On the K-theory of feedback actions on linear systems

Linear Algebra and its Applications 440 (2014) 233–242

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Linear Algebra and its Applications www.elsevier.com/locate/laa

On the K -theory of feedback actions on linear systems ✩ Miguel V. Carriegos a,∗ , Ángel Luis Muñoz Castañeda b a b

Departamento de Matemáticas, Universidad de León, Spain Institut für Mathematik, Freie Universität, Berlin, Germany

a r t i c l e

i n f o

Article history: Received 19 July 2013 Accepted 28 October 2013 Available online 14 November 2013 Submitted by E. Zerz MSC: 93B10 15A21 13C10

a b s t r a c t A categorical approach to linear control systems is introduced. Feedback actions on linear control systems arise as symmetric monodical category S R . Stable feedback isomorphisms generalize dynamic enlargement of pairs of matrices. Subcategory of locally Brunovsky linear systems B R is studied. We prove that the stable feedback isomorphisms of locally Brunovsky linear systems are characterized by the Grothendieck group K 0 ( B R ). Hence a link from linear dynamical systems to algebraic K -theory is established. © 2013 Elsevier Inc. All rights reserved.

Keywords: Feedback equivalence Commutative ring Dynamic enlargement Stable equivalence K -theory invariant

1. Introduction This paper deals with the study of feedback actions on a linear control system. Concrete description of the feedback classification of constant linear systems by means of sets of invariants and canonical forms goes back to the seminal works by Kalman, Casti and Brunovsky (see the fundamental references [5,7,11]), but the more general framework of parametrized families of linear systems [8,14] is proved to be a hard task in [4]. We restrict ourselves to the class of locally Brunovsky systems because a complete description of feedback invariants is available (see [6]).

✩

*

The research is partially supported by Ministerio de Ciencia y Tecnología and Ministerio de Industria (INTECO), Spain. Corresponding author. E-mail addresses: [email protected] (M.V. Carriegos), [email protected] (Á.L. Muñoz Castañeda).

0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.10.048

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On the other hand we also are interested in the so-called dynamic feedback equivalence (see [3,9,10] as main references). This dynamic study is based on the addition of some suitable ancillary variables to systems [2, Ch. 4]. We introduce the notion of stable feedback equivalence and show that it is a generalization of both feedback and dynamic feedback equivalence. This generalization does not go too far, because if the base ring is a field, then feedback, dynamic feedback and stable feedback equivalence are the same notion. We think an adequate tool to study the above subjects is category theory. First of all, the definition of the category S R of linear systems over a commutative ring and feedback actions arises in a natural way, moreover feedback equivalences are the isomorphisms in the category. Dynamic enlargements and stabilization of linear systems are consequence of some bi-product (both product and coproduct) in the category and hence a symmetric monoidal structure in S R is obtained. Finally the stable equivalences are the stable isomorphisms in the category. Consequently the invariants characterizing the stable equivalence will be collected in the K 0 group of the K -theory of the category, which is just the Grothendieck group completion of the monoidal structure when possible (i.e. when the isomorphism classes in the category are sets). The paper is organized as follows. Section 2 is devoted to review main definitions used in the paper: linear system, feedback isomorphism, direct sum of linear systems and dynamic isomorphism. These notions generalize respectively the standard notions of pair of matrices, feedback equivalence, dynamic enlargement and dynamic feedback equivalence. We also define stable isomorphism of linear systems as adequate generalization of both feedback and dynamic isomorphism. Section 3 is the core section of the paper devoted to the stable classification of linear systems. We prove that the pair

