The Stability Of A Repairable System With Commom-Cause Failure

In this paper, we mainly investigate a repairable system with a cold standby unit and common-cause failure which is consisting of two identical parallel redundant active units and a cold standby unit. We mainly study the exponential stability of the system by use of cofinal theory and resolvent positive operator theory. Inspired by the above methods, we transport these theories into studying a transport equation in slab geometry with reflecting boundary condition, and get the corresponding properties of the transport operator.First of all, we obtain the calculus equations of the repairable system and define the domain space as L1 space. Besides, under some reasonable assumption, we transfer the equations into an Abstract Cauchy Problem in Banach space by defining main operator and system operator.Secondly, we discuss the properties of the two defined operators. In this part, we prove that both of the operators are densely defined resolvent positive operators, meanwhile, we estimate the spectral bound of the main operator and get its adjoint operator and the domain of the adjoint operator.Thirdly, we prove that both of the two operators generate positive Co-semigroups and the spectral bound of the main operator is equal to its growth bound, so is the system op-erator. So that, we gain the existence and uniqueness of the nonnegative time-dependence solution of the system. Besides, by calculating the growth bound of the system operator, analysising the number of point spectrum qualitatively and studying the algebraic multiplic-ity of the dominant eigenvalue, we obtain the spectral distribution of the system operator and get the exponential stability of the system using expansion theorem of semigroup.Finally, inspired by the above research, we study a transport equation in slab geometry with reflecting boundary condition. By choosing appropriate domain space and defining corresponding operator, we prove that the transport operator is a densely defined resolvent positive operator and generates a positive Co-semigroup whose spectral bound is equal to its growth bound in L1 space.