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