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First- and Second-Order Necessary Optimality Conditions for Optimal Control Problems Governed by Stationary Navier–Stokes Equations with Pure State Constraints

Vietnam Journal of Mathematics volume 44, pages 103–131 (2016)Cite this article

Abstract

Based on tools of variational analysis and the diffuse variation method, we derive the Pontryagin maximum principle and second-order necessary optimality conditions for optimal control problems governed by stationary Navier–Stokes equations with pure state constraints. Our results are established under a smallness assumption on the control and the optimality conditions are of Fritz–John type.

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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. This research was partially supported by the Korea Science and Engineering Foundation NRL program grant of the Korea government (MEST)(No.ROA-2008-000-20010-0), the Alexander von Humboldt Foundation and the NAFOSTED period 2015–2017

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Authors and Affiliations

  1. Department of Optimization and Control Theory, Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet road, Hanoi, Vietnam

    Bui Trong Kien

  2. Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea

    Gue Myung Lee

  3. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam

    Nguyen Hai Son

Authors
  1. Bui Trong Kien
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  2. Gue Myung Lee
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  3. Nguyen Hai Son
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Correspondence to Bui Trong Kien.

Additional information

Dedicated to Professor Eberhard Zeidler on the occasion of his 75th birthday.

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Kien, B.T., Lee, G.M. & Son, N.H. First- and Second-Order Necessary Optimality Conditions for Optimal Control Problems Governed by Stationary Navier–Stokes Equations with Pure State Constraints. Vietnam J. Math. 44, 103–131 (2016). https://doi.org/10.1007/s10013-015-0173-8

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  • DOI: https://doi.org/10.1007/s10013-015-0173-8

Keywords

  • Optimal control
  • The stationary Navier–Stokes equations
  • Pontryagin’s principle
  • Pure state constraint
  • First-order optimality condition
  • Second-order optimality condition

Mathematics Subject Classification (2010)

  • 49K20
  • 76D05
  • 65J15