本文選題：攝動項　切入點：變分李雅普諾夫函數(shù)　出處：《山東師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：眾所周知,脈沖現(xiàn)象作為一種瞬時突變的現(xiàn)象普遍存在于現(xiàn)代科技各領(lǐng)域的實際問題中,其數(shù)學(xué)模型往往可歸結(jié)為脈沖微分系統(tǒng).隨著現(xiàn)代科學(xué)技術(shù)的發(fā)展,人們更加認(rèn)識到脈沖微分系統(tǒng)在各領(lǐng)域的重要性并感受到了它的廣泛應(yīng)用.這是由于,脈沖微分系統(tǒng)比相應(yīng)的不帶脈沖的微分系統(tǒng)更能深刻精確地描述許多事物的變化規(guī)律,比如:醫(yī)學(xué)領(lǐng)域中的神經(jīng)網(wǎng)絡(luò),遺傳和流行病的研究,人口生物物種經(jīng)過饑荒,突然捕撈的存在形式等,這些現(xiàn)象所涉及的動態(tài)系統(tǒng)有一個共同特點,那就是系統(tǒng)的狀態(tài)往往都是在某一時刻發(fā)生突然變化,而這種連續(xù)和離散共存的現(xiàn)象可以用脈沖微分系統(tǒng)來加以描述.鑒于脈沖微分系統(tǒng)在現(xiàn)代諸多領(lǐng)域中有著重要的理論意義和廣泛的應(yīng)用價值,從上世紀(jì)90年代開始,國內(nèi)外許多專家學(xué)者都對其進行了定性研究并取得了很大的進展[3-13],[23-24],短短幾十年間得到了許多研究成果.然而,在實際建立脈沖微分系統(tǒng)的過程中,常常會出現(xiàn)某些無法估計的微小干擾力,這些干擾力對系統(tǒng)的運行軌跡將產(chǎn)生瞬時的或者持續(xù)性的影響,我們稱這種干擾力為微分系統(tǒng)的攝動項,相應(yīng)的微分系統(tǒng)稱為攝動微分系統(tǒng),由此也引起了人們對脈沖攝動微分系統(tǒng)的廣泛關(guān)注[18,32].本文是在以往的研究結(jié)果基礎(chǔ)上,主要利用變分李雅普諾夫函數(shù)方法來研究脈沖攝動微分系統(tǒng)關(guān)于兩個測度的穩(wěn)定性,并得到了若干新的結(jié)果,全文共分為三章.在本文的第一章中,我們介紹了脈沖微分?jǐn)z動系統(tǒng)的研究背景,說明了本文研究的主要意義,同時闡述了本文進行研究時所應(yīng)用的核心思想——變分李雅普諾夫函數(shù)思想.在文章的第二章中,我們用比較方法討論了具有限次脈動的脈沖攝動微分系統(tǒng)的穩(wěn)定性.首先給出了系統(tǒng)關(guān)于兩個測度穩(wěn)定的基本定義,然后將錐值李雅普諾夫函數(shù)方法與變分方法相結(jié)合,用錐值變分李雅普諾夫函數(shù)的基本思想來建立一個新的比較原理,使得比較系統(tǒng)的右端函數(shù)只需在合適的錐上滿足相應(yīng)條件,而不需它在整個Rn上滿足擬單調(diào)遞減的條件,這在應(yīng)用上有極大的便利.而后,在這個比較原理的基礎(chǔ)上,得到了一系列系統(tǒng)關(guān)于兩個測度最終穩(wěn)定和實際穩(wěn)定的判別準(zhǔn)則.在本文的第三章中,我們首先給出具依賴狀態(tài)脈沖攝動微分系統(tǒng)關(guān)于兩個測度的完全穩(wěn)定的定義,然后研究了系統(tǒng)關(guān)于兩個測度完全穩(wěn)定的直接結(jié)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.
【學(xué)位授予單位】：山東師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

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本文編號：1582559