Exponential Stability Analysis Of A Four-State Repairable System

Repairable system is one of the most important systems discussed in the reliability math-ematics theory. Many scholars, both domestic and abroad, have done a lot of work about this kind of questions, they have gave proves to the existence and uniqueness of the solu-tion,asymptotic stability,the eigenvalue and the reliability of the system. In this paper, we'll prove the exponential stability of the system under certain condition.The reliability of the complex system we studied in this article passes through a gradual deterioration until at some critical level, the system fails completely. In this article, the existence and uniqueness of the system solution was firstly obtained by transforming the system into Volterra operator equation under Banach space. We also show that the system operator generates a positive C0 semigroup in Banach space, and the steady-state solution of system is exist and asymptotic. Secondly, the existence of strictly dominant eigenvalue is studied by applying functional analysis and, particularly, the theory of linear operators in Banach space. Through analyzing the variation of the essential spectral radius of semigroups before and after perturbation, it is shown that the dynamic solution of the system converges to the steady-state solution of the system exponentially under certain condition. At last, it is proved that the system is reliable, that is, the instantaneous reliability is more than or equal to its stable reliability.