Numerical analysis of state-constrained optimal control problems for PDEs

该方法对具有点态约束的非线性偏微分方程系统的最优控制有一定的贡献。工作集中在两个方面的相关数值方法和他们的分析。在第一个主题中，将研究Lavrentiev型正则化技术来解决状态约束问题。举例来说，特别强调的是半线性抛物方程的边界控制和状态约束的领域。项目的第二部分致力于这种情况，其中控制是由有限多个ansatz函数的线性组合给出的，其中系数是常数或可能取决于时间。这种情况是大多数实际应用的特点，其中非线性偏微分方程的耦合系统对问题进行了建模。在许多情况下，需要点态约束。这部分集中在半无限优化方面，如二阶最优性条件和适应的数值方法。本文研究了一类迄今为止在数值分析中被广泛忽视的最优控制问题。关键词：最优控制，偏微分方程，点态约束，Lavrentiev正则化，半无限优化

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