The Exponential Stability And Reliability Analysis Of A Kind Of System

Reparable system is one of the most important systems discussed in the reliability theory, and also the primary subject studied in reliabilty mathematics. This paper mainly investigates the exponential stability of the reparable systems described by generalized Markov models, which are formulated by supplementary variable technique.It is significant for the reliability of the systems. Many scholars have done a lot of work about the well-posedness and asymptotic stability of the systems.However, the exponential stability of systems is no well solved. In this paper,under the reasonable assumption that the mean of the reparable rate exists, we prove the solution of the system,with or without integral boundary, is exponentially stable.Taking a further step, we could consider the numerical simulation and optimal control.In this paper,we first discuss the four-state system, which derived from the single-component repairable system.First of all, we make up the model of the four-state system and deal with it by functional analysis theory,then we make some reasonable supposition.So the system can be written as Abstract Chuchy Problem in Banach space. Then we abstract the operator of the system and prove the operator generates a positive C0-semigroup of contractions by semigroups of linear operators theory. So the dynamic solution of the system has property of the existence and the uniqueness.But the initial value po(0)(?)D(A+B),so the solution is a mild solution.From the established result, we know that the classic solution has the property of the existence and the uniqueness. So it can be sure that the mild solution is the classic solution.What's more,we extract the steady-state solution.We prove that dynamic solution of the system exponentially converges to the steady-state solution by the result that essential spectrum bound does not change under compact perturbation.At last,we discuss the the monotonicity of the instantaneous availability of the single-component repairable system and obtain the condition by which the instantaneous availability can decrease whenμ(x) is a positive step function.And prove that the supremum of the third section ofμ(x)is less than the supremum of the second section ofμ(x).