Stochastic Maximum Principle for Optimal Control of a Class of Nonlinear SPDEs with Dissipative Drift

We prove a version of the stochastic maximum principle, in the sense of Pontryagin, for the finite horizon optimal control of a stochastic partial differential equation driven by an infinite-dimensional additive noise. In particular, we treat the case in which the nonlinear term is of Nemytskii type, dissipative, and with polynomial growth. The performance functional to be optimized is fairly general and may depend on point evaluation of the controlled equation. The results can be applied to a large class of nonlinear parabolic equations such as reaction-diffusion equations.