On the necessary conditions of optimal controls for stochastic partial differential equations

This paper concerns optimal control of systems governed by stochastic partial differential equations in which drift and diffusion terms are second- and first-order differential operators, respectively. Necessary conditions for an optimal control are derived for both nondegenerate and degenerate systems, and all the coefficients appearing in the equations are allowed to depend on the control variables. Furthermore, the results obtained in the paper can also be used to derive necessary conditions of optimality for partially observed diffusions with correlation between the signals and the observation noises.