Optimal Control of a Viscous Two-Field Damage Model with Fatigue

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