Exponentially stable stationary solutions for stochastic evolution equations and their perturbation

Applied Mathematics and Optimization

Volumn 50, Issue 3, 2004, Pages 183-207

  Caraballo, Tomás ^a   Kloeden, Peter E ^b   Schmalfuß, Björn ^c  

^a UNIVERSITY OF SEVILLE   (Spain)

^b GOETHE UNIVERSITY   (Germany)

^c UNIVERSITY OF PADERBORN   (Germany)

Author keywords

Exponential stability; Random dynamical systems; Stabilization; Stationary solutions

Indexed keywords

DIFFERENTIAL EQUATIONS; ELECTRIC EQUIPMENT PROTECTION; EVOLUTIONARY ALGORITHMS; PARAMETER ESTIMATION; PERTURBATION TECHNIQUES; RANDOM PROCESSES; STABILIZATION; THEOREM PROVING;

EXPONENTIAL STABILITY; RANDOM DYNAMICAL SYSTEMS; STATIONARY SOLUTIONS; WIENER SHIFT;

STOCHASTIC CONTROL SYSTEMS;

EID: 10044273281     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00245-004-0802-1     Document Type: Article

References (26)

1. L. Arnold. Random Dynamical Systems. Springer-Verlag, New York, 1998.
2. H. Brezis. Analyse Fonctionnelle. Masson, Paris, 1983.
3. T. Caraballo and J. Langa. Comparison of the long-time behavior of linear Ito and Stratonovich partial differential equations. Stochastic Analysis and Applications, 19(2):183-195, 2001.
4. T. Caraballo, K. Liu, and X. Mao. On stabilization of partial differential equations by noise. Nagoya Mathematical Journal, 161:155-170, 2001.
5. T. Caraballo and J. Real. On the pathwise exponential stability of nonlinear stochastic partial differential equations. Stochastic Analysis and Applications, 12(5):517-525, 1994.
6. C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions. LNM 580. Springer-Verlag, Berlin, 1977.
7. P.L. Chow. Stability of nonlinear stochastic-evolution equations. Journal of Mathematical Analysis and Applications, 89(2):400-419, 1982.
8. H. Crauel, A. Debussche, and F. Flandoli. Random attractors. Journal of Dynamics and Differential Equations, 9:307-341, 1997.
9. G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, 1992.
10. R. Dautray and J.L. Lions. Analyse mathématique et calcul numérique pour les sciences et les techniques. Volume 1 of Collection du Commissariat à l'Énergie Atomique: Série Scientifique. Masson, Paris, 1984.
11. F. Flandoli and H. Lisei. Stationary conjugation of flows for parbolic SPDEs with multiplicative noise and some applications. Manuscript, 2002.
12. F. Flandoli and B. Schmalfuß. Random attractors for the stochastic 3-D Navier-Stokes equation with multiplicative white noise. Stochastics and Stochastics Reports, 59:21-45, 1996.
13. W. Grecksch and C. Tudor. Stochastic Evolution Equations - a Hilbert Space Approach. Volume 85 of Mathematical Research. Akademie Verlag, Berlin, 1995.
14. R. Z. Hasḿinskiǐ. Stochastic Stability of Differential Equations. Sijthoff & Nordhoff, Alphen aan den Rijn, 1980.
15. U.G. Haussmann. Asymptotic stability of the linear Itô equation in infinite dimensions. Journal of Mathematical Analysis and Applications, 65(1):219-235, 1978.
16. A. Ichikawa. Stability of semilinear stochastic evolution equations. Journal of Mathematical Analysis and Applications, 90(1): 12-44, 1982.
17. N.V. Krylov and B.L. Rozovskii. Ob evoljucionnych stochastičeskich uravneniach. Volume 14 of Sovremennych problemy matematiki, pages 71-146. Nauka, Moskva, 1979.
18. A.A. Kwiecinska. Stabilization of evolution equations by noise. Proceedings of the American Mathematical Society, 130(10):3067-3074, 2001.
19. K. Liu and X. Mao. Exponential stability of non-linear stochastic evolution equations. Stochastic Processes and their Applications, 78(2):173-193, 1998.
20. X. Mao. Exponential Stability of Stochastic Differential Equations. Marcel Dekker, New York, 1994.
21. S.-E. A. Mohammed, T. Zhang, and H. Zhao. The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations. Preprint, 2003.
22. B. Øksendal. Stochastic Differential Equations, third edition. Springer-Verlag, Berlin, 1992.
23. B. Schmalfuß. Backward cocycles and attractors of stochastic differential equations. In V. Reitmann, T. Riedrich, and N. Koksch, editors, Proceedings of the International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, pages 185-192, 1992.
24. B. Schmalfuß. Lyapunov functions and non-trivial stationary solutions of stochastic differential equations. Dynamical Systems, 19(4):303-317, 2001.
25. R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second edition. Springer-Verlag, Berlin, 1997.
26. E. Zeidler. Nonlinear Functional Analysis and its Applications, volume I. Springer-Verlag, New York, 1985.