generated torsionfree modules.

Proof. Suppose thatA is torsionfree and non-zero. By Lemma 3.7 all finitely gener-ated torsionfree modules areA-torsionless. If on the other handM isA-torsionless,then by the last claim of Theorem 3.9,(0 : P)A /= 0 for all associated primesP ofM. SinceA is torsionfree this implies that the set of associated primes ofM iseither{0} or { }, and soM is torsionfree.

Conversely, suppose that the finitely generatedA-torsionless modules are pre-cisely the finitely generated torsionfree modules. Now forM = D/J , J a non-zeroideal,M/kerθM must beA-torsionless, so torsionfree, and therefore kerθM = M.So J⊥ = HomD(D/J,A) = 0 for any non-zeroJ. HenceA contains no non-ze-ro torsion elements. Since howeverD is A-torsionless, HomD(D,A) /= 0 and soA /= 0. �

Corollary 3.11 applies in particular to the spaces of compactly supported distri-butions, compactly supported smooth functions, and rapidly decreasing functionsover the ringD = R[z1, . . . , zn]; these are torsionfreeD-modules. Thus over sucha signal space, a moduleN ⊆ D1,q is Willems closed if and only ifD1,q/N istorsionfree. Moreover, the Willems closure ofN is the smallest Willems closedsubmodule containingN, which must be theN′ ⊆ D1,q such thatN′/N =t (D1,q/N). Thus we obtain a reconfirmation of the results of [27,35] for these signalspaces.

Corollary 3.12. LetD be a commutative Noetherian domain. If all finitely gener-ated modules areA-torsionless, then every maximal ideal is an associated prime ofA. The converse holds when(0 : P∞)A is injective for all prime ideals P.

Proof. If all finitely generated modules areA-torsionless, then by the last claimof Theorem 3.9(0 : P)A /= 0 for all primesP. For maximal idealsP this requiresthatP be an associated prime ofA. Conversely if all maximal ideals are associatedprimes ofA then(0 : P)A /= 0 for all maximal ideals, and therefore for all primes.The last claim of Theorem 3.9 again completes the proof.�

This completes the description of Willems closed modules over a commutativedomain when the signal spaceA satisfies certain conditions as discussed above. Un-fortunately we cannot expect to be able to extend Theorem 3.9 to the case whereD isnon-commutative. The first difficulty in such an extension is that a primary decompo-sition does not generally exist (a “tertiary decomposition” does [38, Chapter VII.1],but the tertiary components do not have certain properties which seem essential


to the proof of Theorem 3.9). More importantly, the non-commutative rings ofprincipal interest are the Weyl algebraR[t1, . . . , tn, z1, . . . , zn] (wherezi = �/�ti),and certain quotient rings as mentioned in Section 2. The Weyl algebra is well knownto be simple, i.e. it has no non-zero prime ideals, and the same property must holdfor any quotient ring. Prime ideals are therefore of no use in describing the algebraicstructure of these rings and their modules.

4. The elimination problem

The elimination problem is as follows. Given a behaviour with two sets of vari-ablesw, l described by the equations

Bw,l = kerA(−R M) ={(w

l

)∈ Aq+h

∣∣∣∣ Rw = Ml}, (23)

whereR ∈ Dg,q andM ∈ Dg,h, does there exist a kernel representation for the be-haviour

Bw ={w ∈ Aq

∣∣∣∣ ∃ l ∈ Ah with

(w

l

)∈ Bw,l

}, (24)

and if so, how do we construct one? We term this thegeneral elimination problem,and we say that it issolvable for(−R M) if Bw in (24) has a kernel representation.Thespecial elimination problemis the same, but for the caseR = I ; that is, given abehaviour with an image representation

Bw = {w ∈ Aq |w = Ml for somel ∈ Ah

}, (25)

doesBw have a kernel representation, and if so, how do we find one? IfBw doeshave a kernel representation, then we say that the special elimination problem issolvable for M. Note that we are not concerned here with the “formal eliminationproblem” discussed for example in [30], but rather with the problem of eliminatingvariables which take values in some given function space.

4.1. Elimination and injectivity

We begin with the following elementary result, which states that if the specialelimination problem is solvable forM, then the general elimination problem is solv-able for(−R M).

Lemma 4.1. LetBw,l andBw be as specified in(23) and(24). If imAM = kerACfor someD-matrix C, thenBw = kerACR.

Proof. Let C be as specified. ThenBw = {w ∈ Aq |Rw ∈ imAM

} = {w ∈

Aq |CRw = 0}. �


Let us therefore restrict our attention to the special elimination problem

Bw = imAM = {w ∈ Aq |w = Ml for somel ∈ Ah

}.

What are the conditions onw to be inBw? Clearly any relation (“syzygy”) on therows ofM gives us a condition on such aw, i.e.

w ∈ Bw ⇒ vw = 0 for anyv ∈ D1,q with vM = 0. (26)

Thefundamental principleof Ehrenpreis–Palamodov [3,26] states that this conditionbecomes necessary and sufficient whenA = C∞(Rn,R) or D′(Rn,R) for linearPDEs with real coefficients. Thus in this case to test whetherw = Ml for somel itis sufficient to test thatw ∈ kerAC, whereC is a universal left annihilator ofM. Sowe obtain the solution to the special elimination problem: imAM = kerAC, whereC is any universal left annihilator ofM for these signal spaces.

