Inventory control problem

The inventory control problem is a type of problem encountered within the field of optimal control. It exists in several variations, and is typically formulated as follows:

A store has, at time $k$, $x_{k}$ items in stock. It then orders (and receives) $u_{k}$ items, and sells $w_{k}$ items, where $w$ follows a given probability distribution. Thus

$x_{k+1}=x_{k}+u_{k}-w_{k}$.

The store has costs that are related to the number of items in store and the number of items ordered:

$c_{k}=c(x_{k},u_{k})$.

The store wants to select $u_{k}$ in an optimal way, i.e. to minimize

$\sum _{k=0}^{\infty }c_{k}$.