#### On the Optimal Control of Partially Observed Inventory Systems

**Abstract: **This
Note introduces recent developments in the analysis of inventory systems with
partial observations. The states of these systems are typically
conditional distributions, which evolve in infinite dimensional spaces over
time. Our analysis involves introducing unnormalized probabilities to
transform nonlinear state transition equations to linear ones. With the
linear equations, the existence of the optimal feedback policies are proved for
two models where demand and inventory are partially observed. In a third model
where the current inventory is not observed but a past inventory level is fully
observed, a sufficient statistic is provided to serve as a state. The last
model serves as an example where a partially observed model has a finite
dimensional state. In that model, we also establish the optimality of the
basestock and (*s,S*) policies, hence generalizing the corresponding
classical models with full information.