It is demonstrated that if P is a continuous optimal control problem whose system of differential equations is linear in the control and the state variables, and whose control and state variable constraint sets are convex, a direct method of determining an optimal solution of P exists. It is demonstrated that such a “continuous” problem can be replaced by a sequence of finite-dimensional, “discrete” optimization problems in which the control and state variable constraints are treated directly. The approximation obtained relates the respective optimal solutions.