An a posteriori error analysis of an elliptic optimal control problem in measure space
Provides a theoretical error analysis for a specific class of optimal control problems, but the contribution is incremental as it extends existing a posteriori techniques to measure-valued controls.
The paper develops an a posteriori error estimator for sparse optimal control problems where the control is a Borel measure, discretized via Dirac measures. The estimator is proven locally efficient and reliable in 2D and 3D, with numerical validation.
We propose an a posteriori error estimator for a sparse optimal control problem: the control variable lies in the space of regular Borel measures. We consider a solution technique that relies on the discretization of the control variable as a linear combination of Dirac measures. The proposed a posteriori error estimator can be decomposed into the sum of two contributions: an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the $L^2$-norm that accounts for the approximation of the state equation. We prove that the designed error estimator is locally efficient and we explore its reliability properties. The analysis is valid for two and three-dimensional domains. We illustrate the theory with numerical examples.