Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations
Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations
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DOI：
10.1007/s00009-017-0867-0
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发表时间：
2017-03
影响因子：
1.1
通讯作者：
Jinrong Wang;Michal Feckan;Ying Tian
中科院分区：
文献类型：
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作者：
Jinrong Wang;Michal Feckan;Ying Tian
对于一类新的线性非瞬时脉冲微分方程，引入了非瞬时脉冲Cauchy矩阵的概念，并通过脉冲点与连接点的距离分析了其特征值的指数结构.得到了线性非瞬时脉冲微分方程渐近稳定性的许多有用的判据，这些判据使我们能够在较弱的充分条件下建立一个统一的框架来处理线性非瞬时脉冲微分方程的渐近稳定性.特别地，给出了两个例子来说明线性部分的理论结果的应用。此外，我们还研究了非线性非瞬时脉冲方程，分别在指数增长约束和非瞬时脉冲Cauchy矩阵稳定条件下，研究了非线性非瞬时脉冲方程解的存在性、唯一性和Ulam-Hyers-Rassias稳定性，为非线性非瞬时脉冲方程在Ulam-Hyers-Rassias稳定性意义下的近似解的求解提供了一种途径.所得结果覆盖了瞬时脉冲微分方程的标准结果，实质上推广了前人的理论。
For a new linear non-instantaneous impulsive differential equations, we introduce the notation of non-instantaneous impulsive Cauchy matrix and analyze its exponential structure in terms of eigenvalues of matrix via the distance between impulsive points and junction points. Many useful criteria for asymptotic stability of linear non-instantaneous impulsive problems are derived, which allow us to establish a uniform framework to deal with asymptotic stability of linear non-instantaneous impulsive differential equations with perturbation under mild sufficient conditions. In particular, two examples are given to demonstrate the application of theoretical results for linear part. In addition, we study nonlinear non-instantaneous impulsive equations and investigate existence, uniqueness of their solutions and Ulam–Hyers–Rassias stability under the restriction of exponential growth or stable conditions for non-instantaneous impulsive Cauchy matrix, respectively, which provide an approach to find approximate solution to nonlinear non-instantaneous impulsive equations in the sense of Ulam–Hyers–Rassias stability. The obtained results cover the standard results for instantaneous impulsive differential equations, which are essentially extend the theory in the previous literatures.