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The Small-Gain Condition for Infinite Networks Modeled on $\ell^{\infty}$-Spaces

Christoph Kawan

TL;DR

This work develops a unifying small-gain framework for networks with countably many subsystems modeled on \(\ell^{\infty}\) by introducing gain operators built from interconnection gains and monotone aggregations. A central feature is the path of strict decay, including the \(\oplus\)-MBI property, which links to UGAS and provides constructive methods to obtain ISS Lyapunov functions for the entire network. The paper establishes necessary and sufficient conditions for path existence across several operator classes (max-type, homogeneous/subadditive, finite-dimensional) and derives a range of equivalences among NJI, UGAS, and fixed-point properties, thereby extending finite-network results to infinite settings. It also presents three concrete path-construction techniques and discusses obstacles toward a complete infinite-case theory, laying groundwork for scalable stability analysis of large or growing networks.

Abstract

In recent years, attempts have been made to extend nonlinear small-gain theorems for input-to-state stability (ISS) from finite networks to countably infinite networks with finite indegrees. Under specific assumptions about the interconnection gains and the ISS formulation, corresponding infinite-dimensional small-gain results have been proven. However, concerning these assumptions, the results are still too narrow to be considered a full extension of the state-of-the-art for finite networks. We take a step to closing this gap by developing a general technical framework within which the small-gain condition for both finite and infinite networks can be analyzed. This includes a thorough investigation of various monotone operators associated with a network and a specific ISS formulation. Our results extend and generalize the existing theory for finite networks, yield complete characterizations of the small-gain condition for specific ISS formulations, and show which obstacles still have to be overcome to obtain a complete theory for the most general infinite case.

The Small-Gain Condition for Infinite Networks Modeled on $\ell^{\infty}$-Spaces

TL;DR

This work develops a unifying small-gain framework for networks with countably many subsystems modeled on by introducing gain operators built from interconnection gains and monotone aggregations. A central feature is the path of strict decay, including the -MBI property, which links to UGAS and provides constructive methods to obtain ISS Lyapunov functions for the entire network. The paper establishes necessary and sufficient conditions for path existence across several operator classes (max-type, homogeneous/subadditive, finite-dimensional) and derives a range of equivalences among NJI, UGAS, and fixed-point properties, thereby extending finite-network results to infinite settings. It also presents three concrete path-construction techniques and discusses obstacles toward a complete infinite-case theory, laying groundwork for scalable stability analysis of large or growing networks.

Abstract

In recent years, attempts have been made to extend nonlinear small-gain theorems for input-to-state stability (ISS) from finite networks to countably infinite networks with finite indegrees. Under specific assumptions about the interconnection gains and the ISS formulation, corresponding infinite-dimensional small-gain results have been proven. However, concerning these assumptions, the results are still too narrow to be considered a full extension of the state-of-the-art for finite networks. We take a step to closing this gap by developing a general technical framework within which the small-gain condition for both finite and infinite networks can be analyzed. This includes a thorough investigation of various monotone operators associated with a network and a specific ISS formulation. Our results extend and generalize the existing theory for finite networks, yield complete characterizations of the small-gain condition for specific ISS formulations, and show which obstacles still have to be overcome to obtain a complete theory for the most general infinite case.

Paper Structure

This paper contains 21 sections, 51 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gain operators
  3. Preliminaries
  4. Definition and elementary properties
  5. Associated operators
  6. Stability properties
  7. Existence of paths of strict decay
  8. The max-MBI property
  9. Necessary conditions
  10. Three path construction methods
  11. The minimal path
  12. The linear interpolation method
  13. A combined method
  14. Particular classes of gain operators
  15. Max-type operators
  16. ...and 6 more sections

Key Result

Proposition 2.1

Any gain operator $\Gamma$ has the following properties:

Figures (1)

  • Figure 1: The dynamics of $\bar{\Gamma}_b$ in the weak$^*$-topology under the $\oplus$-MBI property. The order interval $[s_*(b),s^*(b)]$ is an invariant set which is a weak$^*$ attractor. (Observe that any $s$ not satisfying $s \leq s_*(b)$ or $s \geq s^*(b)$ does satisfy $s^1 \leq s \leq s^2$ for some $s^1 \leq s_*(b)$ and $s^*(b) \leq s^2$.)

Theorems & Definitions (117)

  • Definition 2.1
  • Example 2.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 107 more