Dynamic analysis of one-unit repairable systems with imperfect repairs

This paper conducts a dynamic analysis of a one-unit repairable system with two different failure modes and imperfect repairs. Compared to classical models with single failure modes and perfect repair mechanisms, the proposed system more accurately reflects real-world maintenance scenarios. By employing the $ C_0 $-semigroup theory of linear operators, we prove the existence and uniqueness of a non-negative time-dependent solution (T-DS) for the system. Furthermore, we investigate the asymptotic behavior of the T-DS, demonstrate its exponential convergence to the steady-state solution (S-SS), and also derive explicit asymptotic expressions for the T-DS. Numerical examples illustrate how key parameters influence transient reliability metrics and their convergence characteristics. This study offers theoretical insights into the dynamics of repairable systems with complex failures and imperfect repair mechanisms, while providing practical guidelines for optimizing system design and reliability assessment.