论文标题
在剪切厚的液体建模中产生的非平滑PDE的最佳控制
Optimal control of a nonsmooth PDE arising in the modeling of shear-thickening fluids
论文作者
论文摘要
本文重点介绍了对最佳控制问题的分析，该问题由非平滑的准部分偏微分方程，该方程对固定的不可压缩剪切厚的液体进行建模。我们首先研究状态方程中非平滑项的定向可区分性，这是证明解决方案操作员的定向可不同性的先前步骤。此后，我们建立了一个原始的一阶必要最佳条件（Bouligand（b）平稳性），该条件是从解决方案操作员的定向可不同性中得出的。通过使用非平滑项的局部正规化并进行渐近分析，我们严格地为局部最小值提供了弱的平稳性系统。通过结合B和弱的平稳性条件，并使用Lagrange乘数的规律性，我们能够获得一个强大的平稳性系统，该系统包括国家和Lagrange乘数之间标量产品的不平等。
This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (Bouligand (B) stationarity), which is derived from the directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a weak stationarity system for local minima. By combining the B- and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a strong stationarity system that includes an inequality for the scalar product between the symmetrized gradient of the state and the Lagrange multiplier.
