📝 SummarXiv

Abstract

In recent years, attempts have been made to extend ISS small-gain theorems 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 a thorough investigation of various monotone operators associated with an infinite network and a specific ISS formulation. Our results 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 case.