The Conditions Of Morital Context Ring To Be (Pincipally) Quasi-Baer Rings

Reparable system is one of the most important systems discussed in the reliability theory, and also the primary subject studied in reliability mathematics. Many scholars have done a lot of work about the well-posedness and asymptotic stability for the system. In this paper, we prove the exponential stability for this system in theory.The parallel repairable system with two parallel units and one warm standby is studied in this paper. To begin with, according to Laplace transformation, we set the system with supplement variable technique, and in the Banach space X, we describe it as an abstract Cauchy problem. The, we show that 0 is a simple eigenvalue, and we discuss the existence of the steady-state solution of the system, which is the eigenvector of 0. After this, we proof the existence and uniqueness of the nonnegative solution of the system is obtained by use of functional analysis and semi-group theory. And by the method of strong continuous semi-group, the paper analyzed the restriction of essential spectral growth bound of the system operator. The essential spectral radius of the system operator is also discussed before and after perturbation. The results show that the dynamic solution of the system is exponential stability and tends to the steady solution of the system.