Optimal control of a two-dimensional elliptic equation with exponential nonlinearity and Dirac measure data

TL;DR

This paper develops an optimal control framework for a 2D elliptic PDE with exponential nonlinearity and Dirac measures, addressing ill-posedness and deriving first-order optimality conditions through regularization.

Contribution

It introduces a regularization approach to handle the ill-posed control problem with singular measures and nonlinearities, establishing necessary optimality conditions.

Findings

Existence of a unique very weak solution under certain control thresholds

Regularized problems yield smooth control-to-state operators within specific bounds

Limit analysis provides necessary optimality conditions for the original problem

Abstract

This work addresses an optimal control problem for a semilinear elliptic equation in two-dimensional space, characterized by an exponential nonlinearity and a singular source term. The source is modeled as a finite linear combination of Dirac measures concentrated at a fixed set of distinct points. The control variable is a finite-dimensional vector whose components represent the masses assigned to these point sources. Due to the interplay between the exponential nonlinearity and the singular measure data, the state equation is generally ill-posed and admits a unique very weak solution only when the largest component of the control vector does not surpass a certain critical threshold. Consequently, the control-to-state operator might be continuously differentiable only on an open subset of the control space. To derive first-order optimality conditions for the original problem, we introduce a family of regularized problems by imposing box constraints on the control variables. These constraints are chosen such that the admissible control sets of the regularized problems lie entirely within the open subset where the control-to-state operator is smooth. By analyzing the optimality systems associated with the regularized problems and passing to the limit, we obtain necessary optimality conditions for the original, unregularized problem.