Optimal Control of Elliptic and Parabolic Quasi-Variational Inequalities

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Quasi-variational inequalities (QVIs) often arise in applications where non-smooth and nonlinear phenomena lead to complex state-dependent constraints. They can be used to describe, for instance, the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, or the chemotactic behavior of bacteria in competition. Mathematically, the solutions to these problems are not unique and their dependence on input quantities (data, controls, etc.) is non-smooth.This project is devoted to analyzing and numerically solving optimal control problems associated with elliptic and parabolic QVIs. The research work is organized as follows:(a) It starts with the development of function-space based solution algorithms for QVIs tailored to constraints of obstacle- or gradient-type. In particular, we aim at path-following semi smooth Newton schemes which exhibit fast local mesh-independent convergence.(b) Then it focuses on an enhanced solution theory for the underlying QVIs. More specifically, properties of the minimal and maximal solutions will be studied along with associated (differential) stability and numerical approximation schemes.(c) Then, in a two progressively more demanding research steps, stationary conditions for optimal control problems for the QVIs of interest will be derived. These optimization problems fall into the realm of mathematical programs with equilibrium constraints (MPECs) in function space. In technical terms, in our stationarity considerations two smoothing approaches will be pursued, one utilizing a Moreau-Yosida technique and the the other one relying on a technique modifying the underlying differential operators. (d) Finally, bundle-free implicit programming methods for the numerical solution of the MPEC under consideration are pursued. These also involve relaxation and path-following techniques, and advanced discretization schemes. The analytical as well as numerical advance in the project work will be validated against prototypical applications. These involve in particular the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, and the chemotactic behavior of bacteria in competition.