Optimal distributed control of a stochastic Cahn-Hilliard equation

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 a relaxed optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analyzed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.