  We study an optimal distributed control problem associated to a stochastic
Cahn-Hilliard equation with a classical double-well potential and Wiener
multiplicative noise, where the control is represented by a source-term in the
definition of the chemical potential. By means of probabilistic and analytical
compactness arguments, existence of an optimal control is proved. Then the
linearized system and the corresponding backward adjoint system are analysed
through monotonicity and compactness arguments, and first-order necessary
conditions for optimality are proved.
