The vanishing viscosity method in infinite dimensions

Piermarco Cannarsa; Giuseppe Da Prato

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1989)

  • Volume: 83, Issue: 1, page 79-84
  • ISSN: 1120-6330

Abstract

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The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.

How to cite

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Cannarsa, Piermarco, and Da Prato, Giuseppe. "The vanishing viscosity method in infinite dimensions." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 83.1 (1989): 79-84. <http://eudml.org/doc/287506>.

@article{Cannarsa1989,
abstract = {The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.},
author = {Cannarsa, Piermarco, Da Prato, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Hamilton-Jacobi equations; Infinite dimensions; Viscosity solution; Optimal control; infinite dimensions; vanishing viscosity method; parabolic equations in Hilbert spaces; approximation},
language = {eng},
month = {12},
number = {1},
pages = {79-84},
publisher = {Accademia Nazionale dei Lincei},
title = {The vanishing viscosity method in infinite dimensions},
url = {http://eudml.org/doc/287506},
volume = {83},
year = {1989},
}

TY - JOUR
AU - Cannarsa, Piermarco
AU - Da Prato, Giuseppe
TI - The vanishing viscosity method in infinite dimensions
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1989/12//
PB - Accademia Nazionale dei Lincei
VL - 83
IS - 1
SP - 79
EP - 84
AB - The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.
LA - eng
KW - Hamilton-Jacobi equations; Infinite dimensions; Viscosity solution; Optimal control; infinite dimensions; vanishing viscosity method; parabolic equations in Hilbert spaces; approximation
UR - http://eudml.org/doc/287506
ER -

References

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  1. BARBU, V. and DA PRATO, G., 1983. Solution of the Bellman equation associated with an infinite dimensional Stochastic control problem and synthesis of optimal control. SIAM J. Control Opt., 21, 4: 531-550. Zbl0511.93072MR704473DOI10.1137/0321032
  2. CANNARSA, P. and DA PRATO, G.Some results on nonlinear optimal control problems and Hamilton-Jacobi equations in infinite dimensions. J. Funct. Anal., (to appear). Zbl0717.49022MR1047576DOI10.1016/0022-1236(90)90079-Z
  3. CANNARSA, P. and DA PRATO, G., 1989. Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations. SIAM J. Control Opt., 27, 4: 861-875. Zbl0682.49033MR1001924DOI10.1137/0327046
  4. CRANDALL, M.G. and LIONS, P.L., 1983. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 277: 183-186. Zbl0599.35024MR690039DOI10.2307/1999343
  5. CRANDALL, M.G. and LIONS, P.L., 1985. Hamilton-Jacobi equations in infinite dimensions Part I. Uniqueness of Viscosity Solutions. J. Funct. Anal., 62: 379-396. Zbl0627.49013MR794776DOI10.1016/0022-1236(85)90011-4
  6. CRANDALL, M.G. and LIONS, P.L., 1986. Hamilton-Jacobi equations in infinite dimensions. Part II. Existence of Viscosity Solutions. J. Funct. Anal., 65: 368-405. Zbl0639.49021MR826434DOI10.1016/0022-1236(86)90026-1
  7. CRANDALL, M.G. and LIONS, P.L., 1986. Hamilton-Jacobi equations in infinte dimensions. Part III. J. Funct. Anal., 68: 368-405. Zbl0739.49015MR852660DOI10.1016/0022-1236(86)90005-4
  8. CRANDALL, M.G. and LIONS, P.L., 1987. Solutions de visconsitê pour les équations de Hamilton-Jacobi en dimension infinie intervenant dans le contrôle optimal des problèmes d'évolution. C.R. Acad. Sci. Paris, 305: 233-236. Zbl0588.35018MR907950
  9. DALECKII, J.L., 1966. Differential equations with functional derivatives and stochastic equations for generalized random processes. Dokl. Akad. Nauk SSSR, 166: 1035-1038. Zbl0305.35084MR214943
  10. DA PRATO, G., 1987. Some Results on Parabolic Evolution Equations with Infinitely Many Variables. J. Differential Equations, 68, 2: 281-297. Zbl0628.35044MR892028DOI10.1016/0022-0396(87)90196-3
  11. DA PRATO, G., 1985. Some results on Bellman equation in Hilbert spaces and applications to infinite dimensional control problems. In «Stochastic Differential Systems, Filtering and Control», Lecture Notes in Control and Information Sciences n° 69, Proceedings of the IFIP-WG 7/1 Working Conference, Marseille-Luminy, France, March 12-17, 1984. MATIVIER M. and PARDOUX E. Editors: 270-280. Zbl0581.93068MR798330DOI10.1007/BFb0005082
  12. FLEMING, W.H., 1969. The Cauchy problem for a nonlinear first order partial differential equation. J. Differential Equations, 5: 515-530. Zbl0172.13901MR235269
  13. GROSS, L., 1967. Potential theory in Hilbert space. J. Func. Anal., 1: 123-181. Zbl0165.16403MR227747
  14. HAVARNEANU, T., 1985. Existence for the Dynamic Programming equations of control diffusion processes in Hilbert spaces. Nonlinear Anal. T.M.A, 9, n° 6: 619-629. MR794831DOI10.1016/0362-546X(85)90045-8
  15. LIONS, P.L., 1982. Generalized solutions of Hamilton-Jacobi equations. Pitman, Boston. Zbl0497.35001MR667669
  16. PIECH, M.A., 1969. A Fundamental Solution of the Parabolic Equations in Hilbert spaces. J. Funct. Analysis, 3: 85-114. Zbl0169.47103MR251588

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