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\(L^\infty \)-Stability of a Parametric Optimal Control Problem Governed by Semilinear Elliptic Equations

Abstract

This paper studies local stability of a parametric optimal control problem governed by semilinear elliptic equations with mixed pointwise constraints. We show that if the unperturbed problem satisfies the strictly nonnegative second-order optimality conditions, then the solution map is upper Hölder continuous in \(L^\infty \)-norm of control variable.

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Acknowledgements

The research of the first author and the second author was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2019.308. This work was also supported by the Ministry of Science and Technology, Taiwan (grant No. MOST 108-2115-M-037-001). The authors would like to thank the anonymous referees for their comments and suggestions which improve the paper greatly.

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Correspondence to Ching-Feng Wen.

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Kien, B.T., Tuan, N.Q., Wen, CF. et al. \(L^\infty \)-Stability of a Parametric Optimal Control Problem Governed by Semilinear Elliptic Equations. Appl Math Optim 84, 849–876 (2021). https://doi.org/10.1007/s00245-020-09664-5

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Keywords

  • Solution stability
  • Locally Hölder upper continuity
  • Optimality condition
  • Second-order sufficient optimality condition

Mathematics Subject Classification

  • 49K15
  • 90C29