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

Vu Huu Nhu, Nguyen Hai Son

Published 2023-11-27Version 1

We prove a novel optimality condition in terms of Bouligand generalized differentials for a local minimizer of optimal control problems governed by a nonsmooth semilinear elliptic partial differential equation 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. We also establish the optimality conditions in the form of multiplier existence. There, in addition to the existence of the adjoint state and of the nonnegative multipliers associated with the 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 the optimal control problems without distributed and boundary pointwise constraints to derive the so-called \emph{strong} stationarity conditions, 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.