Exponential Stability of Availability Analysis of System with Two Types of Repair Facilities

摘要

This paper presents a mathematical model of repairable systems. Failed systems repair times are arbitrarily distributed. Two types of repair facilities are needed to repair a failed system. First, we obtain the unique existence of the non-negative time-dependent solution of the system by using the Co-semigroup theory of linear operators in Banach space. Second, we prove the asymptotic stability of solutions of this system. {γ ∈ c|Reγ > 0, or γ = ia, a ∈ R, a ≠ 0} belongs to the resolvent set. 0 is an eigenvalue of the adjoint operator of the operator corresponding to the original system with algebraic multiplicity one. Last, we show that the time-dependent solution of the original system converges exponentially to its steady-state solution by studying the essential spectral radius of the system operator before and after perturbation. 【Keywords】 Co-semigroup; resolvent set; eigenvalue; disturbance; exponential stability