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The defect, the Malgrange functor, and linear control systems

Alex Martsinkovsky

Abstract

The notion of defect of a finitely presented functor on a module category is extended to arbitrary additive functors. The new defect and the contravariant Yoneda embedding form a right adjoint pair. The main result identifies the defect of the covariant Hom modulo projectives with the Bass torsion of the fixed argument. When applied to a linear control systems, it shows that the defect of the Malgrange functor of the system modulo projectives is isomorphic to the autonomy of the system. Furthermore, the defect of the contravariant Hom modulo injectives is shown to be isomorphic to the cotorsion coradical of the fixed argument. Since the Auslander-Gruson-Jensen transform of cotorsion is isomorphic to torsion, the above results raise two important questions: a) what is a control-theoretic interpretation of the covariant Yoneda embedding of the Malgrange module modulo injectives, and b) what is a control-theoretic interpretation of the Auslander-Gruson-Jensen duality?

The defect, the Malgrange functor, and linear control systems

Abstract

The notion of defect of a finitely presented functor on a module category is extended to arbitrary additive functors. The new defect and the contravariant Yoneda embedding form a right adjoint pair. The main result identifies the defect of the covariant Hom modulo projectives with the Bass torsion of the fixed argument. When applied to a linear control systems, it shows that the defect of the Malgrange functor of the system modulo projectives is isomorphic to the autonomy of the system. Furthermore, the defect of the contravariant Hom modulo injectives is shown to be isomorphic to the cotorsion coradical of the fixed argument. Since the Auslander-Gruson-Jensen transform of cotorsion is isomorphic to torsion, the above results raise two important questions: a) what is a control-theoretic interpretation of the covariant Yoneda embedding of the Malgrange module modulo injectives, and b) what is a control-theoretic interpretation of the Auslander-Gruson-Jensen duality?
Paper Structure (5 sections, 14 theorems, 16 equations)

This paper contains 5 sections, 14 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The Malgrange functor of a linear control system
  3. The defect of a finitely presented functor
  4. The defect of an arbitrary additive functor
  5. Concluding remarks and open questions

Key Result

Proposition 3.2

Let $F$ and $G$ be functors. If $F$ is finitely generated, then the class of all natural transformations $g : F \to G$, denoted by $\mathrm{Nat}(F, G)$, is a set.

Theorems & Definitions (36)

  • Remark 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 26 more