本文選題：泛函微分包含　切入點(diǎn)：Filippov解　出處：《湖南大學(xué)》2016年博士論文

【摘要】：據(jù)我們所知,在機(jī)械工程、力學(xué)、神經(jīng)網(wǎng)絡(luò)、自動(dòng)控制以及生物學(xué)等領(lǐng)域,右端不連續(xù)泛函微分方程是大量存在的.一般地,對(duì)右端不連續(xù)泛函微分方程而言,由于其右端函數(shù)不是連續(xù)的,因而經(jīng)典的泛函微分方程理論體系無法適用.為了分析和研究右端不連續(xù)泛函微分方程的解的基本性質(zhì)及其一些動(dòng)力學(xué)行為,我們先通過應(yīng)用Filippov微分包含正規(guī)化方法,將其轉(zhuǎn)化為一個(gè)恰當(dāng)?shù)姆汉⒎职?然后利用該泛函微分包含,給出了右端不連續(xù)的泛函微分方程的Filippov意義下解的定義及其在給定的初始條件下的解的定義.在此基礎(chǔ)上,并利用泛函微分包含理論,進(jìn)一步研究了具可變時(shí)滯和分布時(shí)滯的泛函微分方程的Filippov意義下解的一些基本性質(zhì)和一些動(dòng)力學(xué)行為.主要的研究內(nèi)容包括:Filippov意義下解的局部與整體存在性(延拓性)、解軌線的周期(概周期)動(dòng)力學(xué)行為及其穩(wěn)定性和收斂性行為(例如:全局指數(shù)穩(wěn)定性、同步性、全局耗散性)等等.本文將從以下兩個(gè)方面展開,一是根據(jù)實(shí)際的生產(chǎn)及科學(xué)實(shí)踐中出現(xiàn)的一些不連續(xù)現(xiàn)象,利用右端不連續(xù)泛函微分方程來建立各種數(shù)學(xué)模型對(duì)其進(jìn)行描述.然后通過Filippov正規(guī)化方法,將右端不連續(xù)泛函微分方程轉(zhuǎn)化為相應(yīng)的泛函微分包含.其二是在Filippov泛函微分包含的基本框架內(nèi),討論Filippov意義下解的各種動(dòng)力學(xué)行為.主要研究內(nèi)容包括:周期解與多個(gè)周期解的存在性;周期解與概周期解的存在性和唯一性;Filippov意義下解的各種穩(wěn)定性及其收斂性.主要研究工具與研究方法包括:集值分析中的一些不動(dòng)點(diǎn)理論、集值分析中的拓?fù)涠壤碚�、非光滑分析理論、矩陣分析、矩陣測度理論、廣義Lyapunov泛函方法等等.本學(xué)位論文共分為六章.在第一章中,先簡要介紹了右端不連續(xù)泛函微分方程與泛函微分包含理論的發(fā)展歷史及其研究概況.同時(shí),也簡單介紹當(dāng)前不連續(xù)神經(jīng)網(wǎng)絡(luò)系統(tǒng)和不連續(xù)生物系統(tǒng)的研究概況.最后,就本文的主要研究內(nèi)容與結(jié)構(gòu)安排作了介紹.在第二章中,介紹本文研究所必需的一些基本理論知識(shí).第三章的討論是針對(duì)一類具可變時(shí)滯和分布時(shí)滯的Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)系統(tǒng)展開的,其神經(jīng)元激勵(lì)函數(shù)是一元不連續(xù)函數(shù)(分段連續(xù)函數(shù)).本章所用的工具和方法涉及到泛函微分包含理論,集值分析中的一些不動(dòng)點(diǎn)理論、非光滑分析理論以及廣義Lypunov泛函方法等等.首先,在不要求神經(jīng)元激勵(lì)函數(shù)是有界的且不滿足線性增長假設(shè)的情形下,研究了具不連續(xù)激勵(lì)函數(shù)和具時(shí)滯的CohenGrossberg神經(jīng)網(wǎng)絡(luò)系統(tǒng)周期解與多個(gè)周期解的存在性問題.