S R = (linear systems, feedback morphisms) is a category whose isomorphisms are precisely feedback isomorphisms defined in [6]. Thus feedback classification of linear systems is just given by the classes of isomorphisms ( S R )iso . Reachable systems A R and locally Brunovsky systems B R arise as subcategories of S R equipped with the same homomorphisms: the feedback actions. We define the operation ⊕ on linear systems and show that: (a) ‘sum’ operation ⊕ is both the categorical product and coproduct in the category of linear systems; (b) dynamic enlargement of a linear system now arises as the ‘sum’ of the system with a trivial one; and (c ) category of linear systems equipped with ⊕ operation are symmetric monoidal, see [12] or [15]. Section 3 concludes with a characterization of stable equivalence in B R (locally Brunovsky systems) in terms of first K -theory group K 0 ( B R ) of the category. Finally, Section 4 is devoted to compute effectively K 0 ( B R ) as the Grothendieck group completion of the monoid ( B R )iso of locally Brunovsky systems up to feedback isomorphisms. 2. Stable feedback isomorphisms between linear systems Let R be a commutative ring with 1 = 0. In this section we introduce the dynamic and stable feedback isomorphisms of linear systems over R. Definition 2.1. (See [8].) A linear system is a triple Σ = ( X , f , B ) where X is an R-module, f : X → X an endomorphism and B ⊂ X a submodule. Definition 2.2. (See [6].) Two linear systems Σ1 = ( X 1 , f 1 , B 2 ) and Σ2 = ( X 2 , f 2 , B 2 ) are feedback isomorphic (f.i.) if there exists an isomorphism of R-modules between the space states φ : X 1 → X 2 such that 1. φ( B 1 ) = B 2 , 2. Im( f 2 ◦ φ − φ ◦ f 1 ) ⊂ B 2 . Remark 2.3. Recall that all pairs of matrices ( A , B ) define a linear system

  ( A , B ) → Σ A , B = R n , A , Im( B )

(1)

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235

Note that this correspondence is neither injective nor surjective. Observe also that the feedback isomorphism for linear systems is a generalization of the feedback equivalence for pairs of matrices ( A , B ) in the following sense: Suppose ( A 1 , B 1 ) and ( A 2 , B 2 ) are feedback isomorphic, then there are invertible matrices, P ∈ G L n ( R ) and Q ∈ G L m ( R ) and a matrix K ∈ R n×m such that A 2 = P ( A 1 + B 1 K ) P −1 and B 2 = P B 1 Q . Then it is straightforward to show that the matrix P gives a feedback equivalence between Σ A 1 , B 1 and Σ A 2 , B 2 . Two pairs of matrices ( A 1 , B 1 ) and ( A 2 , B 2 ) are dynamic equivalent if the pairs of matrices of orders ( p + n × p + n) and ( p + n × p + m)



0 0 0 A1

       1 0 0 0 1 0 , , , 0 B1

0 A2

(2)

0 B2

are feedback isomorphic (see [3,10]). This is physically realized by introducing free ancillary variables. Consider the pair of matrices ( A , B ) and let Σ A , B = ( R n , A , Im( B )) be the associated linear system. Consider also the linear system Γ ( p ) = ( R p , 0, R p ). Then the linear system associated to the pair of matrices



0 0 0 A

   1 0 ,

(3)

0 B

is precisely ( R p ⊕ R n , 0 ⊕ A , R p ⊕ Im( B )). This motivates the following definition. Definition 2.4. Let Σi = ( X i , f i , B i ) (i = 1, 2) be linear systems. The direct sum of Σ1 and Σ2 is defined by

Σ1 ⊕ Σ 2 = ( X 1 ⊕ X 2 , f 1 ⊕ f 2 , B 1 ⊕ B 2 )

(4)

Throughout the paper, it will be used Bass’s matrix notation for the direct sum of homomorphisms (see [1]), hence the matrix

 f1

0  0 f2

actually represents the homomorphism f 1 ⊕ f 2 . Elements of the

direct sum of objects, X 1 ⊕ X 2 , will be presented as column vectors the notation consistent.

 x1  y1

∈ X 1 ⊕ X 2 in order to make

Definition 2.5. Linear systems Σ1 and Σ2 are dynamic feedback isomorphic (d.i.) if there exists p ∈ N such that the linear systems Σ1 ⊕ Γ ( p ) and Σ2 ⊕ Γ ( p ) are feedback isomorphic. Definition 2.6. Linear systems Σ1 and Σ2 are stable feedback isomorphic (s.i.) if there exists a linear system Γ such that the linear systems Σ1 ⊕ Γ and Σ2 ⊕ Γ are feedback isomorphic. Of course, the relations f.i., d.i., and s.i. satisfy the axioms for equivalence relations. It is also clear that s.i. is a generalization of d.i. and that the d.i. is a generalization of the f.i. f .i .

Σ Σ

⇒

d .i .

Σ Σ

⇒

s .i .

Σ Σ

(5)

Moreover, if R is a field then it is not hard to prove that the three relations define the same classification problem, that is f .i .

