A Lyapunov-type stability criterion using $L^\alpha$ norms
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- by Meirong Zhang and Weigu Li PDF
- Proc. Amer. Math. Soc. 130 (2002), 3325-3333 Request permission
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.References
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Additional Information
- Meirong Zhang
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
- Email: mzhang@math.tsinghua.edu.cn
- Weigu Li
- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- Email: weigu@math.pku.edu.cn
- Received by editor(s): October 3, 2000
- Received by editor(s) in revised form: June 15, 2001
- Published electronically: March 25, 2002
- Additional Notes: This project was supported by the National Natural Science Foundation of China, The National 973 Project of China, and The Excellent Personnel Supporting Plan of the Ministry of Education of China
- Communicated by: Carmen Chicone
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3325-3333
- MSC (2000): Primary 34L15, 34D20, 34C25
- DOI: https://doi.org/10.1090/S0002-9939-02-06462-6
- MathSciNet review: 1913012