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arXiv:1001.2255v3 [math.OC] 30 Jul 2010

Periodic Behaviors∗

Diego Napp–Avelli†

Marius van der Put‡

Shiva Shankar§

Dedicated to Jan C. Willems on the occasion of his 70th birthday.

Abstract

This paper studies behaviors that are defined on a torus, or equivalently, behaviors defined in

spaces of periodic functions, and establishes their basic properties analogous to classical results

of Malgrange, Palamodov, Oberst et al. for behaviors on Rn. These properties - in particular

the Nullstellensatz describing the Willems closure - are closely related to integral and rational

points on affine algebraic varieties.

Introduction

In classical control theory the structure of a linear lumped dynamical system, considered as an

input-output system, is determined by its frequency response, i.e. its response to periodic inputs.

This idea is the foundation of the subject of frequency domain analysis and the work of Bode,

Nyquist and others, and is also the idea underpinning the theory of transfer functions, including

its generalization to multidimensional systems [5, 7, 11, 15].

The more recent Behavioral Theory of J.C. Willems challenges the notion of an open dynamical

system as an input-output system [13]. Instead, a system is considered to be the collection of

all signals that can occur and which are therefore the signals that obey the laws of the system.

This collection of signals, called the behavior of the system, is the system itself, and is analogous

to Poincar´ e’s notion of the phase portrait of a vector field. Notions of causality and the related

input-output structure are not part of the primary description, but are secondary structures to be

imposed only if necessary. The behavioral theory can be seen as a generalization of the Kalman

State Space Theory, and the ideas of state space theory, as well as those of frequency domain can

be carried over to the more general situation of behaviors. It is the purpose of this paper to initiate

the study of frequency domain ideas in the theory of distributed behaviors.

A second motivation for this paper is the following. The theory of behaviors has so far been

developed for signal spaces that live on the ‘base space’ Rn, or on its convex subsets. The commut-

ing global vector fields ∂1,...,∂ngenerate the algebra C[∂1,...,∂n] of differential operators with

constant coefficients, and distributed behaviors are defined by equations whose terms are from this

∗MSC2000: 93C05, 93C35, 93B25, 93C20, 35B10, 35E20

†RD Unit Mathematics and Applications, Department of Mathematics, University of Aveiro, Portugal,

diego@ua.pt

‡Institute of Mathematics and Computing Science, University of Groningen, P.O. Box 407, 9700 AK Groningen,

The Netherlands, mvdput@math.rug.nl

§Chennai Mathematical Institute, Plot No. H1, SIPCOT, IT Park, Padur P.O., Siruseri, Chennai (Madras)-603103

India, sshankar@cmi.ac.in

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algebra. This paper considers the case where the base space Rnis replaced by a torus Rn/Λ, with

Λ a lattice. Functions on the torus can be identified with Λ-invariant functions on Rn, in other

words, functions which are periodic with respect to Λ. The torus is an example of a parallelizable

manifold; other manifolds of this type, such as the 3-sphere S3, would be of interest for behavior

theory. Another possibly interesting base space for behavior theory is Pn(R), the real n-dimensional

projective space. The vector space of global vector fields on this projective space is isomorphic to

the Lie algebra sln+1and its enveloping algebra acts as a ring of differential operators on the space

of smooth functions on Pn(R).

In this paper we consider the real torus T ∶= Rn/2πZn. Now C∞(T), the space of smooth

functions on the torus T, is identified with the space of smooth functions on Rnhaving the lattice

2πZnas its group of periods. It is a Fr´ echet space under the topology of uniform convergence of

functions and all their derivatives. On it acts the ring of constant coefficient partial differential

operators D ∶= C[∂1,...,∂n], and makes C∞(T) a topological D-module. The aim of this paper

is to develop the basic properties of system theory in this situation. It turns out that behaviors,

contained in C∞(T)q, are related to integral points on algebraic varieties in An. A comparison with

the fundamental paper [3] is rather useful.

Functions which are periodic with respect to the lattice 2πZnremain periodic with respect to

lattices which are integral multiples of this lattice. Thus, one can relax the condition of periodicity

with respect to 2πZnby considering smooth functions on Rnwhich are periodic with respect to a

lattice N2πZnfor some integer N ≥ 1, depending on the function. This space of periodic functions,

denoted by C∞(PT), can be naturally identified with a dense subspace of the space of continuous

functions on the inverse limit PT ∶= lim

is the strict direct limit of the Fr´ echet spaces C∞(Rn/N2πZn); it is therefore a barrelled and

bornological topological vector space, and is also a topological D-module.

In the situation of this protorus PT, behaviors are related to rational points of algebraic varieties

in An. We consider various choices of signal spaces, their injectivity (or their injective envelopes)

as D-modules and make explicit computations of the associated Willems closure for submodules of

Dq. For the 1D case the results are elementary. For the more important nD case (with n > 1) the

Willems closure is explicitly given for various choices of signal spaces. This involves the knowledge

of the existence of (many) rational points or integral points on algebraic varieties over Q or Z. This

connection between periodic behaviors and arithmetic algebraic geometry (diophantine problems)

is rather surprising.

←

Rn/N2πZn, which we call a protorus. Further, C∞(PT)

1 Behaviors and the Willems Closure

As in the introduction let D = C[∂1,...,∂n].

C[D1,...,Dn]. We consider a faithful D-module F, i.e. a module having the property that if

r ∈ D and rF = 0, then r = 0. This module is now taken as the signal space. We recall the usual

set up for behaviors.

