【作者】 刘瑞芹；

【导师】 毛军军；

【作者基本信息】 安徽大学 ， 概率论与数理统计， 2011， 硕士

【摘要】 基于粗糙集理论的智能控制需要大量粗糙函数作为其理论基础,单纯粗糙集仍不能满足这些应用需求,所以建立粗糙函数并且对其性质及应用的研究就显得尤为重要,并且本文还对函数粗糙集和粗糙微分方程进行归纳和研究。本文的主要研究内容和创新工作如下：(1)分别对一元粗糙函数和二元粗糙函数的性质进行概括、归纳和研究,得到了一些性质和定理,如一元粗糙函数的介值定理、积分中值定理等；二元粗糙函数的粗极值必要条件定理、中值定理,二重积分化累次积分定理等。另外本文对有些内容进行图示和举例进行说明,以使内容显得具体易懂。(2)探讨粗糙微分方程的性质和粗糙微分方程的解法。通过对粗糙微分方程的进行分析,得到存在唯一性定理,并且结合前面介绍的粗糙函数的粗糙导数和粗糙积分的性质,得到解粗糙微分方程的几种方法。(3)给出函数粗糙集的定义,并且定义了度量函数粗糙集的不确定性和模糊性的几种方法。然后介绍了函数粗糙集的应用理论分析,最后给出函数粗糙集的应用实例。本文的主要创新工作：对二元粗糙函数进行深入的研究,定义二元粗糙函数的粗极值,并且得到粗极值必要条件定理,中值定理,二重积分化累次积分定理等。对粗糙微分方程进行研究,得到粗糙微分方程的存在唯一性定理和解粗糙微分方程的几种方法。更多还原

【Abstract】 Intelligent control based on rough set theory need many rough functions as its basis of theory, but rough set doesn’t satisfy need, so it is important to found rough function and research on its properties and application.In this paper, function rough set and rough differential equation are also generated and researched.Main contents are as follows:(1) The properties of rough function of one variable and two variables separately generated and researched, such as, intermediate value theorem and mean value theorem of integrals of rough function of one variable, necessary condition for rough extreme value theorem, mean value theorem of integrals, changing of double integral into repeated integral rough function of two variables, and so on. In addition, to make some contents concrete and understood easily, these contents are graphed and exemplified.(2) Discussing the properties and solution of rough differential equation. By analyzing rough differential equation, theorem of existence and uniqueness of rough differential equation are arrived at. Connecting the properties of rough derivative and rough integral, some methods of solving rough differential equation are arrived at.(3) Function rough set is presented; some methods of measuring the uncertainty of function rough set are defined. Introducing application theory analysis of function rough set, one application example of function rough set is presented.Creative results are as follows:Researching on rough function of two variables, defining rough extreme value of rough function of two variables, arriving at necessary condition for rough extreme value theorem、mean value theorem of integrals、changing of double integral into repeated integral rough function of two variables, and so on. Researching on rough differential equation, theorem of existence and uniqueness of rough differential equation and some methods of solving rough differential equation are arrived at.更多还原