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Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations

Abstract

This paper deals with the Pontryagin maximum principle for optimal control problems governed by 3D Navier–Stokes equations with pointwise control constraint. The obtained result is proved by using some results on regularity of solutions of the Navier–Stokes equations and techniques of optimal control theory.

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Acknowledgements

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 Vietnam National Foundation for Science and Technology Development (NAFOSTED) period 2016–2018 under Grant Number 101.01-2015.13. A part of this work was done when the first author worked at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for their support and hospitality.

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Correspondence to B. T. Kien.

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Communicated by Michael Hinze.

Appendix

Appendix

Proposition A.1

([14, Proposition 6.1]) Let \(\varOmega \subset {\mathbb {R}}^n\) be bounded, open and of class \(C^l\). Let \(s_1, s_2, s_3\) be real number, \(0\le s_1\le l\), \(0\le s_2\le l-1\), \(0\le s_3\le l\). Assume that

  • (i) \(s_1+s_2+s_3\ge n/2\) if \(s_i\ne n/2\) for all \(i=1,2,3\) or

  • (ii) \(s_1+s_2+s_3>n/2\) if \(s_i=n/2\) for at least one i.

Then there exists a constant depending on \(s_1, s_2, s_3, \varOmega \), scale invariant such that

$$\begin{aligned} |\mathbf{b }(u, v, w)|&\le c|\varOmega |^{\frac{s_1+s_2+s_3}{n}-\frac{1}{2}}\Vert u\Vert _{[s_1]}^{1+[s_1]-s_1}\Vert u\Vert _{[s_1]+1}^{s_1-[s_1]} \Vert v\Vert _{[s_2]+1}^{1+[s_2]-s_2}\Vert v\Vert _{[s_2]+2}^{s_2-[s_2]} \nonumber \\&\quad \cdot \Vert w\Vert _{[s_3]}^{1+[s_3]-s_3}\Vert w\Vert _{[s_3]+1}^{s_3-[s_3]} \end{aligned}$$
(61)

for all \(u, v, w\in C^\infty ({{\bar{\varOmega }}})^n.\) Here \([s_i]\) denotes the integer part of \(s_i\) and \(\Vert \cdot \Vert _0\) denotes the norm of \(L^2(\varOmega )\).

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Kien, B.T., Rösch, A. & Wachsmuth, D. Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations. J Optim Theory Appl 173, 30–55 (2017). https://doi.org/10.1007/s10957-017-1081-8

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  • DOI: https://doi.org/10.1007/s10957-017-1081-8

Keywords

  • Optimal control
  • Navier–Stokes equation
  • Pontryagin’s principle
  • Strong solution
  • Control constraint

Mathematics Subject Classification

  • 49K20
  • 93B50
  • 93C05
  • 93C20
  • 93C35
  • 76D05
  • 65J15