We study a class of infinite-dimensional singular stochastic control problems with

applications in economic theory and finance. The control process linearly affects an abstract

evolution equation on a suitable partially-ordered infinite-dimensional space X, it takes values in

the positive cone of X, and it has right-continuous and nondecreasing paths. We first provide a

rigorous formulation of the problem by properly defining the controlled dynamics and integrals

with respect to the control process. We then exploit the concave structure of our problem

and derive necessary and sufficient first-order conditions for optimality. The latter are finally

exploited in a specification of the model where we find an explicit expression of the optimal

control. The techniques used are those of semigroup theory, vector-valued integration, convex

analysis, and general theory of stochastic processes.