Control of the stochastic Burgers model of turbulence

SIAM Journal on Control and Optimization

Volumn 37, Issue 4, 1999, Pages 1123-1149

  Da Prato, Giuseppe ^a   Debussche, Arnaud ^b  

^a SCUOLA NORMALE SUPERIORE   (Italy)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; DYNAMIC PROGRAMMING; MATHEMATICAL MODELS;

HAMILTON-JACOBI EQUATIONS; STOCHASTIC BURGERS EQUATION;

STOCHASTIC CONTROL SYSTEMS;

EID: 0032629549     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012996311307     Document Type: Article

References (24)

1. V. BARBU AND G. DA PRATO, Hamilton-Jacobi Equations in Hilbert Spaces, Pitman Res. Notes Math. Ser. 86, Longman, Harlow, UK, 1982.
2. L. BERTINI, N. CANCRINI, AND G. JONA-LASINIO, The stochastic Burgers equation, Comm. Math. Phys., 165 (1994), pp. 211-232.
3. D.H. CHAMBERS, R.J. ADRIAN, P. MOIN, D.S. STEWART, AND H.J. SUNG, Karhunuen-Loeve expansion of Burgers model of turbulence, Phys. Fluids, 31 (1988), p. 2573.
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), pp. 27-47.
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, Pitman Res. Notes Math. Ser. 268, G. Da Prato and L. Tubaro, eds., Longman, Harlow, UK, 1992, pp. 72-85.
6. 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), pp. 509-543.
7. M.G. CRANDALL, H. ISHI, AND P.L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1-67.
8. G. DA PRATO, Some results on Bellman equation in Hilbert spaces, SIAM J. Control Optim., 23 (1985), pp. 61-71.
9. G. DA PRATO, A. DEBUSSCHE, AND R. TEMAM, Stochastic Burgers equation, NoDEA Nonlinear Differential Equations Appl., 1 (1994), pp. 389-402.
10. G. DA PRATO AND D. GA̧TAREK, Stochastic Burgers equation with correlated noise, Stochastics Stochastics Rep., 52 (1995), pp. 29-41.
11. G. DA PRATO AND J. ZABCZYK, Stochastic Evolution Equations in Infinite Dimensions, Cambridge University Press, London, UK, 1992.
12. W.H. FLEMING AND H.M. SONER, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, Berlin, New York, 1993.
13. F. GOZZI, Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem, Comm. Partial Differential Equations, 20 (1995), pp. 775-826.
14. F. GOZZI, Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities, J. Math. Anal. Appl., 198 (1996), pp. 399-443.
15. F. GOZZI AND E. ROUY, Regular solutions of second order stationary Hamilton-Jacobi equations, J. Differential Equations, 130 (1996), pp. 201-234.
16. F. GOZZI, E. ROUY, AND A. SWIECH, Second Order Hamilton-Jacobi Equations in Hilbert Spaces and Stochastic Boundary Control, preprint 46, Scuola Normale Superiore di Pisa, Pisa, Italy, 1996.
17. Y. KIFER, The Burgers Equation with a Random Force and a General Model for Directed Polymers in Random Environments, preprint, 1995.
18. J.L. LIONS, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier - Villars, Paris, 1969.
19. 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), pp. 243-278, Part II: Optimal control of Zakai's equation, in Stochastic Partial Differential Equations and Applications, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro, eds., Springer-Verlag, New York, 1989, Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal., 86 (1989), pp. 1-18.
20. 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), pp. 243-278, Part II: Optimal control of Zakai's equation, in Stochastic Partial Differential Equations and Applications, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro, eds., Springer-Verlag, New York, 1989, Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal., 86 (1989), pp. 1-18.
21. 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), pp. 243-278, Part II: Optimal control of Zakai's equation, in Stochastic Partial Differential Equations and Applications, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro, eds., Springer-Verlag, New York, 1989, Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal., 86 (1989), pp. 1-18.
22. 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), pp. 243-278, Part II: Optimal control of Zakai's equation, in Stochastic Partial Differential Equations and Applications, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro, eds., Springer-Verlag, New York, 1989, Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal., 86 (1989), pp. 1-18.
23. A. SWIECH, Viscosity Solutions of Fully Nonlinear Partial Differential Equations with "Unbounded" Terms in Infinite Dimensions, Ph.D. thesis, University of California, Santa Barbara, 1993 .
24. M. TESSITORE AND J. ZABCZYK, Invariant Measures for Stochastic Heat Equation, preprint 16. Scuola Normale Superiore di Pisa, Pisa, Italy.