其次,在神經(jīng)元激勵(lì)函數(shù)是非單調(diào)的情形下,研究了具不連續(xù)激勵(lì)函數(shù)和具時(shí)滯的Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)系統(tǒng)周期解的存在性、唯一性及其指數(shù)型穩(wěn)定性問題.同時(shí),也討論了該不連續(xù)神經(jīng)網(wǎng)絡(luò)系統(tǒng)輸出解的依測度收斂性問題.最后,在不要求神經(jīng)元激勵(lì)函數(shù)是有界的和單調(diào)非減的情形下,研究了具不連續(xù)激勵(lì)函數(shù)和具時(shí)滯的Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)系統(tǒng)概周期動(dòng)力學(xué)行為.所獲得的關(guān)于具可變時(shí)滯和分布時(shí)滯的不連續(xù)神經(jīng)網(wǎng)絡(luò)系統(tǒng)的這些研究結(jié)果是對(duì)已有結(jié)果的推廣和改進(jìn).第四章討論了一類具可變時(shí)滯的不連續(xù)神經(jīng)網(wǎng)絡(luò)驅(qū)動(dòng)-響應(yīng)系統(tǒng)的同步性.本章所用的工具和方法涉及到泛函微分包含理論,非光滑分析理論以及廣義Lypunov泛函方法,一些不等式技巧等等.利用連續(xù)和不連續(xù)狀態(tài)反饋控制器,得出了具不連續(xù)激勵(lì)函數(shù)的神經(jīng)網(wǎng)絡(luò)驅(qū)動(dòng)-響應(yīng)系統(tǒng)的指數(shù)型同步性.本章的不連續(xù)激勵(lì)函數(shù)可能是非單調(diào)、超線性的、甚至是指數(shù)型的,所得結(jié)果推廣并改進(jìn)了一些相關(guān)結(jié)果.第五章的討論是針對(duì)具二元不連續(xù)激勵(lì)函數(shù)的時(shí)滯BAM神經(jīng)網(wǎng)絡(luò)系統(tǒng)展開的.首先,通過定義恰當(dāng)?shù)腇ilippov包含,給出其Filippov意義下解的定義.并通過泛函微分包含理論,研究了具二元不連續(xù)激勵(lì)函數(shù)的時(shí)滯BAM神經(jīng)網(wǎng)絡(luò)系統(tǒng)Filippov解的局部存在性和整體存在性及其全局耗散性.其次,通過設(shè)計(jì)不連續(xù)狀態(tài)反饋控制器,得到了具二元不連續(xù)激勵(lì)函數(shù)的時(shí)滯BAM神經(jīng)網(wǎng)絡(luò)系統(tǒng)的指數(shù)型同步性.最后,應(yīng)用集值分析中的拓?fù)涠壤碚?研究了具二元不連續(xù)激勵(lì)函數(shù)的時(shí)滯BAM神經(jīng)網(wǎng)絡(luò)系統(tǒng)的周期解的存在性問題.第六章針對(duì)可再生資源的開發(fā)與管理,先提出了更為一般的不連續(xù)收獲策略,并考慮了具該不連續(xù)收獲策略的Lotka-Volterra競爭系統(tǒng).利用泛函微分包含理論、集值分析中的不動(dòng)點(diǎn)定理、一些分析技巧和方法,本章研究了Filippov解的局部存在性和整體存在性,正周期解的存在性.最后,通過一些數(shù)值例子來說明我們的主要結(jié)果的正確性與有效性.通過對(duì)這些問題的探討,一方面,在一定程度上加深和完善了右端不連續(xù)泛函微分方程理論以及泛函微分包含理論;另一方面,也為分析和解決神經(jīng)網(wǎng)絡(luò)、生物學(xué)等科學(xué)與工程領(lǐng)域中的一些實(shí)際問題提供了一些方法和理論支持.
