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On forward invariance in Lyapunov stability theorem for local stability (2006.04280v1)
Published 7 Jun 2020 in math.OC, cs.GT, cs.SY, econ.TH, eess.SY, and math.DS
Abstract: Forward invariance of a basin of attraction is often overlooked when using a Lyapunov stability theorem to prove local stability; even if the Lyapunov function decreases monotonically in a neighborhood of an equilibrium, the dynamic may escape from this neighborhood. In this note, we fix this gap by finding a smaller neighborhood that is forward invariant. This helps us to prove local stability more naturally without tracking each solution path. Similarly, we prove a transitivity theorem about basins of attractions without requiring forward invariance. Keywords: Lyapunov function, local stability, forward invariance, evolutionary dynamics.