Optimal control of a non-smooth elliptic PDE with non-linear term acting on the control

This paper continues the investigations from 7 and it is concerned with the derivation of first-order conditions for a control constrained optimization problem governed by a nonsmooth elliptic PDE. The control appears in the state equation as the argument of a regularization of the Heaviside function, while the nonsmoothness is locally Lipschitz-continuous and directionally differentiable. Because of the lack of G\ᵃteaux-differentiability, the application of standard adjoint calculus is excluded. We derive conditions under which a strong stationary optimality system can be established, i. e. , a system that is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. For this, two assumptions are made on the unknown optimizer. These are fulfilled if the non-smoothness is locally convex around its non-differentiable points and if an estimate involving only the given data is true. The presented findings open the door to future research regarding limit optimality systems for non-smooth shape optimization problems 7.