# Small gain theorems for general networks of heterogeneous infinite-dimensional systems

## TL;DR

This paper establishes small-gain theorems for complex networks of heterogeneous infinite-dimensional systems, including differential equations, providing new tools for stability analysis in control theory.

## Contribution

It introduces small-gain theorems applicable to various ISS formulations for heterogeneous infinite-dimensional systems, extending existing stability analysis methods.

## Key findings

- Proved small-gain theorems for nonlinear heterogeneous systems
- Derived stability results for different ISS properties
- Discussed tightness and introduced new stability-related properties

## Abstract

We prove a small-gain theorem for interconnections of $n$ nonlinear heterogeneous input-to-state stable (ISS) control systems of a general nature, covering partial, delay and ordinary differential equations. Furthermore, for the same class of control systems, we derive small-gain theorems for asymptotic gain, uniform global stability and weak input-to-state stability properties. We show that our technique is applicable for different formulations of ISS property (summation, maximum, semimaximum) and discuss tightness of achieved small-gain theorems. Finally, we introduce variations of uniform asymptotic gain and uniform limit properties, which are particularly useful for small-gain arguments and characterize ISS in terms of these notions.

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Source: https://tomesphere.com/paper/1901.03747