arXiv subjects

Nguyen Hai Son

Publications and source records attributed to Nguyen Hai Son.

4 recordsLinked to original sources

arXiv2026

Generalized Differentiability and Second-Order Necessary Optimality Conditions for an Elliptic Optimal Control Problem with Exponential Nonlinearity and Discrete Measures

This paper deals with generalized differentiability and second-order necessary optimality conditions for a box-constrained optimal control problem governed by an exponential semilinear elliptic equation with discrete measures as sources, where the control belongs to the space of absolutely summable sequences. The presence of the exponential nonlinearity and discrete measures makes the analysis particularly challenging. In particular, the control-to-state operator may fail to be directionally differentiable. To address this issue, we first establish finite-dimensional directional differentiability of the control-to-state operator; that is, the operator is directionally differentiable along directions contained in finite-dimensional subspaces of the control space. We then introduce a notion of generalized derivative defined as the limit of the associated finite-dimensional directional derivatives as the dimension of these subspaces tends to infinity. Based on this concept, together with estimates for first- and second-order Taylor-type expansions of the exponential Nemytskii operator associated with the control-to-state mapping, we derive first- and second-order generalized differentiability of the reduced objective functional. This leads to first- and second-order necessary optimality conditions for the optimal control problem.

math.OC↗

arXiv2025

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

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.

math.OC↗

arXiv2022

Locally Hölder continuity of the solution map to a boundary control problem with finite mixed control-state constraints

The local stability of the solution map to a parametric boundary control problem governed by semilinear elliptic equations with finite mixed pointwise constraints is considered in this paper. We prove that the solution map is locally Hölder continuous in $L^\infty-$norm of control variable when the strictly nonnegative second-order optimality conditions are satisfied for the unperturbed problem.

math.OC↗

arXiv2020

Upper semicontinuity of the solution map to a parametric boundary optimal control problem with unbounded constraint sets

We would like to study the solution stability of a parametric control problem governed by semilinear elliptic equations with a mixed state-control constraint, where the cost function is nonconvex and the admissible set is unbounded. The main goal of this paper is to give some sufficient conditions under which the solution map is upper semicontinuous and continuous with respect to parameters.

math.OC↗