We investigate in this article Pontryagin's maximum principle for a class of control problems associated with a Cahn–Hilliard–Navier–Stokes model in a two dimensional bounded domain. The model consists of the Navier–Stokes equations for the velocity v, coupled with a Cahn–Hilliard model for the order (phase) parameter . We derive Pontryagin's maximum principle for the control problems assuming that a solution exists. Let us note that the coupling between the Navier–Stokes and the Cahn–Hilliard systems makes the analysis of the control problem more involved.

中文翻译：

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具有状态约束的Cahn-Hilliard-Navier-Stokes模型的最优控制的最大原理

我们在本文中研究庞特里亚金关于与二维有界域中的Cahn–Hilliard–Navier–Stokes模型相关的一类控制问题的最大原理。该模型由Navier-Stokes方程为速度v，再加上的Cahn-Hilliard的模型的顺序（相）参数。假设存在解决方案，我们推导庞特里亚金关于控制问题的最大原理。让我们注意到，Navier-Stokes系统与Cahn-Hilliard系统之间的耦合使控制问题的分析更加复杂。