Optimal Control of Elliptic and Parabolic Quasi-Variational Inequalities

椭圆和抛物型拟变分不等式的最优控制

• 批准号: 314216459
• 负责人: Professor Dr. Michael Hintermüller
• 金额: --
• 依托单位: Department of Mathematical Sciences
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314216459?language=en
• 关键词: Optimal; Control; Elliptic; Parabolic; Quasi

Quasi-variational inequalities (QVIs) often arise in applications where non-smooth and nonlinear phenomena lead to complex state-dependent constraints. They can be used to describe, for instance, the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, or the chemotactic behavior of bacteria in competition. Mathematically, the solutions to these problems are not unique and their dependence on input quantities (data, controls, etc.) is non-smooth.This project is devoted to analyzing and numerically solving optimal control problems associated with elliptic and parabolic QVIs. The research work is organized as follows:(a) It starts with the development of function-space based solution algorithms for QVIs tailored to constraints of obstacle- or gradient-type. In particular, we aim at path-following semi smooth Newton schemes which exhibit fast local mesh-independent convergence.(b) Then it focuses on an enhanced solution theory for the underlying QVIs. More specifically, properties of the minimal and maximal solutions will be studied along with associated (differential) stability and numerical approximation schemes.(c) Then, in a two progressively more demanding research steps, stationary conditions for optimal control problems for the QVIs of interest will be derived. These optimization problems fall into the realm of mathematical programs with equilibrium constraints (MPECs) in function space. In technical terms, in our stationarity considerations two smoothing approaches will be pursued, one utilizing a Moreau-Yosida technique and the the other one relying on a technique modifying the underlying differential operators. (d) Finally, bundle-free implicit programming methods for the numerical solution of the MPEC under consideration are pursued. These also involve relaxation and path-following techniques, and advanced discretization schemes. The analytical as well as numerical advance in the project work will be validated against prototypical applications. These involve in particular the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, and the chemotactic behavior of bacteria in competition.

拟变分不等式（QVI）经常出现在非光滑和非线性现象导致复杂的状态依赖约束的应用中。例如，它们可以用来描述超导体的磁化、扭转中的热塑性效应、颗粒材料的行为或竞争中细菌的趋化行为。从数学上讲，这些问题的解决方案不是唯一的，它们依赖于输入量（数据，控件等）。本计画致力于分析与数值求解与椭圆型与抛物型QVI相关的最佳控制问题。研究工作的组织如下：（a）它开始与开发的功能空间为基础的解决方案算法的QVI定制的障碍或梯度型的约束。特别是，我们的目标是路径跟踪半光滑牛顿计划，表现出快速的局部网格无关收敛。(b)然后，它集中在一个增强的解决方案理论的基础QVI。更具体地说，最小和最大的解决方案的性质将研究沿着与相关的（微分）稳定性和数值逼近方案。(c)然后，在两个逐步要求更高的研究步骤中，将导出感兴趣的QVI的最优控制问题的平稳条件。这些优化问题属于函数空间中具有平衡约束的数学规划（MPEC）的范畴。在技术方面，在我们的平稳性考虑两个平滑的方法将被追求，一个利用Moreau-Yosida技术，另一个依赖于一种技术修改的基础微分算子。(d)最后，无故障隐式编程方法的数值解的MPEC正在考虑追求。这些还涉及松弛和路径跟踪技术，以及先进的离散化方案。项目工作中的分析和数值方面的进展将在原型应用中得到验证。这些特别涉及超导体的磁化、扭转中的热塑性效应、颗粒材料的行为以及细菌在竞争中的趋化行为。