Let e1,...,eq be the standard basis of Dq. Associate to a submodule M ⊂ Dqits behavior

M⊥⊂ Fqconsisting of all elements (f1,...,fq) ∈ Fqsatisfying ∑rj(fj) = 0 for all ∑rjej∈ M. In

other words, M⊥is the image of the map HomD(Dq/M,F) → Fq, given by ℓ ↦ (ℓ(¯ e1),...,ℓ(¯ eq)),

where ¯ ejis the class of ejin Dq/M. The above defines the set of behaviors B ⊂ Fq. For a behavior

B, define B⊥∶= {r = ∑rjej∈ Dq∣ ∑rj(fj) = 0 for all (f1,...,fq) ∈ B}.

For any behavior B it follows that B⊥⊥= B. The Willems closure of a submodule M ⊂ Dqwith

Let Dj =

1

ı∂j, j = 1,...,n, so that also D =

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respect to F is, by definition, M⊥⊥⊂ Dq[9]. Clearly M ⊂ M⊥⊥. It is well known that M⊥⊥= M

holds if the signal space F is an injective cogenerator. For more general signal spaces one has the

following.

Lemma 1.1 M⊥⊥/M = {ξ ∈ Dq/M ∣ ℓ(ξ) = 0 for all ℓ ∈ HomD(Dq/M,F)}. Moreover, M⊥⊥/M

is a submodule of the torsion module (Dq/M)tor of Dq/M (where (Dq/M)tor∶= {η ∈ Dq/M ∣∃r ∈

D, r ≠ 0, rη = 0}.

Proof. By the above definition, η = ∑ηjej ∈ M⊥⊥if and only if ∑ηjℓ(¯ ej) = 0 for every ℓ in

HomD(Dq/M,F). The latter is equivalent to ℓ(∑ηj¯ ej) = 0 for all ℓ ∈ HomD(Dq/M,F).

Define the torsion free module Q by the exact sequence

0 → (Dq/M)tor→ Dq/M → Q → 0.

To show that M⊥⊥/M ⊂ (Dq/M)tor amounts to showing that for every non zero element ξ ∈ Q

there exists a homomorphism ℓ ∶ Q → F with ℓ(ξ) ≠ 0. As Q is torsion free it is a submodule of

Drfor some r, and it therefore suffices to verify the above property for D itself. This amounts to

showing that for every r ∈ D, r ≠ 0, there exists an element f ∈ F with r(f) ≠ 0. But this is just

the assumption that F is a faithful D-module.

Corollary 1.1 Suppose either that the signal space F is injective, or that the exact sequence 0 →

(Dq/M)tor→ Dq/M → Q → 0 splits. Then M⊥⊥/M consists of the elements ξ ∈ (Dq/M)tor such

that ℓ(ξ) = 0 for every ℓ ∈ HomD((Dq/M)tor,F).

Proof. In both the cases, every homomorphism ℓ ∶ (Dq/M)tor → F extends to an element of

HomD(Dq/M,F).

Corollary 1.2 Consider two signal spaces F0⊂ F. Assume that for every a ∈ F, a ≠ 0, there exists

a homomorphism m ∶ F → F0such that m(a) ≠ 0. Then the Willems closure of M with respect to

F0equals that with respect to F.

Proof. Consider ξ ∈ Dq/M. If there exists a homomorphism ℓ ∶ Dq/M → F with ℓ(ξ) ≠ 0, then, by

assumption, there exists a homomorphism˜ℓ ∶ Dq/M → F0with˜ℓ(ξ) ≠ 0. Since the converse of this

statement is obvious, the two Willems closures of M coincide.

See also [14] for related results.

◻

◻

◻

2Periodic Functions and the Protorus

We consider, as in the introduction, the torus T ∶= Rn/2πZn. An element f ∶ T → C of C∞(T) is

represented by its Fourier series: f(x) = ∑a∈Zn caeı<a,x>, where a = (a1,...,an), x = (x1,...,xn)

and < a,x >= ∑ajxj. Further, the coefficients ca∈ C are required to satisfy the property: for every

integer k ≥ 1 there exists a constant Ck> 0 such that ∣ca∣ ≤

space of distributions on T has a similar description, however with different requirements on the

absolute values ∣ca∣.)

The vector space C∞(T) = C∞(Rn/2πZn) has the natural structure of a Fr´ echet space, moreover

it is a topological D-module. For positive integers N1dividing N2, the natural D-module morphism

Ck

j=1∣aj∣)kfor all a. (We note that the

(1+∑n

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C∞(Rn/2πN1Zn) → C∞(Rn/2πN2Zn) identifies the first linear topological space with a closed sub-

space of the second one. We define C∞(PT) ∶= lim

of Fr´ echet spaces, and is a locally convex bornological and barrelled topological vector space. The

elements of D act continuously on it so that C∞(PT) is also a topological D-module. An element f

in it is represented by the series f(x) = ∑a∈Qn caeı<a,x>, where the support of f, i.e., {a ∈ Qn∣ ca≠ 0},

is a subset of

of rapid decrease on the absolute values ∣ca∣ as above.

As in the Introduction, call the inverse limit PT ∶= lim

topological group. The map PT → Rn/2πNZnembeds C∞(Rn/2πNZn) in the space C(PT) of

continuous functions on the protorus (which is a Banach space with respect to the sup norm) for

every N. The exact sequence

→

C∞(Rn/2πNZn). This is a strict direct limit

1

NZnfor some integer N ≥ 1, depending on f. Further, there is the same requirement

←

Rn/2πNZna protorus. PT is a compact

0 → 2πZn/2πNZn→ Rn/2πNZn→ Rn/2πZn→ 0

for each N, gives upon taking inverse limits the exact sequence

0 →̂Zn→ PT → Rn/2πZn→ 0

where the group lim

̂Znsits inside the protorus PT as a compact subgroup and is totally disconnected. This implies

that any continuous map̂Zn→ C(PT) is the uniform limit of locally constant maps.

