Weak and Semi-Contraction for Network Systems and Diffusively Coupled Oscillators

We develop two generalizations of contraction theory, namely, semi-contraction and weak-contraction theory. First, using the notion of seminorm, we propose a geometric framework for semi-contraction theory. We introduce matrix semimeasures and characterize their properties. We show that the spectral abscissa of a matrix is the infimum over weighted semimeasures. For dynamical systems, we use the semimeasure 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.