可修复模型的系统分析
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摘要
可修复系统是可靠性理论及可靠性数学的主要研究对象.由于可修复系统模型的系统分析涉及可修复系统数学模型构建的各个环节和各种因素,涉及包括系统工程、可靠性理论、随机过程理论、微分方程理论、线性算子半群理论等多种学科的理论与方法的综合运用,涉及包括数值计算、数值模拟等计算机技术的应用,因此,从某种意义上讲,可修复系统模型的系统分析过程是一项系统工程,必须根据总体协调的需要,将它们有机地结合起来并加以研究.为此,本文基于实际研究背景的需求和对从定性到定量综合集成方法的认识,以及对可修复系统模型研究现状和存在问题的思考,提出了运用从定性到定量综合集成方法,开展可修复模型的系统分析研究课题.
围绕上述研究课题,本文首先分析了可修复系统模型的研究现状及存在问题,介绍了可修复系统的相关概念和相关理论,并在此基础上提出了基于综合集成思想的可修复系统模型的系统分析方法；其次介绍一类特殊的线性算子半群—C0半群的相关内容,它们在讨论可修复系统模型的适定性和渐近性等方面发挥重要作用,是研究可修复系统模型的数学工具和理论基础；此后利用随机过程理论和增补变量方法,建立了一类两不同部件并联可修复系统的数学模型,并根据系统分析的需要,将系统模型转换成为卷积型Volterra积分方程以及Banach空间上的抽象Cauchy问题.针对可修复系统的数学模型,本文利用Volterra积分方程理论和C0半群理论,讨论了系统非负解的存在性和唯一性、系统解的适定性和渐近稳定性,并在一定附加条件下证明了系统解的指数稳定性；与此同时,利用微分方程理论和Banach空间的相关理论,研究了系统的可靠性、可控性、最优控制和系统模型的半离散化等问题,并利用数值计算和数值模拟的方法对本文所得部分理论结果进行了数值实验.
本文的研究目的是通过对两不同部件并联可修复系统模型的实际研究,寻求一种行之有效的可修复系统模型的系统分析方法,并为实现系统的、总体的、科学的研究可修复系统尤其是复杂可修复系统提供可资借鉴的案例.
本文的研究意义是为可修复系统模型的研究和应用提供了新的方法,既有利于丰富可修复系统的理论,又有利于促进可靠性理论及可靠性数学的应用,并有利于为军事、航空、航天、计算机技术等生产经营决策提供科学依据和技术支持.
本文的主要创新之处包括：
(1)应用从定性到定量的综合集成方法,对一类两不同部件并联可修复系统模型进行系统分析,以期丰富和完善可修复系统的理论和方法,并为实现系统的、总体的、科学的研究可修复系统提供工具.
(2)利用泛函分析理论,对一类具有耗散特征的线性算子进行深入研究,以期不断得到新的生成定理,并为研究可修复系统提供理论基础.
(3)利用数值计算的方法,对可修复系统模型进行数值模拟,以期获得系统的动态可靠度、可用度、可维护度等可靠性数量指标,并为进一步开展可靠性分析提供依据.
Repairable system is the main research object of reliability theory and reliability mathematics. Because of the system analysis of repairable system model involving all kinds of links and factors of the established mathematical model, including the com-prehensive utilization of the theories and methods of many disciplines such as systems engineering, reliability theory, the theory of stochastic process, the theory of differen-tial equation, the linear operator semigroup theory, etc., touching upon the application of computer technology such as numerical calculation, numerical simulation and so on, therefore, in a sense, the system analysis process of repairable system model is a systems engineering, must be based on the needs of overall coordination, to combine them or-ganically and study them separately. For this reason, based on the needs of the actual research background and the cognitions of the meta synthesis approach from qualitative to quantitative, and the thinking of the current researches and existent problems of re-pairable system model, proposed the application of the meta synthesis approach from qualitative to quantitative, developed the system analysis research topic of repairable system model.
