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

Abstract

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.