A Generalized Nonuniform Contraction and Lyapunov Function

A Generalized Nonuniform Contraction and Lyapunov Function

For nonautonomous linear equations x′=A(t)x, we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform (D,μ)-contractions. As an applic...

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Main Authors: Fang-fang Liao, Yongxin Jiang, Zhiting Xie
Format: Article
Language:English
Published: Hindawi Limited 2012-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2012/613038
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spelling doaj-556d6ca31787493eb70a3559df4e1e122020-11-24T22:07:40ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/613038613038A Generalized Nonuniform Contraction and Lyapunov FunctionFang-fang Liao0Yongxin Jiang1Zhiting Xie2Nanjing College of Information Technology, Nanjing 210046, ChinaDepartment of Mathematics, College of Science, Hohai University, Nanjing 210098, ChinaDepartment of Mathematics, College of Science, Hohai University, Nanjing 210098, ChinaFor nonautonomous linear equations x′=A(t)x, we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform (D,μ)-contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.http://dx.doi.org/10.1155/2012/613038
collection DOAJ
language English
format Article
sources DOAJ
author Fang-fang Liao
Yongxin Jiang
Zhiting Xie
spellingShingle Fang-fang Liao
Yongxin Jiang
Zhiting Xie
A Generalized Nonuniform Contraction and Lyapunov Function
Abstract and Applied Analysis
author_facet Fang-fang Liao
Yongxin Jiang
Zhiting Xie
author_sort Fang-fang Liao
title A Generalized Nonuniform Contraction and Lyapunov Function
title_short A Generalized Nonuniform Contraction and Lyapunov Function
title_full A Generalized Nonuniform Contraction and Lyapunov Function
title_fullStr A Generalized Nonuniform Contraction and Lyapunov Function
title_full_unstemmed A Generalized Nonuniform Contraction and Lyapunov Function
title_sort generalized nonuniform contraction and lyapunov function
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2012-01-01
description For nonautonomous linear equations x′=A(t)x, we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform (D,μ)-contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.
url http://dx.doi.org/10.1155/2012/613038
work_keys_str_mv AT fangfangliao ageneralizednonuniformcontractionandlyapunovfunction
AT yongxinjiang ageneralizednonuniformcontractionandlyapunovfunction
AT zhitingxie ageneralizednonuniformcontractionandlyapunovfunction
AT fangfangliao generalizednonuniformcontractionandlyapunovfunction
AT yongxinjiang generalizednonuniformcontractionandlyapunovfunction
AT zhitingxie generalizednonuniformcontractionandlyapunovfunction
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