Dynamic Programming for the stochastic Navier-Stokes equations

Giuseppe da Prato; Arnaud Debussche

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 2, page 459-475
  • ISSN: 0764-583X

Abstract

top
We solve an optimal cost problem for a stochastic Navier-Stokes equation in space dimension 2 by proving existence and uniqueness of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation.

How to cite

top

da Prato, Giuseppe, and Debussche, Arnaud. "Dynamic Programming for the stochastic Navier-Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 459-475. <http://eudml.org/doc/197463>.

@article{daPrato2010,
abstract = { We solve an optimal cost problem for a stochastic Navier-Stokes equation in space dimension 2 by proving existence and uniqueness of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation. },
author = {da Prato, Giuseppe, Debussche, Arnaud},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Stochastic Navier-Stokes equations; dynamic programming; optimal control; Hamilton-Jacobi-Bellmann equations.; optimal cost problem; stochastic Navier-Stokes equation; existence; uniqueness; smooth solution; Hamilton-Jacobi-Bellman equation},
language = {eng},
month = {3},
number = {2},
pages = {459-475},
publisher = {EDP Sciences},
title = {Dynamic Programming for the stochastic Navier-Stokes equations},
url = {http://eudml.org/doc/197463},
volume = {34},
year = {2010},
}

TY - JOUR
AU - da Prato, Giuseppe
AU - Debussche, Arnaud
TI - Dynamic Programming for the stochastic Navier-Stokes equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 2
SP - 459
EP - 475
AB - We solve an optimal cost problem for a stochastic Navier-Stokes equation in space dimension 2 by proving existence and uniqueness of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation.
LA - eng
KW - Stochastic Navier-Stokes equations; dynamic programming; optimal control; Hamilton-Jacobi-Bellmann equations.; optimal cost problem; stochastic Navier-Stokes equation; existence; uniqueness; smooth solution; Hamilton-Jacobi-Bellman equation
UR - http://eudml.org/doc/197463
ER -

References

top
  1. F. Abergel and R. Temam, On some control problems in fluid mechanics. Theor. and Comp. Fluid Dynamics1 (1990) 303-325.  Zbl0708.76106
  2. V. Barbu and S. Sritharan, H∞-control theory of fluids dynamics. Proc. R. Soc. Lond. A454 (1998) 3009-3033.  Zbl0919.93026
  3. T. Bewley, P. Moin and R. Temam, Optimal and robust approaches for linear and nonlinear regulartion problems in fluid mechanics, AIAA 97-1872, 28th AIAA Fluid Dynamics Conference and 4th AIAA Shear Flow Control Conference (1997).  
  4. P. Cannarsa and G. da Prato, Some results on nonlinear optimal control problems and Hamilton-Jacobi equations in infinite dimensions. J. Funct. Anal.90 (1990) 27-47.  Zbl0717.49022
  5. P. Cannarsa and G. da Prato, Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces, in: Stochastic partial differential equations and applications, G. da Prato and L. Tubaro Eds, Pitman Research Notes in Mathematics Series n.268 (1992) pp. 72-85.  Zbl0805.49016
  6. S. Cerrai, Optimal control problem for stochastic reaction-diffusion systems with non Lipschitz coefficients (to appear).  Zbl0987.60073
  7. H. Choi, R. Temam, P. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation. J. Fluid Mech.253 (1993) 509-543.  Zbl0810.76012
  8. G. da Prato and A. Debussche, Differentiability of the transition semigroup of stochastic Burgers equation. Rend. Acc. Naz. Lincei, s.9, v. 9 (1998) 267-277.  Zbl0931.37036
  9. G. da Prato and A. Debussche, Dynamic Programming for the stochastic Burgers equations. Annali di Mat. Pura ed Appl. (to appear).  Zbl1016.49024
  10. G. da prato and J. Zabczyk, Differentiability of the Feynman-Kac semigroup and a control application. Rend. Mat. Acc. Lincei. s.9, v. 8 (1997) 183-188.  Zbl0910.93025
  11. H. Fattorini and S. Sritharan, Existence of optimal controls for viscous flow problems. Proc. R. Soc. Lond. A439 (1992) 81-102.  Zbl0786.76063
  12. F. Gozzi, Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem. Commun. in partial differential equations20 (1995) 775-826.  Zbl0842.49021
  13. F. Gozzi, Global Regular Solutions of Second Order Hamilton-Jacobi Equations in Hilbert spaces with locally Lipschitz nonlinearities. J. Math. Anal. Appl.198 (1996) 399-443.  Zbl0858.35129
  14. P.L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolution. Acta Math.161 (1988) 243-278.  Zbl0757.93082
  15. S. Sritharan, Dynamic programming of the Navier-Stokes equations. Syst. Cont. Lett.16 (1991) 299-307.  Zbl0737.49021
  16. S. Sritharan, An introduction to deterministic and stochastic control of viscous flow, in Optimal control of viscous flows, p. 1-42, SIAM, Philadelphia, S. Sritharan Ed.  
  17. A. Swiech, Viscosity solutions of fully nonlinear partial differential equations with "unbounded'' terms in infinite dimensions, Ph.D. thesis, University of California at Santa Barbara (1993).  
  18. R. Temam, T. Bewley and P.Moin, Control of turbulent flows, Proc. of the 18th IFIP TC7, Conf. on system modelling ond optimization, Detroit, Michigan (1997).  Zbl0925.93417
  19. R. Temam, The Navier-Stokes equation, North-Holland (1977).  Zbl0383.35057

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.