Static contact problems in the regime of finite strain elasticity are an important class of mechanical problems with nonlinear, non-smooth behaviour. Finite strains occur if the considered materials are soft, for example for rubber or biological soft tissue. Aim of this project is the development and analysis of algorithms for the optimal control of these problems.

We construct and analyse a regularization and homotopy approach. The non-penetration condition of the contact problem and the local injectivity condition of elasticity are relaxed. Properties of the regularized problem and convergence of the homotopy are studied.

The resulting regularized optimal control problems will still be nonconvex and will be solved by a function space oriented composite step method. This method will exploit problem structure of finite strain, in particular the variational structure of elasticity and the group structure of deformations. To finally approximate solutions of the original problem we will develop and analyse an affine invariant path-following method tailored for this class of problems.