A Lyapunov-type stability criterion using $L^\alpha $ norms

Abstract

Let $q(t)$ be a $T$-periodic potential such that $\int _0^T q(t) dt< 0$. The classical Lyapunov criterion to stability of Hill’s equation $-\ddot x+ q(t) x=0$ is $\|q_-\|_1=\int _0^T|q_-(t)|dt \le 4/T$, where $q_-$ is the negative part of $q$. In this paper, we will use a relation between the (anti-)periodic and the Dirichlet eigenvalues to establish some lower bounds for the first anti-periodic eigenvalue. As a result, we will find the best Lyapunov-type stability criterion using $L^\alpha$ norms of $q_-$, $1\le \alpha \le \infty$. The numerical simulation to Mathieu’s equation shows that the new criterion approximates the first stability region very well.