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This paper deals with a control constrained optimization problem governed by a nonsmooth elliptic PDE in the presence of a non-differentiable objective. The nonsmooth non-linearity in the state equation is locally Lipschitz continuous and directionally differentiable, while one of the nonsmooth terms appearing in the objective is convex. Since these mappings are not necessarily Gâteaux-differentiable, the application of standard adjoint calculus is excluded. Based on their limited differentiability properties, we derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. [ABSTRACT FROM AUTHOR]

This paper examines optimal control problems governed by elliptic variational inequalities of the second kind with bounded and unbounded operators. To tackle the bounded case, we employ the polyhedricity of the test set appearing in the dual formulation of the governing variational inequality. Based thereon, we are able to prove the directional differentiability of the associated solution operator, which leads to a strong stationary optimality system. The second part of the paper deals with the unbounded case. Due to the non-smoothness of the variational inequality and the unboundedness of the governing elliptic operator, the directional differentiability of the solution operator becomes difficult to handle. Our strategy is to apply the Yosida approximation to the unbounded operator, while the non-smoothness of the variational inequality is still preserved. Based on the developed strong stationary result for the bounded case, we are able to derive optimality conditions for the unbounded case by passing to the limit in the Yosida approximation. Finally, we apply the developed results to Maxwell-type variational inequalities arising in superconductivity. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem governed by a nonsmooth, semilinear parabolic PDE. The nonlinearity in the state equation is only directionally differentiable, locally Lipschitz continuous and is allowed to have infinitely many non-differentiable points. By employing its limited properties, Bouligand-differentiability of the control-to-state map is shown (in an extended sense). This enables us to establish second-order sufficient optimality conditions. We provide concrete settings where these reduce to the first-order necessary optimality condition. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with an optimal control problem governed by a nonsmooth coupled system of equations. The nonsmooth nonlinearity is Lipschitz continuous and directionally differentiable, but not Gâteaux differentiable. We derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. The abstract result is then applied to prove strong stationarity for optimal control of a coupled system consisting of a viscous evolutionary variational inequality (EVI) and an elliptic PDE. To this end, we show that EVIs with viscosity can be formulated as nonsmooth ODEs in Hilbert space in a general setting. The nonsmooth nonlinearity appearing in the ODE turns out to be the solution operator of an elliptic variational inequality, for which we can give an explicit formula. [ABSTRACT FROM AUTHOR]

In this note, we consider an optimal control problem governed by an elliptic variational inequality (VI) of the second kind involving an unbounded operator. We present our main ideas for deriving optimality conditions. [ABSTRACT FROM AUTHOR]