Abstract: In some inventory control contexts, such as Vendor Managed Inventories, inventory with spoilage, misplacement, or theft, inventory levels may not always be observable to the decision makers. However, when shortages occur, inventory levels receive more attention and they may become completely observed. We study such an inventory control context where the unmet demand is lost and orders must be decided on the basis of partial information to minimize the total discounted costs over a infinite horizon. This problem has an infinite-dimensional state space, and for it we establish the existence of a feedback policy when one period costs are bounded or when the discount factor is sufficiently small. We provide an approximately optimal feedback policy that uses a finite state representation.