Abstract
In this paper, we study singularly perturbed impulsive stochastic delay differential systems (SPISDDSs). By establishing an L-operator delay differential inequality and using the stochastic analysis technique, we obtain some sufficient conditions ensuring the exponential p-stability of any solution of SPISDDSs for sufficiently small ɛ > 0. The results extend and improve the earlier publications. An example is also discussed to illustrate the efficiency of the obtained results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. W. Derstine, H. M. Gibbs, F. A. Hopf, and D. L. Kaplan, “Bifurcation gap in a hybrid optical system,” Phys. Rev. A, vol. 26, no. 6, pp. 3720–3722, December 1982.
M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287–289, July 1977.
J. H. Cruz and P. Z. Táboas, “Periodic solutions and stability for a singularly perturbed linear delay differential equation,” Nonlinear Anal., vol. 67, no. 6, pp. 1657–1667, September 2007.
X. Z. Liu, X. M. Shen, and Y. Zhang, “Exponential stability of singularly perturbed systems with time delay,” Appl. Anal., vol. 82, no. 2, pp. 117–130, February 2003.
H. J. Tian, “The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag,” J. Math. Anal. Appl., vol. 270, no. 1, pp. 143–149, June 2002.
H. J. Tian, “Dissipativity and exponential stability of theta-method for singularly perturbed delay differential equations with a bounded lag,” J. Comput. Math., vol. 21, no. 6, pp. 715–726, November 2003.
D. Y. Xu, Z. G. Yang, and Z. C. Yang, “Exponential stability of nonlinear impulsive neutral differential equations with delays,” Nonlinear Anal., vol. 67, no. 5, pp. 1426–1439, September 2007.
D. Y. Xu, W. Zhu, and S. J. Long, “Global exponential stability of impulsive integro-differential equation,” Nonlinear Anal., vol. 64, no. 12, pp. 2805–2816, June 2006.
D. D. Bainov and P. S. Simenov, Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood Limited, Chichester, 1989.
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
P. S. Simeonov and D. D. Bainov, “Stability of the solutions of singularly perturbed systems with impulse effect,” J. Math. Anal. Appl., vol. 136, no. 2, pp. 575–588, December 1988.
P. S. Simeonov and D. D. Bainov, “Exponential stability of the solutions of singularly perturbed systems with impulse effect,” J. Math. Anal. Appl., vol. 151, no. 2, pp. 462–487, September 1990.
W. Zhu, D. Y. Xu, and C. D. Yang, “Exponential stability of singularly perturbed impulsive delay differential equations,” J. Math. Anal. Appl., vol. 328, no. 2, pp. 1161–1172, April 2007.
Z. G. Yang, D. Y. Xu, and L. Xiang, “Exponential p-stability of impulsive stochastic differential equations with delays,” Phys. Lett. A, vol. 359, no. 2, pp. 129–137, November 2006.
S.-E.A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1986.
X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publication, Chichester, 1997.
X. R. Mao, “Attraction, stability and boundedness for stochastic differential delay equations,” Nonlinear Anal., vol. 47, no. 7, pp. 4795–4806, August 2001.
X. R. Mao, “Razumikihin-type theorems on exponential stability of stochastic functional differential equations,” Stochast. Proc. Appl., vol. 65, no. 2, pp. 233–250, December 1996.
L. G. Xu and D. Y. Xu, “Mean square exponential stability of impulsive control stochastic systems with time-varying delay,” Phys. Lett. A, vol. 373, no. 3, pp. 328–333, January 2009.
L. Socha, “Exponential stability of singularly perturbed stochastic systems,” IEEE Trans. Automat. Contr., vol. 45, no. 3, pp. 576–580, March 2000.
M. El-Ansary, “Stochastic feedback design for a class of nonlinear singularly perturbed systems,” Int. J. Syst. Sci., vol. 22, no. 10, pp. 2013–2023, October 1991.
M. El-Ansary and H. K. Khalil, “On the interplay of singular perturbations and wide-band stochastic fluctuations,” SIAM J. Contr. Optim., vol. 24, no. 1, pp. 83–94, January 1986.
E. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, New York, 1961.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Editor Young Il Lee. The work is supported by National Natural Science Foundation of China under Grants 11101367 and 11026140. The author thanks the reviewers for their constructive suggestions and helpful comments.
Liguang Xu received his Ph.D. degree in Operational Research and Cybernetics from the College of Mathematics, Sichuan University, China. His current research interests include qualitative theory of stochastic and impulsive systems, delay differential systems, switched systems and neural networks.
Rights and permissions
About this article
Cite this article
Xu, L. Exponential p-stability of singularly perturbed impulsive stochastic delay differential systems. Int. J. Control Autom. Syst. 9, 966–972 (2011). https://doi.org/10.1007/s12555-011-0518-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-011-0518-3