This paper concentrates on the reliability of a discrete-time controlled
Markov system with finite states and actions, and aims to give an efficient
algorithm for obtaining an optimal (control) policy that makes the system have
the maximal reliability for every initial state. After establishing the
existence of an optimal policy, for the computation of optimal policies, we
introduce the concept of an absorbing set of a stationary policy, and find some
characterization and a computational method of the absorbing sets. Using the
largest absorbing set, we build a novel optimality equation (OE), and prove the
uniqueness of a solution of the OE. Furthermore, we provide a policy iteration
algorithm of optimal policies, and prove that an optimal policy and the maximal
reliability can be obtained in a finite number of iterations. Finally, an
example in reliability and maintenance problems is given to illustrate our
results