Abstract
Stabilization for multi-group coupled models stochastic by delay feedback control and nonlinear impulsive control are considered in this paper. Using graph theory and Lyapunov method, some sufficient conditions are acquired by some control methods. Those criteria are easier to verify and no need to solve any linear matrix inequalities. These results can be designed more easily in practice. At last, the effectiveness and advantage of the theoretical results are verified by an example.
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References
Harary F. Graph Theory. Boston: Addison-Wesley, 1969
West D B. Introduction to Graph Theory. Upper Saddle River: Prentice Hall, 1996
Guo H B, Li M Y, Shuai Z S. A graph-theoretic approach to the method of global Lyapunov functions. Proc Amer Math Soc, 2008, 136: 2793–2802
Zhang C M, Li W X, Wang K. Graph-theoretic approach to stability of multi-group models with dispersal. Discrete Contin Dyn Syst-Ser B, 2015, 20: 259–280
Fotsin H B, Daafouz J. Adaptive synchronization of uncertain chaotic colpitts oscillators based on parameter identification. Phys Lett A, 2005, 339: 304–315
Huang D. Simple adaptive-feedback controller for identical chaos synchronization. Phys Rev E, 2005, 71: 037203
Lu J Q, Cao J D. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. Chaos, 2005, 15: 043901
Park J H. Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters. J Comput Appl Math, 2008, 213: 288–293
Agiza H N, Yassen M T. Synchronization of Rossler and Chen chaotic dynamical systems using active control. Phys Lett A, 2001, 278: 191–197
Lei Y M, Xu W, Zheng H C. Synchronization of two chaotic nonlinear gyros using active control. Phys Lett A, 2005, 343: 153–158
Wang C C, Su J P. A new adaptive variable structure control for chaotic synchronization and secure communication. Chaos Soliton Fract, 2004, 20: 967–977
Park J H. Synchronization of Genesio chaotic system via backstepping approach. Chaos Soliton Fract, 2006, 27: 1369–1375
Chen W M, Xu S Y, Zou Y. Stabilization of hybrid neutral stochastic differential delay equations by delay feedback control. Syst Control Lett, 2016, 88: 1–13
Mao X R, Lam J, Huang L R. Stabilisation of hybrid stochastic differential equations by delay feedback control. Syst Control Lett, 2008, 57: 927–935
Zhu Q X, Zhang Q Y. pth moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete-time state observations with a time delay. IET Control Theory Appl, 2017, 11: 1992–2003
Fotsin H B, Woafo P. Adaptive synchronization of a modified and uncertain chaotic Van der Pol-Duffing oscillator based on parameter identification. Chaos Soliton Fract, 2005, 24: 1363–1371
Heagy J F, Carroll T L, Pecora L M. Experimental and numerical evidence for riddled basins in coupled chaotic systems. Phys Rev Lett, 1994, 73: 3528–3531
Huang L L, Feng R P, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A, 2004, 320: 271–275
Kakmeni F, Bowong S, Tchawoua C. Nonlinear adaptive synchronization of a class of chaotic systems. Phys Lett A, 2006, 355: 47–54
Deng F Q, Luo Q, Mao X R. Stochastic stabilization of hybrid differential equations. Automatica, 2012, 48: 2321–2328
Zhu Q X. Razumikhin-type theorem for stochastic functional differential equations with L´evy noise and Markov switching. Int J Control, 2017, 90: 1703–1712
Wang B, Zhu Q X. Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. Syst Control Lett, 2017, 105: 55–61
Li C D, Liao X F, Zhang X F, et al. Impulsive stabilization and synchronization of a class of chaotic delay systems. Chaos, 2005, 15: 023104
Peng S G, Zhang Y, Yu S M. Global mean-square exponential stabilization of stochastic system with time delay via impulsive control. Asian J Control, 2012, 14: 288–299
Xu L G, He D H, Ma Q. Impulsive stabilization of stochastic differential equations with time delays. Math Comput Model, 2013, 57: 997–1004
Liu J, Liu X Z, Xie W C. Impulsive stabilization of stochastic functional differential equations. Appl Math Lett, 2011, 24: 264–269
Cheng P, Deng F Q, Peng Y J. Robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay. Commun Nonlinear Sci Numer Simul, 2012, 17: 4740–4752
Zhu Q. pth moment exponential stability of impulsive stochastic functional differential equations with Markovian switching. J Franklin Institute, 2014, 351: 3965–3986
Mao X R. Stochastic Differential Equations and Their Applications. 2nd ed. Chichester: Horwood Publishing, 2007
Yang X S, Yang Z C. Synchronization of TS fuzzy complex dynamical networks with time-varying impulsive delays and stochastic effects. Fuzzy Sets Syst, 2014, 235: 25–43
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grants Nos. 61573156, 61273126, 61503142, 11372107), Fundamental Research Funds for the Central Universities (Grant No. x2zdD2153620), Ph.D. Start-up Fund of Natural Science Foundation of Guangdong Province (Grant No. 2014A030310388), and Science and Technology Plan Foundation of Guangzhou (Grant No. 201704030131).
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Zhang, C., Deng, F. & Luo, Y. Stabilization for multi-group coupled stochastic models by delay feedback control and nonlinear impulsive control. Sci. China Inf. Sci. 61, 70212 (2018). https://doi.org/10.1007/s11432-017-9281-3
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DOI: https://doi.org/10.1007/s11432-017-9281-3