对流冷却的最优控制及其数值实现
Optimal Control of Convection-Cooling and Numerical Implementation
摘要：
本文研究了通过对稳态扩散-对流模型描述的不可压缩速度场进行主动控制来增强对流冷却的问题。这实质上导致了双线性最优控制问题。给出了最优控制存在性的严格证明，并利用变分不等式导出了求解该控制的一阶最优性条件。此外，还建立了刻画局部极小的二阶充分条件。最后，利用有限元方法与非线性迭代格式进行数值实验，以证明和验证我们的控制设计的有效性。
英文摘要：
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.
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