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Dynamical systems that are contracting on a subspace are said to besemicontracting. Semicontraction theory is a useful tool in the study ofconsensus algorithms and dynamical flow systems such as Markov chains. Todevelop a comprehensive theory of semicontracting systems, we investigateseminorms on vector spaces and define two canonical notions: projection anddistance semi-norms. We show that the well-known lp ergodic coefficients areinduced matrix seminorms and play a central role in stability problems. Inparticular, we formulate a duality theorem that explains why theMarkov-Dobrushin coefficient is the rate of contraction for both averaging andconservation flows in discrete time. Moreover, we obtain parallel results forinduced matrix log seminorms. Finally, we propose comprehensive theorems forstrong semicontractivity of linear and non-linear time-varying dynamicalsystems with invariance and conservation properties both in discrete andcontinuous time.

Dual Seminorms, Ergodic Coefficients and Semicontraction Theory