  We consider an inventory system in which inventory level fluctuates as a
Brownian motion in the absence of control. The inventory continuously
accumulates cost at a rate that is a general convex function of the inventory
level, which can be negative when there is a backlog. At any time, the
inventory level can be adjusted by a positive or negative amount, which incurs
a fixed cost and a proportional cost. The challenge is to find an adjustment
policy that balances the holding cost and adjustment cost to minimize the
long-run average cost. When both upward and downward fixed costs are positive,
our model is an impulse control problem. When both fixed costs are zero, our
model is a singular or instantaneous control problem. For the impulse control
problem, we prove that a four-parameter control band policy is optimal among
all feasible policies. For the singular control problem, we prove that a
two-parameter control band policy is optimal.
  We use a lower-bound approach, widely known as "the verification theorem", to
prove the optimality of a control band policy for both the impulse and singular
control problems. Our major contribution is to prove the existence of a
"smooth" solution to the free boundary problem under some mild assumptions on
the holding cost function. The existence proof leads naturally to numerical
algorithms to compute the optimal control band parameters. We demonstrate that
the lower-bound approach also works for Brownian inventory model in which no
inventory backlog is allowed. In a companion paper, we will show how the
lower-bound approach can be adapted to study a Brownian inventory model under a
discounted cost criterion.
