Graph-theoretic approach to exponential stability of delayed coupled systems on networks under periodically intermittent control
DOI: https://doi.org/10.3846/mma.2018.004Abstract
In this paper, the exponential stability of delayed coupled systems on networks (DCSNs) is investigated via periodically intermittent control. By utilizing graph-theoretic approach and Lyapunov function method, a novel method for stability analysis of DCSNs is developed. Moreover, some useful and easily verifiable sufficient conditions are presented in the form of Lyapunov-type theorem and coefficients-type criterion. These laws reveal that the stability has a close relationship with the topology structure of the networks. In addition, as a subsequent result, the obtained theory is successfully applied to study the exponential stability of delayed coupled oscillators on networks under periodically intermittent control. Finally, a numerical example is given to validate the effectiveness of theoretical results.
Keywords:
delayed coupled systems, periodically intermittent control, graph-theoretic method, exponential stabilityHow to Cite
Guo, B., Xiao, Y., & Zhang, C. (2018). Graph-theoretic approach to exponential stability of delayed coupled systems on networks under periodically intermittent control. Mathematical Modelling and Analysis, 23(1), 44-63. https://doi.org/10.3846/mma.2018.004
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Copyright (c) 2018 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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C. Hu, J. Yu, H. Jiang and Z. Teng. Exponential lag synchronization for neural networks with mixed delays via periodically intermittent control. Chaos, 20(2):023108, 2010. <a href= "https://doi.org/10.1063/1.3391900" target="_blank" rel="noopener"> https://doi.org/10.1063/1.3391900</a>
T. Huang, C. Li, S. Duan and J. Starzyk. Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects. IEEE Trans. Neural Netw. Learn. Syst., 23(6):866–875, 2012. <a href= "https://doi.org/10.1109/TNNLS.2012.2192135“ target="_blank" rel="noopener"> https://doi.org/10.1109/TNNLS.2012.2192135</a>
C. Li, X. Liao and T. Huang. Exponential stabilization of chaotic systems with delay by periodically intermittent control. Chaos, 17(1):013103, 2007. <a href= "https://doi.org/10.1063/1.2430394" target="_blank" rel="noopener"> https://doi.org/10.1063/1.2430394</a>
C. Li, X. Yu, T. Huang and X. He. Distributed optimal consensus over resource allocation network and its application to dynamical economic dispatch. IEEE Trans. Neural Netw. Learn. Syst., PP(99):1–12, 2017. <a href= "https://doi.org/10.1109/TNNLS.2017.2691760" target="_blank" rel="noopener"> https://doi.org/10.1109/TNNLS.2017.2691760</a>
H. Li, G. Chen, T. Huang and Z. Dong. High-performance consensus control in networked systems with limited bandwidth communication and time-varying directed topologies. IEEE Trans. Neural Netw. Learn. Syst., 28(5):1043–1054, 2017. <a href= "https://doi.org/10.1109/TNNLS.2016.2519894" target="_blank" rel="noopener"> https://doi.org/10.1109/TNNLS.2016.2519894</a>
H. Li, G. Chen, T. Huang and Z. Dong et al. Event-triggered distributed average consensus over directed digital networks with limited communication bandwidth. IEEE T. Cybern., 46(12):3098–3110, 2016. <a href= "https://doi.org/10.1109/TCYB.2015.2496977" target="_blank" rel="noopener"> https://doi.org/10.1109/TCYB.2015.2496977</a>
J. Li, C. Li, Y. Xu and Z. Dong et al. Noncooperative game-based distributed charging control for plug-in electric vehicles in distribution networks. IEEE Trans. Ind. Inform., PP(99):1–1, 2016. <a href= "https://doi.org/10.1109/TII.2016.2632761" target="_blank" rel="noopener"> https://doi.org/10.1109/TII.2016.2632761</a>
M.Y. Li and Z. Shuai. Global-stability problem for coupled systems of differential equations on networks. J. Differ. Equ., 248(1):1–20, 2010. <a href= "https://doi.org/10.1016/j.jde.2009.09.003" target="_blank" rel="noopener"> https://doi.org/10.1016/j.jde.2009.09.003</a>
W. Li, H. Su, D. Wei and K. Wang. Global stability analysis of coupled nonlinear systems with Markovian switching. Commun. Nonlinear Sci. Numer. Simulat., 17(6):2609–2616, 2012. <a href= "https://doi.org/10.1016/j.cnsns.2011.09.039" target="_blank" rel="noopener"> https://doi.org/10.1016/j.cnsns.2011.09.039</a>
W. Li, X. Zhang and C. Zhang. Exponential stability of delayed multi-group model with reaction-diffusion and multiple dispersal based on Razumikhin technique and graph theory. Commun. Nonlinear Sci. Numer. Simulat., 27(1–3):237–253, 2015. <a href= "https://doi.org/10.1016/j.cnsns.2015.03.012" target="_blank" rel="noopener"> https://doi.org/10.1016/j.cnsns.2015.03.012</a>
X. Mao and C. Yuan. Stochastic differential equations with Markovian switching. Imperial College Press, 2006. <a href= "https://doi.org/10.1142/p473" target="_blank" rel="noopener"> https://doi.org/10.1142/p473</a>
D. Reddy, A. Sen and G. Johnston. Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett., 80(23):5109–5112., 1998. <a href= "https://doi.org/10.1103/PhysRevLett.80.5109" target="_blank" rel="noopener"> https://doi.org/10.1103/PhysRevLett.80.5109</a>
H. Su, W. Li, K. Wang and X. Ding. Stability analysis for stochastic neural network with infinite delay. Neurocomputing, 74(10):1535–1540, 2011. <a href= "https://doi.org/10.1016/j.neucom.2010.12.027" target="_blank" rel="noopener"> https://doi.org/10.1016/j.neucom.2010.12.027</a>
H. Su, P. Wang and X. Ding. Stability analysis of discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its applications. Discrete Contin. Dyn. Syst-Ser. B, 21:253–269, 2016.
J. Suo, J. Sun and Y. Zhang. Stability analysis for impulsive coupled systems on networks. Neurocomputing, 99:172–177, 2013. <a href= "https://doi.org/10.1016/j.neucom.2012.06.002" target="_blank" rel="noopener"> https://doi.org/10.1016/j.neucom.2012.06.002</a>
P. Venkataram, S. