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On an extension of the Lyapunov criterion of stability for quasi-linear systems via integral inequalities methods


Author: Nguyen Huu Du
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 70 (2004).
Journal: Theor. Probability and Math. Statist. 70 (2005), 29-40
MSC (2000): Primary 60H10; Secondary 34F05, 93E15
DOI: https://doi.org/10.1090/S0094-9000-05-00628-9
Published electronically: August 5, 2005
MathSciNet review: 2109820
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Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we concern ourselves with a new concept for comparing the stability degree of two dynamical systems. By using the integral inequality method, we give a criterion which allows us to compare the growth rate of two Itô quasi-linear differential equations. It can be viewed as an extension of the Lyapunov criterion to the stochastic case.


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  • 1. L. Arnold, Random Dynamical Systems, Springer, New York-Berlin, 1998. MR 1723992 (2000m:37087)
  • 2. L. Arnold and H. Crauel, Random Dynamical Systems (Lyapunov Exponents; Proceedings, Oberwolfach 1990), Lecture Notes in Mathematics, vol. 1486, Springer, New York-Berlin, 1991. MR 1178943 (93m:60116)
  • 3. A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge-New York, 1992. MR 1191160 (93i:93001)
  • 4. Nguyen Dinh Cong, Lyapunov spectrum of nonautonomous linear stochastic differential equations, Stoch. Dyn. 1 (2001), no. 1, 127-157. MR 1837159 (2002e:60098)
  • 5. V. F. Bylov, R. E. Vinograd, D. M. Grobman, and D. M. Neminskii, Theory of Lyapunov Exponents, ``Nauka'', Moscow, 1966. (Russian) MR 0206415 (34:6234)
  • 6. B. P. Demidovich, Lectures on the Mathematical Theory of Stability, ``Nauka'', Moscow, 1967. (Russian) MR 0226126 (37:1716)
  • 7. I. I. Gihman and A. V. Skorokhod, Stochastic Differential Equations, Springer-Verlag, 1972. MR 0346904 (49:11625)
  • 8. I. Ya. Goldsheid and G. A. Margulis, Lyapunov exponent of random matrices product, Russian Math. Surveys 44 (1989), no. 5(269), 11-41. MR 1040268 (91j:60014)
  • 9. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. MR 0637061 (84b:60080)
  • 10. R. S. Khasminskii, Stability of Systems of Differential Equations with Random Perturbations of their Parameters, ``Nauka'', Moscow, 1969. (Russian) MR 0259283 (41:3925)
  • 11. R. S. Liptser and A. N. Shiryaev, Statistics of Stochastic Processes, ``Nauka'', Moscow, 1974. (Russian) MR 0431365 (55:4365)
  • 12. V. M. Millionshchikov, Formulae for Lyapunov exponent of a linear systems of differential equations, Transactions of the I. N. Vekua Institute of Applied Mathematics, vol. 22, Tbilisi State University (USSR), 1987, pp. 150-179. MR 0946666 (90a:58088)
  • 13. V. Lakshmikantam, A. A. Martynyuk, and S. Lila, Stability of Motion: The Comparison Method, ``Naukova Dumka'', Kiev, 1991. (Russian) MR 1180322 (93i:34095)
  • 14. N. H. Du, On the comparison of the stability and control problem of differential systems, Stochastic Anal. Appl. 16 (1998), no. 3, 533-551. MR 1619780 (99f:93105)

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Additional Information

Nguyen Huu Du
Affiliation: Faculty of Mathematics, Mechanics, and Informatics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
Email: nhdu2001@yahoo.com

DOI: https://doi.org/10.1090/S0094-9000-05-00628-9
Keywords: Lyapunov exponent, It\^o's stochastic process, Bihari's inequality
Received by editor(s): December 14, 2002
Published electronically: August 5, 2005
Additional Notes: This work was supported by VNCR program # QT 01.01.
Article copyright: © Copyright 2005 American Mathematical Society

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