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A dual notion to BIBO stability

Felix L. Schwenninger, Alexander A. Wierzba

TL;DR

The paper develops a duality framework between BIBO stability and $L^1$-to-$L^1$ stability (LILO) for infinite-dimensional system nodes. It shows that while finite-dimensional input/output spaces yield equivalence between BIBO and LILO and allow stability preservation under duality, the general infinite-dimensional setting breaks these correspondences, though the two notions remain dual. A complete LILO theory is established in the SISO case, including a sufficient criterion and duality results that connect LILO and BIBO stability under suitable assumptions. The results clarify when stability properties transfer through duality in infinite dimensions and highlight the nuanced landscape beyond the SISO setting, where LILO provides a natural and technically advantageous dual viewpoint.

Abstract

In this paper we consider BIBO stability of infinite-dimensional linear state-space systems and the related notion of $L^1$-to-$L^1$ input-output stability (abbreviated LILO). We show that in the case of finite-dimensional input and output spaces, both are equivalent and preserved under duality transformations. In the general case, neither of these properties is satisfied, but BIBO and LILO stability remain dual to each other.

A dual notion to BIBO stability

TL;DR

The paper develops a duality framework between BIBO stability and -to- stability (LILO) for infinite-dimensional system nodes. It shows that while finite-dimensional input/output spaces yield equivalence between BIBO and LILO and allow stability preservation under duality, the general infinite-dimensional setting breaks these correspondences, though the two notions remain dual. A complete LILO theory is established in the SISO case, including a sufficient criterion and duality results that connect LILO and BIBO stability under suitable assumptions. The results clarify when stability properties transfer through duality in infinite dimensions and highlight the nuanced landscape beyond the SISO setting, where LILO provides a natural and technically advantageous dual viewpoint.

Abstract

In this paper we consider BIBO stability of infinite-dimensional linear state-space systems and the related notion of -to- input-output stability (abbreviated LILO). We show that in the case of finite-dimensional input and output spaces, both are equivalent and preserved under duality transformations. In the general case, neither of these properties is satisfied, but BIBO and LILO stability remain dual to each other.
Paper Structure (8 sections, 12 theorems, 57 equations, 1 figure)

This paper contains 8 sections, 12 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. System nodes and BIBO stability
  4. BIBO stability under duality
  5. LILO stability
  6. A sufficient criterion for LILO stability
  7. Duality between BIBO and LILO stability
  8. Outlook

Key Result

Proposition 3.3

Let $\Sigma(A, B, C, \mathbf{G})$ be a $L^\infty$-BIBO stable system node with finite-dimensional input and output spaces and assume that $A^*$ generates a $C_0$-semigroup. Then $\Sigma(A^*,C^*,B^*,\overline{\mathbf{G}}(\overline{\mkern2mu\cdot\mkern2mu}))$ is $L^\infty$-BIBO stable.

Figures (1)

  • Figure 1: Relationship between different BIBO and LILO stability notions

Theorems & Definitions (33)

  • Definition 2.1
  • Example 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Corollary 4.4
  • ...and 23 more