Optimality conditions in terms of Bouligand generalized differentials for a nonsmooth semilinear elliptic optimal control problem with distributed and boundary control pointwise constraints

Abstract: This paper is concerned with an optimal control problem governed by nonsmooth semilinear elliptic partial differential equations with both distributed and boundary unilateral pointwise control constraints, in which the nonlinear coefficient in the state equation is not differentiable at one point. Therefore, the Bouligand subdifferential of this nonsmooth coefficient in every point consists of one or two elements that will be used to construct the two associated Bouligand generalized derivatives of the control-to-state operator in any admissible control. These Bouligand generalized derivatives appear in a novel optimality condition, which extends the purely primal optimality condition saying that the directional derivative of the reduced objective functional in admissible directions in nonnegative. We then establish the optimality conditions in the form of multiplier existence. There, in addition to the existence of the adjoint states and of the nonnegative multipliers associated with the unilateral pointwise constraints as usual, other nonnegative multipliers exist and correspond to the nondifferentiability of the control-to-state mapping. The latter type of optimality conditions shall be applied to an optimal control satisfying the so-called \emph{constraint qualification} to derive a \emph{strong} stationarity, where the sign of the associated adjoint state does not vary on the level set of the corresponding optimal state at the value of nondifferentiability. Finally, this strong stationarity is also shown to be equivalent to the purely primal optimality condition.