论文标题
随机对流Cahn-Hilliard方程的分析和最佳速度控制
Analysis and optimal velocity control of a stochastic convective Cahn-Hilliard equation
论文作者
论文摘要
考虑了具有随机乘法噪声和随机对流项的Cahn-Hilliard方程。该模型描述了在移动流体中发生的等温相分离，并解释了相位分离本身和流动诱导过程中显微镜水平出现的随机性。对流项中随机组件的呼吁自然源于应用，因为流体的搅拌程序通常是由机械设备或磁性设备引起的。解决了状态系统的适合性，并研究了相对于速度控制的标准跟踪类型成本的优化。证明了最佳控制的存在，并显示了控制对状态图的Gâteaux-Fréchet可不同性。最后，分析了相应的伴随向后问题，并根据涉及固有的伴随变量的变异不等式来得出一阶必要条件。
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 first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.
