Optimal Control For Stable Solution Of Repairable Human-machine Systems

The repairable systems were studied in this paper, which have common-cause failure, hardware failure and critical human-error. Repairable systems are one of the important systems about reliable theory and the main subject in reliable math-ematic. Firstly, based on the mathematic model established by increasing variable, the Volterra integral equation were applied.To study weak solution of the repairable system,the system equation was transform into a abstract differential equation in Banach space.The operator Generate C0-semigroup and the existence of the weak solution was proved.The weak solution is strong solution,thus the existence and uniqueness of the repairable system was proved. Secondly, it is proved that 0 is the single eigenvalue and the eigenvector is the stable solution of the system. The solution for the system converged its stable solution. And the asymptotic stability of the system was also proved.As perturbation theorem,the solution was studied with index form in some strict condition. Finally,μ(x) was control variable and Banach-Saks-Mazur Theorem was applied.The optimal control of the stable solution is studied by using minimizing sequence and standard of norm index function.This method was important for the repairable system.