For f ∈ C(PT) and z ∈̂Zn, define the function fz by fz(t) = f(z + t).

is continuous and therefore a uniform limit of locally constant maps.

limit of functions fiin C(PT), where z ↦ (fi)z is locally constant. This implies that each fiis

invariant under the shift N̂Znfor some integer N ≥ 1, depending on i; in other words fi is an

element of C(Rn/2πNZn), the space of continuous complex valued functions on Rn/2πNZn. As

C∞(Rn/2πNZn) is dense in C(Rn/2πNZn), it follows that C∞(PT) is a dense subspace of C(PT).

As the partial sums of a Fourier series expansion converge uniformly, it follows that for L(D)

in D,

L(D)( ∑

←2πZn/2πNZnequalŝZn,̂Z being the well known profinite completion lim

←Z/NZ.

The map z ↦ fz

Thus f is the uniform

a∈Qncaeı<a,x>) = ∑

a∈QncaL(a)eı<a,x>

The basic observation, leading to the computation of the Willems closure is that L(D) is injective

on C∞(PT) if and only if the polynomial equation L(a) = L(a1,...,an) = 0 has no solutions in Qn.

(We note, in passing, that the condition L(a1,...,an) ≠ 0 for (a1,...,an) ∈ Qndoes not imply that

L(D) is surjective; see Theorem 2.1.)

Another observation is that C∞(PT) is not an injective D-module, not even a divisible module.

Indeed, the image of the morphism D1∶ C∞(PT) → C∞(PT) consists of those elements f whose

support is contained in {(a1,...,an) ∈ Qn∣ a1≠ 0}. The kernel of D1is the subspace of C∞(PT)

consisting of those elements f whose support lies in {(a1,...,an) ∈ Qn∣ a1= 0}. The cokernel of

the morphism D1is represented by this same subspace of C∞(PT), it is therefore not surjective.

We also consider the subalgebra C∞(PT)[x1,...,xn] of C∞(Rn) obtained by adjoining the ele-

ments x1,...,xn, that is the coordinate functions, to C∞(PT), and similarly C∞(T)[x1,...,xn] etc.

Yet another observation is

Lemma 2.1 C∞(PT)[x1,...,xn] = ⊕a∈Nn C∞(PT)xa1

C∞(T)[x1,...,xn] = ⊕a∈Nn C∞(T)xa1

1...xan

n, where a = (a1,...,an); and similarly

1...xan

n.

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Proof. Clearly C∞(PT)[x1,...,xn] = ∑a∈Nn C∞(PT)xa1

is direct.

We first observe that C∞(PT)[x1] = ⊕a∈NC∞(PT)xa

∑a∈Nfaxa

relation implies that f0= −∑a>0faxa

sum on the right hand side is not. Thus f0= 0. This implies that the relation above is of the

form x1(∑a>0faxa−1

∑a>0faxa−1

1

= 0, leading to a contradiction just as above.

Suppose now by induction that C∞(PT)[x1,...,xn−1] = ⊕a∈Nn−1 C∞(PT)xa1

pose that C∞(PT)[x1,...,xn−1,xn] = C∞(PT)[x1,...,xn−1][xn] is not a direct sum. Then there

is a relation ∑a∈Nfaxa

again leads to a contradiction as above. Thus C∞[x1,...,xn] = ⊕a∈NC∞(PT)[x1,...,xn−1]xa

⊕a∈Nn C∞(PT)xa1

This lemma allows us to write an element in C∞(PT)[x1,...,xn] uniquely as a polynomial in the

xi’s with coefficients in C∞(PT).

Define C∞(PT)finto be the D-submodule of C∞(PT) consisting of those elements f with finite

support, i.e. those elements whose Fourier series expansion is a finite sum. Just as above, C∞(PT)fin

is not an injective D-module. However, the following proposition gives an explicit expression for

its injective envelope.

1...xan

n, so it remains to show that the sum

1, for if not, there would be a relation

Suppose f0 is nonzero; then the above

1= 0, with finitely many of the fa nonzero.

1. This is a contradiction because f0is in C∞(PT) while the

1

) = 0. As the function x1is zero only on a set of measure 0, it follows that

1...xan−1

n−1, and sup-

n= 0, with finitely many of the fa(in C∞(PT)[x1,...,xn−1]) nonzero. This

n=

1...xan

n.

◻

Proposition 2.1 The D-module C∞(PT)fin[x1,...,xn] is an injective envelope of C∞(PT)fin. Sim-

ilarly, C∞(T)fin[x1,...,xn] is an injective envelope of C∞(T)fin.

Proof. The Fundamental Principle of Malgrange - Palamodov states that C∞(Rn) is an injective

D-module. It is also a cogenerator (Oberst [3]). From this it follows that its submodule MIN ∶=

C[{eı<a,x>}a∈Cn,x1,...,xn] is the direct sum of the injective envelopes E(D/m) of the modules D/m,

where m varies over the set {(D1− a1,...,Dn− an),a = (a1,...,an) ∈ Cn} of maximal ideals of D.