By combining the fundamental principle with Lemma 4.1, we obtain the generalelimination algorithm reported in [24, Corollary 2.38] and [17].

It may be instructive to express the relationships between modules above usingexact sequences. The condition thatC ∈ Dc,q is a universal left annihilator ofM ∈Dq,h is expressed by exactness of the sequence

D1,c C−→D1,g M−→D1,h −→M −→ 0. (27)

Now by applying the functor HomD(−,A) to this exact sequence, we obtain a com-plex

Ah M−→Ag C−→Ac (28)

(stating only that imAM ⊆ kerAC). However, when the moduleA is injective, thefunctor HomD(−,A) becomes exact, so that exactness of (27) guarantees exactnessof (28), and therefore the fundamental principle and elimination algorithm. Con-versely, if the fundamental principle holds, then in particular it holds whenh = 1and the entries ofM generate an idealJ. In this case, the statement of the fundamen-tal principle can easily be seen to be equivalent to the well-known Baer’s criterionfor injectivity (e.g. [18, (3.7)]). Thus we obtain the following simple result, alreadyimplicit in [24]:

Lemma 4.2. The following are equivalent:1. A is an injectiveD-module.2. The special elimination algorithm is solvable for any M, and furthermore the

solution isimAM = kerAC, where C is any universal left annihilator of M.3. The general elimination algorithm is solvable for any(−R M), and furthermore

the solution isBw = kerACR, where C is any universal left annihilator of M.

Example 4.3.1. Take the case of linear PDEs with constant (real) coefficients, i.e.D = R[z1, . . . ,zn], wherezi = �/�ti . As already discussed, the fundamental principle due toEhrenpreis/Palamodov [3,26] shows that the spacesA = C∞(Rn,R) andA =


D′(Rn,R) are injectiveD-modules, and so the general and special eliminationproblems are always solvable over theseA. More generally, Ref. [26, Corol-lary VII.8.4] shows thatC∞(U,R) andD′(U,R) are injective for any open setU ⊆ Rn whose connected components are convex.

2. For linear ordinary differential equations with variable coefficients, Fröhler andOberst [10] have shown that neitherC∞(U,R) norA = D′(U,R) is an injectivemodule over any of the ringsR[t, z], R(t)[z], or (R(t) ∩ C�(U,R))[z], wherezis the derivative operator andU ⊆ Rn is an open set. However, the module ofhyperfunctions is injective, and so elimination can be done in this space [10]. Thesituation for linear PDEs with variable coefficients is to the best of our knowledgestill unknown.

3. The delay-differential signal spaceA = C∞(R,R) is not injective when the op-erator ring is taken to beD = R[s, z] (s = derivative operator,z = unit shift).Consider for example the problem of eliminating the variablel from the behaviourdefined by the equation(

10

)w =

(z− 1s

)l.

Application of the usual elimination algorithm (constructing an appropriate uni-versal left annihilator and applying Lemma 4.1) gives us the answer

Bw = kerA(s),

which is wrong. It is easy to see that the correct answer isBw = kerA(1) = 0.Thus the general elimination algorithm does not apply, and by Lemma 4.2Acannot be an injectiveD-module. As we will soon see, this implies that thereexist general and special elimination problems which are not solvable over thissignal space.

Notice that Lemma 4.2 does not claim that if we can always solve the specialelimination problemby some methodthenA is injective. However this is in fact thecase, as shown by the following simple new result.

Lemma 4.4. Suppose thatannA = 0, and thatR1 andR2 are D-matrices withimAR1 = kerAR2. LetR′

1 be a universal right annihilator ofR2, andR′2 a univer-

sal left annihilator ofR1. Then we also have thatimAR1 = kerAR′2 and imAR

′1 =

kerAR2. Thus if the special elimination problem is solvable for M, then a solution isimAM = kerAC, where C is any universal left annihilator of M.

Proof. Let A, R1, R2, R′1 andR′

2 be as stated. Letg be the number of columnsof R1 andh the number of rows ofR2. ThenR2R1 : Ag �→ Ah is the zero map,which means that every entry ofR2R1 must be in annA = 0. HenceR2R1 is thezero matrix. This implies that the rows ofR2 are syzygies on the rows ofR1, and thecolumns ofR1 are syzygies on the columns ofR2. Therefore there exist matricesLandQ with R2 = LR′

2 andR1 = R′1Q. We now find


kerAR2 = imAR1 ⊆ imAR′1

and

kerAR′2 ⊆ kerAR2 = imAR1.

The reverse inclusions hold sinceR2R′1 = 0 andR′

2R1 = 0. �

Lemma 4.4 establishes that, provided annA = 0, injectivity of A is not onlysufficient, but also necessary for the special elimination problem to be solvable forall M.

