【作者】 李晓迪；

【导师】 傅希林；

【作者基本信息】 山东师范大学 ， 应用数学， 2008， 硕士

【摘要】 近年来，脉冲泛函微分系统已被广泛应用于神经网络，光学控制，人口动力学，生物技术，经济学等领域．对这类系统解的性质的研究已经成为许多数学工作者的热门研究课题，并已经取得了一些好的研究成果．但在这些方面的研究还是不够的，例如，在脉冲扰动下不变性原理能否拓广到非自治系统?在超前型含混合常数变元脉冲泛函微分系统振动性方面，是否有更一般结果?在无穷延滞脉冲泛函微分系统定性理论方面，是否能找到保证这类系统解全局稳定的充分条件?等等．因此，在这个领域还有很多工作要我们去做．本文主要的研究工作就是着重于脉冲泛函微分系统的动力学分析，对上述部分问题做了肯定回答．全文分三章．在第一章，我们研究了滞后型脉冲泛函微分系统的一致渐近稳定性，全局指数稳定性及W-稳定性．将一维时滞Halanay微分不等式推广到Dini导数脉冲微分不等式，用它研究了系统(1)的弱指数渐近稳定性与全局指数稳定性．利用Lyapunov函数和Razumikhin技巧，我们又得到了保证系统(1)零解一致稳定和一致渐近稳定的若干新结果，并在第三章中给出其在神经网络中的相关应用．此外，利用Lyapunov函数和Razumikhin技巧，我们还讨论了不变性原理在非自治系统中拓广,研究了系统(1)的W-稳定性及W-一致稳定性．初步建立了脉冲泛函微分系统W-稳定性与零解一致稳定性，渐近稳定性的关系．在第二章，我们研究了超前型含混合常数变元的脉冲时滞微分系统的振动性和渐近性，其中Z_+表示全体正整数集合，[·]表示取整函数，σ是任意的正奇数比，m>0，m∈Z_+.通过巧妙构造辅助函数，我们对系统(2)的系数a_i，p，e非正、非负的情形分别做了相应的讨论和分析：当a_i，p，e非负时，我们得到了保证系统(2)所有有界解振动的若干充分条件；当a_i．p．e非正时，我们得到了保证系统f2)所有有界解或者振动或者趋于零的一个充分条件．同时，我们还对导数的次数σ的不同范围给出了详细的分类和讨论．本章所得的部分结果改进并推广了部分已有文献的结果．在第三章，通过建立适当的Lyapunov泛函并结合第一章得到的Lyapunov稳定性结果，研究了脉冲时滞Hopfield型神经网络平衡解一致渐近稳定性．同时，利用线性矩阵不等式(LMI)比较技术，我们还得到了保证脉冲时滞Hopfield型神经网络平衡解全局指数稳定的充分条件．值得一提的是，本章得到的部分结果推广了一些已有的结论,有利于实际应用．更多还原

【Abstract】 In recent years. impulsive functional differential systems have been widely used in neural networks, optical control, population dynamics, biotechnology, economics and other fields, It has become a hot research topic and also attracts many mathematician’s attention. and various results for the impulsive functional differential system are obtained . However, the research in those fields is not enough, for instance, does the invariance principle can extend to non-autonomous system under impulsive disturbance? Are there some more general results on oscillation of impulsive functional differential system of mixed type with advanced constant argument? Are there some sufficient conditions that guarantee the global stability of impulsive functional differential system with infinite delays? and so on. Hence, there are a lot of work we need to do in this field. In this paper. we focus on the research work on dynamics analysis of impulsive functional differential systems and give some affirmative answers for above problems. This paper is divided into three parts.In chapter one. we study the uniform asymptotic stability, global exponential stability and W-stability of impulsive functional differential systems with finite delays as follows:We investigate the weak exponential stability and global exponential stability of system (1) by establishing the extended Halanay’ s 1-dimension delay differential inequality. Besides, we study the stability of system (1) by using Lyapunov functions and Razumikhin technique. Some new Razumikhin type theorems on uniform stability and uniform asymptotic stability are obtained here. Also, some application on neural networks is given to illustrate the advantages of the obtained results on chapter three. On the other hand, we discuss the invariance principle that can extend to non-autonomous system under the effects of impulses and delays. We investigate the W- stability and W-uniform stability of impulsive functional differential systems and preliminary establish the relation between uniform stability, asymptotic stability and W-stability. In chapter two. we study the oscillation and asymptotic property of impulsive delay differential equation with piecewise constant argument of advanced type as follows:where Z_+ is the set of all positive integers, [·] denotes set of maximum integers,σis any quotient of positive odd integers, m > 0, m∈Z_+. By establishing auxiliary function and using analysis technique, we analyze and investigate the oscillation and asymptotic property of system (2) with non-positive coefficient or non-negative coefficient. There are two sections: when a_i,e,p are non-negative functions, we obtain some results ensuring the bounded solution to be oscillatory; when a_i, e, p are non-positive functions, we obtain some results ensuring the bounded solution either to oscillate or to tend to zero. Besides, we analyze and investigate above results with the change ofσ. Our results in this chapter generalize and improve several known results.In chapter three, we obtain a criterion for the uniform asymptotic stability of the equilibrium point of impulsive delayed Hopfield neural networks by using Lyapunov functions and some obtained results in chapter one. Besides, some new sufficient conditions ensuring exponential stability of the equilibrium point of impulsive delayed Hopfield neural networks are obtained by using linear matrix inequality approach (LMI). Our results in this chapter generalize and improve several known results. which is beneficial to practical application.更多还原