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Applied Mathematics and Optimization

, Volume 57, Issue 3, pp 329–348 | Cite as

On the Dynamic Programming Approach for the 3D Navier–Stokes Equations

  • Luigi MancaEmail author
Article

Abstract

The dynamic programming approach for the control of a 3D flow governed by the stochastic Navier–Stokes equations for incompressible fluid in a bounded domain is studied. By a compactness argument, existence of solutions for the associated Hamilton–Jacobi–Bellman equation is proved. Finally, existence of an optimal control through the feedback formula and of an optimal state is discussed.

Keywords

Navier–Stokes equations Dynamic programming Hamilton–Jacobi–Bellman equations 

Rèsumè

Nous étudions la programmation dynamique pour le contrôle d’un flux tridimensionnel gouverné par les équations de Navier–Stokes stochastiques pour un fluide incompressible dans un domaine borné. Nous démontrons l’existence de solutions pour l’équation associée de Hamilton–Jacobi–Bellman par un argument de compacticité. Enfin nous examinons l’existence d’un contrôle optimal et d’un état optimal au moyen de la formule de feedback.

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPaduaItaly

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