The condition annA = 0 in Lemma 4.4 is a minor one; it states that there shouldbe no single operator in the ring which kills every element of the signal space. Thiscondition will be fulfilled for the vast majority of signal spaces of interest in systemstheory (an exception being signals on a cylinder-shaped lattice, under shift opera-tors). Furthermore, ifJ := annA /= 0, then the signal space is a left module overD̂ := D/J , and we can do our systems theory over this ring instead. The defini-tions of universal left/right annihilators over̂D can easily be expressed in terms ofthe original ringD, thus giving us a mechanism which does not require explicitconsideration of̂D, but for brevity we will omit further details.

The following result says that we have the same situation for the general elimina-tion problem as for the special elimination problem; that is, if a kernel representationexists, then one can be found by the algorithm outlined earlier. The only caveat hereis that, before applying that algorithm, it is first necessary to compute the Willemsclosure of the given equation module, as defined in Section 3.

Lemma 4.5. Let A be a signal space withannA = 0, and (−R M) ∈ Dg,q+h.Suppose thatImD(−R M) is Willems closed with respect toA. Then the behaviour

Bw = {w ∈ Aq |Rw = Ml for somel ∈ Ah

}has a kernel representation if and only ifBw = kerACR, where C is a universal leftannihilator of M.

Proof. Since ImD(−R M) is Willems closed, it contains all equations onBw,l =kerA(−R M), including all those which involvew only. Such equations must there-fore be of the formxRwith xM = 0. So, withC a universal left annihilator ofM, wefind

B⊥w={

v = xR for somex ∈ D1,g with xM = 0}

={v = xR for somex ∈ D1,g with x = yC for somec

}= ImDCR.

Now if Bw has a kernel representation, thenB⊥⊥w = Bw, so we have

Bw = (ImDCR)⊥ = kerACR

as required. The converse is obvious.�


The Willems closure condition in Lemma 4.5 is essential, as it guarantees thatthe module of system equations ofBw is embedded in the row span of(−R M).Moreover, we see from the proof that the Willems closure condition guarantees thatB⊥⊥w = (ImDCR)

⊥ = kerACR, in other words that the smallest behaviour with akernel representation and containingBw is kerACR. This must hold for any signalspaceA.

Notice that, unlike in Lemma 4.5, the Willems closure condition does not ap-pear in the statement of Lemma 4.4 (the special elimination case). This is becauseImD(−I M) is always Willems closed when annA = 0.

4.2. Elimination without injectivity

We know from Lemmas 4.2 and 4.4 that, except in the unlikely event that annA /=0, when adjustments to the results have to be made, injectivity is not only sufficientbut also necessary to a general solution of the general elimination problem. However,injectivity is a strong condition on the signal space (although it is met in many casesof practical interest, as mentioned in the last section), and for any given eliminationproblem, injectivity is not necessary.

Example 4.6. ConsiderA = C∞(U,R), whereU is R3 with the t3 axis removed,and the ring of linear partial differential operators with real coefficients,D = R

[z1, . . . , zn]. Look at the two behavioursB1 = imAR1, B2 = imAR2 defined bythe images of the grad and curl operators onR3:

R1 =z1z2z3

, R2 =

0 −z3 z2z3 0 −z1

−z2 z1 0

.

We can identifyB1 andB2 with the spaces of smooth closed 1-forms and smoothclosed 2-forms onU, respectively. Since the fundamental group ofU is non-trivial,the image ofR1 is not equal to the kernel of the universal left annihilatorR2. Further,as annA = 0, elimination of the variablesl from w = R1l is not possible. That is,there exists no set of linear PDEs with real coefficients, the smooth solutions to whichare the smooth closed 1-forms. However, by standard algebraic topology theory, theimage ofR2 is equal to the kernel of the div operator, which isR3 = (z1 z2 z3),a universal left annihilator ofR2. In the case whereU = R3\{0}, the opposite situ-ation occurs. Thus for neither of these open sets isC∞(U,R) injective (if it were,elimination would be solvable in both cases).

We can express the ability to eliminate in a given case through the algebraic con-cept of the extension functor Ext (see Definition 2.6). This functor has appearedbefore in thenD systems literature; see [29,31,45], and also [19] for extensive use ofExt1D in a 1D systems context.


From the definition of Ext1D together with Eqs. (27)–(28) we have the followingcorollary to Lemma 4.4:

Corollary 4.7. Let A be a signal space withannA = 0, and M aD-matrix withM := CokerDM. Then the following are equivalent:1. The special elimination problem is solvable for M.

2. Ext1D(M,A) = 0.In particular, the existence of a solution to the special elimination problem for agiven M depends only uponCokerDM and onA.

Of course, we can generalize Corollary 4.7 to the case where annA /= 0 by re-placingD by the ringD/(annA). Also, the solution of the special elimination prob-lem, when it exists, is given by constructing a universal left annihilator as shownin Lemma 4.4. Injectivity ofA is clearly an unnecessarily strong condition for agiven elimination problem. Using the condition of the vanishing of Ext1

D(M,A),we are able to identify some weaker conditions onA under which certain classes ofelimination problems can be solved.

