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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Meirong Zhang and Weigu Li
Journal: Proc. Amer. Math. Soc. 130 (2002), 3325-3333
MSC (2000): Primary 34L15, 34D20, 34C25
Published electronically: March 25, 2002
MathSciNet review: 1913012
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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 $\Vert q_-\Vert _1=\int_0^T\vert q_-(t)\vert dt \le 4/T$, 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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06462-6
PII: S 0002-9939(02)06462-6
Keywords: Hill's equation, Lyapunov stability, eigenvalue
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
Article copyright: © Copyright 2002 American Mathematical Society