Error estimates for halfspace regularization of state constrained multiobjective elliptic control problems

Abstract

This paper generalizes the halfspace regularization given in (Jadamba et al. in Syst Control Lett 61(6):707–713, 2012) from scalar to multiobjective optimization for an abstract linearly constrained multiobjective optimization problem. We establish quantitative error estimates for the regularization error in terms of the Hausdorff distance of the efficient set of the original problem to the regularized efficient set. We apply these results to state-constrained multiobjective optimal control problems. In particular, we consider two different situations: distributed and finite-dimensional controls, and for each one, we define two different halfspace regularization schemes given in terms of a family of triangulations of the domain. This leads to two regularized problems corresponding to finitely many pointwise and integral constraints. By applying the abstract results, we get regularization error estimates for the four control problems under general conditions.

Keywords:

• Multiobjective optimization
• elliptic optimal control problems
• pointwise state constraints
• set-valued analysis
• stability estimates

2010 Mathematics Subject Classifications:

• 90C29
• 49J53
• 49J20

Acknowledgments

We are grateful to the reviewers for the careful reading that brought substantial improvements to our manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación (Spain) [project number PID2020-112491GB-I00 / AEI / 10.13039/501100011033]. Miguel Sama is also supported by ETSI Industriales, UNED [grant number 2021-MAT11]. Akhtar A. Khan is supported by the FEAD-2022-23 research grant.