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Stability Analysis for Delayed Stochastic GRNs

  • Xian ZhangEmail author
  • Yantao Wang
  • Ligang Wu
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 207)

Abstract

In this chapter we will establish a robust asymptotic mean square stability criterion for delayed stochastic GRNs with parameter uncertainties.

References

  1. 1.
    Balasubramaniam, P., Rakkiyappan, R., Krishnasamy, R.: Stochastic stability of Markovian jumping uncertain stochastic genetic regulatory networks with interval time-varying delays. Math. Biosci. 226(2), 97–108 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balasubramaniam, P., Sathy, R.: Robust asymptotic stability of fuzzy Markovian jumping genetic regulatory networks with time-varying delays by delay decomposition approach. Commun. Nonlinear Sci. Numer. Simul. 16(2), 928–939 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balasubramaniam, P., Sathy, R., Rakkiyappan, R.: A delay decomposition approach to fuzzy Markovian jumping genetic regulatory networks with time-varying delays. Fuzzy Sets Syst. 164(1), 82–100 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Feng, W., Yang, S.X., Wu, H.: On delayed uncertain genetic regulatory networks: robust stability analysis. Int. J. Comput. Math. 88(12), 2448–2463 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gomez, M.M., Sadeghpour, M., Bennett, M.R., Orosz, G., Murray, R.M.: Stability of systems with stochastic delays and applications to genetic regulatory networks. SIAM J. Appl. Dyn. Syst. 15(4), 1844–1873 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    He, G., Fang, J.A., Wu, X.: Robust stability of Markovian jumping genetic regulatory networks with mode-dependent delays. Math. Prob. Eng. 2012 (Article ID 504378, 2012)Google Scholar
  7. 7.
    He, Y., Fu, L.Y., Zeng, J., Wu, M.: Stability of genetic regulatory networks with interval time-varying delays and stochastic perturbation. Asian J. Control 13(5), 625–634 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Krishnasamy, R., Balasubramaniam, P.: Stochastic stability analysis for switched genetic regulatory networks with interal time-varying delays based on average dwell time approach. Stoch. Anal. Appl. 32(6), 1046–1066 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lakshmanan, S., Rihan, F.A., Rakkiyappan, R., Park, J.H.: Stability analysis of the differential genetic regulatory networks model with time-varying delays and Markovian jumping parameters. Nonlinear Anal.: Hybrid Syst. 14, 1–15 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Li, C., Chen, L., Aihara, K.: Stochastic stability of genetic networks with disturbance attenuation. IEEE Trans. Circuits Syst. Video Technol. 54(10), 892–896 (2007)CrossRefGoogle Scholar
  11. 11.
    Li, J., Chesi, G., Hung, Y.S.: Robust stochastic stability of genetic regulatory networks with time delays and parametric uncertainties. Asian J. Control 13(5), 635–644 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, J., Hu, M., Cao, J., Yang, Y., Jin, Y.: Stability of uncertain impulsive stochastic genetic regulatory networks with time-varying delay in the leakage term. Abstr. Appl. Anal. 2014 (Article ID 706720, 2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li, X.D., Rakkiyappan, R.: Delay-dependent global asymptotic stability criteria for stochastic genetic regulatory networks with Markovian jumping parameters. Appl. Math. Modell. 36(4), 1718–1730 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li, X.D., Rakkiyappan, R., Pradeep, C.: Robust \(\mu \)-stability analysis of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 17(10), 3894–3905 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Li, Y.R., Zhu, Y.Z., Zeng, N.Y., Du, M.: Stability analysis of standard genetic regulatory networks with time-varying delays and stochastic perturbations. Neurocomputing 74(17), 3235–3241 (2011)CrossRefGoogle Scholar
  16. 16.
    Li, Z., Chen, K.: Exponential stability of stochastic genetic regulatory networks with interval uncertainties and multiple delays. Arab. J. Sci. Eng. 39(8), 6507–6520 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, J., Xue, D.: Asymtotic stability of Markovian jumping genetic regulatory networks with random delays. Chin. J. Electron. 22(2), 263–268 (2013)Google Scholar
  18. 18.
    Lou, X., Ye, Q., Cui, B.: Exponential stability of genetic regulatory networks with random delays. Neurocomputing 73, 759–769 (2010)CrossRefGoogle Scholar
  19. 19.
    Ma, C., Zeng, Q., Zhang, L., Zhu, Y.: Passivity and passification for Markov jump genetic regulatory networks with time-varying delays. Neurocomputing 136, 321–326 (2014)CrossRefGoogle Scholar
  20. 20.
    Meng, Q., Jiang, H.J.: Robust stochastic stability analysis of Markovian switching genetic regulatory networks with discrete and distributed delays. Neurocomputing 74(1), 362–368 (2010)CrossRefGoogle Scholar
  21. 21.
    Pan, W., Wang, Z., Hu, J.: Robust stability of delayed genetic regulatory networks with different sources of uncertainties. Asian J. Control 13(5), 645–654 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rakkiyappan, R., Balasubramaniam, P.: Delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with mixed time-varying delays: an LMI approach. Nonlinear Anal.: Hybrid Syst. 4(3), 600–607 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Rakkiyappan, R., Lakshmanan, S., Balasubramaniam, P.: Delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with time-varying delays. Circuits Syst. Signal Process. 32, 1147–1177 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ratnavelu, K., Kalpana, M., Balasubramaniam, P.: Asymtotic stability of Markovian switching genetic regulatory networks with leakage and mode-dependent time delays. J. Franklin Inst. 353(7), 1615–1638 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sakthivel, R., Raja, R., Anthoni, S.M.: Asymptotic stability of delayed stochastic genetic regulatory networks with impulses. Phys. Scr. 82(5) (Article ID 055009, 2010)CrossRefGoogle Scholar
