Summary

The aim of shape optimization is to reduce the costs and improve the performance of complex systems, like reducing the drag of aircrafts, cars and boats, increasing the stiffness of plates, reducing the weight of radiators, but also improving the precision in electrical impedance tomography, or tackling some questions in image processing. It could be understood as a part of optimal control theory but the mathematical challenge lies in the fact that the control variable is no longer a set of functions and/or parameters, but the structure or shape of a geometric object.

As it is common in optimal control, the goal functional does not only depend directly on the shape of considered objects, but also on the state of some shape-dependent quantities. These quantities might be the flow velocity, deformation of the elastic body, temperature distribution in the object etc. Typically, mathematical models for such quantities are given by partial differential equations. Therefore, a development of modelling techniques, analytical framework, and numerical methods for partial differential equations will be of the highest importance in this project. However, some phenomena are not well described by partial differential equations, and part of the project will be devoted to the study of nonlocal equations, or a more general framework of pseudodifferential operators, as well as some applications.

An important goal of the project is further development of shape calculus, with particular attention paid to multiple state problems, resolving some robustness questions, and conducting uncertainty analysis. The first and second order optimality conditions in shape optimization problems will be studied in order to achieve more efficient numerical algorithms, but also theoretically in a simplified setting to encompass erratic occurrences of local extrema.