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
TL;DR
The paper develops a Bouligand-differentiability framework for an optimal control problem governed by a nonsmooth semilinear elliptic PDE with distributed and boundary unilateral constraints, where the nonlinearity coefficient $d$ is nondifferentiable at a break point $\bar t$. By constructing two Bouligand generalized derivatives $G_{-}^{u,v}$ and $G_{+}^{u,v}$ from left and right differentiability sequences, it derives a novel multiplier-based optimality system in which nonnegative multipliers associated with the nondifferentiability have supports on the level set $\{y=\bar t\}$. Under a constraint qualification, the authors obtain strong stationarity, with adjoint states and sign conditions on the adjoint on the level set, and show this strong stationarity is equivalent to the purely primal first-order condition. The approach avoids regularization and relies directly on Bouligand subdifferentials, providing a principled way to handle nondifferentiability in the control-to-state map for PDE-constrained optimization and yielding a rigorous link between strong and C-stationarity notions in this nonsmooth setting.
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.