论文标题
对Ericksen的耗散溶液的近似和最佳控制 - 莱斯利系统
Approximation and optimal control of dissipative solutions to the Ericksen--Leslie system
论文作者
论文摘要
我们分析了配备了Oseen的Ericksen-LESLIE系统，该系统在三个空间维度中。最近，作者介绍了耗散解决方案的概念。与早期引入的测量值解决方案相比，这些解决方案显示出几个优点。在本文中，我们认为耗散解决方案可以通过相对简单的方案在数值上近似，该方案在每个步骤中都可以满足导演的规范限制。我们介绍了一个半分化方案，并得出了该方案解决方案的相对能量不等式的近似版本。在半差异化中，我们达到了耗散解决方案。此外，我们引入了最佳控制方案，显示了最佳控制的存在和可能的近似策略。我们证明，对于此近似值的收敛性，成本函数的半连续性较低，并认为在存在额外规律性的解决方案的情况下，实现了最佳控制。
We analyze the Ericksen--Leslie system equipped with the Oseen--Frank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measure-valued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relative simple scheme, which fulfills the norm-restriction on the director in every step. We introduce a semi-discrete scheme and derive an approximated version of the relative-energy inequality for solutions of this scheme. Passing to the limit in the semi-discretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, show the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semi-continuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity.
