Abstract
Linear differential systems, in Willems’ behavioral system theory, are defined to be the solution sets to systems of linear constant coefficient PDEs, and they are naturally parameterized in a bijective way by means of polynomial modules. In this article, introducing appropriate topologies, this parametrization is made continuous in both directions. Moreover, the space of linear differential systems with a given complexity polynomial is embedded into a Grassmannian.
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Appendices
Appendix A: Convergence in Grassmannians
Let W be a finite-dimensional linear space of dimension, say, l. The Grassmannian \(\text {Gr}(W)\) is a collection of all linear subspaces of W. For \(1\le k\le l\), one defines the \(\text {Gr}_k(W)\) to be the set of all k-dimensional linear subspaces of W. The set \(\text {Gr}(W)\) is the disjoint union of its k-dimensional parts.
Let us denote by \(Mon({\mathbb {F}}^k,W)\) the set of all monomorphisms from \({\mathbb {F}}^k\) to W. This is an open subset of \(Hom({\mathbb {F}}^k,W)\), which is a finite-dimensional Hausdorff topological space.
There is a surjective map
The topology in \(\text {Gr}_k(W)\) is the quotient topology; namely, U is an open set of \(\text {Gr}_k(W)\) if and only if \(\pi _k^{-1}(U)\) is open in \(Mon({\mathbb {F}}^k,W)\). The topology can be described in terms of convergence as follows. A family \(X_\varepsilon \) (\(\varepsilon \ne 0\)) of k-dimensional subspaces of W converges to a k-dimensional subspace X if and only if there exist injective homomorphisms \(f_\varepsilon :{\mathbb {F}}^k\rightarrow W\ (\varepsilon \ne 0)\) and \(f:{\mathbb {F}}^k\rightarrow W\) such that
Lemma 9
Let R be a full row rank (scalar) matrix, and assume that \(R=[R_1\ | -R_2]\), where \(R_1\) is square nonsingular. Then,
Proof
Follows from the theory of linear equations. \(\square \)
Lemma 10
If \(X_\varepsilon \rightarrow X\), then
Proof
We certainly may assume that \(W={\mathbb {F}}^l\).
Choose full column rank matrices \(M_\varepsilon \) and M so that \(M_\varepsilon \rightarrow M\) and \(Im(M_\varepsilon )=X_\varepsilon \), \(Im(M)=X\). We then have
and
(It is a standard fact that if R is a (scalar) matrix, then \(Im(R)^\perp =Ker(R^\mathrm{tr})\).) Without loss of generality, we may assume that M and \(M_\varepsilon \) have the forms
where \(M_1\) and \(M_{1,\varepsilon }\) are square nonsingular. By the above lemma,
The inverse of a nonsingular matrix is a continuous function of the entries of the matrix, and so
This completes the proof. \(\square \)
Lemma 11
Assume that \(X_\varepsilon \rightarrow X\) and \(Y_\varepsilon \rightarrow Y\).
(a) If \(dim(X_\varepsilon \cap Y_\varepsilon )=dim(X\cap Y)\) for all sufficiently small \(\varepsilon \), then
(b) If \(dim(X_\varepsilon + Y_\varepsilon )=dim(X + Y)\) for all sufficiently small \(\varepsilon \), then
Proof
Left to the reader. \(\square \)
Appendix B: F.g. graded modules and coherent sheaves
A graded module (over T) is a T-module M together with a decomposition into \({\mathbb {F}}\)-linear spaces
such that \(s_iM_d\subseteq M_{d+1}\) for \(i\in [0,n]\) and \(d\in {\mathbb {Z}}\). An element \(x\in M\) is called homogeneous of degree d if \(x\in M_d\). A submodule \(N\subseteq M\) is graded if \(N=\bigoplus (N\cap M_d)\).
A homomorphism of graded modules \(M\rightarrow N\) is a module homomorphism \(u:M\rightarrow N\) such that \(u(M_{d})\subseteq N_{d}\) for all d.
For a graded T-module M and an integer a, one denotes by M(a) the graded T-module whose homogeneous components are defined by
The module \(T(-a)\) is a free graded module of rank 1 generated by an element of degree a. A f.g. graded free module is a one that is isomorphic to a graded module of the form
If M is a f.g. graded module, then by Hilbert’s syzygy theorem, there exists an exact sequence
where \(0\le l\le n\), \(F_0, \ldots ,F_l\) are graded free T-modules of finite rank, and \(F_0/Im\phi _1\simeq M\). Such a sequence is called a free graded resolution. The number l is called the length of the resolution, and the ranks of the free modules are called the Betti numbers. A free graded resolution is said to be minimal if, for each \(i\ge 1\),
A minimal free graded resolution exists and is unique up to isomorphism. The length of the minimal free resolution is called the projective dimension of M; the ranks of the free modules are called the Betti numbers. Supposing that \(F_i\simeq \oplus _jT(-a_{i,j})\) in the minimal resolution of M, the number
is called the Castelnuovo–Mumford regularity (see Ch.4 in Eisenbud [1]). The projective dimension and the Betti numbers measure the size of the minimal resolution; the Castelnuovo–Mumford regularity measures its complexity.
