文章研究具有小修和一般型更换策略的多状态退化 系统,该系统由$2n+1$\;个带有边界条件的微积分方程表示.运 用$C_0$-半群理论及算子理论证明该系统非负动态解的存在唯一性.并通过对 系统算子谱特征的研究证明该系统的动态解强收敛于其稳态解.进一步利用共尾及 豫解正算子的相关理论证明该系统的指数稳定性.

In this paper, we investigate a multi-state deteriorating system with minimal repairs and the replacement policy in general distribution, which is described by 2n+1 differential-integral equations with boundary conditions. By using the $C_0$-semigroup theory and the operator theory, we prove the existence of a unique positive dynamic solution. And then, by analyzing the spectral characteristics of the system operator, we prove the dynamic solution of the system converges strongly to its steady-state solution. Furthermore, by using the theory of cofinal and resolvent positive operator, we prove the exponential stability of the system.