  Consider a storage system where the content is driven by a Brownian motion
absent control. At any time, one may increase or decrease the content at a cost
proportional to the amount of adjustment. A decrease of the content takes
effect immediately, while an increase is realized after a fixed lead time
$\lt$. Holding costs are incurred continuously over time and are a convex
function of the content. The objective is to find a control policy that
minimizes the expected present value of the total costs. Due to the positive
lead time for upward adjustments, one needs to keep track of all the
outstanding upward adjustments as well as the actual content at time $t$ as
there may also be downward adjustments during $[t,t+\lt)$, i.e., the state of
the system is a function on $[0,\ell]$. To the best of our knowledge, this is
the first paper to study instantaneous control of stochastic systems in such a
functional setting. We first extend the concept of $L^\natural$-convexity to
function spaces and establish the $L^\natural$-convexity of the optimal cost
function. We then derive various properties of the cost function and identify
the structure of the optimal policy as a state-dependent two-sided reflection
mapping making the minimum amount of adjustment necessary to keep the system
states within a certain region.