The first such condition involves the torsionfree degree of the moduleM =CokerDM (recall Definition 2.12). Note that this quantity is interesting from a sys-tems-theoretical point of view, as explained in [29]. The following new result statesthat the special elimination problem is solvable when the torsionfree degree ofA isat least as high as the injective dimension ofA.

Theorem 4.8. LetA be a leftD-module andM a finitely generated leftD-module.If

tf(M) � id(A), (29)

then Ext1D(M,A) = 0 and so the special elimination problem is solvable for M,whereM = CokerDM.

Proof. Let l = id(A) andd = tf(M). Suppose thatd � l, and choose exact se-quences of leftD-modules having the form

0−→Aφ0−→ I0

φ1−→ I1φ2−→ I2 −→ · · · φl−→ Il −→ 0,

0−→Mψ1−→P1

ψ2−→P2 −→ · · · ψd−→Pd

with I0, . . . , Il injective andP1, . . . , Pd projective (free). DefineAi := im φi , i =0, . . . , l, andMj := cokerψj , j = 1, . . . , d, with M0 := M. Now for two exactsequences of leftD-modules

0 −→ A−→B −→C −→ 0, 0 −→ A′ −→B ′ −→C′ −→ 0

with B projective andB ′ injective, ExtD1 (A,A′) = 0 if and only if Ext1D(C,C

′) = 0;see [18, Theorem 5.50]. Therefore we find


Ext1D(M0,A0) = 0 ⇔ Ext1D(M1,A1) = 0

⇔ · · · ⇔ Ext1D(Ml ,Al ) = 0.

Note that the moduleMl exists asl � d. SinceAl = Il is injective, we have Ext1D

(Ml ,Al ) = 0. Therefore Ext1D(M,A) = Ext1D(M0,A0) = 0 as required. �

Example 4.9. TakeD to be the ring of linear partial differential operators withreal coefficients, andA to be the set of smooth functions vanishing on some closedconvex setT ⊆ Rn. Then we have an exact sequence

0 −→ A−→C∞(Rn,R)−→C∞(T ,R) −→ 0.

Now C∞(T ,R) is the direct limit of the family of setsC∞(Ui,R), for Ui open,convex and containingT. Each of these is injective, and therefore so is their directlimit (e.g. [18, Theorem (3.46)]). Thus the exact sequence above establishes thatAhas injective dimension at most 1. Elimination overA is therefore possible wheneverM is at least torsionfree.

The conditions onA in Theorem 4.8 are in fact unnecessarily strong; they guaran-tee not only that Ext1

D(M,A) = 0 for M with sufficiently high torsionfree degree,but also ExtiD(M,A) = 0 for i � 2 (see e.g. [2,33] for a definition of these). Itis shown in [26, Theorem VII.10.3] that for linear PDEs with constant coefficients(D = R[z1, . . . , zn], wherezi = �/�ti), if tf (M) = l and ifU ⊆ Rn is an open setwith a locally finite convex covering consisting of regions having no more than(l +1)-fold mutual intersections, then forA = C∞(U,R) or A = D′(U,R) we haveExtiD(M,A) = 0, i � 1.

Corollary 4.10. Let M be aD-matrix andM = CokerDM. If M is torsionfree andA has injective dimension at most1, then the special elimination problem is solvablefor M. Also, if M is projective then the special elimination problem is solvable forM, regardless ofA.

Proof. This is immediate from Theorem 4.8, though it is instructive to prove thesecond claim directly. Assume therefore thatM is projective, so that there is anexact sequence

D1,g M−→D1,h φ−→M −→ 0,

whereDg,h andφ is the natural projection. SinceM is projective, the mapφ splits,and it follows that the mapM does also. It is now easy to construct a mapQ : D1,h �→D1,g, necessarily representable by aD-matrix such thatMQM = M. Therefore foranyw = imAM, sayw = Ml for somel ∈ Ah, we have

M(Qw) = MQMl = Ml = w. (30)


Thus imAM ⊆ kerA(I −MQ). Conversely, if(I −MQ)w = 0 thenw = M(Qw) ∈ imAM. So(I −MQ) is a kernel representation of imAM. �

The result for torsionfreeM is likely to be of some systems-theoretic significance,since torsionfree modules appear to be heavily tied up with controllability in thebehavioural sense (e.g. [8,27,28,42]).

The discussion of Section 4 should be in particular demonstrate that elimination isnot a purely formal algebraic problem, in the sense that we cannot eliminate variablesfrom a system of equations without consideration of the signal space and its analyticproperties. Theonly exceptionto this occurs when the moduleM = CokerDM isprojective. The result, reported in Corollary 4.10, that projectivity implies that spe-cial elimination (and therefore general elimination) is solvable independently ofA,is elementary. The proof of Corollary 4.10 demonstrates that, whenM is projec-tive, given aw ∈ imAM we can always reconstruct a possible value ofl such thatw = Ml. The converse result, that ifM is such that the special elimination prob-lem is solvable forM independently ofA thenM must be projective, follows fromLemma 2.7 together with Corollary 4.7.

