Optimal Control of Elliptic Variational Inequalities

Prof. Dr. Christian Meyer

Abstract

We consider optimal control problem governed by elliptic variational inequalities (VIs). In contrast to classical optimal control problems subject to elliptic partial differential equations (PDEs), these class of problems provides a certain nonsmooth character which complicates their theoretical and numerical treatment. On the other hand there are various important application problems which are modeled by means of VIs. We only mention elastoplastic deformation or piezo electricity. Thus there is a strong demand for specialized techniques to optimize processes modeled by VIs.

Within the short course we will mainly focus on the obstacle problem, which is probably the VI with the simplest structure. We will first discuss the VI itself and show existence, uniqueness, and continuous dependence of its solution on the data. Afterwards we will turn to the optimal control problem and prove existence of optimal solutions. The associated proof is similar to the analysis of classical optimal control problems governed by PDEs. In contrast to this the derivation of first-order necessary conditions is much more delicate, since the solution mapping of xthe VI is not Gateaux- differentiable. Therefore we will apply a regularization approach to establish stationarity condition by a limit analysis. As we will see these conditions are in some cases not the most rigorous optimality conditions that can be proven. The course ends with the derivation of second-order sufficient optimality condition and some numerical examples.

Date and Place

• 07.11.2012: 09:00 - 12:00, TUM MW 2701m (Maschinenwesen)
• 08.11.2012: 14:00 - 17:00, TUM MI 02.04.011
• 09.11.2012: 09:00 - 12:00, TUM MI 03.04.011