分数阶时滞微分方程的稳定性与控制

发布时间：2018-08-24 10:41

【摘要】：本文介绍了分数阶时滞微分方程的渐近稳定性及其控制。鉴于对时滞的分类,本文编排成三个主要的部分：Laplace变换对分数阶微分方程的可行性,单时滞分数阶微分方程的稳定性理论及应用,多时滞分数阶微分方程的稳定性理论及应用。论文的第一部分绪论介绍了课题的背景及研究意义,及分数阶微分方程的的基本定义,和分数阶微分方程稳定性理论发展历史及研究现状。第二部分讲述了分数阶时滞微分方程Laplace变换的可行性。根据Laplace变换的理论,通过一系列的分析,主要是包括对分数阶时滞微分方程的解的估计分析和分数阶导数的Laplace变换的分析,得到了一类可以进行Laplace变换的分数阶微分方程,并用线性分数阶微分方程的解来佐证该理论的正确性。文章的第三章写的是分数阶单时滞微分方程的(全局)渐近稳定性。通过与线性的分数阶模型对比,得出了非线性单时滞分数阶微分方程稳定性判定的依据,并将这些结论分别运用到复杂网络模型中。同时,数值模拟的结果验证了理论的正确性。文章的第四章写的是分数阶多时滞微分方程的(全局)渐近稳定性。首先给出了两个Mittag-Leffler函数是非负函数的结论,随后,给出了两个不同类型分数阶多时滞微分方程的比较定理,最后得出了非线性多时滞分数阶微分方程稳定性判定的依据,并将这些结论分别运用到神经细胞网络模型中。同时,数值模拟的结果验证了理论的正确性。
[Abstract]:In this paper, the asymptotic stability and control of fractional delay differential equations are introduced. In view of the classification of delay, this paper is organized into three main parts: Laplace transform, the feasibility of fractional differential equation, the stability theory of fractional differential equation with single delay and its application. Stability Theory and Application of Fractional differential equations with multiple delays. The first part of the thesis introduces the background and significance of the research, the basic definition of fractional differential equation, and the history and research status of stability theory of fractional differential equation. In the second part, the feasibility of Laplace transform for fractional delay differential equations is discussed. According to the theory of Laplace transform, through a series of analyses, including the estimation of the solution of fractional delay differential equation and the analysis of Laplace transform of fractional derivative, a kind of fractional differential equation which can be transformed by Laplace is obtained. The solution of linear fractional differential equation is used to prove the correctness of the theory. The third chapter of this paper deals with the (global) asymptotic stability of fractional order single delay differential equations. By comparing with the linear fractional order model, the basis for judging the stability of nonlinear fractional differential equation with single delay is obtained, and these conclusions are applied to the complex network model. At the same time, the numerical simulation results verify the correctness of the theory. The fourth chapter of this paper deals with the (global) asymptotic stability of fractional order differential equations with multiple delays. In this paper, we first give the conclusion that two Mittag-Leffler functions are nonnegative, then we give the comparison theorems of two different types of fractional multi-delay differential equations, finally, we obtain the basis for judging the stability of nonlinear multi-delay fractional differential equations. These conclusions are applied to neural network models. At the same time, the numerical simulation results verify the correctness of the theory.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O231

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