Around the above research subject, this paper firstly analyzed the current researches and existent problems of repairable system model, introduced the related concepts and theories of repairable system, and on this basis, the system analysis method of repairable system model which uses meta synthesis was put forward; Secondly, introduced the re-lated contents of a special kind of linear operator semigroup-C0 semigroup, these contents play an important role on the discussions of the well-posedness and asymptotic property of the repairable system model, are the mathematical tools and theoretical foundations of studying the repairable system model; Afterwards, by the theory of stochastic process and supplemented variables, a parallel repairable system with two non-identical unit was established, and converted the system model into convolution Volterra integral equation and abstract Cauchy problem in Banach space according to the needs of system anal-ysis. Be aimed at the repairable system model, using the theory of Volterra integral equation and C0 semigroup, the existence and uniqueness of the nonnegative solution of the system, the well-posedness and asymptotic property of the solution of the system were discussed, and proved the exponential asymptotic property of the solution of the system under some additional conditions; In the meantime, the problems of reliability, controllability, optimal control and the semi-discretization of the system model by using the theory of differential equation and Banach space were studied, and according to the methods of numerical calculation and the numerical simulation, some conclusions drawn from this thesis have been proved.
This dissertation is intended to study the actual research of a parallel repairable system with two non-identical unit, try to find out a effective way of the system analysis for the repairable system model, and so as to provide reference for realizing the systemic, overall, scientific research on the repairable system especially the complex repairable system.
The meaning of this dissertation shows that it can provide a new method for the research and the application of the repairable system model, it would not only benefits to enrich the repairable system theory, but also to promote the application of reliability theory and reliability mathematics, and to provide the scientific evidence and technical support for the production and management decision-making on military, aviation, space-flight, computer technology.
The major contributions and innovations of this dissertation are as follows:
(1) Utilizing the meta synthesis approach from qualitative to quantitative, so as to analyze a parallel repairable system model with two non-indentical units systematically, to wish can be rich and perfect the theory and method of repairable system, and in order to provide implementation for realizing the systemic, overall, scientific research on the repairable system.
(2) Using the theory of functional analysis, to study deeply about linear operator with dissipative characteristics, in order to get new generation theorems constantly, and provide theoretical basis for studying the repairable system.
(3) By the method of numerical calculation, so as to carry on numerical simulation for repairable system model, in order to obtain the quantity index of reliability such as the reliability, availability and maintainability of system dynamics and so on, and to provide the evidence for further developing the reliability analysis.
围绕上述研究课题,本文首先分析了可修复系统模型的研究现状及存在问题,介绍了可修复系统的相关概念和相关理论,并在此基础上提出了基于综合集成思想的可修复系统模型的系统分析方法；其次介绍一类特殊的线性算子半群—C0半群的相关内容,它们在讨论可修复系统模型的适定性和渐近性等方面发挥重要作用,是研究可修复系统模型的数学工具和理论基础；此后利用随机过程理论和增补变量方法,建立了一类两不同部件并联可修复系统的数学模型,并根据系统分析的需要,将系统模型转换成为卷积型Volterra积分方程以及Banach空间上的抽象Cauchy问题.针对可修复系统的数学模型,本文利用Volterra积分方程理论和C0半群理论,讨论了系统非负解的存在性和唯一性、系统解的适定性和渐近稳定性,并在一定附加条件下证明了系统解的指数稳定性；与此同时,利用微分方程理论和Banach空间的相关理论,研究了系统的可靠性、可控性、最优控制和系统模型的半离散化等问题,并利用数值计算和数值模拟的方法对本文所得部分理论结果进行了数值实验.
本文的研究目的是通过对两不同部件并联可修复系统模型的实际研究,寻求一种行之有效的可修复系统模型的系统分析方法,并为实现系统的、总体的、科学的研究可修复系统尤其是复杂可修复系统提供可资借鉴的案例.
本文的研究意义是为可修复系统模型的研究和应用提供了新的方法,既有利于丰富可修复系统的理论,又有利于促进可靠性理论及可靠性数学的应用,并有利于为军事、航空、航天、计算机技术等生产经营决策提供科学依据和技术支持.