Example 4.11. Again takeD = R[z1, z2], treated as a ring of partial differentialoperators, but let the signal space be any vector space of functions onR2 closedunder differentiation. Consider the system of equations

w1= �l�t1

− l, (31)

w2= �l�t2

− l, (32)

w3= �l�t1

+ �l�t2. (33)

By constructing a universal left annihilator of the correspondingD-matrix (i.e. byformal elimination in the usual manner), we obtain

�w1

�t1+ w1 + �w2

�t1− w2 − �w3

�t1+ w3=0,

�w1

�t2− w1 + �w2

�t2+ w2 − �w3

�t2+ w3=0.

Furthermore, to anyw1, w2, w3 satisfying these equations there corresponds anlsuch that (31)–(33) are satisfied: we can reconstruct such anl as

l = 1

2(w3 − w1 − w2).

The reason why we are able to find such a reconstruction rule is that the moduleM = CokerDR = D/(z1 − 1, z2 − 1, z1 + z2) is projective. In this particular


example,M = 0, which implies thatl is uniquely determined byw in (31)–(33).These relationships are entirely independent of the signal space, due to the strongproperties of the algebraic structure of the equations.

We can produce different classifications ofM andA for which Ext1D(M,A)vanishes, by considering different module-theoretic properties ofM andA, thoughthe properties considered may not be so natural and interesting from a systems theorypoint of view. However, we will present two more new results which may be ofinterest.

Lemma 4.12. Suppose thatD is a commutative Noetherian domain, and letA beaD-module withannA = 0. Then the following are equivalent:1. A is a divisible module, i.e. the equationrw = w′ for w′ ∈ A, r ∈ D, r /= 0,

always has a solution forw ∈ A.2. For any full row rank matrixM ∈ Dg,h, the mapM : Ah �→ Ag is surjective,

and so the special elimination problem is solvable for M.3. The special elimination problem is solvable for any M such thatM := CokerDM

has projective dimension� 1.

Proof. (1)⇒ (2). Suppose thatA is divisible, and letM ∈ Dg,h be a full row rankD-matrix. Then by taking a right inverse over the quotient field we see that thereexists aY ∈ Dh,g withMY = rI for somer ∈ D, r /= 0. Now for anyw ∈ Ag, bydivisibility we have that there existsw′ ∈ Ag with w = (rI )w′ = M(Yw′), provingthatM : Ah �→ Ag is surjective.(2)⇒ (3). Suppose that condition 2 holds, and letM be such thatM = CokerDM

has projective dimension at most 1. ThenM = CokerDM ′ for someM ′ with fullrow rank, and by condition 2 the special elimination problem is solvable forM ′.By Corollary 4.7, Ext1D(M,A) = 0, and applying that result again we have that thespecial elimination problem is solvable forM.(3)⇒ (1). Suppose that condition 3 holds, and letr ∈ D, r /= 0 be arbitrary.

Then the idealJ = Dr is principal, and so the moduleD/J has projective di-mension at most 1. By supposition, the special elimination problem is solvable forM = (r), and by Lemma 4.4 the solution is imAM = kerA(0) = A. In other words,w is unconstrained inw = (r)w′. Since this holds for allr ∈ D\0, A isdivisible. �

Divisibility is generally significantly weaker than injectivity, though over a prin-cipal left ideal ring the two are equivalent (e.g. [18, Corollary (3.17)′]).

Example 4.13. Consider the spaceA = C∞(R,R) over the delay-differentialoperator ringR[s, z], wheres = d/dt and z is the unit shift operator. This spaceis known to be divisible [13,15], but is not injective (see Example 4.3.3). Considerthe problem of eliminating the variablel from the equations


w = Rl, R =s(z− 1)

s2

s(s − z)

over this signal space. Since CokerDR ∼= D/(s), it has projective dimension 1 and soin fact by Lemma 4.12 we can solve this in the usual way; we obtain the necessaryand sufficient conditions(−z z− 1 −z+ 1

−s s − 1 −s)w = 0.

The last class of signal spaces which we will consider in this section is theclass of signal spaces which are flatD-modules (see Definition 2.9). This con-dition is satisfied for linear PDEs with constant coefficients (D = R[z1, . . . , zn])by the spacesC∞

0 (Rn,R), E′(Rn,R) [26, Corollary VII.8.4], [36] andS(Rn,R)

[36].

Lemma 4.14. Suppose thatA is flat. Then the special elimination problem is solv-able for aD-matrixM ∈ Dg,h if M is a universal right annihilator of some otherD-matrix, or equivalently if the rightD-moduleDg/MDh is torsionfree.

Proof. Suppose thatA is flat. LetM ∈ Dg,h, and suppose thatDg/MDh is tor-sionfree. By Lemma 2.8, an equivalent condition is that Ext1

D(CokerDM,D) = 0.Thus by Corollary 4.7, the special elimination problem can be solved forM over themoduleD, and by Lemma 4.4, the solution is{

Mv | v ∈ Dh} = {

u ∈ Dg |Cu = 0},

whereC is a universal left annihilator ofM. By flatness ofA, we can have left tensorwith A to obtain

imAM = kerAC

as required. The equivalence of the universal right annihilator condition is from Cor-ollary 2.5. �

The conditions for elimination in Lemma 4.14 have been shown in [36] to be bothnecessary and sufficient for the classical spaces listed above.

