This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion–convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.

本文关注的问题是通过对不可压缩速度场的主动控制来增强对流冷却，用固定扩散对流模型描述。这实质上导致了双线性最优控制问题。给出了最优控制的存在性的严格证明，并推导了一阶最优性条件以使用变分不等式求解控制。此外，建立了二阶充分条件来表征局部极小值。最后，利用有限元方法和非线性迭代方案进行了数值实验，以证明和验证我们的控制设计的有效性。