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Stabilization of evolution equations by noise


Author: Anna A. Kwiecinska
Journal: Proc. Amer. Math. Soc. 130 (2002), 3067-3074
MSC (2000): Primary 35K90, 37L55; Secondary 47D06
DOI: https://doi.org/10.1090/S0002-9939-02-06443-2
Published electronically: March 29, 2002
MathSciNet review: 1908931
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a deterministic equation of evolution

\begin{displaymath}X'(t)=AX(t)dt,\end{displaymath}

in a separable, real Hilbert space. We prove that if $A$generates a $C_0$-semigroup, then this equation can be stabilized, in terms of Lyapunov exponents, by noise.

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  • 1. L. Arnold: Stochastic Differential Equations: Theory and Applications. Wiley-Interscience, Wiley, New York, 1974. MR 56:1456
  • 2. L. Arnold: A new example of an unstable system being stabilized by random parameter noise. Inform. Comm. Math. Chem., 1979, 133-140. MR 81g:60067
  • 3. L. Arnold, H. Crauel and V. Wihstutz: Stabilization of linear systems by noise. SIAM J. Control Optim. 21, 1983, 451-461. MR 84g:93080
  • 4. L. Arnold and P. Kloeden: Lyapunov exponents and rotation number of two-dimensional systems with telegraphic noise. SIAM J. Appl. Math. 49, 1989, 1242-1274. MR 90f:93062
  • 5. T. Caraballo, K. Liu and X. Mao: On stabilization of partial differential equations by noise, Nagoya Math. J. 161, 2001, 155-170.
  • 6. G. Da Prato and J. Zabczyk: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications vol.44, Cambridge University Press, Cambridge, 1992.
  • 7. A. A. Kwiecinska: Stabilization of partial differential equations by noise. Stochastic Process. Appl. 79, 1999, 179-184. MR 2000b:35284
  • 8. G. Leha, B. Maslowski and G. Ritter: Stability of solutions to semilinear stochastic evolution equations. Stochastic Anal. Appl. 17(6), 1999, 1009-1051. MR 2001a:60074
  • 9. E. Pardoux and V. Wihstutz: Lyapunov exponents and rotation number of two-dimensional stochastic systems with small diffusion. SIAM J. Appl. Math. 48, 1998, 442-457. MR 89e:60116
  • 10. E. Pardoux and V. Wihstutz: Lyapunov exponents of linear stochastic systems with large diffusion term. Stochastic Process. Appl. 40, 1992, 289-308. MR 93e:60114
  • 11. A. Pazy: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences vol. 44, Springer-Verlag, New York, 1983. MR 85g:47061

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Additional Information

Anna A. Kwiecinska
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950 Warszawa, Poland
Email: akwiecin@impan.gov.pl

DOI: https://doi.org/10.1090/S0002-9939-02-06443-2
Received by editor(s): April 2, 2001
Received by editor(s) in revised form: June 1, 2001
Published electronically: March 29, 2002
Additional Notes: This research was partially supported by KBN grant 2 P03A 016 16
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2002 American Mathematical Society

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