Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T19:01:05.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic Programming for the stochastic Navier-Stokesequations

Published online by Cambridge University Press:  15 April 2002

Giuseppe da Prato  and
Arnaud Debussche
Giuseppe da Prato
Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy.
Arnaud Debussche
Affiliation:
CNRS et Université de Paris-Sud, 91405 Orsay Cedex, France. (arnaud.debussche@math.u-psud.fr)
Get access

Abstract

We solve an optimal cost problem for a stochasticNavier-Stokes equation in spacedimension 2 by proving existence and uniqueness of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation.

Keywords

Stochastic Navier-Stokes equationsdynamic programmingoptimal controlHamilton-Jacobi-Bellmann equations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 2: Special issue for R. Teman's 60th birthday , March 2000 , pp. 459 - 475
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abergel, F. and Temam, R., On some control problems in fluid mechanics. Theor. and Comp. Fluid Dynamics 1 (1990) 303-325. CrossRef
Barbu, V. and Sritharan, S., H∞-control theory of fluids dynamics. Proc. R. Soc. Lond. A 454 (1998) 3009-3033. CrossRef
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).
Cannarsa, P. and da Prato, G., Some results on nonlinear optimal control problems and Hamilton-Jacobi equations in infinite dimensions. J. Funct. Anal. 90 (1990) 27-47. CrossRef
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.
S. Cerrai, Optimal control problem for stochastic reaction-diffusion systems with non Lipschitz coefficients (to appear).
Choi, H., Temam, R., Moin, P. and Kim, J., Feedback control for unsteady flow and its application to the stochastic Burgers equation. J. Fluid Mech. 253 (1993) 509-543. CrossRef
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.
G. da Prato and A. Debussche, Dynamic Programming for the stochastic Burgers equations. Annali di Mat. Pura ed Appl. (to appear).
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.
Fattorini, H. and Sritharan, S., Existence of optimal controls for viscous flow problems. Proc. R. Soc. Lond. A 439 (1992) 81-102. CrossRef
Gozzi, F., Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem. Commun. in partial differential equations 20 (1995) 775-826. CrossRef
Gozzi, F., Global Regular Solutions of Second Order Hamilton-Jacobi Equations in Hilbert spaces with locally Lipschitz nonlinearities. J. Math. Anal. Appl. 198 (1996) 399-443. CrossRef
Lions, P.L., 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.
Sritharan, S., Dynamic programming of the Navier-Stokes equations. Syst. Cont. Lett. 16 (1991) 299-307. CrossRef
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.
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).
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).
R. Temam, The Navier-Stokes equation, North-Holland (1977).