  Conditions are established under which the optimal control of processes
having both absolutely continuous and singular (with respect to time) controls
are equivalent to linear programs over a space of measures on the state and
control spaces. This paper considers long-term average and discounted criteria
and includes budget and resource constraints. The linear programs optimize over
measures representing the expected occupation measure of the state and
absolutely continuous control processes and a similar expected occupation
measure of the state and control when the singular action of the process
occurs. The evolution of these processes is characterized through an adjoint
equation which the measures must satisfy in relation to the absolutely
continuous and singular generators of the process. Existence of optimal relaxed
controls of feedback type are established in general while existence of an
optimal form of strict control is proven under additional closedness and
compactness conditions.
