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We consider an existence theorem for control systems whose state variables for every t are in C , the set of continuous functions varying over a given set I . The dependence of the state variables upon a ε I is 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 ε I , 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

An existence theorem for optimal control systems with state variable inC, and stochastic control problems