  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 stablisability 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 verified via the Koopman
framework. Furthermore, the converse claims are based on an novel relations
between the Koopman method and construction of a Kazantzis-Kravaris-Luenberger
observer.
