Journal of Applied Mathematics

A Class of Stochastic Nonlinear Delay System with Jumps

Ling Bai, Kai Zhang, and Wenju Zhao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 458306, 11 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1395855295

Digital Object Identifier
doi:10.1155/2014/458306

Mathematical Reviews number (MathSciNet)
MR3166764

Zentralblatt MATH identifier
07010640

Citation

Bai, Ling; Zhang, Kai; Zhao, Wenju. A Class of Stochastic Nonlinear Delay System with Jumps. J. Appl. Math. 2014 (2014), Article ID 458306, 11 pages. doi:10.1155/2014/458306. https://projecteuclid.org/euclid.jam/1395855295


Export citation

References

  • F. Wu and S. Hu, “Stochastic suppression and stabilization of delay differential systems,” International Journal of Robust and Nonlinear Control, vol. 21, no. 5, pp. 488–500, 2011.
  • F. Wu, X. Mao, and S. Hu, “Stochastic suppression and stabilization of functional differential equations,” Systems & Control Letters, vol. 59, no. 12, pp. 745–753, 2010.
  • F. Wu and S. Hu, “Suppression and stabilisation of noise,” International Journal of Control, vol. 82, no. 11, pp. 2150–2157, 2009.
  • X. Mao, Stochastic Differential Equations and their Applications, Ellis Horwood, Chichester, UK, 1997.
  • C. Yuan and X. Mao, “Stability of stochastic delay hybrid systems with jumps,” European Journal of Control, vol. 6, pp. 595–608, 2010.
  • G. Yin and F. Xi, “Stability of regime-switching jump diffusions,” SIAM Journal on Control and Optimization, vol. 48, no. 7, pp. 4525–4549, 2010.
  • D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, UK, 2nd edition, 2009. \endinput