  In this paper, we consider joint drift rate control and impulse control for a
stochastic inventory system under long-run average cost criterion. Assuming the
inventory level must be nonnegative, we prove that a
$\{(0,q^{\star},Q^{\star},S^{\star}),\{\mu^{\star}(x): x\in[0, S^{\star}]\}\}$
policy is an optimal joint control policy, where the impulse control follows
the control band policy $(0,q^{\star},Q^{\star},S^{\star})$, that brings the
inventory level up to $q^{\star}$ once it drops to $0$ and brings it down to
$Q^{\star}$ once it rises to $S^{\star}$, and the drift rate only depends on
the current inventory level and is given by function $\mu^{\star}(x)$ for the
inventory level $x\in[0,S^{\star}]$. The optimality of the
$\{(0,q^{\star},Q^{\star},S^{\star}),\{\mu^{\star}(x): x\in[0,S^{\star}]\}\}$
policy is proven by using a lower bound approach, in which a critical step is
to prove the existence and uniqueness of optimal policy parameters. To prove
the existence and uniqueness, we develop a novel analytical method to solve a
free boundary problem consisting of an ordinary differential equation (ODE) and
several free boundary conditions. Furthermore, we find that the optimal drift
rate $\mu^{\star}(x)$ is firstly increasing and then decreasing as $x$
increases from $0$ to $S^{\star}$ with a turnover point between $Q^{\star}$ and
$S^{\star}$.
