MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract

Sprekels, Jürgen

A deep quench approach to the optimal control of an Allen--Cahn system with dynamic boundary conditions and double-obstacle potentials

We consider optimal distributed and boundary control problems for an Allen--Cahn system with a nonlinear dynamic boundary condition involving the Laplace--Beltrami operator, where both the bulk and surface potentials are of double-obstacle type. Results concerning existence and first-order necessary optimality conditions for optimal controls are derived for this non-differentiable problem involving variational inequalities. The method used is to employ recently established results for the case of (differentiable) potentials of logarithmic type and to perform a so-called "deep quench limit" by approximating the subdifferential of the involved indicator functions by the graphs of logarithmic potentials.