Scarpa Luca
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Department of Mathematics, Politecnico di Milano, Via E. Bonardi 9, 20133 Milan, Italy.
J Nonlinear Sci. 2021;31(2):45. doi: 10.1007/s00332-021-09702-8. Epub 2021 Apr 3.
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.
考虑一个带有随机乘性噪声和随机对流项的Cahn-Hilliard方程。该模型描述了在运动流体中发生的等温相分离，并考虑了在相分离本身以及流动诱导过程中微观层面出现的随机性。对流项中需要随机成分自然源于应用，因为流体的搅拌过程通常是由机械或磁性装置引起的。研究了状态系统的适定性，然后研究了关于速度控制的标准跟踪型代价的优化问题。证明了最优控制的存在性，并展示了控制到状态映射的Gâteaux-Fréchet可微性。最后，分析了相应的伴随反向问题，并根据涉及内在伴随变量的变分不等式导出了最优性的一阶必要条件。
