A sufficient condition for $k$-contraction of the series connection of two systems

Abstract

The flow of contracting systems contracts 1-dimensional parallelotopes, i.e., line segments, 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 generalization of contracting systems is k-contracting systems, where k ∈ {1, . . . , n}. The flow of such systems contracts the volume of k-dimensional parallelotopes at an exponential rate, and in particular they reduce to contracting systems when k = 1. It was shown by Muldowney and Li that time-invariant 2-contracting systems have a well-ordered asymptotic behaviour: all bounded trajectories converge to the set of equilibria.Here, we derive a sufficient condition guaranteeing that the system obtained from the series interconnection of two sub-systems is k-contracting. This is based on a new formula for the kth multiplicative and additive compounds of a block-diagonal matrix, which may be of independent interest. As an application, we find conditions guaranteeing that 2-contracting systems with an exponentially decaying input retain the well-ordered behaviour of time-invariant 2-contracting systems.