[Abstract]:As far as we know, in mechanical engineering, mechanics, neural network, automatic control and other fields of biology, discontinuous functional differential equations is substantial. Generally, the right side is the discontinuous functional differential equations, because of its right side function is not continuous, and the theoretical system of classical functional differential equation cannot in order to apply. Basic properties of solutions to the analysis and study of discontinuous functional differential equations and dynamics, we first include regularization method through the application of Filippov differential, it is transformed into a proper functional differential included. Then the functional differential inclusions, gives the definition of the right end of the solution of functional differential equations the discontinuous Filippov and its significance in the given initial conditions of the definition of the solution. On this basis, and the use of functional differential inclusion theory, further studies with variable delay and Functional differential equation with delay of the cloth under the Filippov some basic properties of solutions and some dynamic behavior. The main research contents include: the sense of Filippov local and global existence of solution (Continuation), periodic trajectories (periodic) dynamics and the stability and convergence behavior (for example: global index stability, synchronization, and so on) global dissipativity. This paper will start from the following two aspects, one is according to some discontinuous phenomenon in the actual production and scientific practice, the use of discontinuous functional differential equations to establish mathematical models are described. Then through the Filippov regularization method, right end of discontinuous functional differential equation into the corresponding functional differential inclusions. The second is the basic framework of Filippov functional differential inclusions, discuss various dynamic behavior under the Filippov solution. The research contents include: the existence of multiple periodic solutions and periodic solutions; existence and uniqueness of periodic solution and almost periodic solutions; stability and convergence of the solution in the Filippov sense. Including the main research tools and research methods: set-valued analysis in some fixed point theory, set-valued analysis theory of topological degree the theory of nonsmooth analysis, matrix analysis, matrix measure theory, generalized Lyapunov function method and so on. This thesis is divided into six chapters. In the first chapter, first the discontinuous functional differential equations and functional differential inclusion of historical development and theoretical research are briefly introduced. At the same time, also introduces the research situation the discontinuous neural network system and discontinuous biological system. Finally, it introduces the main research contents and structures. In the second chapter, this paper introduces some necessary base of research of the theory of knowledge The third chapter is the general discussion. For a class of variable delay and distributed delay Cohen-Grossberg neural network system, the activation function of the neurons is a discontinuous function element (piecewise continuous function). The tools and methods used in this chapter involves the functional differential inclusion theory, set-valued analysis in some fixed point theory the theory of nonsmooth analysis, and generalized Lypunov function method and so on. First of all, in is bounded and does not satisfy the assumption of linear growth does not require the activation function of the neurons, the cycle of CohenGrossberg neural network system with discontinuous activation functions and time delay. The existence of solutions and multiple periodic solutions. Secondly, in the stimulation function is a non monotonic case, study the existence of solutions with Cohen-Grossberg neural network system of discontinuous periodic excitation function and delay, uniqueness and the index type The stability problem is also discussed. At the same time, the output of neural network system solutions according to the problems of convergence in measure. Finally, in is bounded and monotone nondecreasing case does not require the activation function of the neurons, the discontinuous activation functions and time delay Cohen-Grossberg neural network system almost periodic dynamical behavior. Get on with variable delay and distributed delay discontinuous neural network system. These results generalize and improve the previous results. The fourth chapter discusses a class of discontinuous variable delay neural network drive response synchronization system. Tools and methods used in this chapter involves the functional differential contains the theory of nonsmooth analysis theory and generalized Lypunov function method, some inequalities and so on. By using the continuous and discontinuous state feedback controller is obtained with discontinuous incentive function of God The network drive response synchronization index system. This chapter is not continuous incentive function may be non monotonic, superlinear, or even exponential, the results generalize and improve some related results. The fifth chapter is for the delayed BAM neural networks with discontinuous two yuan incentive function expansion at first, through the appropriate definition of Filippov includes definition solution gives its sense of Filippov. And through functional differential inclusion theory, the two yuan is not continuous excitation functions for BAM neural networks Filippov solutions for local and global existence and global dissipativity. Secondly, through the design of discontinuous state a feedback controller, the exponential synchronization of the delayed BAM neural network two yuan discontinuous activation function is obtained. Finally, the application of the set value of topological degree in theoretical analysis, the two yuan is not continuous The existence of periodic solutions of delayed BAM neural network system incentive function. The sixth chapter for the development and management of renewable resources, first proposed a more general continuous harvest strategy, and consider the Lotka-Volterra competition system with the continuous harvesting strategy. Using functional differential inclusion theory, analysis of the fixed point theorem of set-valued analysis, some techniques and methods, this chapter studies the Filippov solution of the local and global existence, the existence of positive periodic solutions. Finally, the correctness and validity through some numerical examples to illustrate our main results. Through the discussion of these issues, on the one hand, deepen and perfect the discontinuous functional differential equations and functional differential inclusion theory to a certain extent; on the other hand, also solve the neural network and analysis, some science and Engineering in the field of biology. The intertemporal problem provides some methods and theoretical support.

【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)博士學(xué)位論文 前1條

1 王佳伏;時(shí)滯微分包含初值問題與穩(wěn)定性的理論及應(yīng)用[D];湖南大學(xué);2009年

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本文編號(hào)：1692263