  In this paper, we first introduce a new spatial-temporal interaction operator
to describe the space-time dependent phenomena. Then we consider the stochastic
optimal control of a new system governed by a stochastic partial differential
equation with the spatial-temporal interaction operator. To solve such a
stochastic optimal control problem, we derive an adjoint backward stochastic
partial differential equation with spatial-temporal dependence by defining a
Hamiltonian functional, and give both the sufficient and necessary
(Pontryagin-Bismut-Bensoussan type) maximum principles. Moreover, the existence
and uniqueness of solutions are proved for the corresponding adjoint backward
stochastic partial differential equations. Finally, our results are applied to
study the population growth problems with the space-time dependent phenomena.

Optimization and Control