Σ Σ

⇔

d .i .

Σ Σ

⇔

s .i .

Σ Σ

(6)

3. Stable classification of locally Brunovsky linear systems This section deals with the classification of linear systems modulo stable feedback isomorphisms. Invariants will be found in some group by using a bit of K -theory, thus we need to start with the categorical properties of linear systems.

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3.1. The category of linear systems and its monoidal structure In order to construct the category of linear systems we need to define the homomorphisms of the category. Definition 3.1. Let Σ1 = ( X 1 , f 1 , B 1 ) and Σ2 = ( X 2 , f 2 , B 2 ) be linear systems. A homomorphism between Σ1 and Σ2 is a homomorphism of R-modules φ : X 1 → X 2 such that 1. φ( B 1 ) ⊂ B 2 , 2. Im( f 2 ◦ φ − φ ◦ f 1 ) ⊂ B 2 . The composition of homomorphisms of linear systems is defined as the composition of homomorphisms of R-modules. It is routine to check that the composition of homomorphisms of systems is a homomorphism of systems. It is also straightforward to check that identities are homomorphisms of systems, as well as zero maps. Associative laws follow from associative laws of composition of homomorphisms of R-modules. Definition 3.2. Let S R denote the category whose objects are linear systems and whose morphisms are homomorphisms of linear systems. The following result shows that the category S R is the natural framework to address the problem of the feedback classification of linear systems. Proposition 3.3. Isomorphisms in the category S R are exactly the feedback isomorphisms. Proof. It is routine to check that (a) every isomorphism of systems (invertible homomorphism of systems) is a feedback isomorphism and (b) the inverse of a feedback isomorphism is a homomorphism of systems. This gives the result. 2 After introducing the category of linear systems it is natural to ask for the algebraic structure induced by the direct sum of linear systems ⊕ in this category. We will describe this structure and we will see that ⊕ descends to the isomorphism classes of linear systems. Lemma 3.4. The direct sum of two homomorphisms of linear systems is a homomorphism of linear systems. Proof. Consider the linear systems Σi = ( X i , f i , B i ) and Γi = (Y i , g i , C i ) (i = 1, 2) and homomorphisms φ : Σ1 → Σ2 , ψ : Γ1 → Γ2 . Let us take the direct sum of φ and ψ as homomorphisms of R-modules



φ 0 0 ψ



: X1 ⊕ Y 1 → X2 ⊕ Y 2

Since φ( B 1 ) ⊂ B 2 and ψ(C 1 ) ⊂ C 2 we have that

 x1  y1

φ

0 0 ψ

 ( B 1 ⊕ C 1 ) ⊂ B 2 ⊕ C 2 . On the other hand, if

∈ X 1 ⊕ Y 1 then



      φ 0 φ 0 x1 f1 0 x1 − y1 y1 0 ψ 0 ψ 0 g1          = f 2 φ(x1 ) − φ f 1 (x1 ) , g 2 ψ( y 1 ) − ψ g 1 ( y 1 ) ∈ B 2 ⊕ C 2 f2 0 0 g2



by hypothesis, so Im systems. 2

 f 2

0  φ 0 0 ψ 0 g2



−

φ

0 0 ψ

 f 1

0  0 g1

⊂ B 2 ⊕ C 2 and

φ

0 0 ψ



is a homomorphism of linear

M.V. Carriegos, Á.L. Muñoz Castañeda / Linear Algebra and its Applications 440 (2014) 233–242

237

Define the zero system as Z = (0, 0, 0). It is clear that Z satisfies the identity property for the direct sum of linear systems. Obviously Z is both an initial and final object in S R . Now we can consider the category of linear systems with an extra structure ( S R , ⊕, Z ). Lemma 3.5. The direct sum of linear systems ⊕ is both a categorical product and co-product in S R , that is to say ⊕ is a bi-product in the category S R . Proof. The universal property of product (see [12]) arises from the following picture: Given a system Γ and homomorphisms ψi : Γ → Σi , dotted line Γ → Σ1 ⊕ Σ2 always exists making the following diagram commutative and it is given by the adequate Bass’s matrix whose entries are the ψi ψ1

