The Exponential Stability Analysis Of A Repairable System With Two Types Of Repair Facilities

Reliability system study began as early as the 1950's and reliability technology has been used extensively in electronics systems design, power systems, mechanical systems design, aerospace and military combat problems. In recent years, as the development of science and technology, electronproductions and network are used everywhere. So the reliability analysis of the system becomes more and more important. Reparable system is one of the most important systems discussed in the reliability theory, and also the primary subject studied in reliabilty mathematics. This paper mainly investigates the exponential stability of the reparable systems described by generalized Markov models, which are formulated by supplementary variable technique. It is significant for the reliability of the systems. Many scholars have done a lot of work about the well-posedness and asymptotic stability of the systems. However, the exponential stability of systems is no well solved.In this paper, a repairable system with two types of repair facilities is studied. By the method of functional analysis and strong continuous semi-group, to begin with, the group of integro-differential equations can be written as an abstract Cauchy problem in the Banach Space X. Moreover, the existence and uniqueness of the nonnegative solution of the system is obtained by use of the Volterra operator equation in a Banach Space and elementary techniques.And we proved the existence and uniqueness of the nonnegative mild solution of the system, then, we get the strong solution is the mild solution. Finally, the paper proved the existence of strictly dominant eigenvalue. The essential spectral bound of the system operator is also discussed before and after perturbation. Under certain condition the results show that the dynamic solution of the system is exponential stability and tends to the steady solution of the system. And this paper presents reliability analysis based on a class of repairable system model, and we prove that p0(t) is monotone decreasing when the repair rateμ(x) under certain condition, as a result, we obtain the system steady state solution is credibility.