Thus this module is again injective, and is in fact a minimal injective cogenerator over D, unique up

to isomorphism (see [4] for more details). The elements of MIN are finite sums ∑a∈Cn pa(x)eı<a,x>,

where the pa(x) are polynomials in x1,...,xn. Define the map π ∶ MIN → C∞(PT)fin[x1,...,xn] by

π( ∑

a∈Cnpa(x)eı<a,x>) = ∑

a∈Qnpa(x)eı<a,x>

Clearly π is a C-linear projection; it also commutes with the operators Dj, j = 1,...,n. Thus π

is a morphism of D-modules which splits the inclusion i ∶ C∞(PT)fin[x1,...,xn] → MIN. It follows

that C∞(PT)fin[x1,...,xn] is a direct summand of MIN, hence an injective D-module. Moreover,

the extension of modules C∞(PT)fin⊂ C∞(PT)fin[x1,...,xn] is essential. Indeed, consider a term f =

xm1

an)mn(f) = ceı<a,x>, for some nonzero constant c. Thus we conclude that C∞(PT)fin[x1,...,xn] is

an injective envelope of C∞(PT)fin.

Observations 2.1 (1) C∞(PT) ⊂ C∞(PT)[x1,...,xn] is not an essential extension. Indeed, con-

sider f = x1∑a∈Zn caeı<a,x>in C∞(PT)[x1,...,xn] with ca∈ C and all ca≠ 0. For any L(D) ∈ D,

L(D)f = x1∑caL(a1,...,an)eı<a,x>+ (an element of C∞(PT)). Thus L(D)f ∈ C∞(PT) implies

L = 0 (no nonzero polynomial can vanish at every integral point).

(2) The polynomials in x1,...,xnhave no interpretation as functions on the protorus PT, but are

functions on the space Rn, which can be seen as the universal covering of the protorus.

1⋯xmn

neı<a,x>with a ∈ Qn. As (Dj−aj)(xjeı<a,x>) =1

ıeı<a,x>, it follows that (D1−a1)m1⋯(Dn−

◻

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Lemma 2.2 Let n = 1. Then C∞(T)[x] is an injective D = C[D]-module, where D =1

for a / ∈ Z, the map D − a is bijective on C∞(T)[x]. For a ∈ Z the kernel of D − a on C∞(T)[x] is

Ceıax.

There is exactly one injective envelope of C∞(T) in C∞(T)[x], and it consists of the elements

∑j≥0fjxjsuch that fjhas finite support for j ≥ 1.

Similar statements hold for C∞(PT) replacing C∞(T) and Q replacing Z.

Proof. Since n = 1, injectivity is equivalent to divisibility. Thus it suffices to show that (D − a) ∶

C∞(T)[x] → C∞(T)[x] is surjective for every a ∈ C. But if g = ∑k

where the gj are in C∞(T), then an f such that (D − a)f = g is, by the ‘variation of constants’

formula, given by f(x) = eıax∫

Now, the theory of Matlis [1] applied to the case of this injective module C∞(T)[x], states that

it admits a decomposition

C∞(T)[x] = ⊕

a∈Z

where the torsion module tor(C∞(T)[x]) of C∞(T)[x] equals ⊕a∈ZC[x]eıaxand where the module

V ≃ C∞(T)[x]/tor(C∞(T)[x]) is injective and torsion free (see also [4]). In general V is not unique,

and one can only speak of an injective envelope of C∞(T) in C∞(T)[x]; nonetheless it turns out for

the case at hand that there is exactly one injective envelope as described in the statement.

This follows from the fact that an injective envelope of Ceıaxis C[x]eıax; thus as Ceıaxis

contained in C∞(T), the above decomposition implies that any injective envelope E of C∞(T) in

C∞(T)[x] must satisfy

⊕

a∈Z

ı

d

dx. Thus

j=1gjxjis an element of C∞(T)[x],

x

0e−ıatg(t)dt, which is again in C∞(T)[x].

C[x]eıax⊕V

C[x]eıax+ C∞(T) ⊆ E = ⊕

a∈Z

C[x]eıax⊕(V ⋂E)

But if an element f = ∑k

D. Now suppose that k ≥ 1. Since L(D)f = (L(D)fk)xk+ (terms of lower degree in x), it follows that

L(D)fk= 0 and therefore that fkhas finite support {a1,...,as}. Then M(D) ∶= (D−a1)⋯(D−as)

satisfies M(D)fk = 0. After replacing f by M(D)f, induction with respect to k implies that

f1,...,fkall have finite support. Thus

j=0fjxjin C∞(T)[x] belongs to E, then 0 ≠ L(D)f ∈ C∞(T) for some L(D) in

⊕

a∈Z

C[x]eıax+ C∞(T) ⊆ E ⊆ ⊕

a∈Z

C[x]xeıax⊕C∞(T)

which implies equality throughout. This proves the second statement.

The corresponding statements for the protorus follow from the fact that C∞(PT) is the union

of its subspaces C∞(R/N2πZ),N ≥ 1.

Proposition 2.2 The spaces C∞(T)fin[x1,...,xn] ⊂ C∞(T)[x1,...,xn] define the same Willems

closure. The same holds for the inclusion of the two signal spaces C∞(T)fin ⊂ C∞(T). These

statements remain valid for PT replacing T.

◻

Proof. For a b ∈ Zn, define the homomorphism

mb∶ C∞(T)[x1,...,xn] → C∞(T)fin[x1,...,xn]

by mb(∑a∈Zn pa(x)eı<a,x>) = pb(x)eı<b,x>. The first statement now follows from Corollary 1.2. The

other cases are similar.

◻

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Theorem 2.1 For n > 1, the D-modules C∞(T)[x1,...,xn] and C∞(PT)[x1,...,xn] are not divis-

ible (and therefore not injective).