  26. 26.
    Salimpour, A., Majd, V.J., Sojoodi, M.: Comment on: robust stability of stochastic genetic regulatory networks with discrete and distributed delays. Symp. (Int.) Combust. 15(4), 769–770 (2010)CrossRefGoogle Scholar
  27. 27.
    Wang, G., Cao, J.: Robust exponential stability analysis for stochastic genetic networks with uncertain parameters. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3369–3378 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, W., Zhong, S.: Stochastic stability analysis of uncertain genetic regulatory networks with mixed time-varying delays. Neurocomputing 82, 143–156 (2012)CrossRefGoogle Scholar
  29. 29.
    Wang, W., Zhong, S., Liu, F., Cheng, J.: Robust delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with random discrete delays and distributed delays. Int. J. Robust Nonlinear Control 24(16), 2574–2596 (2014).  https://doi.org/10.1002/rnc.3011MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, W., Zhong, S., Nguang, S.K., Liu, F.: Robust deley-probability-distribution-dependent stability of uncertain genetic regulatory networks with time-varying delays. Neurocomputing 119(SI), 153–164 (2013)CrossRefGoogle Scholar
  31. 31.
    Wang, W.Q., Nguang, S.K., Zhong, S.M., Liu, F.: Robust stability analysis of stochastic delayed genetic regulatory networks with polytopic uncertainties and linear fractional parametric uncertainties. Commun. Nonlinear Sci. Numer. Simul. 19(5), 1569–1581 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang, Y., Cao, J., Li, L.: Global robust power-rate stability of delayed genetic regulatory networks with noise perturbations. Cogn. Neurodyn. 4(1), 81–90 (2010)CrossRefGoogle Scholar
  33. 33.
    Wang, Y., Wang, Z., Liang, J.: On robust stability of stochastic genetic regulatory networks with time delays: a delay fractioning approach. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 40(3), 729–740 (2010)CrossRefGoogle Scholar
  34. 34.
    Wang, Y.T., Yu, A.H., Zhang, X.: Robust stability of stochastic genetic regulatory networks with time-varying delays: a delay fractioning approach. Neural Comput. Appl. 23(5), 1217–1227 (2013)CrossRefGoogle Scholar
  35. 35.
    Wang, Z., Liao, X., Guo, S., Wu, H.: Mean square exponential stability of stochastic genetic regulatory networks with time-varying delays. Inf. Sci. 181(4), 792–811 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wang, Z., Liao, X., Mao, J., Liu, G.: Robust stability of stochastic genetic regulatory networks with discrete and distributed delays. Symp. (Int.) Combust. 13(12), 1199–1208 (2009)CrossRefGoogle Scholar
  37. 37.
    Wang, Z.X., Liu, G.D., Sun, Y.H., Wu, H.L.: Robust stability of stochastic delayed genetic regulatory networks. Cogn. Neurodyn. 3(3), 271–280 (2009)CrossRefGoogle Scholar
  38. 38.
    Wu, H., Liao, X., Guo, S., Feng, W., Wang, Z.: Stochastic stability for uncertain genetic regulatory networks with interval time-varying delays. Neurocomputing 72(13–15), 3263–3276 (2009)CrossRefGoogle Scholar
  39. 39.
    Yu, T.T., Wang, J., Zhang, X.: A less conservative stability criterion for delayed stochastic genetic regulatory networks. Math. Prob. Eng. 2014 (Article ID 768483, 11 pages, 2014)Google Scholar
  40. 40.
    Zhang, B., Xu, S., Chu, Y., Zong, G.: Delay-dependent stability for Markovian genetic regulatory networks with time-varying delays. Asian J. Control 14(5), 1403–1406 (2012)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Zhang, D., Yu, L.: Passivity analysis for stochastic Markovian switching genetic regulatory networks with time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 16(8), 2985–2992 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zhang, W., Fang, J., Tang, Y.: Stochastic stability of Markovian jumping genetic regulatory networks with mixed time delays. Appl. Math. Comput. 217(17), 7210–7225 (2011)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Zhang, W., Tang, T., Fang, J.A., Wu, X.: Stochastic stability of genetic regulatory networks with a finite set of delay characterization. Chaos 22(2) (Article ID 023106, 2012)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zhang, W.B., Fang, J.A., Tang, Y.: New robust stability analysis for genetic regulatory networks with random discrete delays and distributed delays. Neurocomputing 74(14–15), 2344–2360 (2011)CrossRefGoogle Scholar
  45. 45.
    Zhang, W.B., Tang, Y., Wu, X.T., Fang, J.A.: Stochastic stability of switched genetic regulatory networks with time-varying delays. IEEE Trans. Nanobiosci. 13(3), 336–342 (2014)CrossRefGoogle Scholar
  46. 46.
    Zhou, Q., Xu, S.Y., Chen, B., Li, H.Y., Chu, Y.M.: Stability analysis of delayed genetic regulatory networks with stochastic disturbances. Phys. Lett. A 373(41), 3715–3723 (2009)CrossRefGoogle Scholar
  47. 47.
    Zhu, Y., Zhang, Q., Wei, Z., Zhang, L.: Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties. Neurocomputing 110, 44–50 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical ScienceHeilongjiang UniversityHarbinChina
  2. 2.School of AstronauticsHarbin Institute of TechnologyHarbinChina

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