  In this paper we prove new connections between two frameworks for analysis
and control of nonlinear systems: the Koopman operator framework and
contraction analysis. Each method, in different ways, provides exact and global
analyses of nonlinear systems by way of linear systems theory. The main results
of this paper show equivalence between contraction and Koopman approaches for a
wide class of stability analysis and control design problems. In particular:
stability or stablizability in the Koopman framework implies the existence of a
contraction metric (resp. control contraction metric) for the nonlinear system.
Further in certain cases the converse holds: contraction implies the existence
of a set of observables with which stability can be verified via the Koopman
framework. We provide results for the cases of autonomous and time-varying
systems, as well as orbital stability of limit cycles. Furthermore, the
converse claims are based on a novel relation between the Koopman method and
construction of a Kazantzis-Kravaris-Luenberger observer. We also provide a
byproduct of the main results, that is, a new method to learn contraction
metrics from trajectory data via linear system identification.
