具预警状态的单模块可修复系统的指数稳定性

Exponential Stability of Single-Module Repairable System with Warning State

研究了具有预警状态的单模块可修复系统,将其动态变化过程用一组微分方程描述.通过选取适当的状态空间和系统算子的定义域,将方程化为Banach空间中的抽象柯西问题.利用泛函分析和线性算子半群理论证明了系统具有严格占优的单重的0本征值,说明系统的解满足渐近稳定性,并求出其稳态解.随后通过研究系统主算子的谱分布,证明了系统主算子的本质谱界为负.最后讨论了系统主算子在紧扰动下的本质谱界变化情况,结果表明系统的动态解是指数稳定的.

A kind of single-module repairable system with warning state was investigated,which can be expressed as a set of differential equations. By selecting the appropriate state space and the definition domain of the system operator,the equations were transformed into an abstract Cauchy problem in a Banach space. With the functional analysis method and linear operator semigroup theory,the eigenvalue 0 was proved to be the strictly dominant simple eigenvalue of the system. It shows the solution of the system is asymptotically stable. Then by studying the spectrum distribution of the main operator of the system,it was proved that the essential spectrum bound of the main operator of the system is negative. Finally,the variation of essential spectrum bound of the main operator under compact perturbation were discussed. The results demonstrate that the dynamic solution of the system is exponentially stable.