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

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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.

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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Authors and Affiliations

  1. Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Luigi Manca

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  1. Luigi Manca
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Correspondence to Luigi Manca.

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This paper has been written at Scuola Normale Superiore di Pisa and at École Normale Supérieure de Cachan, Antenne de Bretagne.

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Manca, L. On the Dynamic Programming Approach for the 3D Navier–Stokes Equations. Appl Math Optim 57, 329–348 (2008). https://doi.org/10.1007/s00245-007-9024-7

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  • DOI: https://doi.org/10.1007/s00245-007-9024-7

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