Optimization and Optimal Control

Group Kunisch

The research of this group focuses on optimal control of infinite dimensional systems typically with constraints given by partial differential equations. The mathematical analysis of particular problems as well as their numerical treatment are investigated. Open loop optimal control problems typically involve large systems of coupled partial differential equations. Recently significant progress has been made in their solution based on both, improved concepts from optimisation theory (SQP-methods and semi-smooth Newton methods in function space) and numerical analysis (hierarchical methods, adaptivity). These new techniques now allow the control of problems in fluid dynamics which were unsolvable only a few years ago. Future work will encompass the control of coupled systems (control by magnetic fields, fluid-structure interaction), control of quantum mechanical systems, as well as closed-loop control. The latter requires to consider the Hamilton-Jacobi-Bellman equation which is still untractable for systems of practical size. Therefore model reduction techniques, including proper orthogonal systems, balanced truncation and the center manifold theorem (nonlinear Galerkin methods) will be investigated. These methods are important also in their own right, for instance for real-time optimal control.