可修复人机系统的指数稳定性

Exponential Stability Analysis of Repairable Human-Machine System

研究了两相同部件温储备可修的人机系统,运用C_0半群的相关理论,对系统主算子的谱界进行估值.估算系统的算子产生的半群的增长界,然后运用了共尾的概念及相关的理论,得到了系统算子A+B的谱界与系统算子产生的半群的增长界相同.进而运用相关代数知识证得,0为系统算子的简单本征值,并分析了系统算子的谱分布,得到系统的指数稳定性.并研究了系统算子预解式的特性.对任意给定的δ>0,γ=a+bi,-μ+δ<a_1≤a≤a_2,得到lim|b|→∞‖R(γ;A+B)‖=0.进而得到在~sRγ≥a_1的右半平面内相应于系统算子A+B的谱点由有限个本征值组成.

In this paper,we investigate the asymptotic behavior of a warm standby repairable human-machine system with two identical units.Using Co semigroup theory,we estimate the spectrum bound of the system＇s host operator and the growth bound of semigroup which is produced by the system operator.Then using the concept of cofinal and related theory.We prove that the spectral bound of A ＋ B shares the same value of the growth bound of system operator semigroup.Further more,wo show that 0 is the simple eigenvalue of A ＋ B and analysis the spectral distribution of the system operator,so,we obtain the exponential stability of the system.The Property of operator＇s resolvent is discussed.We randomly giveδ 0,and γ = a ＋ bi,Let＇s fix a1 and a2,which is satisfying to-μ ＋ δ a1 ≤ a ≤ a2,so,we get a conclusion is lim｜b｜→∞｜｜ R（γ;A ＋ B） ｜｜= 0.Consequently,we obtain that in the right place of -sRγ ≥ a1 is composed of the finite isolating eigenvalue corresponding spectrum of operator A ＋ B system.