Numerical analysis of state-constrained optimal control problems for PDEs

偏微分方程状态约束最优控制问题的数值分析

• 批准号: 25290113
• 负责人: Professor Dr. Fredi Tröltzsch
• 金额: --
• 依托单位: Fachgebiet Optimierung bei partiellen Differentialgleichungen
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2006
• 资助国家: 德国
• 起止时间: 2005-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/25290113?language=en
• 关键词: Numerical; analysis; state; constrained; optimal

The proposal is a contribution to the optimal control of nonlinear systems of PDEs with pointwise state-constraints. The work is focussed on two aspects of associated numerical methods and their analysis. In a first topic, regularization techniques of Lavrentiev type will be studied to solve state-constrained problems. Exemplarily, special emphasis is placed on semilinear parabolic equations with boundary control and state constraints in the domain. A second part of the project is devoted to the case, where the controls are given by a linear combination of finitely many ansatz functions, where the coefficients are constant or may depend on time. This situation is characteristic for the majority of applications in practice, where coupled systems of nonlinear PDEs model the problem. In many of them, pointwise state constraints are required. This part concentrates on aspects of semi-infinite optimization such as second-order optimality conditions and adapted numerical methods. It is devoted to a class of optimal control problems that so far has been widely disregarded in the numerical analysis, Keywords: Optimal control, partial differential equation, pointwise state constraint, Lavrentiev regularization, semi-infinite optimization

该提案是对具有点状状态约束的PDE的非线性系统的最佳控制的贡献。这项工作的重点是相关数值方法的两个方面及其分析。在第一个主题中，将研究Lavrentiev类型的正则化技术，以解决受国家约束的问题。示例性的是，特别重点放在半连接抛物线方程，并具有边界控制和域中的状态约束。该项目的第二部分专门针对该案例，其中控件是通过有限的许多ANSATZ函数的线性组合给出的，其中系数是恒定或可能取决于时间的。这种情况是实践中大多数应用的特征，在这种情况下，非线性PDES的耦合系统对问题进行了建模。在其中许多情况下，需要点状状态约束。该部分集中在半无限优化的各个方面，例如二阶最佳条件和改编的数值方法。它专门用于一类最佳控制问题，到目前为止，这些问题在数值分析，关键字中已被广泛忽略：最佳控制，部分微分方程，点状状态约束，Lavrentiev正则化，半无限制优化