The Stability Of Warning Repairable System And Application Analysis

In recent years, reliability theory has been widely applied and play a vital role in many field, such as production, life, military, and scientific research. Therefore, whether in practice or in theory, the study of reliability and its related problems has important significance.In reliability theory, the study of repairable system is an important part of it. Since the last century fifties and sixties, many scholars have obtained a large number of results for various repairable system, made lots of innovation in regard to researchmethods, and even took great achievement. Earlier, the reliability problems of repairable hardware system whose maintenance follows general distribution, was solved by introducing a supplementary variable method. After that, the existence and uniqueness, asymptotic behavior as well as the relevant issues of the system solution is gived with the theory of semigroups.For repairable systems, however, the exponential stability of conclusion is not only less, but also rare. In addition, another major research direction of repairable systema is optimal problem, and the optimal maintenance rate of research regarding the system has wide prospect. Besides, as a useful tool for system analysis, system semi-discretization of the repairable system is greatly convenient especially complex repairable system research and processing, and has the vital significance on the study of repairable system.Based on the above reasons, we discuss a repairable system with warning device and repairman vacation as a research object. By using Co semigroup theory and functional analysis methods, we discuss the exponential stability of the solution of the system. The optimal control problem is studied with Banach space related theory. Finally using semigroup theory and the theory of integral equation,we analysis the semi-discrete problem in the system.The full text is divided into six chapters, as follows:In chapter 1, research ongoings, existed Problems, significance of the theory and practices, main contents and methods in this dissertation are briefly introduced.In chapter 2, the system model of interest is established. By the probability analysis method, the system is first transformed into a group of integro-differential equations.In chapter 3, the preliminary knowledge required are gived. They play a key role in the exponential stability of the system, optimization, and even semi-discretization.In chapter 4, we discuss the exponential stability of the solution of the system. By constructing special operators and using Co semigroup theory and functional anal-ysis methods, we prove that the system operator generates a positive compression Co semigroup. And then the exponential stability is obtained with the analysis of quasi-compactness.In chapter 5, The optimal control problem is studied with Banach space related theory. That is, the optimal steady-state availability, and the optimal maintenance rate which makes the system availability of maximum and minimum system cost.In chapter 6, we analyze the semi-discrete problem of system problem. Through converting the system model into the Volterra integral equation and selecting the ap-propriate step function approximation method of repair rate, the semi-discrete model is obtained. Then combining semigroup theory and integral equation theory, we prove the rationality of the semi-discrete method.