Proof. It suffices to show that C∞(T)[x1,x2] is not divisible. Towards this let ℓ be any Liouville

number, and consider L = D1+ ℓD2 in D. Let g = ∑a∈Z2 caeı<a,x>be any element in C∞(T), so

that for every integer k ≥ 1, there is a constant Cksuch that ∣ca∣ ≤ Ck(1+∣a1∣+∣a2∣)−kholds for all

a ∈ Z2. If C∞(T)[x1,x2] were divisible, then L would define a surjective morphism on it, and so

there would be an element f = ∑a∈Z2 pa(x)eı<a,x>in it such that L(f) = g. Thus

∑

a∈Z2(D1pa(x)+ ℓD2pa(x)+ (a1+ℓa2)pa(x))eı<a,x>= ∑

which implies by Lemma 2.1 that pa(x) is a constant for all a in Z2, and that (a1+ ℓa2)pa= ca.

As ℓ is Liouville, it is irrational, hence a1+ ℓa2≠ 0 for all a = (a1,a2) ≠ (0,0). It follows that

the paare equal to

a1+ℓa2for a ≠ 0.

By assumption this solution belongs to C∞(T)[x1,x2] for every g in C∞(T) and thus for every

choice of the {ca} that are rapidly decreasing. It would then follow that ∣a1+ℓa2∣ ≥ c(1+∣a1∣+∣a2∣)−N

for some N ≥ 1, some c > 0 and all (a1,a2) ∈ Z2. This is a contradiction, for since ℓ is a Liouville

number, there cannot be such a bound.

a∈Z2caeı<a,x>

ca

◻

3Signal spaces for periodic 1D systems

In this section T = R/2πZ and D = C[D] with D =1

the Willems closure M⊥⊥of a module M ⊂ Dq. Write (Dq/M)tor= ⊕D/(D−ai)ni. By Lemma 1.1,

M⊥⊥/M ⊂ (Dq/M)tor, and using Corollary 1.1 and Lemma 2.1 it follows that:

1. For F = C∞(T),

M⊥⊥/M = (⊕ai/ ∈ZD/(D −ai)ni) ⊕(⊕ai∈Z(D − ai)D/(D − ai)ni)

2. For F = C∞(T)[x], or for the injective envelope of C∞(T) in it,

M⊥⊥/M = ⊕ai/ ∈ZD/(D − ai)ni

and M⊥⊥consists of the elements r ∈ Dqsuch that Lr is in M for an L ∈ D without zeros in

Z.

ı

d

dx. We compute for various signal spaces F

3. For F = C∞(PT),

M⊥⊥/M = (⊕ai/ ∈QD/(D −ai)ni) ⊕(⊕ai∈Q(D −ai)D/(D − ai)ni)

4. For F = C∞(PT)[x], or for the injective envelope of C∞(PT) in it,

M⊥⊥/M = ⊕ai/ ∈QD/(D − ai)ni

and M⊥⊥consists of the elements r ∈ Dqsuch that Lr is in M for an L ∈ D without zeros in

Q.

Case (2) can be rephrased by stating that M = M⊥⊥if and only if the support of the module

(Dq/M)torlies in Z ⊂ A1= C. The signal space F = C∞(T)[x] gives rise to a rather restricted set of

behaviors in Fq. Indeed, the modules M = M⊥⊥corresponding to behaviors in Fqare of the form

L ⋅W ⊂ M ⊂ W, where W is a direct summand of Dqand L ∈ D, L ≠ 0 has all its zeros in Z.

Case (4) can be rephrased by stating that M = M⊥⊥if and only if the support of (Dq/M)tor

lies in Q ⊂ A1= C. This gives rise to a somewhat richer set of behaviors in Fq.

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4Signal spaces for periodic nD systems

In this section T = Rn/2πZnand n ≥ 2. For various choices of signal spaces we investigate the set

of behaviors and the corresponding Willems closure.

4.1

C∞(PT)[x1,...,xn] and C∞(PT)fin[x1,...,xn]

According to Proposition 2.2 we may restrict ourselves in this subsection to the injective signal

space F = C∞(PT)fin[x1,...,xn]. We start with examples illustrating some of the features of the

Willems closure.

Example 4.1 F = C∞(PT)fin[x1,x2], i = (D2

For L ∈ D and pλeıλ⋅x∈ F it follows that, L(pλeıλ⋅x) = {L(D1+ λ1,D2+ λ2)(pλ)}eıλ⋅x. Thus, to

determine the behavior of the ideal i we have to consider the two equations

1,D1D2) ⊂ D = C[D1,D2].

(D1+λ1)2pλ= 0; (D1+λ1)(D2+ λ2)pλ= 0, where pλ∈ C[x1,x2]

If λ1≠ 0, then the only solution is pλ= 0.

If λ1= 0, λ2∈ Q,λ2≠ 0, then the solutions are p(0,λ2)∈ C[x2].

If λ1= λ2= 0, then the solutions are p(0,0)= a0+a1x1with a0∈ C[x2] and a1∈ C.

The behavior B = i⊥is then B0+ B1where B0∶= C + Cx1and B1∶= {∑λ2∈Qp(0,λ2)eıλ2x2∣ all p(0,λ2)∈

C[x2]}.

One easily sees that B⊥

B1are behaviors and correspond to the primary decomposition i = (D2

Note also that the behavior of the ideal (D2

of the behaviors B0and B1. (The lattice structure of behaviors under the operations of sum and

intersection is studied in more detail for the classical spaces in [10].)

0= (D2

1,D2), (D2

1,D2)⊥= B0and B⊥

1= (D1), (D1)⊥= B1. Thus B0and

1,D2) ∩ (D1) of the ideal i.

