Stochastic maximum principle for optimal control of partial differential equations driven by white noise

We prove a stochastic maximum principle of Pontryagin’s type for the optimal control of a stochastic partial differential equation driven by white noise in the case when the set of control actions is convex. Particular attention is paid to well-posedness of the adjoint backward stochastic differential equation and the regularity properties of its solution with values in infinite-dimensional spaces.