OPTIMAL CONTROLS FOR STOCHASTIC PARTIAL-DIFFERENTIAL EQUATIONS

This paper deals with control problems for a state process governed by controlled linear stochastic partial differential equations, whose drift and diffusion coefficients are the second order elliptic and the first order differential operators, respectively. A relaxed system is introduced as a generalization of admissible control and the continuous dependence of state process on a relaxed system, assuming some regularity conditions, is proved. Appealing to the usual compactification method, this continuity result derives the existence of an optimal relaxed system and, under convexity condition of coefficients, an optimal relaxed system provides an optimal admissible control in a wider sense. A relaxed control can be approximated by an admissible control which is Brownian adapted and where Bellman principle holds. As an application, stochastic control of diffusions with partial observation, where the state noise and the observation noise may not be independent, is discussed.