1,D2) +(D1) = (D1,D2) is C, which is the intersection

◻

Example 4.2 F = C∞(PT)fin[x1,x2]. Let p ⊂ D denote the prime ideal generated by the operator

L(D1,D2) = (D2

ideal p are {(1,1),(−1,−1)}. The behavior B ∶= p⊥⊂ F has the form B1⋅eı(x1+x2)⊕B−1⋅e−ı(x1+x2),

where B1and B−1are the kernels of the operators L1∶= L(D1+1,D2+1) = D2

π)D1+ (−2+ π)D2and L2∶= L(D1−1,D2− 1) respectively, acting on C[x1,x2].

Let C[x1,x2]≤ndenote the vector space of the polynomials of total degree ≤ n. Observe that the

map L1∶ C[x1,x2]≤n→ C[x1,x2]≤n−1is surjective. It follows that B1∩ C[x1,x2]≤nhas dimension

n + 1. Thus B1is an infinite dimensional subspace of C[x1,x2] and the same holds for B−1. An

explicit calculation showing B⊥= p is possible. However, the statement p⊥⊥= p follows at once from

Theorem 4.3.

1−D2

2)+π(D1D2−1). The rational points of the variety V(p) ⊂ A2defined by the

1−D2

2+πD1D2+(2+

◻

Proposition 4.1 Let F = C∞(PT)fin[x1,...,xn], and let M be a submodule of Dq. Then the

Willems closure M⊥⊥of M with respect to F consists of the elements x in Dqfor which the ideal

{r ∈ D∣ rx ∈ M} is not contained in any maximal ideal of the form (D1− b1,...,Dn− bn) with

(b1,...,bn) ∈ Qn. In other words M⊥⊥is the largest submodule M+of Dqcontaining M, such that

the support S ⊂ Anof M+/M satisfies S ∩Qn= ∅.

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Proof. M⊥⊥/M consists of the elements ξ such that ℓ(ξ) = 0 for all ℓ ∈ HomD(Dq/M,F). Let

i ∶= {r ∈ D∣ rξ = 0}. Since F is injective one has that ξ ∈ M⊥⊥/M if and only if HomD(D/i,F) = 0.

If i lies in a maximal ideal m ∶= (D1−b1,...,Dn−bn) with (b1,...,bn) ∈ Qn, then HomD(D/i,F) ≠

0 because HomD(D/m,F) ≠ 0.

On the other hand, suppose that ℓ ∈ HomD(D/i,F) is non zero. Then ℓ(1+i) = ∑a∈Qn pa(x)eı<a,x>

has a non zero term t ∶= pb(x)eı<b,x>and rt = 0 for all r ∈ i. If b = (b1,...,bn), then (Dj− bj)t =

(Djpb(x))eı<b,x>. Thus for suitable integers mj≥ 0, t0∶= (D1− b1)m1⋯(Dn− bn)mnt = ceı<b,x>with

c ∈ C∗. Since i ⋅t0= 0, it follows that i ⊂ (D1−b1,...,Dn− bn).

A second formulation of the structure of M⊥⊥uses the notion of primary decomposition of

modules. Let p ⊂ D be a prime ideal. A submodule M of Dqis called p-primary (with respect

to Dq) if the set Ass(Dq/M) of associated primes of Dq/M equals {p}. For a general submodule

M ⊂ Dq, there exists an irredundant (one where no term can be omitted) primary decomposition

M = M1∩ ⋅⋅⋅ ∩ Mtwhere Miis pi-primary and {p1,...,pt} = Ass(Dq/M). For more details we

refer to [2]. We note that the following theorem is an analogue of the Nullstellensatz of [9]. See

also [6, 10, 8] on this topic.

◻

Theorem 4.1 (Nullstellensatz) Let the submodule M ⊂ Dqhave an irredundant primary decom-

position M = M1∩ ⋅⋅⋅ ∩ Mtwhere Miis pi-primary. Let V(pi), the variety defined by pi, contain

a rational point for i = 1,...,r and not for i = r + 1,...,t. Then the Willems closure M⊥⊥with

respect to F = C∞(PT)fin[x1,...,xn] is equal to M1∩⋯∩Mr. Thus M equals M⊥⊥if and only if

every V(pi) contains rational points.

Proof. It is easy to see that M0∶= M1∩⋯∩Mris independent of the primary decomposition (see

[9]). We first claim that the behavior M⊥

suffices to show that M⊥⊂ M⊥

such that m(D)f ≠ 0. However for every r in the ideal (M ∶ m), r(D)(m(D)f) = 0. Taking Fourier

transforms - every element of F is a temperate distribution - gives r(x)

support of̂

and if f = ∑a∈Qn pa(x)eı<a,x>, thenˆf(x) = ∑a∈Qn pa(D)δa - where δa is the Dirac distribution

supported at a - so that the support of̂

On the other hand the ideal (M ∶ m) equals ∩t

the radical ideal

is contained in ∪t

which is a contradiction to the choice of f and m above.

We now show that M0is the largest submodule of Dqwith the same behavior as that of M.

