Abstract and Applied Analysis

Mean Square Exponential Stability of Stochastic Switched System with Interval Time-Varying Delays

Manlika Rajchakit and Grienggrai Rajchakit

Full-text: Open access

Abstract

This paper is concerned with mean square exponential stability of switched stochastic system with interval time-varying delays. The time delay is any continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, a switching rule for the mean square exponential stability of switched stochastic system with interval time-varying delays and new delay-dependent sufficient conditions for the mean square exponential stability of the switched stochastic system are first established in terms of LMIs. Numerical example is given to show the effectiveness of the obtained result.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 623014, 12 pages.

Dates
First available in Project Euclid: 1 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364846440

Digital Object Identifier
doi:10.1155/2012/623014

Mathematical Reviews number (MathSciNet)
MR2947666

Zentralblatt MATH identifier
1246.93081

Citation

Rajchakit, Manlika; Rajchakit, Grienggrai. Mean Square Exponential Stability of Stochastic Switched System with Interval Time-Varying Delays. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 623014, 12 pages. doi:10.1155/2012/623014. https://projecteuclid.org/euclid.aaa/1364846440


Export citation

References

  • M. C. de Oliveira, J. C. Geromel, and L. Hsu, “LMI characterization of structural and robust stability: the discrete-time case,” Linear Algebra and Its Applications, vol. 296, no. 1–3, pp. 27–38, 1999.
  • Y.-J. Sun, “Global stabilizability of uncertain systems with time-varying delays via dynamic observer-based output feedback,” Linear Algebra and Its Applications, vol. 353, pp. 91–105, 2002.
  • I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,” Abstract and Applied Analysis, vol. 2011, Article ID 920412, 14 pages, 2011.
  • D. Y. Khusainov, J. Diblík, Z. Svoboda, and Z. Šmarda, “Instable trivial solution of autonomous differential systems with quadratic right-hand sides in a cone,” Abstract and Applied Analysis, vol. 2011, Article ID 154916, 23 pages, 2011.
  • J. Diblík, D. Y. Khusainov, I. V. Grytsay, and Z. Šmarda, “Stability of nonlinear autonomous quadratic discrete systems in the critical case,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 539087, 23 pages, 2010.
  • J. Diblík, D. Y. Khusainov, and M. R\accent23užičková, “Solutions of Discrete equations with prescrid asymptotic behavior,” Dynamic Systems and Applications, vol. 4, pp. 395–402, 2004.
  • J. Diblík, D. Y. Khusainov, and I. V. Grytsay, “Stability investigation of nonlinear quadratic discrete dynamics systems in the critical case,” Journal of Physics, vol. 96, no. 1, Article ID 012042, 2008.
  • J. Baštinec, J. Diblík, D. Y. Khusainov, and A. Ryvolová, “Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients,” Boundary Value Problems, vol. 2010, Article ID 956121, 20 pages, 2010.
  • J. Diblík and I. Hlavičková, “Combination of Liapunov and retract methods in the investigation of the asymptotic behavior of solutions of systems of discrete equations,” Dynamic Systems and Applications, vol. 18, no. 3-4, pp. 507–537, 2009.
  • J. Baštinec, J. Diblík, and Z. Šmarda, “Existence of positive solutions of discrete linear equations with a single delay,” Journal of Difference Equations and Applications, vol. 16, no. 9, pp. 1047–1056, 2010.
  • J. Diblík, M. R\accent23užičková, and Z. Šmarda, “Ważewski's method for systems of dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1124–e1131, 2009.
  • O. M. Kwon and J. H. Park, “Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 58–68, 2009.
  • H. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009.
  • J. Sun, G. P. Liu, J. Chen, and D. Rees, “Improved delay-range-dependent stability criteria for linear systems with time-varying delays,” Automatica, vol. 46, no. 2, pp. 466–470, 2010.
  • W. Zhang, X.-S. Cai, and Z.-Z. Han, “Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 174–180, 2010.
  • V. N. Phat, “Robust stability and stabilizability of uncertain linear hybrid systems with state delays,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 2, pp. 94–98, 2005.
  • V. N. Phat and P. T. Nam, “Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function,” International Journal of Control, vol. 80, no. 8, pp. 1333–1341, 2007.
  • V. N. Phat and P. Niamsup, “Stability analysis for a class of functional differential equations and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. 6265–6275, 2009.
  • V. N. Phat, Y. Khongtham, and K. Ratchagit, “LMI approach to exponential stability of linear systems with interval time-varying delays,” Linear Algebra and Its Applications, vol. 436, no. 1, pp. 243–251, 2012.
  • K. Ratchagit and V. N. Phat, “Stability criterion for discrete-time systems,” Journal of Inequalities and Applications, vol. 2010, Article ID 201459, 6 pages, 2010.
  • F. Uhlig, “A recurring theorem about pairs of quadratic forms and extensions: a survey,” Linear Algebra and Its Applications, vol. 25, pp. 219–237, 1979.
  • K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proceedings of the 39th IEEE Confernce on Decision and Control, vol. 3, pp. 2805–2810, IEEE Publisher, New York, NY, USA, 2000.
  • Y. Wang, L. Xie, and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992.
  • S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994.