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

The flow of an
$n$
-dimensional
$k$
-contracting system, with
$k\in \lbrace 1,\ldots,n\rbrace$
, 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
$k$
th 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.