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On BIBO stability of infinite-dimensional linear state-space systems

Felix L. Schwenninger, Alexander A. Wierzba, Hans Zwart

Abstract

In this paper we consider BIBO stability of systems described by infinite-dimensional linear state-space representations, filling the so far unattended gap of a formal definition and characterization of BIBO stability in this general case. Furthermore, we provide several sufficient conditions guaranteeing BIBO stability of a particular system and discuss to which extent this property is preserved under additive and multiplicative perturbations of the system.

On BIBO stability of infinite-dimensional linear state-space systems

Abstract

In this paper we consider BIBO stability of systems described by infinite-dimensional linear state-space representations, filling the so far unattended gap of a formal definition and characterization of BIBO stability in this general case. Furthermore, we provide several sufficient conditions guaranteeing BIBO stability of a particular system and discuss to which extent this property is preserved under additive and multiplicative perturbations of the system.
Paper Structure (22 sections, 20 theorems, 56 equations, 1 figure)

This paper contains 22 sections, 20 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. System nodes and their solutions
  4. Classical solutions
  5. Distributional solutions in the sense of Staffans
  6. BIBO stability
  7. Proof of Theorem \ref{['thm:classicalBIBOBoundedVariation']}
  8. From a measure of bounded variation to $C^\infty$-BIBO stability
  9. From $C^\infty$-BIBO stability to a measure of bounded variation
  10. Proof of Theorem \ref{['thm:generalBIBOclassicalBIBO']}
  11. BIBO-inequality for all classical solutions with $x(0) = 0$
  12. BIBO-inequality for generalised distributional solutions
  13. Relation to the results from a_UnserNoteOnBiboStability
  14. Sufficient conditions for BIBO stability
  15. Riesz-spectral systems
  16. ...and 7 more sections

Key Result

Lemma 2.5

\newlabellem:existenceClassicalSolutions0 \newlabellem:TransferFunctionMultiRepCompact0 Let $\Sigma(A,B,C,\mathbf{G})$ be a system node.

Figures (1)

  • Figure 1: Decomposition of the additively perturbed system

Theorems & Definitions (54)

  • Definition 2.1: a_TucsnakWeissWellPosed2014
  • Remark 2.2
  • Definition 2.3: a_TucsnakWeissWellPosed2014
  • Definition 2.4: a_TucsnakWeissWellPosed2014
  • Lemma 2.5
  • Proof 1
  • Definition 2.6: b_Staffans2005
  • Lemma 2.7: b_Staffans2005
  • Definition 3.1: Using distributional solutions
  • Definition 3.2: Using classical solutions
  • ...and 44 more