Lyapunov regularity of impulsive differential equations
Lyapunov regularity of impulsive differential equations
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DOI：
10.1016/j.jde.2010.07.016
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发表时间：
2010-10
影响因子：
2.4
通讯作者：
L. Barreira;C. Valls
中科院分区：
文献类型：
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作者：
L. Barreira;C. Valls
对于线性脉冲微分方程，我们给出了非一致指数二分性的一个简单判据，它包含了一致指数二分性作为一个非常特殊的情况。为此，我们引入了线性脉冲微分方程的Lyapunov正则性的概念，即所谓的正则性系数。然后利用该理论证明了如果Lyapunov指数为非零，则存在非一致指数行为，它可以用微分方程的Lyapunov指数和正则性系数来表示。我们还考虑了只有负Lyapunov指数时非一致指数压缩的特殊情况。考虑到这一关系，给出Lyapunov正则性的另一种刻画，特别是得到正则性系数的精确的下界和上界也是有趣的。特别地，我们得到了用定义脉冲线性系统的矩阵表示的界，并得到了用体积的指数增长率表示的特征。此外，我们还证明了一类线性脉冲微分方程解在足够小的非线性扰动下的持久性。
For linear impulsive differential equations, we give a simple criterion for the existence of a nonuniform exponential dichotomy, which includes uniform exponential dichotomies as a very special case. For this we introduce the notion of Lyapunov regularity for a linear impulsive differential equation, in terms of the so-called regularity coefficient. The theory is then used to show that if the Lyapunov exponents are nonzero, then there is a nonuniform exponential behavior, which can be expressed in terms of the Lyapunov exponents of the differential equation and of the regularity coefficient. We also consider the particular case of nonuniform exponential contractions when there are only negative Lyapunov exponents. Having this relation in mind, it is also of interest to provide alternative characterizations of Lyapunov regularity, and particularly to obtain sharp lower and upper bound for the regularity coefficient. In particular, we obtain bounds expressed in terms of the matrices defining the impulsive linear system, and we obtain characterizations in terms of the exponential growth rate of volumes. In addition we establish the persistence of the stability of a linear impulsive differential equation under sufficiently small nonlinear perturbations.