Σ1 Γ

 ψ1  ψ2

π1

Σ1 ⊕ Σ 2

π2

Σ2

ψ2

In the same way one can check the universal property of coproduct: Given a system Γ and homomorphisms φi : Σi → Γ , dotted line Σ1 ⊕ Σ2 → Γ always exists making the following diagram commutative and it is given by the adequate Bass’s matrix whose entries are the φi φ1

Σ1 ι1

ι2

Σ2

Σ1 ⊕ Σ 2

(φ1 ,φ2 )

Γ

2

φ2

As a consequence of the above result ( S R , ⊕, Z ) is a monoidal category. Moreover, it is a symmetric monoidal category because: (a) ⊕ is coproduct and Z is terminal (see [12]); and (b) Σ1 ⊕ Σ2 ∼ = Σ2 ⊕ Σ1 by means of the R-module homomorphism X 1 ⊕ X 2 → X 2 ⊕ X 1 given, in Bass’s notation, 01 . by 10 Note also that the direct sum in S R descends to the isomorphism classes of linear systems. To be precise, if Σ1 Σ2 and Γ1 Γ2 then Σ1 ⊕ Γ1 Σ2 ⊕ Γ2 . 3.2. Stable classification of locally Brunovsky linear systems and the K 0 group Once we have obtained the symmetric monoidal structure of categories of linear systems and feedback actions we will characterize the stable isomorphism of linear systems in terms of K 0 group of the category. Grothendieck group completion of monoid ( S R )iso of isomorphism classes in the category S R is the K -theory group K 0 ( S R ) (see [15]). But in general ( S R )iso is not even a set. To avoid this obstruction, the subcategory B R of locally Brunovsky linear systems is considered because, in this case, the isomorphisms classes ( B R )iso form a well defined set, in fact [6] proves that ( B R )iso = P( R )∞ is the set of finite support sequences of elements of P( R ), which are isomorphism classes of finitely generated projective R-modules. Let Σ = ( X , f , B ) be a linear system over the ring R. Recall the definition of the invariant modules of Σ (see [6]): 1. N i = B + f ( N i −1 ) for i  1 being N 0 = 0, 2. M i = X / N i ,

238

M.V. Carriegos, Á.L. Muñoz Castañeda / Linear Algebra and its Applications 440 (2014) 233–242 1

3. I i = ker( M i −1 → M i → 0) (I-invariants), f

4. Z i = ker( I i → I i +1 → 0) (Z-invariants). Lemma 3.6. Let Σ1 and Σ2 be linear systems and consider the direct sum Σ1 ⊕ Σ2 . Then Σ1 ⊕Σ2

= N iΣ1 ⊕ N iΣ2 , Σ Σ ∼ 2. M i = Mi 1 ⊕ Mi 2 , Σ ⊕Σ Σ Σ 3. I i 1 2 ∼ = Ii 1 ⊕ Ii 2 , Σ1 ⊕Σ2 ∼ Σ1 Σ 4. Z = Z ⊕ Z 2. 1. N i

Σ1 ⊕Σ2

i

i

i

Proof. Let’s denote the elements of X = X 1 ⊕ X 2 by columns. Homomorphism f 1 ⊕ f 2 is written, in

 f1

0  . 0 f2

Bass’s notation, by

Then:

Σ1 ⊕Σ2

(1) Is clear because N 0 f Ni = ( B 1 ⊕ B 2 ) + 01 Σ1 ⊕Σ2

= (0, 0), N 1Σ1 ⊕Σ2 = B 1 ⊕ B 2 = N 1Σ1 ⊕ N 1Σ2 , and one obtains sequently

0  ( N iΣ−11 f2

⊕ N iΣ−21 ) = N iΣ1 ⊕ N iΣ2 .

x  (2) Since M iΣ1 ⊕Σ2 = X / N iΣ1 ⊕Σ2 , its elements are the classes x12 + N iΣ1 ⊕Σ2 . Consider the (well de x1 +N iΣ1   x1   x1   x1  Σ1 ⊕Σ2 Σ ⊕Σ fined) linear map μ . Since π x = x + N i 1 2 , then the result follows Σ2 = x2 + N i 2 2 x2 + N i

from application of Short-Five-Lemma on the following commutative diagram with exact rows  i1 0 

