A Sufficient Condition for k-Contraction of the Series Connection of Two Systems

The flow of an n-dimensional k-contracting system, with k1, n, contracts k-dimensional parallelotopes. For k=1, this reduces to a standard contracting system. One reason for the usefulness of 1-contracting systems is that many interconnections of contracting subsystems yield an overall 1-contracting system. Here, we derive a new sufficient condition guaranteeing that the system obtained from the series interconnection of two subsystems is k-contracting. This is based on a new formula for the kth multiplicative and additive compounds of a block-diagonal matrix (BDM), 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 behavior of time-invariant 2-contracting systems.