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.

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.