  Motivated by fatigue damage models, this paper addresses optimal control
problems governed by a non-smooth system featuring two non-differentiable
mappings. This consists of a coupling between a doubly non-smooth
history-dependent evolution and an elliptic PDE. After proving the directional
differentiability of the associated solution mapping, an optimality system
which is stronger than the one obtained by classical smoothening procedures is
derived. If one of the non-differentiable mappings becomes smooth, the
optimality conditions are of strong stationary type, i.e., equivalent to the
primal necessary optimality condition.
