Analysis and optimal velocity control of a stochastic convective Cahn-Hilliard equation

A Cahn-Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid's stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved and the G\^ateaux-Fr\'echet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.