Abstract In this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

中文翻译：

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由二维非局部 Cahn-Hilliard-Navier-Stokes 方程控制的最优控制问题的 Pontryagin 最大值原理和二阶最优性条件

摘要 在本文中，我们制定了与二维有界域中两种等温、不可压缩、不混溶流体的演化相关的分布式最优控制问题。分布式最优控制问题被定义为受受控非局部 Cahn-Hilliard-Navier-Stokes 方程约束的合适成本函数的最小化。我们通过庞特里亚金最小原理描述了最优性的一阶必要条件，并证明了问题的二阶最优性充分必要条件。