Peer-to-Peer Networking and Applications

, Volume 12, Issue 6, pp 1716–1725 | Cite as

The express decay effect of time delays for globally exponentially stable nonlinear stochastic systems

  • Kaili Sun
  • Song ZhuEmail author
Part of the following topical collections:
  1. Special Issue on Networked Cyber-Physical Systems


This paper considers the express decay effect of time-varying delays for nonlinear stochastic systems. Given a globally exponentially stable nonlinear stochastic system, we investigate upper bounds of the time delays that the perturbed stochastic system can withstand to sustain its previous stability or decay rate faster than before. The upper bounds of the time delays are obtained directly from the global exponential stability coefficients conditions and expressed by transcendental equations containing adjustable parameters. Finally, a numerical simulation is conducted to illustrate the effectiveness of the theoretical results.


Global exponential stability Time delays Exponential decay Nonlinear stochastic system 



This work was supported by the National Natural Science Foundation of China under Grant 61873271 and the Fundamental Research Funds for the Central Universities 2018XK QYMS15.


  1. 1.
    Hale J (1977) Functional differential equations. Springer-Verlag, New YorkzbMATHGoogle Scholar
  2. 2.
    Appleby JAD, Mao X (2005) Stochastic stabilization of functional differential equations. Syst Control Lett 54:1069–1081zbMATHGoogle Scholar
  3. 3.
    Mao X (2007) Stability and stabilization of stochastic differential delay equations. IET Control Theory Appl 1:1551–1566Google Scholar
  4. 4.
    Zhu Q, Li X (2012) Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen-Grossberg neural networks. Fuzzy Set Syst 203:74–94MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen H (2013) New delay-dependent stability criteria for uncertain stochastic neural networks with discrete interval and distributed delays. Neurocomputing 101:1–9Google Scholar
  6. 6.
    Zhu Q, Cao J (2014) Mean-square exponential input-to-state stability of stochastic delayed neural networks. Neurocomputing 131:157–163Google Scholar
  7. 7.
    Obradović M, Milośević M (2017) Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler-Maruyama method. J Comput Appl Math 309:244–266MathSciNetzbMATHGoogle Scholar
  8. 8.
    Liu Y, Guo Y, Li W (2016) The stability of stochastic coupled systems with time delays and time-varying coupling structure. Appl Math Comput 290:507–520MathSciNetzbMATHGoogle Scholar
  9. 9.
    Pavlović G, Janković S (2012) Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinnite delay. J Comput Appl Math 236:1679–1690MathSciNetzbMATHGoogle Scholar
  10. 10.
    Liu L, Zhu Q (2016) Mean square stability of two classes of theta method for neutral stochastic differential delay equations. J Comput Appl Math 305:55–67MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen Y, Zheng W, Xue A (2010) A new result on stability analysis for stochastic neutral systems. Automatica 46:2100– 2104MathSciNetzbMATHGoogle Scholar
  12. 12.
    Zong X, Wu F, Huang C (2015) Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations. J Comput Appl Math 286:172– 185MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chen W, Zheng W (2010) Robust stability analysis for stochastic netural networks with time-varying delay. IEEE Trans Neural Netw 21:508–514Google Scholar
  14. 14.
    Jiang F, Yang H, Shen Y (2013) On the robustness of global exponential stability for hybrid neural networks with noise and delay perturbations. Neural Comput Appl 24:1497– 1504Google Scholar
  15. 15.
    Shen Y, Wang J (2012) Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances. IEEE Trans Neural Netw Learn Syst 23:87–96Google Scholar
  16. 16.
    Li C, Hu W, Wu S (2011) Stochastic stability of impulsive BAM neural networks with time delays. Comput Math Appl 61:2313–2316MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chen H, Zhong S, Shao J (2015) Exponential stability criterion for interval neural networks with discrete and distributed delays. Appl Math Comput 250:121–130MathSciNetzbMATHGoogle Scholar
  18. 18.
    Song Y, Yin Q, Shen Y, Wang G (2013) Stochastic suppression and stabilization of nonlinear differential systems with general decay rate. J Frankl Inst 350:2084–2095MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mao X (2007) Stochastic differential equations and applications, 2nd edn. Harwood, ChichesterzbMATHGoogle Scholar
  20. 20.
    Marco M, Grazzini M, Pancioni L (2011) Global robust stability criteria for interval delayed full-range cellular neural networks. IEEE Trans Neural Netw 22:666–671Google Scholar
  21. 21.
    Mathiyalagan K, Sakthivel R, Marshal Anthoni S (2012) New robust exponential stability results for discrete-time switched fuzzy neural networks with time delays. Comput Math Appl 64:2926–2938MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zhang Y (2013) Exponential stability analysis for discrete-time impulsive delay neural networks with and without uncertainty. J Franklin Inst 350:737–756MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zeng Z, Wang J, Liao X (2003) Global exponential stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans Circuits Syst I(50):1353– 1358MathSciNetzbMATHGoogle Scholar
  24. 24.