So let m be any element of Dq∖ M0, and consider the exact sequence

0 → D/(M0∶ m)

where the morphism m maps the class of of r to the class of mr, and π is as usual. Applying the

functor HomD(⋅ ,F) gives the exact sequence

0 → HomD(Dq/M0+ (m),F) ?→ HomD(Dq/M0,F)

Observe now that V((M0∶ m)) is the union of some of the varieties V(p1),...V(pr), hence by

assumption there is a rational point, say a on it. Therefore the function eı<a,x>is in the last

0of M0in F equals the behavior M⊥of M. As M ⊂ M0it

0. Suppose it is not. Then there is an f in M⊥and some m in M0∖M

̂

̂

(m(D)f)(x) = 0, hence the

(m(D)f)(x) = m(x)ˆf(x),

m(D)f is contained in V(r)∩Rnfor every r in (M ∶ m). Now

m(D)f is contained in Qnand hence in V((M ∶ m))∩Qn.

i=1(M ∶ m), and as m is in M0∖M it follows that

√(M ∶ m) is equal to the intersection of a subset of pr+1,...,pt. Thus V((M ∶ m))

i=r+1V(pi) whose intersection with Qn, by assumption, is empty. Thus m(D)f = 0,

m

?→ Dq/M0

π

?→ Dq/M0+ (m) → 0

m(D)

?→ HomD(D/(M0∶ m),F) → 0

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term HomD(D/(M0∶ m),F) above and which is therefore nonzero. This implies that the behavior

(M0+ m)⊥is strictly smaller than the behavior M⊥.

A central notion of the subject is that of a controllable behavior [13, 6]. A behavior which

admits an image representation is controllable and the next result characterizes such behaviors.

◻

Theorem 4.2 Let F = C∞(PT)fin[x1,...,xn]. Then the behavior M⊥in Fqof a submodule M ⊂

Dqis the image of some morphism L(D) ∶ Fp→ Fqif and only if the varieties of the nonzero

associated primes of Dq/M do not contain rational points.

Proof. Let M(D) be an r × q matrix whose r rows generate M (so that M⊥equals the kernel of

the morphism M(D) ∶ Fq→ Fr). Let L be the submodule of Dqconsisting of all relations between

the q columns of M(D). Suppose that L is generated by some p elements ℓ1,...,ℓp. Let L(D) be

the matrix whose columns are ℓ1,...,ℓpand which therefore defines a morphism L(D) ∶ Fp→ Fq.

As F is an injective module, its image equals the kernel of a morphism M1(D) ∶ Fq→ Fr1, where

that r1rows of M1(D) generate all relations between the rows of L(D). Let M1be the submodule

of Dqgenerated by the rows of M1(D); then M1/M = (Dq/M)torso that Dq/M1is torsion free.

Thus it follows that M⊥is an image, in fact the image of L(D) ∶ Fp→ Fq, if and only if M⊥= M⊥

i.e if and only if the Willems closure of M equals M1. By the previous theorem this is so if and

only if the variety of every nonzero associated prime of Dq/M does not contain rational points. ◻

1,

4.2

C∞(T)[x1,...,xn] and C∞(T)fin[x1,...,xn]

In this case it suffices to consider the signal space C∞(T)fin[x1,...,xn]. The results of §4.1, as well

as the examples, carry over if everywhere one replaces Q by Z and ‘rational point’ by ‘integral

point’.

4.3

C∞(PT) and C∞(PT)fin

We consider the signal space F = C∞(PT)fin.

Description of i⊥⊥for ideals i ⊂ D and behaviors in F: Recall that the support of a series f(x) =

∑a∈Qn caeı<a,x>is the set {a∣ ca≠ 0}. For a = (a1,...,an) ∈ Cnwe write (D−a) for the maximal ideal

(D1− a1,...,Dn− an). Given an ideal i ⊂ D, let V(i) be its variety in Cn, and let S(i) = V(i)(Q)

(i.e., V(i) ∩ Qnseen as a subset of Cn).

If f(x) = ∑a∈Qn caeı<a,x>∈ i⊥, then each caeı<a,x>∈ i⊥. Thus f ∈ i⊥if and only if the support

of f lies in S(i). Further, i⊥⊥consists of all the polynomials in D which are zero on the set S(i).

In other words i⊥⊥= ⋂a∈S(i)(D − a). Equivalently, i⊥⊥is the reduced ideal of the Zariski closure of

S(i). The behaviors B ⊂ F are in this way in 1-1 correspondence with Zariski closed subsets S of

Cnsatisfying S ∩Qnis Zariski dense in S.

Description of M⊥⊥for submodules M of Dq: The elements of Fqare written in the form f(x) =

∑a∈Qn caeı<a,x>, with ca= (ca1,...,caq)) ∈ Cq. Now m = (m1,...,mq) ∈ Dqapplied to f has the form

∑a∈Qn < m(a),ca> eı<a,x>, with < m(a),ca>= ∑q

we write m(a) = (m1(a),...,mq(a)) ∈ Cq, where as before mi(a) = mi(a1,...,an).

For a fixed a ∈ Qn, the set V (a) ∶= {m(a) ∈ Cq∣ m ∈ M} is a linear subspace of Cq. We conclude

that M⊥consists of the elements f(x) = ∑a∈Qn caeı<a,x>such that < V (a),ca>= 0. It now follows

that M⊥⊥consists of the elements r ∈ Dqsuch that for each a ∈ Qn, r(a) ∈ V (a).

j=1mj(a)caj. (Here, for any m = (m1,...,mq) ∈ Dq

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Example 4.3 n = 2, q = 2 and M ⊂ D2is generated by (D2

for all a ∈ Q2. One finds that M⊥⊂ F2consists of the expressions ∑a∈Q2(ca1,ca2)eı<a,x>satisfying

a2

1,D1D2). Then V (a) = C(a2

1,a1a2)

1ca1+ a1a2ca2= 0. Further M⊥⊥= M.

◻

An ‘algorithm’ computing M⊥⊥for a submodule M of Dq: For every b ∈ Qnone considers the

homomorphism

mb∶ F = C∞(PT)fin→ Ceı<b,x>≅ D/(D −b)

given by mb∶ ∑acaeı<a,x>↦ cbeı<b,x>(where as before (D − b) = (D1− b1,...,Dn− bn)). It fol-

lows at once that ξ ∈ Dq/M belongs to M⊥⊥/M if and only if ℓ(ξ) = 0 for every homomorphism

ℓ ∶ Dq/M → D/(D − b) with b ∈ Qn. As in the proof of Theorem 4.1, we consider an irredundant

primary decomposition ∩Miof M and try to compute the M⊥⊥

i.

