Ergodic Control of Semilinear Stochastic Equations and the Hamilton-Jacobi Equation

Journal of Mathematical Analysis and Applications

Volumn 234, Issue 2, 1999, Pages 592-631

  Goldys, Beniamin ^a   Maslowski, Bohdan ^b  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b INSTITUTE OF MATHEMATICS   (Czech Republic)

Author keywords

Bismut Elworthy formula; Dynamic programming; Ergodic control; Invariant measure; Semilinear stochastic equation; Stationary Hamilton Jacobi equation

Indexed keywords

EID: 0000816867     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1999.6387     Document Type: Article

References (27)

1. A. Bensoussan, "Perturbation Methods in Optimal Control," Wiley, New York, 1988.
2. n, J. Reine Angew. Math. 429 (1992), 125-160.
3. V. S. Borkar, "Optimal Control of Diffusion Processes," Longman, Harlow/New York, 1989.
4. V. S. Borkar and M. K. Ghosh, Ergodic control of multidimensional diffusion I: the existence results, SIAM J. Control Optim. 26 (1988), 112-126.
5. P. M. Cannarsa and G. Da Prato, Second order Hamilton-Jacobi equations in infinite dimensions, SIAM J. Control Optim. 29 (1991), 474-492.
6. P. M. Cannarsa and G. Da Prato, Infinite-dimensional elliptic equation with Hölder-continuous coefficients, Adv. Differential Equations 1 (1996), 425-452.
7. S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum 49 (1994), 349-367.
8. S. Cerrai, Weakly continuous semigroups in the space of functions with polynomial growth, Dynam. Systems Appl. 4 (1995), 351-371.
9. S. Cerrai and F. Gozzi, Strong solutions of Cauchy problems associated to weakly continuous semigroups, Differential Integral Equations 8 (1995), 465-486.
10. A. Chojnowska-Michalik and B. Goldys, Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces, Probab. Theory Related Fields 102 (1995), 331-356.
11. p spaces, Stochastics & Stochastics Rep. 59 (1996), 183-209.
12. A. Chojnowska-Michalik and B. Goldys, "On Ornstein-Uhlenbeck generators," Technical Report S96-12, School of Mathematics, The University of New South Wales, 1996.
13. P. L. Chow and J. L. Menaldi, Infinite dimensional Hamilton-Jacobi equations in Gauss-Sobolev spaces, J Nonlinear Anal., to appear.
14. G. Da Prato, K. D. Elworthy, J. Zabczyk, Strong Feller property for stochastic semilinear equations, Stochastic Anal. Appl. 13 (1995), 35-45.
15. G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, Cambridge, UK, 1992.
16. E. B. Davies, "One Parameter Semigroups," Academic Press, San Diego, 1980.
17. T. Duncan, B. Maslowski, and B. Pasik-Duncan, Ergodic boundary/point control of stochastic semilinear equations, SIAM J. Control Optim. 36 (1998), 1020-1047.
18. T. Duncan, B. Pasik-Duncan, and Ł. Stettner, On ergodic control of stochastic evolution equations, Stochastic Process. Appl. 15 (1997), 723-750.
19. F. Gozzi and E. Rouy, Regular solutions of second-order stationary Hamilton-Jacobi equations, J. Differential Equations 130 (1996), 201-234.
20. F. Gozzi, E. Rouy, and A. Swiech, Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic boundary control, Preprint No. 46, Scuola Normale Superiore, Pisa, 1996.
21. D. Henry, "Geometry Theory of Semilinear Parabolic Equations," Springer, New York, 1981.
22. P. Mandl, On control by non-stopped diffusion processes (Russian), Theory Probab. Appl. 9 (1964), 591-603.
23. B. Maslowski, On probability distributions of solutions of semilinear stochastic evolution equations, Stochastics & Stochastics Rep. 45 (1993), 17-44.
24. J. Seidler, Ergodic behaviour of stochastic parabolic equations, Czechoslovak Math. J. 47 (1997), 277-316.
25. A. Swiech, "Unbounded" second order partial differential equations in infinite-dimensional Hilbert spaces, Comm. Partial Differential Equations 19 (1994), 1999-2036.
26. G. Tessitore, Infinite horizon, ergodic and periodic control for a stochastic infinite dimensional affine equations, J. Math. Systems Estim. Control, to appear.
27. K. Yosida, "Functional Analysis," sixth ed., Springer, New York, 1980.