A linear quadratic control problem for the stochastic heat equation driven by Q-Wiener processes

We consider a control problem for the stochastic heat equation with Neumann boundary condition, where controls and noise terms are defined inside the domain as well as on the boundary. The noise terms are given by independent Q-Wiener processes. Under some assumptions, we derive necessary and sufficient optimality conditions stochastic controls have to satisfy. Using these optimality conditions, we establish explicit formulas with the result that stochastic optimal controls are given by feedback controls. This is an important conclusion to ensure that the controls are adapted to a certain filtration. Therefore, the state is an adapted process as well.