Σ1

Ni

0

Σ2

⊕ Ni

0 i2

 π1 0 

X1 ⊕ X2

 Σ1 ⊕Σ2

π2

Σ1

Mi

⊕ M iΣ2



i

Ni

0

0

π

X

0

μ Σ1 ⊕Σ2

Mi

0

(3) Let μ be the linear maps defined in (2). It is clear that the following diagram with exact rows is commutative  i1 0 

Σ1

Ii

0

Define

⊕ Ii

0 i2

 i1

0

0 i2

i

  i = μ ◦ 01 01 ◦ 01

0 i2

ν as the restriction of μ to I iΣ1

1 0

Σ M i −11

Σ1 ⊕Σ2

Ii

0 Since 1 ◦ μ ◦

Σ2

⊕

Σ M i −21

01

μ



Σ ⊕Σ2

1

M i −11

Σ1

Mi

⊕ M iΣ2

0

μ Σ1 ⊕Σ2

Mi

 i1

0

0

= 0 we deduce that the image of μ ◦ 0 i lies in I iΣ1 ⊕Σ2 . 2  x1 +N iΣ−11   x1 +N iΣ−11  Σ2 ⊕ I i ; that’s to say, ν =μ Σ2 Σ2 , then the result x2 + N i −1

x2 + N i −1

follows from application of Short-Five-Lemma on the following commutative diagram with exact rows  i1 0 

0

0

Σ1

Ii

⊕ Ii

0 i2

ν



Σ2

Σ1 ⊕Σ2

Ii

i

1 0

Σ M i −11

⊕

Σ M i −21

01

μ



Σ ⊕Σ2

1

M i −11

Σ1

Mi

⊕ M iΣ2

0

μ Σ1 ⊕Σ2

Mi

0

M.V. Carriegos, Á.L. Muñoz Castañeda / Linear Algebra and its Applications 440 (2014) 233–242

 f1

0  0 f2

(4) As above, Defining

◦ν◦

 i1

0 0 i2

i 0  = 0, therefore ν ◦ 01 i lies in Z iΣ1 ⊕Σ2 . 2  x1 +N iΣ−11   x1 +N iΣ−11  ⊕ Z iΣ2 , that’s to say, ρ ν = Σ2 Σ2 , we have that

=ν◦

ρ as the restriction of ν to Z iΣ1

 f1

0  0 f2

◦

 i1

0 0 i2

x2 + N i −1

the following diagram is commutative

0

⊕ Zi

0 i2

ρ



Σ1 ⊕Σ2

i

Zi

Σ2

x2 + N i −1

 f1 0 

 i1 0 

Σ1

239

Σ1

Ii

⊕ Ii

0 f2

ν



Σ2

Σ

Σ

I i +11 ⊕ I i +21

0

ν

 f1 0 

Zi

0

Σ ⊕Σ2

0 f2

Σ1 ⊕Σ2

Ii

I i +11

Then the result follows from application of Short-Five-Lemma on the above diagram.

0

2

A linear system Σ is reachable if N s = X or, equivalently, if M s = 0 (see [6,8]). Σ is a locally Brunovsky linear system if the state space is finitely generated and the invariant modules are projective R-modules. Locally Brunovsky linear systems are reachable (see [6]). Let B R be the subcategory of S R whose objects are the locally Brunovsky linear systems and whose homomorphisms are the homomorphisms of linear systems. Since direct sum of finitely generated projectives is again projective it follows that direct sum of locally Brunovsky linear systems is again a locally Brunovsky linear system. In fact, B R is a symmetric monoidal subcategory of S R . Moreover, the isomorphism classes of ∞ of finite support sequences of isomorphism locally Brunovsky linear systems, B iso R , is the set P( R ) classes of finitely generated projective R-modules (see [6]). A typical element of monoid P( R )∞ is a finite support sequence ( P 1 , P 2 , . . . , P k , 0, 0, . . .) of finitely generated projective R-modules. Two sequences are equal if and only if they are componentwise isomorphic, identity element of monoid is of course the zero sequence and the monoid operation is the componentwise ⊕. It is not hard to prove that the triple ( B iso R , ⊕, 0) is a commutative monoid, and therefore we have that monoid structure translates direct sum of linear systems due to Lemma 3.6. Now we characterize the stable feedback isomorphism in terms of the Grothendieck completion K 0 ( B R ) of monoid ( B R )iso . In fact we prove that two locally Brunovsky linear systems (in B R ) are stable feedback equivalent if and only if they lie in the same class in K 0 ( B R ). iso Theorem 3.7. Let us denote by K 0 ( B R ) the Grothendieck group of the monoid B iso R and γ : B R → K 0 ( B R ) the s .i .

