In this work, we model a single-item system, composed a unique unit degrading over time. This constitutes a preliminary work, which will later lead to a more detailled study extended to multi-items systems. In the context of \textit{condition-based maintenance} (CBM), our system is remotely monitored, providing a valuable additional information to the decision-maker in order to finely adapt the maintenance policy.

However, most of the studies in CBM tackling this maintenance optimization program, usually divided between models with continuous-time or discrete-time \cite{elwany2011} processes, make the assumption of perfect sensors. The monitoring performance is an aspect that has not been extensively studied. Nevertheless, in a large variety of applications, such assumption is too strong, either for technical reasons or because of the prohibitive cost of high-performance sensors. Strongly confirmed by our industrial partners involved in the chair Risks and Resilience of Complex Systems, industrial systems will in general be monitored by imperfect sensors, capable of capturing only partially the exact degradation state of the system.