Free graded resolutions were invented by Hilbert in order to compute the dimensions of the components of a graded module. The Hilbert function of a graded module \(M=\bigoplus M_d\) is defined by the formula
It is easily seen that HF(M, d) becomes a polynomial, when \(d\ge reg(M)\). This polynomial is called the Hilbert polynomial and is denoted by HP(M, d).
We pass now to coherent sheaves on the projective space \({\mathbb {P}}^n\). One can give a purely algebraic definition of these objects. (We omit the definition of the projective space itself as it is not necessary for the purposes of this article.)
For each nonempty subset \(\alpha \subseteq [0, n]\) and each \(d\ge 0\), write \(s_\alpha ^d\) for \(\prod _{i\in \alpha }s_i^d\) and define \(T_\alpha \) to be the ring
If \(\alpha \) and \(\beta \) are nonempty subsets of [0, n] and if \(\alpha \subseteq \beta \), then there is a canonical homomorphism
It should be noted that, for each \(k=0,\ldots ,n\), the ring \(T_{\{k\}}\) is a polynomial ring; namely,
(The “hat” = “omission.”) Notice that \(T_{\{0\}}\) is canonically isomorphic to S.
A presheaf (of modules) is a map \({{{\mathcal {F}}}}\) assigning to every \(\alpha \) a \(T_\alpha \)-module \({{{\mathcal {F}}}}(\alpha )\) and to each inclusion \(\alpha \subseteq \beta \) a \(T_\alpha \)-homomorphism
If \(\alpha =\beta \), the homomorphism is required to be the identity map, and if \(\alpha \subseteq \beta \subseteq \gamma \), the composition \( {{{\mathcal {F}}}}(\alpha )\rightarrow {{{\mathcal {F}}}}(\beta )\rightarrow {{{\mathcal {F}}}}(\gamma ) \) is required to be equal to the homomorphism \({{{\mathcal {F}}}}(\alpha )\rightarrow \mathcal{F}(\gamma )\). (In short, a presheaf is a covariant functor from the ordered set of nonempty subsets of [0, n] to the category of modules.)
A presheaf \({{{\mathcal {F}}}}\) is called a sheaf if the canonical homomorphism
is bijective for each inclusion \(\alpha \subseteq \beta \). A sheaf whose modules are finitely generated is called coherent.
A homomorphism of sheaves \(\phi :{{{\mathcal {F}}}}\rightarrow {{{\mathcal {G}}}}\) is a collection of homomorphisms
commuting with the “extension” homomorphisms.
One defines in an obvious way subsheaves and quotient sheaves, tensor products of sheaves. One defines as well, the kernels and the images of homomorphisms of sheaves, and hence, one can speak about exact sequences of sheaves.
Given a sheaf \({{{\mathcal {F}}}}\), for each \(p=0,1,\ldots ,n\), set
The \(\check{\mathrm{C}}\hbox {ech}\) complex of \({{{\mathcal {F}}}}\) is the sequence
where the map \(C^{p-1}({{{\mathcal {F}}}}){\mathop {\rightarrow }\limits ^{\delta }} C^p({{{\mathcal {F}}}})\) is defined by the formula
(The symbol \(\circ (\alpha ,k)\) is taken from Theorem 10.2 in Eisenbud [1]; it denotes the number of elements of \(\alpha \) less than k.) The \(\check{C}\hbox {ech}\) complex is indeed a complex, i.e., the composition of two consecutive arrows is 0. Hence, for each \(p=0,1,\ldots ,n\), one defines
The space \(H^0{{{\mathcal {F}}}}\), the space of 0-dimensional cohomologies or global sections, is especially important. By definition, a global section of a sheaf \({{{\mathcal {F}}}}\) is a family \((f_k)_{0\le k\le n}\) with \(f_k\in {{{\mathcal {F}}}}({\{k\}})\) such that for all i, j
Example
\(H^0{{{\mathcal {O}}}}(d)=T_d\).
If \(M=\bigoplus _{d\ge 0}M_d\) is a graded module over T, one defines its sheafification \({\widetilde{M}}\) by setting for each \(\alpha \)
Notice that if M is finitely generated, then \({\widetilde{M}}\) is a coherent sheaf.