5. Further consequences of injectivity and Willems closure

In this section we will give a new selection of systems-theoretic consequences ofusing an injective signal space and/or a Willems closed equation module.

The first result shows that, when we use Willems closed equation modules, wecan test for the inclusion of one behaviour in another.


Lemma 5.1. LetB1 = kerAR1 andB2 = kerAR2 be two behaviours contained inAq . If there exists aD-matrix L withLR1 = R2, thenB1 ⊆ B2. The converse holdswhenImDR1 is Willems closed with respect toA.

Moreover, if for a givenR1 we have for anyR2 that kerAR1 ⊆ kerAR2 if andonly if there exists L withLR1 = R2, it follows thatImDR1 is Willems closed withrespect toA.

Proof. To prove the first claim, letB1 andB2 be as specified, and suppose thatL ex-ists with LR1 = R2. Then ImDR2 ⊆ ImDR1, from which we deduce that(ImDR1)

⊥ ⊆ (ImDR2)⊥, i.e.B1 ⊆ B2. Conversely, suppose that ImDR1 is Willems

closed, and thatB1 ⊆ B2. Then

ImDR2 ⊆ (ImDR2)⊥⊥ = B⊥

2 ⊆ B⊥1 = (ImDR1)

⊥⊥ = ImDR1,

which ensures the existence ofL.To prove the second claim, suppose thatR1 is such that the given “if and only if”

condition holds for allR2. In particular, it holds when ImDR2 is the Willems closureof ImDR1. Thus kerAR1 = kerAR2, and so by suppositionL exists, proving that

(ImDR1)⊥⊥ = ImDR2 ⊆ ImDR1,

so that ImDR1 must be Willems closed.�

Example 5.2.1. In the case of delay-differential systems, we have the following classic example

[13, Example 2.3]: letD = R[s, z] with s = d/dt andz the unit shift operator,and setA = C∞(R,R). DefineR1 = (s) andR2 = (z − 1). Then kerAR1 ⊆kerAR2, although there exists noL with LR1 = R2. This phenomenon is due tothe fact that ImDR1 is not Willems closed.

2. TakeD = R[z1, z2] with the usual PDE interpretation, andA = C∞0 (R

2,R), thespace of compactly supported smooth functions. Consider the behaviours

B1 = kerAR1, B2 = kerAR2, R1 =(z21 − 1 z2

1 z2 − 1

), R2 = (z1 z2).

Then it is clear that there cannot exist a matrixL with LR1 = R2. Nevertheless,sinceR1 is square and non-singular, kerAR1 = 0 (no non-zero compactly sup-ported function can be annihilated by the determinant ofR1), soB1 is containedin B2.

Combining Lemma 5.1 with Lemma 3.5, we see that if the signal space is a cogen-erator then the given test for inclusion of behaviours can always be applied. Contraryto [24, (2.62)], the conditions of injectivity and “large”ness are not needed for this.

The second claim of Lemma 5.1 demonstrates that Willems closure is not onlysufficient but also necessary in order to detect whether a behaviour is contained inother behaviours.


The following results in this section deal with the characterization of free vari-ables over a general signal space. Recall that, for a behaviourB ⊆ Aq with systemvariables denoted byw1, . . . , wq , a subset of variables{w1, . . . , wm} is said to bea set of free variablesif the mappingB �→ Am, w �→ (w1, . . . , wm), is surjective.This is clearly related to the elimination problem, and is a precursor to the study ofinput/output structures and transfer function matrices.

Let m(B) denote the maximum set of a set of free variables, called thenumberof free variables, and letχ(M), for a finitely generatedD-moduleM, denote themaximum size of aD-linearly independent subset ofM.

Lemma 5.3. Suppose thatannA = 0. LetB be a behaviour with kernel represen-tation R, and defineM = CokerDR. If {w1, . . . , wm} is a set of free variables ofB,then the elementse1 + ImDR, . . . , em + ImDR areD-linearly independent inM. IfA is injective, then the converse also holds.

Proof. For a given submoduleN of D1,q , we can find the subset of equations onthe firstm variables by intersecting with ImD(Im 0). Now if {w1, . . . , wm} is free,then the set of equations which they satisfy is equal to(Am)⊥, which is zero asannA = 0. Thus

B⊥ ∩ ImD(Im 0) = 0. (34)

Since ImDR ⊆ B⊥, we have

ImDR ∩ ImD(Im 0) = 0, (35)

which is precisely the condition fore1 + ImDR, . . . , em + ImDR to beD-linearlyindependent. Conversely, suppose thatA is injective, and thate1 + ImDR, . . . , em +ImDR areD-linearly independent. LetF ⊆ M denote the span of these elements,which is a free submodule ofM, and consider the embedding mapφ : F �→ M.Now for any mapψ : F �→ A, by the definition of injectivity we have that thereis an extensionτ : M �→ A to M, with τ ◦ φ = ψ , so thatτ must agree withψon e1 + ImDR, . . . , em + ImDR. However, asF is free,ψ can take any combi-nation of values on these elements, soτ is similarly unconstrained. Since any mapτ : M �→ A corresponds to a trajectoryw ∈ B as described in Section 2, there existsa trajectoryw ∈ B with w1, . . . , wm arbitrarily chosen withinA. So{w1, . . . , wm}is a set of free variables.�

Notice that the injectivity condition in Lemma 5.3 is unnecessarily strong. It isenough that Ext1

D(M/F,A) = 0, whereF is the submodule ofM introducedin the proof of the lemma (this can be seen from the well-known long exact se-quence in Ext). Equivalently, if we write the original kernel representationR asR = (−Q P), with the Q submatrix corresponding to the variablesw1, . . . , wm,we need that Ext1

D(CokerDP,A) = 0. Even these conditions are stronger than thosestrictly required.


