  We study a class of stochastic control problems where a cost of the form
\begin{equation}\mathbb{E}\int_{[0,\infty)}e^{-\beta s}[\ell(X_s)
ds+h(Y^{\circ}_s) d|Y|_s]\end{equation} is to be minimized over control
processes $Y$ whose increments take values in a cone $\mathbb{Y}$ of
$\mathbb{R}^p$, keeping the state process $X=x+B+GY$ in a cone $\mathbb{X}$ of
$\mathbb{R}^k$, $k\le p$. Here, $x\in\mathbb{X}$, $B$ is a Brownian motion with
drift $b$ and covariance $\Sigma$, $G$ is a fixed matrix, and $Y^{\circ}$ is
the Radon--Nikodym derivative $dY/d|Y|$. Let $\mathcal{L}=-(1/2)trace(\Sigma
D^2)-b\cdot D$ where $D$ denotes the gradient. Solutions to the corresponding
dynamic programming PDE,
\begin{equation}[(\mathcal{L}+\beta)f-\ell]\vee\sup_{y\in\mathbb{Y}:|Gy|=1
}[-Gy\cdot Df-h(y)]=0,\end{equation} on $\mathbb{X}^o$ are considered with a
polynomial growth condition and are required to be supersolution up to the
boundary (corresponding to a ``state constraint'' boundary condition on
$\partial\mathbb{X}$). Under suitable conditions on the problem data, including
continuity and nonnegativity of $\ell$ and $h$, and polynomial growth of
$\ell$, our main result is the unique viscosity-sense solvability of the PDE by
the control problem's value function in appropriate classes of functions. In
some cases where uniqueness generally fails to hold in the class of functions
that grow at most polynomially (e.g., when $h=0$), our methods provide
uniqueness within the class of functions that, in addition, have compact level
sets. The results are new even in the following special cases: (1) The
one-dimensional case $k=p=1$, $\mathbb{X}=\mathbb{Y}=\mathbb{R}_+$; (2) The
first-order case $\Sigma=0$; (3) The case where $\ell$ and $h$ are linear. The
proofs combine probabilistic arguments and viscosity solution methods. Our
framework covers a wide range of diffusion control problems that arise from
queueing networks in heavy traffic.
