Keywords: model reduction, inertial manifold, proper orthogonal decomposition, adaptivity, semi-smooth Newton
Information: The central goal of this project is the advance of analytical and numerical methods for large scale optimal control problems in the context of partial differential equations. It contains method driven and problem driven components alike. Control of partial differential equations has experienced tremendous advances over the last two decades. To have a strong impact on science it must contribute, with increasing vigour, to real world problems. In this respect one of the aims of OPTIM is to strongly collaborate with HEART on deriving optimal, open loop, controls for defibrillation of arrhythmias in the heart.
OPTIM will also derive suboptimal closed loop controls, based on receding horizon techniques. At a later stage closed loop strategies will be based on the Hamilton Jacobi Bellman equation. New methodologies for treating the large scale aspect will be based on model reduction methods. For the first time an analysis of the use of approximate inertial manifold techniques in the context of optimal control will be provided. As an alternative to obtain reduced order models, the method of proper orthogonal decomposition, which was co-initiated by the authors of the proposal, will be significantly advanced by incorporating adaptivity issues related to basis updates, and snapshot location. Semi-smooth Newton methods and related path-following techniques, necessary in cases of low Lagrange multiplier regularity will be investigated for mixed control-state constraints and in the context of time optimal control.