Numerical analysis of state-constrained optimal control problems for PDEs

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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