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