The paper is concerned with the optimal control of a one-dimensional stationary diffusion process on a compact interval. The drift and diffusion coefficients depend upon a stationary control assumed to be a piece-wise continuous function of the state. The costs generated by the process are functions of both the control and the sample path of the process. Mandls concept of a controlled diffusion process is generalized by allowing the controls to be vector-valued with the set of admissible control actions defined by a piecewise continuous set-valued function on the state space. Both single and multi-person problems are considered. The main results include necessary and sufficient conditions for a control to be optimal and conditions assuring the existence of a piecewise continuous optimal control. Applications are given to problems of controlling reservoirs, pollution, queues, investments, welfare, and warfare.