  We develop two generalizations of contraction theory, namely,
semi-contraction and weak-contraction theory. First, using the notion of
semi-norm, we propose a geometric framework for semi-contraction theory. We
introduce matrix semi-measures and characterize their properties. We show that
the spectral abscissa of a matrix is the infimum over weighted semi-measures.
For dynamical systems, we use the semi-measure of their Jacobian to
characterize the contractivity properties of their trajectories. Second, for
weakly contracting systems, we prove a dichotomy for the asymptotic behavior of
their trajectories and novel sufficient conditions for convergence to an
equilibrium. Third, we show that every trajectory of a doubly-contracting
system, i.e., a system that is both weakly and semi-contracting, converges to
an equilibrium point. Finally, we apply our results to various important
network systems including affine averaging and affine flow systems,
continuous-time distributed primal-dual algorithms, and networks of
diffusively-coupled dynamical systems. For diffusively-coupled systems, the
semi-contraction theory leads to a sufficient condition for synchronization
that is sharper, in general, than previously-known tests.