Corollary 5.4. With notation as in Lemma5.3,

m(B) � χ(M) (36)

with equality whenA is injective.

Example 5.5. Consider the curl or rot operator

R = 0 −z3 z2z3 0 −z1

−z2 z1 0

over the ringD = R[z1, . . . , zn], treated as a ring of linear partial differential opera-tors. Over the injective spacesA = C∞(R3,R) orA = D′(R3,R), it is known thatthe number of free variables in the behaviourB = kerAR is the number of variablesminus the rank ofR [24, Theorem 2.69], that is, 1, and by symmetry we see that anyone of the variables of the system can be chosen freely. Now consider the situation forA = C∞

0 (R3,R), which is a flatD-module ([26, Corollary VII.8.4] and [36]) and

satisfies the condition annA = 0. By Lemmas 4.14 and 4.4,B = kerAR admitsthe image representationB = imAC, whereC is the (universal right annihilator)matrix representing the grad operator. ThusB has a free variable only if one ofthe equationsw = z1l, w = z2l, w = z3l is solvable for any compactly supportedsmoothw. However we can see by settingw to be a non-negative compactly support-ed smooth function that in this case none of these equations is solvable for compactlysupportedl, and soB = imAC has no free variable. Thus overC∞

0 (R3,R) we have

0 = m(B) /= χ(M) = 1.

We next present a general condition for a behaviour to have an image representa-tion. Here we use the Tor functor from Definition 2.6.

Lemma 5.6. Assume thatannA = 0, and letB be a behaviour with kernel repre-sentationR ∈ Dg,q . ThenB has an image representation if and only ifTorD1 (D

g/

RDq,A) = 0.

Proof. Write L = Dg/RDq . By Lemma 4.4, ifB has an image representation atall thenB = imAC, whereC is a universal right annihilator ofR. ThusB has animage representation if and only if kerAR/imAC = 0, which from the definition ofTorD1 gives us the desired condition.�

When B = kerAR has an image representation, Lemma 4.4 immediatelyenables us to identify one:B = imAC, whereC is a universal right annihilatorof R. The comments in Section 4.1 on the case where annA /= 0 apply equallyhere.

It is immediate from Lemma 5.6 that whenA is flat (which guarantees the con-dition annA = 0), any behaviour with a kernel representation also has an image


representation. This is also shown and discussed for the classical spacesA = C∞0

(Rn,R), E′(Rn,R), S(Rn,R) in the context of linear PDEs with real coefficients in[36].

Of course, for generalA the condition of Lemma 5.6 is not directly testable, asthe Tor module will be infinitely generated. However in the case whereA is injectiveit is possible to derive from this an explicit and algorithmically testable condition. Wemake the additional minor assumption thatA is not a torsion module (this includesthe condition annA = 0).

Theorem 5.7. Suppose thatA is injective and not torsion, and letM = D1,q/Nbe some finitely generated leftD-module andB = HomD(M,A) the associatedbehaviour. The following are equivalent:1. B has an image representation.2. Some moduleM′ withB = HomD(M

′,A) is torsionfree.3. The moduleD1,q/N⊥⊥ is torsionfree.4. If N′ ⊆ D1,q is such thatN′/N = t (M), then(N′)⊥ = N⊥, i.e. the module

M/t (M) defines the same behaviour asM.

Proof. (1)⇒ (4). If B has an image representation, then by Lemma 5.6 we haveTorD1 (L,A) = 0, whereL = Dg/RDq . Now TorD1 (L,A) is the homology atAq

of the complex obtained by right tensoring a finite free resolution ofL by A. Weobtain the same complex by applying the functor HomD(−,D) and then the functorHomD(−,A) to this finite free resolution. Denote byS the complex obtained by ap-plying HomD(−,D) to the resolution. SinceA is injective, HomD(−,A) is an exact(contravariant) functor, and such a functor commutes with taking homology (e.g. [33,Exercise 6.4]). Thus TorD1 (L,A) equals HomD(H,A), whereH is the homologyof S at D1,q . ThusH = Ext1D(L,D), and we have derived the isomorphism (ofAbelian groups)

TorD1 (L,A) ∼= HomD(Ext1D(L,D),A) (37)

(which is very similar to [33, Theorem 9.51], but with left/right modules changed).Thus TorD1 (L,A) vanishes precisely when Ext1

D(L,D) isA-torsion. However, byLemma 2.8, Ext1D(L,D)

