分数阶微分系统解的存在唯一性、稳定性和可容许性问题

发布时间：2018-03-23 13:22

  本文选题：分数阶微积分　切入点：退化系统　出处：《安徽大学》2017年硕士论文

【摘要】：随着现代社会科技和工业的快速发展和需求,不断提高数学模型的精确度和真实性成为最为重要的课题之一。事实上,大多实际系统的动态过程本质上是分数阶的,比如:电磁波传导系统、粘弹性系统、流体力学系统、生物系统等。对于这些系统而言,要想对其进行更准确地分析和综合,就必须用分数阶微分系统来描述。与此同时,人们也发现时滞和退化也是实际系统的普遍现象。因此,研究含有时滞和退化的分数阶微分系统具有重要的理论价值和现实意义。全文共分为五章。第一章为绪论,对本文的研究背景进行了简要的介绍。第二章为预备知识,本章主要介绍了分数阶微积分的基本概念和退化系统理论的基础知识,并且给出了本文所需的常用引理。第三章讨论了分数阶阻尼系统的解的存在唯一性,给出了分数阶阻尼系统解的存在唯一的充分条件。第四章讨论了分数阶退化时滞系统的有限时间稳定性,给出了分数阶退化时滞微分系统的有限时间稳定的充分条件。第五章讨论了分数阶退化系统的可容许性,给出了阶数为1到2的分数阶退化系统可容许的充分条件。
[Abstract]:With the rapid development and demand of science, technology and industry in modern society, increasing the accuracy and authenticity of mathematical models has become one of the most important topics. For example, electromagnetic wave conduction systems, viscoelastic systems, hydrodynamic systems, biological systems, etc. For these systems to be more accurately analyzed and synthesized, they must be described by fractional differential systems. It is also found that delay and degeneracy are common phenomena in practical systems. Therefore, it is of great theoretical value and practical significance to study fractional differential systems with delay and degenerate. The paper is divided into five chapters. In the second chapter, the basic concepts of fractional calculus and the basic knowledge of degenerate system theory are introduced. In chapter 3, we discuss the existence and uniqueness of the solution of the fractional damping system. Sufficient conditions for the existence and uniqueness of solutions for fractional damped systems are given. In chapter 4, the finite time stability of fractional degenerate time-delay systems is discussed. The sufficient conditions for finite time stability of fractional degenerate delay differential systems are given. In chapter 5, the admissibility of fractional degenerate systems is discussed, and the sufficient conditions for the admissibility of fractional degenerate systems of order 1 to 2 are given.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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