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Second-Order Necessary Optimality Conditions for a Class of Optimal Control Problems Governed by Partial Differential Equations with Pure State Constraints

  • B. T. Kien
  • V. H. Nhu
  • A. Rösch
Article

Abstract

Based on some tools of variation analysis, we deal with first- and second-order necessary optimality conditions for a class of optimal control problems governed by semilinear elliptic equations and stationary Navier–Stokes equations with pure state constraints. To do this, we first derive optimality conditions for an abstract optimal control problem and then apply the obtained results to derive second-order necessary optimality conditions for semilinear elliptic optimal control problems as well as optimal control problems governed by stationary Navier–Stokes equations.

Keywords

Second-order necessary optimality condition Regularity condition Semilinear elliptic equation Stationary Navier–Stokes equation Pure state constraint 

Mathematic Subject Classification

49K20 35J25 

Notes

Acknowledgments

The authors wish to express their sincere thanks to the anonymous referees for their helpful suggestions and comments which improved the original manuscript greatly. The first author would like to thank the Alexander von Humboldt Foundation and the University of Duisburg-Essen for financial support and hospitality.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Optimization and Control TheoryInstitute of Mathematics, Vietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Faculty of Information TechnologyNational Institute of Education ManagementHanoiVietnam
  3. 3.Faculty of MathematicsUniversity of Duisburg-EssenEssenGermany

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