∼= t (M), sot (M) = kerθt(M) ⊆ kerθM by Corollary 3.6.Therefore HomD(M/t (M),A) = HomD(M/kerθM,A), which equals HomD(M,A) by Corollary 3.6.(4)⇒ (3). Suppose that condition 4 holds. ThenN′ ⊆ (N′)⊥⊥ = N⊥⊥, so

t (M) = N′/N ⊆ N⊥⊥/N, which equals kerθM by Lemma 3.5. Sot (M) ⊆kerθM. However the reverse inclusion also holds by Lemma 3.7. HenceD1,q/N⊥⊥ = M/kerθM = M/t (M) is torsionfree.(3)⇒ (2). TakeM′ = D1,q/N⊥⊥.(2)⇒ (1). If M′ is torsionfree withB = HomD(M

′,A), then by Corollary 2.5B = kerAR′ for someR′ which is a universal right annihilator of some matrixC.SinceA is injective, imAC = kerAR = B as required. �


Theorem 5.7 in particular proves that (whenA is injective and not torsion) fora Willems closed equation moduleN ⊆ D1,q , the behaviourN⊥ has an imagerepresentation if and only ifD1,q/N is torsionfree. If the given equation moduleN is arbitrary, then to test for the existence of an image representation it is firstnecessary to compute the Willems closure. Of course, ifD1,q/N is itself torsionfreethen we know that an image representation exists, but in this case by Lemma 3.7 wesee that in fact we already have a Willems closedN! The use of a Willems closedequation module is therefore essential for this test.

An interesting special case of Theorem 5.7 is for the classical spaceS′(Rn,R) inthe context of linear PDEs with real coefficients (which is injective and non-torsion).In this case, the condition for the existence of an image representation is given byShankar [36] and matches that in the theorem.

Aside from their intrinsic interest, image representations are also very significantdue to their connection with controllability. The existence of an image representationwill guarantee controllability in the behavioural sense over any signal space suffi-ciently rich to satisfy the controllability property itself. The reason for this is that toconcatenate or “patch” trajectories of the formw = Ml we need only concatenate thecorresponding freely chosen trajectoriesl. The converse is open for linear variablecoefficient ordinary/partial differential equations, but has been well-established inthe constant coefficient case [8,27,32,43]. Thus for a suitable signal space, the al-gebraic condition of torsionfreeness is sufficient for behavioural controllability, andmay well turn out also to be necessary.

In the special case whereD is commutative, we can derive further conditions onthe existence of an image representation by combining Lemma 5.6 with Theorem 3.9.The advantage of this further test is that it does not require explicit computation ofthe Willems closure.

Corollary 5.8. Let D be a commutative Noetherian domain, M a finitely gener-ated leftD-module andA an injective leftD-module withannA = 0. ThenB =HomD(M,A) has an image representation if and only if no associated prime oft (M) annihilates any non-zero element ofA.

Proof. Let R ∈ Dg,q be a kernel representation ofB, so thatM = CokerDR, anddefineL = Dg/RDq as in the proof of Theorem 5.7. By Lemma 5.6 and Eq. (37)(which requires thatA be injective), a necessary and sufficient condition forB tohave an image representation is that Ext1

D(L,D) isA-torsion. However by Lemma2.8 Ext1D(L,D) = t (M), so the condition is thatt (M) be A-torsion, i.e. thatkerθt(M) = t (M). The result now follows on applying Theorem 3.9 tot (M). �

6. Conclusions

We have discussed what we consider to be the two most important fundamentalproblems in extending the current duality between finitely generated modules and


behaviours to other signal spaces, and in particular to the case of systems definedby linear PDEs with variable coefficients. The first problem, that of constructingthe Willems closure of an equation module, is crucial in order to be able to directlydescribe the algebraic structure of the system. The second problem of eliminationis also of central importance. We have also demonstrated some of the consequentresults of a solution of one or both of these problems, including a characterization ofwhen a given behaviour admits an image representation.

We have also been able to provide full or partial solutions to these problems interms of certain algebraic conditions on the signal space. For example, the full state-ment of Theorem 3.9, which characterizes Willems closure in the case of commuta-tive D, holds when(0 : P∞)A is injective for certain primesP. Theorem 4.8 iden-tifies a large class of combinations of signal space and equation module for whichelimination is possible, but requires knowledge of the injective dimension ofA.

A major open problem in any new context such as the case of linear variablecoefficient PDEs is to identify some specific signal spaces for which the algebraicconditions appearing in the paper hold (for example, injectivity, divisibility, or lowinjective dimension). We anticipate that this will be a difficult problem, requiringsome heavy analytic tools. It also remains to develop algorithms for the construc-tive solution of these problems, for example using the methods of Janet/Spencer[11,16,29,37].

The majority of the results in this paper apply to arbitrary non-commutative leftand right Noetherian domains. They may therefore have applications to systems de-fined by equations of a different type than partial differential equations or differenceequations.

Acknowledgements

I would like to thank Ulrich Oberst and Harish Pillai for very useful discussions,Shiva Shankar for making [35,36] available to me before their publication, and Rod-ney Sharp for helping me to strengthen Theorem 3.9.

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