【作者】 孙光辉；

【导师】 傅希林；

【作者基本信息】 山东师范大学 ， 应用数学， 2003， 硕士

【摘要】 在这篇论文中，我们主要研究如下脉冲摄动微分系统：文中主要利用变分李雅普诺夫函数方法和李雅普诺夫直接方法的思想，讨论了脉冲摄动微分系统（?）两个测度的有界性和稳定性，建立了一些关于有界性，稳定性的判别准则． 文章的前半部分我们主要借助于变分李雅普诺夫函数方法的思想，在比较定理的基础上讨论了脉冲摄动微分系统（1）两个测度的有界性和稳定性。建立了一些关于有界性和稳定性的判别准则，通过无摄动作用的脉冲微分系统两个测度的有界性及稳定性来判断脉冲摄动微分系统（1）两个测度的有界性及稳定性。通过与先前使用李雅普诺夫方法得出的结果加以比较，我们不难发现，变分李雅普诺夫方法是李雅普诺夫方法的一种推广，先前的一些结果可以作为本文结果的特殊情形．后半部分主要运用了李雅普诺夫直接方法，并对李雅普诺夫函数导数的控制函数加以限制，得到了脉冲微分系统稳定性及不稳定性的若干充分条件．全文分为五章。 在第一章中，我们介绍了本文的背景和此类问题的研究现状，并且简单介绍了本文的主要内容． 在第二章中，给出了一个比较定理和几个推论，从而将脉冲摄动微分系统的解与无摄动作用的脉冲微分系统的解联系起来．本章是全文内容的基础． 在第三章中，讨论了脉冲摄动微分系统（1）两个测度的有界性．本章初始部分给出了一种新的有界性定义—（h₀，h₀）-有界，并在比较定理的基础上给出了在已知无摄动作用的脉冲微分系统具有某种（h₀，h₀）-有界性质的条件下，判断脉冲摄动微分系统（1）是否具有相应的（h₀，h₀）-有界性质的若干准则．在本章最后，通过一个例子来说明定理的应用． 在第四章中，研究了脉冲摄动微分系统＊)两个测度的稳定性．首先给出了一 种新的稳定性定义一恤山人)－稳定，并在比较定理的基础上给出了在已知无摄动作 用的脉冲微分系统具有某种…。，h＊－稳定性质的条件下，判断脉冲摄动微分系统＊) 是否具有相应的恤。人)－稳定性质的若干脚u．并给出了一个判断脉冲摄动微分系 统…。人)－稳定性的例子． 在第五章中，运用李雅普诺夫直接方法并对李雅普诺夫函数导数的控制函数加 以限制，得到了脉冲微分系统稳定性及不稳定性的若干充分条件．本章最后，给出 了两个例子来判断脉冲微分系统uD人)－稳定和恤。人)－不稳定．更多还原

【Abstract】 In this dissertation, we discuss the perturbed differential system with impulsive effects as follows:Mainly by using variational Lyapunov method and Lyapunov direct method, we discuss the boundedness and stability in terms of two measures of the perturbed impulsive differential equations (1) and establish some criteria on boundedness and stability.In the former part of this paper, mainly by virtue of variational Lyapunov method ,under the comparison theorem, we discuss the boundedness and stability in terms of two measures of the perturbed differential equations with impulsive effects and establish some criteria on the properties of solutions of the perturbed differential equations with impulsive effects, such as boundedness in terms of two measures, stability in terms of two measures . Compared with the former results obtained by using Lyapunov method, we can see variational Lyapunov method is an extension of Lyapunov method and the former results can be viewed as the special cases of the theorems here. In the latter part of this paper, by using Lyapunov direct method and adding some conditions on the controlling function of the derivate of Lyapunov function , we get some sufficient conditions of stability and unstability for impulsive differential system. There are five chapters in this paper.In the first chapter, we introduce the background, the current situation of studies in this field and main contents of this paper.In the second chapter, we give a comparison theorem and several corollaries, therefore we connect the solution of the perturbed differential system with the solution of the unperturbed differential system. This chapter is the base of this paper.In the third chapter, we discuss the boundedness in terms of two measures of the perturbed differential system(l). First,we give a new concept on boundedness-(h0, h0)-bounded and establish some criteria to decide that the perturbed differential system(1) is (h0, h)-bounded under the condition that the unperturbed system is (h0, h0)-bounded. Finally, we indicate the application of the theorems through an example.In the fourth chapter, we discuss the stability in terms of two measures of the perturbed differential system(l). First, we give a new concept on stability-(h0, h0)-sisible and establish some criteria to decide that the perturbed differential system(l) is (h0,hi)-stable under the condition that the unperturbed system is (h0, h0)-stable and give an example to decide the stability in terms of two measures of the perturbed differential system with impulsive effects.In the fifth chapter , by using the direct Lyapunov method and adding some conditions on the controlling function of the derivate of Lyapunov function , we get some sufficient conditions on stability and unstability of the impulsive differential system. Finally, we give two examples to decide the (h0,h)-stability and (h0,h)-unstability of the impulsive differential system.更多还原