# An Optimal Control Problem in Polyconvex Elasticity

Anton Schiela^{*}, Lars Lubkoll and Martin Weiser

We consider an implant shape design problem arising in the context of facial surgery. The aim is to find the shape of an implant that deforms the soft tissue of the skin in a desired way. Assuming sufficient regularity, we introduce a reformulation as an optimal control problem where the control acts as a boundary force. The solution of that problem can be used to recover the implant shape from the optimal state.\newline For a simplified problem, in the case where the state can be modelled as a minimizer of a polyconvex hyperelastic energy functional, we show existence of optimal solutions.

Mathematics Subject Classification: 49J20

Keywords: polyconvex elasticity; implant design; optimal control

Minisymposion: Stability, Sensitivity and Error Analysis for Optimal Control Problems