We analyze a distributed optimal control problem where the state equation is governed by the coupling of the two-dimensional Cahn–Hilliard and Oberbeck–Boussinesq systems modelling incompressible viscous two-phase flows with convective heat transfer. Pointwise constraints are imposed on the controls that act as external sources in the fluid and convection–diffusion equations. The objective functional is of tracking-type that consists of a weighted energy of the difference between the state and a desired target. We establish the existence of optimal controls, the differentiability of the control-to-state operator, and the necessary and sufficient optimality conditions. For initial and target data with finite energy norms, limited space–time regularity of the adjoint states arises due to convection and surface tension.

我们分析了分布最优控制问题，其中状态方程由二维Cahn-Hilliard和Oberbeck-Boussinesq系统的耦合控制，该系统建模了对流换热的不可压缩粘性两相流。在流体和对流扩散方程中，作为控制源的控制施加了点向约束。客观功能是跟踪型的，它由状态和所需目标之间的差的加权能量组成。我们建立了最优控制的存在，状态控制器的可微性以及必要和充分的最优性条件。对于具有有限能量范数的初始数据和目标数据，由于对流和表面张力，伴随状态的时空规则性有限。