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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

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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Correspondence to Bui Trong Kien.

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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