    Zhao W, Zhu Q (2010) New results of global robust exponential stability of neural networks with delays. Nonlinear Anal: Real World Appl 11:1190–1197MathSciNetzbMATHGoogle Scholar
  25. 25.
    Zhu S, Yang Q, Shen Y (2016) Noise further expresses exponential decay for globally exponentially stable time-varying delayed neural networks. Neural Netw 77:7–13zbMATHGoogle Scholar
  26. 26.
    Shen Y, Wang J (2013) Robustness analysis of global exponential stability of non-linear systems with time delays and neutral terms. IET Control Theory Appl 7:1227–1232MathSciNetGoogle Scholar
  27. 27.
    Li L, Jian J (2015) Exponential convergence and lagrange stability for impulsive Cohen-Grossberg neural networks with time-varying delays. J Comput Appl Math 277:23–35MathSciNetzbMATHGoogle Scholar
  28. 28.
    Chen L, Zhao H, New LMI (2009) conditions for global exponential stability of cellular neural networks with delays. Neural Netw 10:287–297MathSciNetzbMATHGoogle Scholar
  29. 29.
    Chen H, Zhang Y, Hu P (2010) Novel delay-dependent robust stability criteria for neutral stochastic delayed neural networks. Neurocomputing 73:2554–2561Google Scholar
  30. 30.
    Chen H, Zhao Y (2015) Delay-dependent exponential stability for uncertain neutral stochastic neural networks with interval time-varying delay. Int J Syst Sci 46:2584–2597MathSciNetzbMATHGoogle Scholar
  31. 31.
    Li D, Zhu Q (2014) Comparison principle and stability of stochastic delayed neural networks with Markovian switching. Neurocomputing 123:436–442Google Scholar
  32. 32.
    Ito H, Nishimurra Y (2015) Stability of stochastic nonlinear systems in cascade with not necessarily unbounded decay rates. Automatica 62:51–64MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zhou B, Luo W (2018) Improved Razumikhin and Krasovskii stability criteria for time-varying stochastic time-delay systems. Automatica 89:382–391MathSciNetzbMATHGoogle Scholar
  34. 34.
    Li M, Liu L, Deng F (2018) Input-to-state stability of switched stochastic delayed systems with Lvy noise. J Franklin Inst 355:314–331MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hou M, Fu F, Duan G (2013) Global stabilization of switched stochastic nonlinear systems in strict-feedback form under arbitrary switchings. Automatica 49:2571–2575MathSciNetzbMATHGoogle Scholar
  36. 36.
    Zhu S, Shen Y (2013) Robustness analysis for connection weight matrices of global exponential stability of stochastic recurrent neural networks. Neural Netw 38:17–22zbMATHGoogle Scholar
  37. 37.
    Khasminskii R (1980) Stochastic stability of differential equations. Sijthoff and NoordhoffGoogle Scholar
  38. 38.
    Mao X (1994) Stochastic stabilisation and destabilization. Syst Control Lett 23:279–290zbMATHGoogle Scholar
  39. 39.
    Zhang H, Qi Y, Wu J, Fu L, He L (2018) DoS attack energy management against remote state estimation. IEEE Trans Control Netw Syst 5:383–394MathSciNetzbMATHGoogle Scholar
  40. 40.
    Zhang H, YQi H, Zhou J, Zhang J (2017) Sun, testing and defending methods against DoS attack in state estimation. Asian J Control 19:1295–1305MathSciNetzbMATHGoogle Scholar
  41. 41.
    Zhang H, Zheng W (2018) Denial-of-service power dispatch against linear quadratic control via a fading channel. IEEE Trans Autom Control 63:3032–3039MathSciNetzbMATHGoogle Scholar
  42. 42.
    Zhang H, Meng W, Qi J, Wang X, Zheng W (2019) Distributed load sharing under false data injection attack in inverter-based microgrid. IEEE Trans Ind Electron 66:1543–1551Google Scholar
  43. 43.
    Yang C, Shi Z, Han K, Zhang J, Gu Y, Qin Z (2018) Optimization of particle CBMeMBer filters for hardware implementation. IEEE Trans Veh Technol 67:9027–9031Google Scholar
  44. 44.
    Yang G, He S, Shi Z (2017) Leveraging crowdsourcing for efficient malicious users detection in large-scale social networks. IEEE Internet Things J 4:330–339Google Scholar
  45. 45.
    Yang G, He S, Shi Z, Chen J (2017) Promoting cooperation by social incentive mechanism in mobile crowdsensing. IEEE Commun Mag 55:86–92Google Scholar
  46. 46.
    Zhu Y, Zhong Z, Basin MV, Zhou D (2018) A descriptor system approach to stability and stabilization of discrete-time switched PWA systems. IEEE Trans Autom Control 63:3456–3463MathSciNetzbMATHGoogle Scholar
  47. 47.
    Zhu Y, Zhong Z, Zheng W, Zhou D (2018) HMM-based H-∞ filtering for discrete-time Markov jump LPV systems over unreliable communication channels. IEEE Trans Syst, Man, and Cybernetics: Syst 48:2035–2046Google Scholar
  48. 48.
    Zhu Y, Zhang L, Zheng W (2016) Distributed H-∞ filtering for a class of discrete-time Markov jump lure systems with redundant channels. IEEE Trans Ind Electron 63:1876–1885Google Scholar
  49. 49.
    Wang L, Ge M, Zeng Z, Hu J (2018) Finite-time robust consensus of nonlinear disturbed multiagent systems via two-layer event-triggered control. Inform Sci 466:270–283MathSciNetGoogle Scholar
  50. 50.
    Chen H, Shi P, Lim C (2017) Exponential synchronization for Markovian stochastic coupled neural networks of neutral-type via adaptive feedback control. IEEE Trans Neural Netw Learn Syst 28:1618–1632MathSciNetGoogle Scholar
  51. 51.
    Chen H, Shi P, Lim C (2017) Stability of neutral stochastic switched time delay systems: an average dwell time approach. Int J Robust Nonlinear Control 27:512–532MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouChina

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