本文的主要创新之处包括：
(1)应用从定性到定量的综合集成方法,对一类两不同部件并联可修复系统模型进行系统分析,以期丰富和完善可修复系统的理论和方法,并为实现系统的、总体的、科学的研究可修复系统提供工具.
(2)利用泛函分析理论,对一类具有耗散特征的线性算子进行深入研究,以期不断得到新的生成定理,并为研究可修复系统提供理论基础.
(3)利用数值计算的方法,对可修复系统模型进行数值模拟,以期获得系统的动态可靠度、可用度、可维护度等可靠性数量指标,并为进一步开展可靠性分析提供依据.
Repairable system is the main research object of reliability theory and reliability mathematics. Because of the system analysis of repairable system model involving all kinds of links and factors of the established mathematical model, including the com-prehensive utilization of the theories and methods of many disciplines such as systems engineering, reliability theory, the theory of stochastic process, the theory of differen-tial equation, the linear operator semigroup theory, etc., touching upon the application of computer technology such as numerical calculation, numerical simulation and so on, therefore, in a sense, the system analysis process of repairable system model is a systems engineering, must be based on the needs of overall coordination, to combine them or-ganically and study them separately. For this reason, based on the needs of the actual research background and the cognitions of the meta synthesis approach from qualitative to quantitative, and the thinking of the current researches and existent problems of re-pairable system model, proposed the application of the meta synthesis approach from qualitative to quantitative, developed the system analysis research topic of repairable system model.
Around the above research subject, this paper firstly analyzed the current researches and existent problems of repairable system model, introduced the related concepts and theories of repairable system, and on this basis, the system analysis method of repairable system model which uses meta synthesis was put forward; Secondly, introduced the re-lated contents of a special kind of linear operator semigroup-C0 semigroup, these contents play an important role on the discussions of the well-posedness and asymptotic property of the repairable system model, are the mathematical tools and theoretical foundations of studying the repairable system model; Afterwards, by the theory of stochastic process and supplemented variables, a parallel repairable system with two non-identical unit was established, and converted the system model into convolution Volterra integral equation and abstract Cauchy problem in Banach space according to the needs of system anal-ysis. Be aimed at the repairable system model, using the theory of Volterra integral equation and C0 semigroup, the existence and uniqueness of the nonnegative solution of the system, the well-posedness and asymptotic property of the solution of the system were discussed, and proved the exponential asymptotic property of the solution of the system under some additional conditions; In the meantime, the problems of reliability, controllability, optimal control and the semi-discretization of the system model by using the theory of differential equation and Banach space were studied, and according to the methods of numerical calculation and the numerical simulation, some conclusions drawn from this thesis have been proved.
This dissertation is intended to study the actual research of a parallel repairable system with two non-identical unit, try to find out a effective way of the system analysis for the repairable system model, and so as to provide reference for realizing the systemic, overall, scientific research on the repairable system especially the complex repairable system.
The meaning of this dissertation shows that it can provide a new method for the research and the application of the repairable system model, it would not only benefits to enrich the repairable system theory, but also to promote the application of reliability theory and reliability mathematics, and to provide the scientific evidence and technical support for the production and management decision-making on military, aviation, space-flight, computer technology.
The major contributions and innovations of this dissertation are as follows:
(1) Utilizing the meta synthesis approach from qualitative to quantitative, so as to analyze a parallel repairable system model with two non-indentical units systematically, to wish can be rich and perfect the theory and method of repairable system, and in order to provide implementation for realizing the systemic, overall, scientific research on the repairable system.
(2) Using the theory of functional analysis, to study deeply about linear operator with dissipative characteristics, in order to get new generation theorems constantly, and provide theoretical basis for studying the repairable system.
(3) By the method of numerical calculation, so as to carry on numerical simulation for repairable system model, in order to obtain the quantity index of reliability such as the reliability, availability and maintainability of system dynamics and so on, and to provide the evidence for further developing the reliability analysis.
引文
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