关于脉冲摄动微分系统的稳定性分析

发布时间：2018-03-08 05:18

本文选题：摄动项　切入点：变分李雅普诺夫函数　出处：《山东师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：众所周知,脉冲现象作为一种瞬时突变的现象普遍存在于现代科技各领域的实际问题中,其数学模型往往可归结为脉冲微分系统.随着现代科学技术的发展,人们更加认识到脉冲微分系统在各领域的重要性并感受到了它的广泛应用.这是由于,脉冲微分系统比相应的不带脉冲的微分系统更能深刻精确地描述许多事物的变化规律,比如:医学领域中的神经网络,遗传和流行病的研究,人口生物物种经过饥荒,突然捕捞的存在形式等,这些现象所涉及的动态系统有一个共同特点,那就是系统的状态往往都是在某一时刻发生突然变化,而这种连续和离散共存的现象可以用脉冲微分系统来加以描述.鉴于脉冲微分系统在现代诸多领域中有着重要的理论意义和广泛的应用价值,从上世纪90年代开始,国内外许多专家学者都对其进行了定性研究并取得了很大的进展[3-13],[23-24],短短几十年间得到了许多研究成果.然而,在实际建立脉冲微分系统的过程中,常常会出现某些无法估计的微小干扰力,这些干扰力对系统的运行轨迹将产生瞬时的或者持续性的影响,我们称这种干扰力为微分系统的摄动项,相应的微分系统称为摄动微分系统,由此也引起了人们对脉冲摄动微分系统的广泛关注[18,32].本文是在以往的研究结果基础上,主要利用变分李雅普诺夫函数方法来研究脉冲摄动微分系统关于两个测度的稳定性,并得到了若干新的结果,全文共分为三章.在本文的第一章中,我们介绍了脉冲微分摄动系统的研究背景,说明了本文研究的主要意义,同时阐述了本文进行研究时所应用的核心思想——变分李雅普诺夫函数思想.在文章的第二章中,我们用比较方法讨论了具有限次脉动的脉冲摄动微分系统的稳定性.首先给出了系统关于两个测度稳定的基本定义,然后将锥值李雅普诺夫函数方法与变分方法相结合,用锥值变分李雅普诺夫函数的基本思想来建立一个新的比较原理,使得比较系统的右端函数只需在合适的锥上满足相应条件,而不需它在整个Rn上满足拟单调递减的条件,这在应用上有极大的便利.而后,在这个比较原理的基础上,得到了一系列系统关于两个测度最终稳定和实际稳定的判别准则.在本文的第三章中,我们首先给出具依赖状态脉冲摄动微分系统关于两个测度的完全稳定的定义,然后研究了系统关于两个测度完全稳定的直接结果.
[Abstract]:As we all know, the phenomenon of pulse, as a kind of transient sudden change, exists generally in various fields of modern science and technology, and its mathematical model is usually reduced to impulsive differential system. With the development of modern science and technology, People are more aware of the importance of impulsive differential systems in various fields and feel their wide application. This is because impulsive differential systems can describe the changing laws of many things more profoundly and accurately than the corresponding differential systems without impulses. For example, neural networks in the field of medicine, genetic and epidemiological research, famines among living species, forms of sudden fishing, and so on, and the dynamic systems involved in these phenomena have a common characteristic. That is, the state of the system tends to change suddenly at some point, However, this phenomenon of continuous and discrete coexistence can be described by impulsive differential systems. In view of the important theoretical significance and extensive application value of impulsive differential systems in many fields in modern times, since -10s, Many experts and scholars at home and abroad have made great progress in qualitative research [3-13], [23-24], and got a lot of research results in just a few decades. However, in the process of establishing impulsive differential system in practice, There are often small, incalculable disturbances that have an instantaneous or persistent effect on the trajectory of the system, which we call the perturbation term for the differential system. The corresponding differential systems are called perturbed differential systems, which have aroused widespread attention to impulsive perturbed differential systems [1832]. The stability of impulsive perturbed differential systems with respect to two measures is studied by means of the variational Lyapunov function method, and some new results are obtained, which are divided into three chapters. In this paper, we introduce the research background of impulsive differential perturbation system, explain the main significance of this study, and expound the core idea applied in this paper-variational Lyapunov function. In the second chapter, In this paper, we discuss the stability of impulsive perturbed differential systems with limited pulsation by means of comparative method. Firstly, we give the basic definitions of the stability of two measures of the system, and then combine the cone-valued Lyapunov function method with the variational method. A new comparison principle is established by using the basic idea of the cone-valued variational Lyapunov function, so that the right end function of the comparison system only needs to satisfy the corresponding conditions on the proper cone, but not the quasi-monotone decline condition on the whole rn. Then, on the basis of the comparison principle, a series of criteria for the final and practical stability of two measures are obtained. In the third chapter of this paper, We first give the definition of complete stability of impulsive perturbed differential systems with dependent states on two measures, and then we study the direct results on the complete stability of systems with respect to two measures.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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