Optimal distributed control of a stochastic Cahn-Hilliard equation. (English) Zbl 1425.35090
Summary: 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.
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 80A22 | Stefan problems, phase changes, etc. |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
Keywords:
stochastic Cahn-Hilliard equation; phase separation; optimal control; linearized state system; adjoint state system; first-order optimality conditionsReferences:
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