右端不连续泛函微分方程研究

发布时间：2018-03-31 20:09

本文选题：泛函微分包含　切入点：Filippov解　出处：《湖南大学》2016年博士论文

【摘要】：据我们所知,在机械工程、力学、神经网络、自动控制以及生物学等领域,右端不连续泛函微分方程是大量存在的.一般地,对右端不连续泛函微分方程而言,由于其右端函数不是连续的,因而经典的泛函微分方程理论体系无法适用.为了分析和研究右端不连续泛函微分方程的解的基本性质及其一些动力学行为,我们先通过应用Filippov微分包含正规化方法,将其转化为一个恰当的泛函微分包含.然后利用该泛函微分包含,给出了右端不连续的泛函微分方程的Filippov意义下解的定义及其在给定的初始条件下的解的定义.在此基础上,并利用泛函微分包含理论,进一步研究了具可变时滞和分布时滞的泛函微分方程的Filippov意义下解的一些基本性质和一些动力学行为.主要的研究内容包括:Filippov意义下解的局部与整体存在性(延拓性)、解轨线的周期(概周期)动力学行为及其稳定性和收敛性行为(例如:全局指数稳定性、同步性、全局耗散性)等等.本文将从以下两个方面展开,一是根据实际的生产及科学实践中出现的一些不连续现象,利用右端不连续泛函微分方程来建立各种数学模型对其进行描述.然后通过Filippov正规化方法,将右端不连续泛函微分方程转化为相应的泛函微分包含.其二是在Filippov泛函微分包含的基本框架内,讨论Filippov意义下解的各种动力学行为.主要研究内容包括:周期解与多个周期解的存在性;周期解与概周期解的存在性和唯一性;Filippov意义下解的各种稳定性及其收敛性.主要研究工具与研究方法包括:集值分析中的一些不动点理论、集值分析中的拓扑度理论、非光滑分析理论、矩阵分析、矩阵测度理论、广义Lyapunov泛函方法等等.本学位论文共分为六章.在第一章中,先简要介绍了右端不连续泛函微分方程与泛函微分包含理论的发展历史及其研究概况.同时,也简单介绍当前不连续神经网络系统和不连续生物系统的研究概况.最后,就本文的主要研究内容与结构安排作了介绍.在第二章中,介绍本文研究所必需的一些基本理论知识.第三章的讨论是针对一类具可变时滞和分布时滞的Cohen-Grossberg神经网络系统展开的,其神经元激励函数是一元不连续函数(分段连续函数).本章所用的工具和方法涉及到泛函微分包含理论,集值分析中的一些不动点理论、非光滑分析理论以及广义Lypunov泛函方法等等.首先,在不要求神经元激励函数是有界的且不满足线性增长假设的情形下,研究了具不连续激励函数和具时滞的CohenGrossberg神经网络系统周期解与多个周期解的存在性问题.其次,在神经元激励函数是非单调的情形下,研究了具不连续激励函数和具时滞的Cohen-Grossberg神经网络系统周期解的存在性、唯一性及其指数型稳定性问题.同时,也讨论了该不连续神经网络系统输出解的依测度收敛性问题.最后,在不要求神经元激励函数是有界的和单调非减的情形下,研究了具不连续激励函数和具时滞的Cohen-Grossberg神经网络系统概周期动力学行为.所获得的关于具可变时滞和分布时滞的不连续神经网络系统的这些研究结果是对已有结果的推广和改进.第四章讨论了一类具可变时滞的不连续神经网络驱动-响应系统的同步性.本章所用的工具和方法涉及到泛函微分包含理论,非光滑分析理论以及广义Lypunov泛函方法,一些不等式技巧等等.利用连续和不连续状态反馈控制器,得出了具不连续激励函数的神经网络驱动-响应系统的指数型同步性.本章的不连续激励函数可能是非单调、超线性的、甚至是指数型的,所得结果推广并改进了一些相关结果.第五章的讨论是针对具二元不连续激励函数的时滞BAM神经网络系统展开的.首先,通过定义恰当的Filippov包含,给出其Filippov意义下解的定义.并通过泛函微分包含理论,研究了具二元不连续激励函数的时滞BAM神经网络系统Filippov解的局部存在性和整体存在性及其全局耗散性.其次,通过设计不连续状态反馈控制器,得到了具二元不连续激励函数的时滞BAM神经网络系统的指数型同步性.最后,应用集值分析中的拓扑度理论,研究了具二元不连续激励函数的时滞BAM神经网络系统的周期解的存在性问题.第六章针对可再生资源的开发与管理,先提出了更为一般的不连续收获策略,并考虑了具该不连续收获策略的Lotka-Volterra竞争系统.利用泛函微分包含理论、集值分析中的不动点定理、一些分析技巧和方法,本章研究了Filippov解的局部存在性和整体存在性,正周期解的存在性.最后,通过一些数值例子来说明我们的主要结果的正确性与有效性.通过对这些问题的探讨,一方面,在一定程度上加深和完善了右端不连续泛函微分方程理论以及泛函微分包含理论;另一方面,也为分析和解决神经网络、生物学等科学与工程领域中的一些实际问题提供了一些方法和理论支持.
[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.

【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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