natural homomorphism of monoids. Then γ ([Σ]) = γ ([Γ ]) if and only if Σ Γ . iso iso Proof. The Grothendieck group (see [13]) of B iso R is K 0 ( B R ) = ( B R × B R )/ ∼ where equivalence relation ∼ is as follows



   [Σ1 ], [Γ1 ] ∼ [Σ2 ], [Γ2 ] such that

⇔

∃[U ] ∈ B iso R

[Σ1 ] + [Γ2 ] + [U ] = [Σ2 ] + [Γ1 ] + [U ]

We will denote by Σ, Γ  the class of ([Σ], [Γ ]) in K 0 ( B R ). Then the natural homomorphism of monoids γ is defined by γ ([Σ]) = Σ, 0. Now suppose that γ ([Σ1 ]) = γ ([Σ2 ]), then Σ1 , 0 = Σ2 , 0 and by definition there exists a linear system U such that [Σ1 ] + [U ] = [Σ2 ] + [U ]. But the equalities in B iso R are the feedback isomorphisms in the category B R , so f .i .

Σ1 ⊕ U Σ2 ⊕ U s .i .

and therefore Σ1 Σ2 are stable feedback equivalent.

2

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M.V. Carriegos, Á.L. Muñoz Castañeda / Linear Algebra and its Applications 440 (2014) 233–242

Corollary 3.8. The commutative submonoid Im(γ ) ⊂ K 0 ( B R ) collects the stable feedback equivalence classes of locally Brunovsky linear systems over R. 4. The K 0 group of locally Brunovsky linear systems The bijective correspondence between the monoid of isomorphisms classes of locally Brunovsky linear systems and the monoid of finite support sequences of isomorphisms classes of finitely generated projective modules Z

∞ B iso R −→ P( R )

  [Σ] → Z 1Σ , Z 2Σ , . . . , Z sΣ , 0, 0, . . .

(7)

is in fact a monoid isomorphism by Lemma 3.6. This gives a precise description of the monoid of stable equivalence classes of locally Brunovsky linear systems. On the other hand note that universal properties of K 0 yield the computation of K 0 ( B R ): Theorem 4.1. Let P( R ) be the monoid of isomorphisms classes of projective finitely generated R-modules and K 0 ( R ) the Grothendieck group of the monoid P( R ). Then ∞ 1. B iso R P( R ) , 2. K 0 ( B R ) K 0 ( R )∞ .

Proof. (1) It is already proven in [6]. In order to prove (2) we will prove that the Grothendieck completion K 0 (P( R )∞ ) of the monoid P( R )∞ of finite support sequences of isomorphism classes of finitely generated projective R-modules is in fact the group K 0 ( R )∞ of finite support sequences of elements of the group K 0 ( R ) = K 0 (P( R )). Recall that if M is a monoid then K 0 ( M ) has the following universal property (see [13]): K 0 ( M ) is the unique abelian group (up to isomorphisms) such that for any other abelian group G and any monoid homomorphism g : M → G there exists a unique group homomorphism f : K 0 ( M ) → G such that the diagram is commutative g

M γ

G

f

K 0 (M ) where γ is the completion homomorphism γ : M → K 0 ( M ). Let γ : P( R ) → K 0 ( R ) be the completion homomorphism and consider the induced monoid homomorphism γ : P( R )∞ → K 0 ( R )∞ . Consider an abelian group and a homomorphism of monoids g : P( R )∞ → G. Then, by the universal property  of the direct sum, there is a family of monoid homomorphisms g i : P( R ) → G such that g = g i . Because of the universal property of K 0 ( R ) there exists, for each g i , a unique group homomorphism, f i : K 0 ( R ) → G, making the following diagram commutative

P( R ) γ

gi

G

fi

K0(R) Now, it is clear that the following diagram is commutative

P( R )∞ γ

K 0 ( R )∞

g

⊕ fi

G

M.V. Carriegos, Á.L. Muñoz Castañeda / Linear Algebra and its Applications 440 (2014) 233–242

