Abstract
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.
Similar content being viewed by others
References
Hale J (1977) Functional differential equations. Springer-Verlag, New York
Appleby JAD, Mao X (2005) Stochastic stabilization of functional differential equations. Syst Control Lett 54:1069–1081
Mao X (2007) Stability and stabilization of stochastic differential delay equations. IET Control Theory Appl 1:1551–1566
Zhu Q, Li X (2012) Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen-Grossberg neural networks. Fuzzy Set Syst 203:74–94
Chen H (2013) New delay-dependent stability criteria for uncertain stochastic neural networks with discrete interval and distributed delays. Neurocomputing 101:1–9
Zhu Q, Cao J (2014) Mean-square exponential input-to-state stability of stochastic delayed neural networks. Neurocomputing 131:157–163
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 309:244–266
Liu Y, Guo Y, Li W (2016) The stability of stochastic coupled systems with time delays and time-varying coupling structure. Appl Math Comput 290:507–520
Pavlović G, Janković S (2012) Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinnite delay. J Comput Appl Math 236:1679–1690
Liu L, Zhu Q (2016) Mean square stability of two classes of theta method for neutral stochastic differential delay equations. J Comput Appl Math 305:55–67
Chen Y, Zheng W, Xue A (2010) A new result on stability analysis for stochastic neutral systems. Automatica 46:2100– 2104
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– 185
Chen W, Zheng W (2010) Robust stability analysis for stochastic netural networks with time-varying delay. IEEE Trans Neural Netw 21:508–514
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– 1504
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 23:87–96
Li C, Hu W, Wu S (2011) Stochastic stability of impulsive BAM neural networks with time delays. Comput Math Appl 61:2313–2316
Chen H, Zhong S, Shao J (2015) Exponential stability criterion for interval neural networks with discrete and distributed delays. Appl Math Comput 250:121–130
Song Y, Yin Q, Shen Y, Wang G (2013) Stochastic suppression and stabilization of nonlinear differential systems with general decay rate. J Frankl Inst 350:2084–2095
Mao X (2007) Stochastic differential equations and applications, 2nd edn. Harwood, Chichester
Marco M, Grazzini M, Pancioni L (2011) Global robust stability criteria for interval delayed full-range cellular neural networks. IEEE Trans Neural Netw 22:666–671
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 64:2926–2938
Zhang Y (2013) Exponential stability analysis for discrete-time impulsive delay neural networks with and without uncertainty. J Franklin Inst 350:737–756
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– 1358
Zhao W, Zhu Q (2010) New results of global robust exponential stability of neural networks with delays. Nonlinear Anal: Real World Appl 11:1190–1197
Zhu S, Yang Q, Shen Y (2016) Noise further expresses exponential decay for globally exponentially stable time-varying delayed neural networks. Neural Netw 77:7–13
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 7:1227–1232
Li L, Jian J (2015) Exponential convergence and lagrange stability for impulsive Cohen-Grossberg neural networks with time-varying delays. J Comput Appl Math 277:23–35
Chen L, Zhao H, New LMI (2009) conditions for global exponential stability of cellular neural networks with delays. Neural Netw 10:287–297
Chen H, Zhang Y, Hu P (2010) Novel delay-dependent robust stability criteria for neutral stochastic delayed neural networks. Neurocomputing 73:2554–2561
Chen H, Zhao Y (2015) Delay-dependent exponential stability for uncertain neutral stochastic neural networks with interval time-varying delay. Int J Syst Sci 46:2584–2597
Li D, Zhu Q (2014) Comparison principle and stability of stochastic delayed neural networks with Markovian switching. Neurocomputing 123:436–442
Ito H, Nishimurra Y (2015) Stability of stochastic nonlinear systems in cascade with not necessarily unbounded decay rates. Automatica 62:51–64
Zhou B, Luo W (2018) Improved Razumikhin and Krasovskii stability criteria for time-varying stochastic time-delay systems. Automatica 89:382–391
Li M, Liu L, Deng F (2018) Input-to-state stability of switched stochastic delayed systems with Lvy noise. J Franklin Inst 355:314–331
Hou M, Fu F, Duan G (2013) Global stabilization of switched stochastic nonlinear systems in strict-feedback form under arbitrary switchings. Automatica 49:2571–2575
Zhu S, Shen Y (2013) Robustness analysis for connection weight matrices of global exponential stability of stochastic recurrent neural networks. Neural Netw 38:17–22
Khasminskii R (1980) Stochastic stability of differential equations. Sijthoff and Noordhoff
Mao X (1994) Stochastic stabilisation and destabilization. Syst Control Lett 23:279–290
Zhang H, Qi Y, Wu J, Fu L, He L (2018) DoS attack energy management against remote state estimation. IEEE Trans Control Netw Syst 5:383–394
Zhang H, YQi H, Zhou J, Zhang J (2017) Sun, testing and defending methods against DoS attack in state estimation. Asian J Control 19:1295–1305
Zhang H, Zheng W (2018) Denial-of-service power dispatch against linear quadratic control via a fading channel. IEEE Trans Autom Control 63:3032–3039
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 66:1543–1551
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 67:9027–9031
Yang G, He S, Shi Z (2017) Leveraging crowdsourcing for efficient malicious users detection in large-scale social networks. IEEE Internet Things J 4:330–339
Yang G, He S, Shi Z, Chen J (2017) Promoting cooperation by social incentive mechanism in mobile crowdsensing. IEEE Commun Mag 55:86–92
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 63:3456–3463
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 48:2035–2046
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 63:1876–1885
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 466:270–283
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 28:1618–1632
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 27:512–532
Acknowledgment
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the Topical Collection: Special Issue on Networked Cyber-Physical Systems
Guest Editors: Heng Zhang, Mohammed Chadli, Zhiguo Shi, Yanzheng Zhu, and Zhaojian Li
Rights and permissions
About this article
Cite this article
Sun, K., Zhu, S. The express decay effect of time delays for globally exponentially stable nonlinear stochastic systems. Peer-to-Peer Netw. Appl. 12, 1716–1725 (2019). https://doi.org/10.1007/s12083-019-00735-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12083-019-00735-1