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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 =
2
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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.
◻
◻
◻
2 Periodic 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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4 Signal 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
9
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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
11
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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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