241

which proves the existence. If there is another group homomorphism f  : K 0 ( R )∞ → G making the diagram commutative then, by the universal property of the direct sum and Grothendieck group, it must be equal to f . 2 Thus the problem has reduced the study of the monoid of stable isomorphism classes of locally Brunovsky linear systems to the study of the sub-monoid given by the image of natural map γ : P( R ) → K 0 ( R ) for the base ring. Corollary 4.2. The commutative sub-monoid Im(γ¯ : P( R )∞ → K 0 ( R )∞ ) is precisely the stable equivalence classes of locally Brunovsky linear systems over R. Hence the main characterizations for locally Brunovsky linear systems are as follows: 1. Two linear systems Σi in B R are feedback isomorphic if and only if their images under Z -map agree on P( R )∞ , i.e. Z (Σ1 ) = Z (Σ2 ). 2. Two linear systems Σi in B R are stable feedback isomorphic if and only if their images under Z -map agree on K 0 ( R )∞ , i.e. γ¯ (Z (Σ1 )) = γ (Z (Σ2 )). To conclude we review an example of [6]. Let R = R[x, y , z]/(x2 + y 2 + z2 − 1) be the coordinate ring of unit sphere S2R ⊆ R3 immersed into 3-dimensional space. Consider the state-space R 4 and fix the standard basis {e i }. Fix B = span{e 1 , e 2 , e 3 } and let the linear systems Σ = ( R 4 , f , B ) and Σ  = ( R 4 , f  , B ); where, in the standard basis,

⎛

0 ⎜0 f =⎜ ⎝0 1

0 0 0 0

0 0 0 0

⎞

0 0⎟ ⎟, 0⎠ 0

⎛

0 ⎜ 0 f =⎜ ⎝0 x

0 0 0 y

0 0 0 z

⎞

0 0⎟ ⎟ 0⎠ 0

It is proven in [6] that the above linear systems are not feedback isomorphic, but on the other hand they lie in the same class in K 0 ( B R ) hence they are stable isomorphic. In fact both systems became feedback isomorphic by adding the trivial ancillary system Γ (1) = ( R , 0, R ). The reader can check that systems Γ (1) ⊕ Σ and Γ (1) ⊕ Σ  are feedback isomorphic by means of the isomorphism of R 5 given, in standard basis by

⎛

1 ⎜x ⎜ φ=⎜ y ⎝z 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

⎞

0 0⎟ ⎟ 0⎟ ⎠ 0 1

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

H. Bass, Algebraic K-Theory, Benjamin, 1968. J.W. Brewer, J.W. Bunce, F.S. VanVleck, Linear Dynamical Systems over Commutative Rings, Marcel Dekker, New York, 1986. J.W. Brewer, L. Klingler, Dynamic feedback over commutative rings, Linear Algebra Appl. 98 (1988). J.W. Brewer, L. Klingler, On feedback invariants for linear dynamical systems, Linear Algebra Appl. 325 (2001). P.A. Brunovsky, A classification of linear controllable systems, Kibernetika 3 (1970). M.V. Carriegos, Enumeration of classes of linear systems via equations and via partitions in a ordered abelian monoid, Linear Algebra Appl. 438 (2013). J.L. Casti, Linear Dynamical Systems, Academic Press, 1987. M.I.J. Hautus, E.D. Sontag, New results on pole-shifting for parametrized families of systems, J. Pure Appl. Algebra 40 (1986). J.A. Hermida-Alonso, M.T. Trobajo, The dynamic feedback equivalence over principal ideal domains, Linear Algebra Appl. 368 (2003). J.A. Hermida-Alonso, M.M. López-Cabeceira, M.T. Trobajo, When are dynamic and static feedback equivalent?, Linear Algebra Appl. 405 (2005). R.E. Kalman, Kronecker invariants and feedback, in: Ordinary Differential Equations, Academic, 1972, pp. 459–471. S. Mac Lane, Categories for the Working Mathematician, Springer, New York, 1971.

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[13] J. Rosemberg, Algebraic K -Theory and Its Applications, 1994. [14] W.V. Vasconcelos, C.A. Weibel, Bcs rings, J. Pure Appl. Algebra 52 (1988) 173–185. [15] C.A. Weibel, The K -book, an introduction to algebraic K -theory, β -edition (2013/02/23), http://www.math.rutgers.edu/~ weibel/Kbook/Kbook.pdf.