On Convergence Properties of Semi-contracting Nonlinear Systems

Contraction analysis is a method for stability analysis of nonlinear systems, which focuses on the distance between trajectories rather than that from an equilibrium or a target trajectory. In earlier studies, relationships among the three characteristics, i.e., contraction, incrementally exponential stability (IES), and exponential convergence, have been investigated. This paper shows the equivalency among semi-contraction, incremental stability, and convergence, all of which are weaker versions of contraction, IES, and exponential convergence, respectively, without exponential decay. Finally, through a numerical example, we show that the equivalency holds for a nonlinear system having a limit cycle, to which the properties with exponential decay cannot be applied.