Numerical analysis of state-constrained optimal control problems for PDEs

该建议是一个贡献的非线性系统的逐点状态约束的偏微分方程的最优控制。工作集中在两个方面的相关数值方法和分析。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域。该项目的第二部分是专门的情况下，在那里的控制是由一个线性组合的1000多个animals功能，其中的系数是常数或可能取决于时间。这种情况是在实践中，耦合系统的非线性偏微分方程模型的问题，大多数应用程序的特点。在许多情况下，逐点状态约束是必需的。这一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。它致力于一类最优控制问题，迄今为止，已被广泛忽视的数值分析，关键词：最优控制，偏微分方程，逐点状态约束，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