On the Existence of Optimal Relaxed Controls of Stochastic Partial Differential Equations

Abstract

This paper is concerned with control problem of systems governed by stochastic partial differential equations, the drift and diffusion terms of which are second- and first-order differential operators, respectively. The existence of an optimal relaxed control is studied in both cases where the systems are degenerate and nondegenerate. It is shown that the higher regularity conditions on the initial state, as required in the existing results, can be dispensed with if the Wiener process is one-dimensional. Some special cases of multidimensional Wiener process are also discussed, which in particular leads to an improvement of a recent result of Bensoussan and Nisio. The method is based on an analysis of the group generated by the first-order differential operator. As an application, an existence theorem of the optimal relaxed control is proved for partially observed diffusions with correlation between the controlled states and the observation noises.