On forward invariance in Lyapunov stability theorem for local stability
Optimization and Control
2020-06-09 v1 Computer Science and Game Theory
Systems and Control
Theoretical Economics
Systems and Control
Dynamical Systems
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.
Cite
@article{arxiv.2006.04280,
title = {On forward invariance in Lyapunov stability theorem for local stability},
author = {Dai Zusai},
journal= {arXiv preprint arXiv:2006.04280},
year = {2020}
}