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Maximum Principle for Some Optimal Control Problems Governed by 2D Nonlocal Cahn–Hillard–Navier–Stokes Equations
Journal of Mathematical Fluid Mechanics ( IF 1.3 ) Pub Date : 2020-06-06 , DOI: 10.1007/s00021-020-00493-8
Tania Biswas , Sheetal Dharmatti , Manil T. Mohan

This work concerns some optimal control problems associated with the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The Cahn–Hilliard–Navier–Stokes model consists of a Navier–Stokes equation governing the fluid velocity field coupled with a convective Cahn–Hilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study an another optimal control problem of finding the unknown optimal initial data. Optimal data initialization problem is also known as the data assimilation problems in meteorology, where determining the correct initial condition is very crucial for the future predictions.

中文翻译:

由二维非局部Cahn-Hillard-Navier-Stokes方程控制的某些最优控制问题的最大原理

这项工作涉及与二维有界区域中的两种等温,不可压缩,不混溶流体的演化有关的一些最佳控制问题。Cahn–Hilliard–Navier–Stokes模型由控制流体速度场的Navier–Stokes方程和对流其中一种流体的相对浓度的Cahn–Hilliard方程组成。分布式最优控制问题被表述为对受控非局部Cahn–Hilliard–Navier–Stokes方程的成本函数最小化。我们通过证明经典的Ekeland变分原理,证明了Pontryagin极大值原理来对该系统进行最优控制,从而建立了最优性的一阶必要条件。最佳控制的特征是使用伴随变量。我们还研究了另一个未知未知最优初始数据的最优控制问题。最佳数据初始化问题在气象学中也称为数据同化问题,其中确定正确的初始条件对于未来的预测非常重要。
更新日期:2020-06-06
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