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 Gâteaux–Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.

考虑了具有随机乘性噪声和随机对流项的 Cahn-Hilliard 方程。该模型描述了运动流体中发生的等温相分离，并解释了相分离本身和流动诱导过程中出现在微观水平上的随机性。对流术语中对随机分量的要求自然源于应用，因为流体的搅拌过程通常是由机械或磁性装置引起的。解决了状态系统的适定性问题，然后研究了关于速度控制的标准跟踪类型成本的优化。证明了最优控制的存在，并显示了控制到状态映射的 Gâteaux-Fréchet 可微性。最后，分析相应的伴随后向问题，