Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterization of input-to-output stability for infinite-dimensional systems

Patrick Bachmann, Sergey Dashkovskiy, Andrii Mironchenko

TL;DR

This work extends input-to-output stability (IOS) theory to nonlinear infinite-dimensional systems with outputs, developing IOS superposition theorems that extend existing ISS results from finite dimensions and full-state outputs. It introduces and interrelates a broad set of stability and attractivity notions (e.g., OUAG, OULS, OCAG, OLIM, OOULIM, OBORS) and proves multiple equivalences that characterize IOS and IOS with OL; it also provides a sufficient condition for OL and shows how IOS can be decomposed into IOS and IOSS to recover ISS results. The finite-dimensional treatment clarifies when IOS results collapse to familiar ISS theory, and counterexamples delineate the boundaries of these equivalences in infinite dimensions. Overall, the paper lays a foundational meta-toolkit for establishing Lyapunov-based and small-gain results for interconnections of infinite-dimensional IOS subsystems, with implications for time-delay systems, PDEs, and distributed parameter networks.

Abstract

We prove a superposition theorem for input-to-output stability (IOS) of a broad class of nonlinear infinite-dimensional systems with outputs including both continuous-time and discrete-time systems. It contains, as a special case, the superposition theorem for input-to-state stability (ISS) of infinite-dimensional systems and the IOS superposition theorem for systems of ordinary differential equations known from the literature. To achieve this result, we introduce and examine several novel stability and attractivity concepts for infinite-dimensional systems with outputs: We prove criteria for the uniform limit property for systems with outputs, several of which are new already for systems with full-state output, we provide superposition theorems for systems which satisfy both the output Lagrange stability (OL) and IOS, give a sufficient condition for OL and characterize ISS in terms of IOS and input/output-to-state stability. Finally, by means of counterexamples, we illustrate the challenges appearing on the way of extension of the superposition theorems from the literature to infinite-dimensional systems with outputs.

Characterization of input-to-output stability for infinite-dimensional systems

TL;DR

This work extends input-to-output stability (IOS) theory to nonlinear infinite-dimensional systems with outputs, developing IOS superposition theorems that extend existing ISS results from finite dimensions and full-state outputs. It introduces and interrelates a broad set of stability and attractivity notions (e.g., OUAG, OULS, OCAG, OLIM, OOULIM, OBORS) and proves multiple equivalences that characterize IOS and IOS with OL; it also provides a sufficient condition for OL and shows how IOS can be decomposed into IOS and IOSS to recover ISS results. The finite-dimensional treatment clarifies when IOS results collapse to familiar ISS theory, and counterexamples delineate the boundaries of these equivalences in infinite dimensions. Overall, the paper lays a foundational meta-toolkit for establishing Lyapunov-based and small-gain results for interconnections of infinite-dimensional IOS subsystems, with implications for time-delay systems, PDEs, and distributed parameter networks.

Abstract

We prove a superposition theorem for input-to-output stability (IOS) of a broad class of nonlinear infinite-dimensional systems with outputs including both continuous-time and discrete-time systems. It contains, as a special case, the superposition theorem for input-to-state stability (ISS) of infinite-dimensional systems and the IOS superposition theorem for systems of ordinary differential equations known from the literature. To achieve this result, we introduce and examine several novel stability and attractivity concepts for infinite-dimensional systems with outputs: We prove criteria for the uniform limit property for systems with outputs, several of which are new already for systems with full-state output, we provide superposition theorems for systems which satisfy both the output Lagrange stability (OL) and IOS, give a sufficient condition for OL and characterize ISS in terms of IOS and input/output-to-state stability. Finally, by means of counterexamples, we illustrate the challenges appearing on the way of extension of the superposition theorems from the literature to infinite-dimensional systems with outputs.
Paper Structure (17 sections, 21 theorems, 115 equations, 5 figures, 1 table)

This paper contains 17 sections, 21 theorems, 115 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability properties
  4. Attractivity properties
  5. Weak attractivity properties
  6. Superposition theorems
  7. IOS superposition theorem
  8. IOS $\wedge$ OL superposition theorem
  9. Sufficient condition for OL
  10. finite-dimensional IOS theory
  11. OLIM, OULIM and OGULIM on finite-dimensional spaces
  12. IOS on finite-dimensional spaces
  13. Characterizations of ISS
  14. ISS superposition theorem
  15. ISS as superposition of IOS and IOSS
  16. ...and 2 more sections

Key Result

Lemma II.10

The output map $h$ is bounded on bounded sets if and only if there exist $\sigma_1, \gamma_1 \in \mathcal{K}$ and $c \geq 0$ such that for all $x\in X$ , $u \in \mathcal{U}$ and $t \in I$ we have

Figures (5)

  • Figure 1: Diagram of implications.
  • Figure 2: Diagram of implications for the proof of Theorem \ref{['thm:IOSequivalences']}.
  • Figure 3: Diagram of elementary implications summarized in Lemma \ref{['lem:IOStoBORSandOCEPandOGUAGandOGULIM']}.
  • Figure 4: Diagram of implications for the proof of Theorem \ref{['prop:OCAGequivalences']}.
  • Figure 5: Plot for several components $\phi_n$ with initial condition $x_n = c$. Local lower bound for $\phi(n) \leq n$ in black and upper bound for global attractivity in red.

Theorems & Definitions (76)

  • Definition II.1
  • Remark II.2
  • Remark II.3
  • Remark II.4
  • Definition II.5
  • Definition II.6
  • Definition II.7
  • Definition II.8
  • Definition II.9
  • Lemma II.10
  • ...and 66 more