Optimal control of a two-dimensional elliptic equation with exponential nonlinearity and Dirac measure data
Vu Huu Nhu
TL;DR
This work analyzes an optimal control problem for a two-dimensional semilinear elliptic equation with exponential nonlinearity and Dirac measure data. The authors establish that weak solvability of the state equation requires the Dirac masses to stay below a critical threshold $4\pi$ and construct a regularization-based path to first-order optimality conditions, yielding an adjoint system and pointwise optimality relations at the Dirac locations. They prove the control-to-state map is $C^1$ on suitable open sets where the threshold is respected, derive a reduced objective with a tractable gradient, and handle irregular (critical) cases through a limiting regularization procedure. The analysis hinges on exponential-integrability estimates for Poisson equations with measure data and careful handling of the exponential Nemytskii operator, supported by Green function bounds and openness results for the constraint sets. The results provide a rigorous framework for sparse actuator placement in PDEs with exponential nonlinearities and measure data, with potential applications in sensing and control of mean-field-type models.
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.