Exponetial Stability Of A Compound Redundant System With Repair

Since the 1950s, reliability theory has played a major role in many fields, such as aviation, aerospace, etc. The project of bombs and one satelite containes a number of reliability theory and its application. Repairable systems are an important class of system in reliability theory, repairable redundant system is the most critical in repairable system, moreover it is the most common type of system in the actual production.A large number of results have been obtained by domestic and foreign scholars, research methods also made a lot of innovation. In the 20st century, the system solution is obtained by the Laplace transform; Since entering the 21st century, the research method changes to choose an appropriate state space and operator semigroup; however, in the method of the stability of the solution have not been resolved very well, especially the prove of the posedness" is complex. In this paper, research the posedness and the stability of a compound redundant system with repair.Firstly, from the perspective of actual production to make reasonable assumptions, and then use linear time-invariant systems, by the appropriate operator and the state space take equations into Abstract Cauchy Problem in Banach space, pave the way for the operator properties discussed below.Secondly, I prove the domain of the main system operators and the system operator are densely defined in the state space; and then estimate the spectrum bound of the system operator, by the estimating process attained the main system operator and the system operator are densely defined resolvent positive operators; by using cofinal theory show the main system operator generate a positive Co semigroup and the growth bound of the semigroup is equal to the spectral bound of the system main operator, so obtain that the spectral bound of the main operator and the minimum of the mean of repair rate functions are opposite. Thus using the same method like before prove the system operator also generates a positive Co semigroup,. so obtain the posedness of the system, where cofinal theory applications greatly simplifies the past proved. Because the relation of the growth bound of the operator and the spectral bound of the operator is the core issue in cybernetics, so it has great significance here.Finally, using the linear system theory obtain the system dynamic solution; then analysis the spectral distribution of the system operator in the complex plane, the complex plane is divided into three parts, successively prove the right part of the Y-axis are regular points of the system operator, zero is a point spectrum of Algebraic number 1 of the system operator, there are only a finite number of isolated point spectrums of the system operator between zero and the minimum of the mean of repair rate functions; then using the theory of functional analysis obtained the best stability of the system that is exponential stability.