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

Abstract

Let q(t) be a T-periodic potential such that ?0T q(t) dt < 0. The classical Lyapunov criterion to stability of Hill's equation -? + q(t)x = 0 is ?q-?1 = ?0T |q-(t)|dt ? 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? norms of q-, 1 ? ? ? ?. The numerical simulation to Mathieu's equation shows that the new criterion approximates the first stability region very well.