Associate Professor, Graduate School of Economics and Management, Tohoku University

We present a version of Lyapunov’s direct method for stability of a set under a differential inclusion. We pay careful attention to the assumption of forward invariance of a basin of attraction, which is often overlooked when applying the method to local stability. Even if the value of a local Lyapunoov function monotonically changes in some neighborhood of the limit set, this alone does not prevent a trajectory from escaping from this particular neighborhood. In this note, we verify that we can construct a smaller but forward invariant neighborhood. As a corollary, we obtain a transitivity theorem on basins of attractions without requiring forward invariance.