Let M be p-primary for its embedding in Dq, then M⊥⊥⊃ M + pDqand we may replace M

by the p-primary module M1∶= M + pDqsince M⊥⊥= M⊥⊥

a module over D/p, no torsion and therefore is a submodule of (D/p)rfor some r ≥ 1. Now

M⊥⊥

all homomorphisms ℓ.

1. We observe that Dq/M1 has, as

1/M1= ∩(Ker(Dq/M1

ℓ→ D/(D−b)), where the intersection is taken over all b ∈ V(p)∩Qnand

Suppose that the set V(p) ∩ Qnis Zariski dense in V (p) (this holds in particular for p = (0)).

Then ∩(Ker((D/p)rℓ→ D/(D −b)), b ∈ V(p) ∩ Qnand all ℓ, equals (0). It follows that M⊥⊥

Suppose that the set V(p) ∩Qnis empty, then M⊥⊥

Suppose that the set S ∶= V(p) ∩ Qnis not empty and is not dense in V(p). The radical ideal

i ∶= ∩b∈S(D − b) defines V(i) ⊂ Cn, which is the closure of S. Now ∩(Ker(Dq/M1

where the intersection is taken over all b ∈ V(i) ∩ Qnand all ℓ, contains iDq. Thus we may as well

continue with the module M2∶= M1+iDqsince M⊥⊥

In general, M2is not primary and we have to replace M2again by the elements of an irredun-

dant primary ∩(M2)idecomposition of M2. The minimal prime ideals q containing i are associated

primes of Dq/M2. For the corresponding primary factor (M2)ione has (M2)⊥⊥

V(q) ∩ Qnis dense in V(q). If there are no more primary factors (or if the other primary factors

belong to prime ideals r such that V(r)∩Qnis dense in V(r)), then M⊥⊥

However, if M2has a primary factor M3corresponding to a prime ideal r such that V(r) ∩ Qnis

not dense in V(r), then we have to repeat the above process. The Noether property guaranties that

the process ends. Except for the problem of finding rational points on irreducible subspaces of An,

the above is really an algorithm.

1= M1.

1= Dq.

ℓ→ D/(D − b)),

2= M⊥⊥

1.

i

= (M2)ibecause

2= M2, and we are finished.

Example 4.4 Behaviors related to rational points on algebraic varieties.

(1) n = 2. i = (D2

(2) n = 2. i = (D2

defines an affine elliptic curve. Now these are the following possibilities (see [12]):

(a) The elliptic curve has no rational point other than its infinite point. Then i⊥⊥= D.

(b) The elliptic curve has finitely many rational points. Then i⊥⊥⊂ D is the intersection of the

finitely many maximal ideals (D − a) with a ∈ Q2lying on the elliptic curve.

1+D2

1− (D3

2−1) ⊂ D yields i⊥= {∑a∈Q2, a2

1+ aD2

1+a2

2=1caeı<a,x>} and i⊥⊥= i.

1+ bD1+ c)) ⊂ D. We suppose that a,b,c ∈ Q and that the equation

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(c) The rank of the elliptic curve is positive and i⊥⊥= i.

(3) n = 3. Let the principal prime ideal p ⊂ D define an irreducible affine surface S ⊂ A3over Q.

The following possibilities occur:

(a) S(Q) = ∅ and p⊥⊥= D.

(b) S(Q) is finite (and non empty); then p⊥⊥is the intersection of the maximal ideals (D−a) with

a ∈ S(Q).

(c) S(Q) is infinite and the Zariski closure of this set is a curve on S. Then p⊥⊥is the (radical)

ideal of this curve.

(d) S(Q) is Zariski dense in S; then p⊥⊥= p.

◻

4.4

C∞(T) and C∞(T)fin

We consider the signal space F = C∞(T)fin. As in §4.3, there is a 1-1 relation between the behaviors

B ⊂ F and the Zariski closed subsets S of Cnsuch that S ∩ Znis dense in S. For an ideal i ⊂ D,

the ideal i⊥⊥is the intersection of the maximal ideals (D − a) ⊃ i with a ∈ Zn. The descriptions of

M⊥⊥for a submodule M of Dqare the ones given in §4.3 with Z replacing Q.

Example 4.5 (1) Let i ⊂ D be an ideal. Let j ⊂ D denote the smallest ideal containing i which

is generated by elements in Z[D1,...,Dn]. Then i⊥= j⊥. Indeed, i⊥⊥is generated by elements in

Z[D1,...,Dn]. Consider for example the ideal i ⊂ C[D1,D2,D3] generated by (D2

D3

3) + π2(D1D2D3− 1). The ideal j is generated by (D2

i⊥= j⊥and S(i) = {(1,−1,−1),(−1,1,−1)}.

(2) Let i ⊂ C[D1,D2,D3] be generated by D2

{(a1,a2,a3) ∈ Z3∣ a2

Acknowledgement: We are grateful to the referees for their careful reading of the manuscript.

1−D2

2)+π(D2

1+

1− D2

2),(D2

1+ D3

3),(D1D2D3− 1). Then

1+ D2

2− D2

3. Then i⊥⊥= i because the set S(i) =

1+a2

1+ a2

2− a2

3= 0} is Zariski dense in {(a1,a2,a3) ∈ C3∣ a2

2− a2

3= 0}.

◻

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