C-Stationarity for Optimal Control of Static Plasticity with Linear Kinematic Hardening

An optimal control problem is considered for the variational inequality representing the stress-based (dual) formulation of static elastoplasticity. The linear kinematic hardening model and the von Mises yield condition are used. Existence and uniqueness of the plastic multiplier is rigorously proved, which allows for the re-formulation of the forward system using a complementarity condition. In order to derive necessary optimality conditions, a family of regularized optimal control problems is analyzed, wherein the static plasticity problems are replaced by their viscoplastic approximations. By passing to the limit in the optimality conditions for the regularized problems, necessary optimality conditions of C-stationarity type are obtained.