An Existence Theorem for Optimal Control Systems with State Variable in C, and Stochastic Control Problems

We consider an existence theorem for controi systems whose state variables for every t are in C, the set of continuous functions varying over a given set L The dependence of the state variables upon a ~ / i s induced by their dependence upon the initial state and the state equation governing the system. In contrast, the control u ---u(t) is taken as a measurable function of t alone. The usual space constraints and boundary conditions are also allowed to vary over a ~ l , and the cost functional is now taken to be a continuous functional over a suitable class of continuous functions. We also discuss an application of these results to control systems with stochastic boundary conditions.