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 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.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase separation, optimal control, linearized state system, adjoint state system, first-order optimality condition

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