文章基本信息

标题：Uniform asymptotic stability implies exponential stability for nonautonomous half-linear differential systems
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作者：Masakazu Onitsuka ; Tomomi Soeda
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2015
卷号：2015
期号：1
页码：158
DOI：10.1186/s13662-015-0494-7
语种：English
出版社：Hindawi Publishing Corporation
摘要：The present paper is considered a two-dimensional half-linear differential system: x ′ = a 11 ( t ) x + a 12 ( t ) ϕ p ∗ ( y ) $x' = a_{11}(t)x+a_{12}(t)\phi_{p^{*}}(y)$ , y ′ = a 21 ( t ) ϕ p ( x ) + a 22 ( t ) y $y' = a_{21}(t)\phi_{p}(x)+a_{22}(t)y$ , where all time-varying coefficients are continuous; p and p ∗ $p^{*}$ are positive numbers satisfying 1 / p + 1 / p ∗ = 1 $1/p + 1/p^{*} = 1$ ; and ϕ q ( z ) = z q − 2 z $\phi_{q}(z) = z ^{q-2}z$ for q = p $q = p$ or q = p ∗ $q = p^{*}$ . In the special case, the half-linear system becomes the linear system x ′ = A ( t ) x $\mathbf{x}' = A(t)\mathbf {x}$ where A ( t ) $A(t)$ is a 2 × 2 $2 \times2$ continuous matrix and x is a two-dimensional vector. It is well known that the zero solution of the linear system is uniformly asymptotically stable if and only if it is exponentially stable. However, in general, uniform asymptotic stability is not equivalent to exponential stability in the case of nonlinear systems. The aim of this paper is to clarify that uniform asymptotic stability is equivalent to exponential stability for the half-linear differential system. Moreover, it is also clarified that exponential stability, global uniform asymptotic stability, and global exponential stability are equivalent for the half-linear differential system. Finally, the converse theorems on exponential stability which guarantee the existence of a strict Lyapunov function are presented.
关键词：uniform asymptotic stability ; exponential stability ; half-linear differential system ; Lyapunov function ; converse theorem