Exponential Stability of a Multi-state System with Two Repair Distributions

The well-possedness and exponential stability of the solution to a multi-state system with two repair distributions were discussed by applying the C₀ semigroups theory of function analysis. First, the integro-differential equations described the system was transformed into some abstract Cauchy problem on some suitable Banach space. Second, by Hill-Yosida Theorem, the system operator generates a positive contractive C₀ semigroups and the result of well-possedness of the system was obtained. Last, the spectral distribution of the system operator was analyzed. It was proven that 0 is strictly dominant eigenvalue of the system operator and the result of the dynamic solution of the system exponentially converges to the steady one was got since the spectral growth bound is less than zero.