Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

MEAN SQUARE EXPONENTIAL DISSIPATIVITY OF SINGULARLY PERTURBED STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

  • Xu, Liguang (Department of Applied Mathematics Zhejiang University of Technology) ;
  • Ma, Zhixia (College of Computer Science & Technology Southwest University for Nationalities) ;
  • Hu, Hongxiao (College of Science Shanghai University for Science and Technology)
  • Received : 2013.01.18
  • Published : 2014.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper investigates mean square exponential dissipativity of singularly perturbed stochastic delay differential equations. The L-operator delay differential inequality and stochastic analysis technique are used to establish sufficient conditions ensuring the mean square exponential dissipativity of singularly perturbed stochastic delay differential equations for sufficiently small ${\varepsilon}$ > 0. An example is presented to illustrate the efficiency of the obtained results.

Keywords

References

  1. H. C. Chang and M. Aluko, Multi-scale analysis of exotic dynamics in surface catalyzed reactions-I, Chemical Engineering Science 39 (1984), 37-50. https://doi.org/10.1016/0009-2509(84)80128-1
  2. J. H. Chow, Time Scale Modelling of Dynamic Networks, Springer-Verlag, New York, 1982.
  3. P. D. Christofides and P. Dsoutidis, Feedback control of two-time-scale nonlinear systems, Internat. J. Control 63 (1996), no. 5, 965-994. https://doi.org/10.1080/00207179608921879
  4. J. H. Cruz and P. Z. Taboas, Periodic solutions and stability for a singularly perturbed linear delay differential equation, Nonlinear Anal. 67 (2007), 1657-1667. https://doi.org/10.1016/j.na.2006.08.004
  5. D. Da and M. Corless, Exponential stability of a class of nonlinear singularly perturbed systems with marginally sable boundary layer systems, In Proceedings of the American Control Conference, 3101-3106, San Francisco, CA, 1993.
  6. M. El-Ansary, Stochastic feedback design for a class of nonlinear singularly perturbed systems, Internat. J. Systems Sci. 22 (1991), no. 10, 2013-2023. https://doi.org/10.1080/00207729108910767
  7. M. El-Ansary and H. K. Khalil, On the interplay of singular perturbations and wide-band stochastic fluctuations, SIAM J. Control Optim. 24 (1986), no. 1, 83-94. https://doi.org/10.1137/0324004
  8. E. Fridman, Effects of small delays on stability of singularly perturbed systems, Automatica J. IFAC 38 (2002), no. 5, 897-902. https://doi.org/10.1016/S0005-1098(01)00265-5
  9. V. B. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992.
  10. X. Z. Liu, X. M. Shen, and Y. Zhang, Exponential stability of singularly perturbed systems with time delay, Appl. Anal. 82 (2003), no. 2, 117-130. https://doi.org/10.1080/0003681031000063775
  11. X. R. Mao, Razumikhin-type theorems on exponential stability of stochastic functional-differential equations, Stochastic Process. Appl. 65 (1996), no. 2, 233-250. https://doi.org/10.1016/S0304-4149(96)00109-3
  12. X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publication, Chichester, 1997.
  13. X. R. Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations 153 (1999), no. 1, 175-195. https://doi.org/10.1006/jdeq.1998.3552
  14. S.-E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1986.
  15. J. J. Monge and C. Georgakis, The effect of operating variables on the dynamics of catalytic cracking processes, Chemical Engineering Communications 60 (1987), 1-15. https://doi.org/10.1080/00986448708912007
  16. N. Prljaca and Z. Gajic, A method for optimal control and filtering of multitime-scale linear singularly-perturbed stochastic systems, Automatica J. IFAC 44 (2008), no. 8, 2149-2156. https://doi.org/10.1016/j.automatica.2007.12.001
  17. L. Socha, Exponential stability of singularly perturbed stochastic systems, IEEE Trans. Automat. Contr. 45 (2000), no. 3, 576-580. https://doi.org/10.1109/9.847748
  18. H. J. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl. 270 (2002), no. 1, 143-149. https://doi.org/10.1016/S0022-247X(02)00056-2
  19. H. J. Tian, Dissipativity and exponential stability of ${\theta}$-method for singularly perturbed delay differential equations with a bounded lag, J. Comput. Math. 21 (2003), no. 6, 715-726.
  20. H. J. Tian, Numerical and analytic dissipativity of the ${\theta}$-method for delay differential equations with a bounded variable lag, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), no. 5, 1839-1845. https://doi.org/10.1142/S0218127404010096
  21. L. G. Xu, Exponential p-stability of singularly perturbed impulsive stochastic delay differential systems, Int. J. Control. Autom. 9 (2011), no. 5, 966-972. https://doi.org/10.1007/s12555-011-0518-3
  22. D. Y. Xu, Z. G. Yang, and Y. M. Huang, Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Differential Equations 245 (2008), no. 6, 1681-1703. https://doi.org/10.1016/j.jde.2008.03.029

Cited by

  1. Lagrange $$p$$ p -Stability and Exponential $$p$$ p -Convergence for Stochastic Cohen–Grossberg Neural Networks with Time-Varying Delays vol.43, pp.3, 2016, https://doi.org/10.1007/s11063-015-9433-6