  We consider a class of infinite-dimensional singular stochastic control
problems. These can be thought of as spatial monotone follower problems and
find applications in spatial models of production and climate transition. Let
$(D,\mathcal{M},\mu)$ be a finite measure space and consider the Hilbert space
$H:=L^2(D,\mathcal{M},\mu; \mathbb{R})$. Let then $X$ be an $H$-valued
stochastic process on a suitable complete probability space, whose evolution is
determined through an SPDE driven by a self-adjoint linear operator
$\mathcal{A}$ and affected by a cylindrical Brownian motion. The evolution of
$X$ is controlled linearly via an $H$-valued control consisting of the
direction and the intensity of action, a real-valued nondecreasing
right-continuous stochastic process, adapted to the underlying filtration. The
goal is to minimize a discounted convex cost-functional over an infinite
time-horizon. By combining properties of semiconcave functions and techniques
from viscosity theory, we first show that the value function of the problem $V$
is a $C^{1,Lip}(H)$-viscosity solution to the corresponding dynamic programming
equation, which here takes the form of a variational inequality with gradient
constraint. Then, by allowing the decision maker to choose only the intensity
of the control and requiring that the given control direction $\hat{n}$ is an
eigenvector of the linear operator $\mathcal{A}$, we establish that the
directional derivative $V_{\hat{n}}$ is of class $C^1(H)$, hence a second-order
smooth-fit principle in the controlled direction holds for $V$. This result is
obtained by exploiting a connection to optimal stopping and combining results
and techniques from convex analysis and viscosity theory.
