Vietnam Journal of Mathematics

, Volume 44, Issue 1, pp 103–131 | Cite as

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

  • Bui Trong KienEmail author
  • Gue Myung Lee
  • Nguyen Hai Son


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.


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 



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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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  • Bui Trong Kien
    • 1
    Email author
  • Gue Myung Lee
    • 2
  • Nguyen Hai Son
    • 3
  1. 1.Department of Optimization and Control Theory, Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Department of Applied MathematicsPukyong National UniversityBusanKorea
  3. 3.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam

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