One defines the structure sheaf \({{{\mathcal {O}}}}\) as the sheafification of T. More generally, for every \(d\in {\mathbb {Z}}\), one sets
The sheaves \({{{\mathcal {O}}}}(d)\) are called the twisting sheaves.
Given a sheaf \({{{\mathcal {F}}}}\), the twist \({{{\mathcal {F}}}}(k)\) of \({{{\mathcal {F}}}}\) by k is defined to be \({{{\mathcal {F}}}}\otimes {{{\mathcal {O}}}}(k)\).
The great thing is that every sheaf can be obtained by the tilde construction. Namely, as the following lemma states
is a “surjective” functor.
Lemma 12
(Serre) Let \({{{\mathcal {F}}}}\) be a sheaf. Then, there is a canonical isomorphism
where \(H({{{\mathcal {F}}}})=\bigoplus _k H^0{{{\mathcal {F}}}}(k)\).
Proof
See Proposition 5.15 in Ch. II of Hartshorne [3]. \(\square \)
The sheafification functor does not yield an equivalence between the category of graded modules and the category of sheaves. It is easily seen that a graded module generates the same sheaf as its “tails.” If M is a graded module and \(d_0\) an integer, one denotes by \(M_{\ge d_0}\) the graded module whose component in the degree d is \(M_d\) if \(d\ge d_0\) and is zero otherwise. It is clear that
The following lemma says that this is the only reason why the “tilde” fails to be an equivalence.
Lemma 13
(Serre) Let M and N be two graded modules. Then
Proof
See Exercise 5.9 in Ch. II of Hartshorne [3]. \(\square \)
The following statement is known as Serre’s vanishing theorem.
Lemma 14
(Serre) Let \({{{\mathcal {F}}}}\) be a coherent sheaf. Then,
Proof
See Theorem 5.2 in Ch. III of Hartshorne [3]. \(\square \)
The Euler characteristic \(\chi ({{{\mathcal {F}}}})\) of a coherent sheaf \({{{\mathcal {F}}}}\) is an integer defined by the formula
The function
is a polynomial, which is called the Hilbert polynomial of \({{{\mathcal {F}}}}\). By Serre’s vanishing theorem, we have
If \(M=\bigoplus _{d\ge 0}M_d\) is a f.g. graded module, then the canonical linear map
is bijective for all large enough d. (This is immediate from Lemmas 11 and 12.) It follows that
Appendix C: Mumford’s bounding result
The following “strange” notion is extremely important and useful. It is due to Mumford [6].
Given a coherent sheaf \({{{\mathcal {F}}}}\) and an integer m, one says that \({{{\mathcal {F}}}}\) is m-regular if
The Castelnuovo–Mumford regularity of \({{{\mathcal {F}}}}\), denoted by \(reg({{{\mathcal {F}}}})\), is the smallest integer m such that it is m-regular.
Lemma 15
If \({{{\mathcal {F}}}}\) is an m-regular sheaf, then the following statements hold:
(a) The canonical map
is surjective for \(k\ge m\);
(b) Whenever \(i \ge 1\) and \(k \ge m - i\),
In other words, if \({{{\mathcal {F}}}}\) is m-regular, then it is \(m'\)-regular for all \(m'\ge m\).
Proof
See Lemma 2.1 in Nitsure [7]. \(\square \)
The following corollary provides an example where the notion of Castelnuovo–Mumford regularity is useful.
Corollary 3
If \({{{\mathcal {F}}}}\) is a coherent sheaf, then
Proof
By (b), when \(k\ge reg({{{\mathcal {F}}}})\), \(H^i({{{\mathcal {F}}}}(k)) = 0\) for every \(i\ge 1\). \(\square \)
The following result of Mumford played a key role in Sect. 4.
Lemma 16
Let \(\theta \in {\mathbb {Q}}[t]\). There exists a sufficiently big integer m, which has the following property: If \({{\mathcal {F}}}\) is a subsheaf of \({{{\mathcal {O}}}}^q\) with the Hilbert polynomial \(\theta \), then
Proof
See Theorem 2.3 in Nitsure [7]. \(\square \)
Remark
Mumford’s result led to a significant simplification in the proof of Grothendieck’s theorem on existence of Hilbert and Quot schemes [2].
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Lomadze, V. Continuous dependence of linear differential systems on polynomial modules. Math. Control Signals Syst. 32, 385–409 (2020). https://doi.org/10.1007/s00498-020-00263-x
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DOI: https://doi.org/10.1007/s00498-020-00263-x