Fixed-Point Properties of the Bellman Operator in Discounted Stochastic Maintenance Optimization

This paper investigates an infinite-horizon discounted stochastic maintenance optimization problem within the framework of dynamic programming. The system degradation is modeled by a discrete-time stochastic process affected by maintenance actions and random disturbances, while the objective is to minimize the expected discounted maintenance and degradation costs. The analysis is carried out on the Banach space of continuous functions equipped with the supremum norm. It is proved that the associated Bellman operator is well defined, maps the function space into itself, and is a contraction with contraction modulus equal to the discount factor. Consequently, the existence and uniqueness of the optimal value function follow from the Banach Fixed Point Theorem, and the convergence of value iteration is established. In addition, rigorous a priori and a posteriori error estimates are derived, providing theoretical stopping criteria for numerical computation. The theoretical results are complemented by numerical experiments illustrating the optimal stationary maintenance policy, the stability of the computed solution under grid refinement, and the influence of the discount factor on both the optimal policy and the convergence rate of value iteration.