Finite Elements for optimal control with singular phenomena

Optimization of technological processes plays a more and more important role. Many of such processes are described by partial differential equations. Moreover, additional restrictions have to be fulfilled guaranteeing the stable behavior of such processes. Such restrictions often occur as pointwise inequality constraints in the mathematical model. This project is devoted to the optimal control of systems of elliptic and parabolic differential equations. Problems with all kinds of singularities such as reentrant corners, nonsmooth coefficients, and small parameters are of particular interest. Moreover, pointwise inequality constraints generate additional singularities where the localization is a priori unknown. The project strives for two strategic goals: First, starting from a priori error estimates, families of meshes are generated that ensure optimal approximation rates. Second, a posteriori error estimator are developed working reliable in theory and practice. Such error estimators are the base of adaptive mesh refinement. A new challenge in this project is the incorporation of pointwise inequality constraints in adaptive strategies. An essential aspect is that the boundary between active and inactive parts of the inequality constraints generates new singularities. These singularities are real challenges for theory and numerical methods especially in the case of pointwise state constraints. Both strategies can ensure efficient and reliable numerical results. New and efficient discretization methods for optimal control problems will be developed in this project. Moreover, the project contributes to the ultimative aim of solving practical optimal control problems with given accuracy at low costs.

Optimization of technological processes plays a more and more important role. Many of such processes are described by partial differential equations. Moreover, additional restrictions have to be fulfilled guaranteeing the stable behavior of such processes. Such restrictions often occur as pointwise inequality constraints in the mathematical model. This project is devoted to the optimal control of systems of elliptic and parabolic differential equations. Problems with all kinds of singularities such as reentrant corners, nonsmooth coefficients, and small parameters are of particular interest. Moreover, pointwise inequality constraints generate additional singularities where the localization is a priori unknown. The project strives for two strategic goals: First, starting from a priori error estimates, families of meshes are generated that ensure optimal approximation rates. Second, a posteriori error estimator are developed working reliable in theory and practice. Such error estimators are the base of adaptive mesh refinement. A new challenge in this project is the incorporation of pointwise inequality constraints in adaptive strategies. An essential aspect is that the boundary between active and inactive parts of the inequality constraints generates new singularities. These singularities are real challenges for theory and numerical methods especially in the case of pointwise state constraints. Both strategies can ensure efficient and reliable numerical results. New and efficient discretization methods for optimal control problems will be developed in this project. Moreover, the project contributes to the ultimative aim of solving practical optimal control problems with given accuracy at low costs.