Regularity of multipliers in second-order optimality conditions for semilinear elliptic control problems

Abstract

A class of semilinear elliptic optimal control problems with mixed pointwise constraints is considered. Some criteria under which Lagrange multipliers are functions in $L^p$-spaces are given. Based on the criteria, we establish first- and second-order necessary optimality conditions of KKT-type as well as second-order sufficient optimality conditions for the problem.

Keywords:

• Regularity of multipliers
• second-order optimality conditions
• semilinear elliptic optimal control problems

2010 Mathematics Subject Classifications:

• 49K15
• 90C29

Acknowledgments

The research of the second author was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2019.308. The research of the third author was supported by the Feng Tay project of National Yunlin University of Science and Technology, Taiwan. The research of the fourth author was supported by the Ministry of Science and Technology, Taiwan under grant number 108-2115-M-037-001. The authors would like to thank the anonymous referees for their suggestions and comments which improved the paper greatly.

Disclosure statement

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

Additional information

Funding

The research of the second author was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.01-2019.308]. The research of the third author was supported by the Feng Tay project of National Yunlin University of Science and Technology, Taiwan. The research of the fourth author was supported by the Ministry of Science and Technology, Taiwan [grant number 108-2115-M-037-001].