A Lyapunov-type stability criterion using $L^\alpha $ norms
A Lyapunov-type stability criterion using $L^\alpha $ norms
Let $q(t)$ be a $T$-periodic potential such that $\int _0^T q(t) dt< 0$. The classical Lyapunov criterion to stability of Hillâs 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 …