Serial interconnections of 1-contracting and 2-contracting systems

TL;DR

This paper investigates how serial interconnections of 1- and 2-contracting systems behave asymptotically, providing conditions for their predictable long-term dynamics and illustrating these with examples.

Contribution

It offers new theoretical conditions ensuring well-ordered asymptotic behavior in serial interconnections of 1- and 2-contracting systems.

Findings

Interconnections can preserve contraction properties under certain conditions

Conditions for predictable asymptotic behavior are established

Examples demonstrate the theoretical results

Abstract

The flow of contracting systems contracts 1-dimensional polygons (i.e. lines) at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an overall contracting system. A recent generalization of contracting systems is called $k$-contracting systems, where $k\in\{1,\dots,n\}$. The flow of such systems contracts $k$-dimensional polygons at an exponential rate, and in particular they reduce to contracting systems when $k=1$. Here, we analyze serial interconnections of $1$-contracting and $2$-contracting systems. We provide conditions guaranteeing that such interconnections have a well-ordered asymptotic behaviour, and demonstrate the theoretical results using several examples.