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

作　　者：刘东旭 王兰豪 贾瑶 张菁雯 LIU Dong-xu;WANG Lan-hao;JIA Yao;ZHANG Jing-wen(State Key Laboratory of Synthetical Automation for Process Industries,Northeastern University,Shenyang 110819,China;College of Science,Yanbian University,Yanji 133002,China.)

机构地区：[1]东北大学流程工业综合自动化国家重点实验室,辽宁沈阳110819 [2]延边大学理学院,吉林延吉133002

出　　处：《东北大学学报（自然科学版）》2018年第7期954-958,共5页Journal of Northeastern University(Natural Science)

基　　金：国家自然科学基金资助项目(11461074)

摘　　要：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.

关 键 词：C0半群 指数稳定性 本质谱界 紧算子 扰动 

分 类 号：O231.4[理学—运筹学与控制论]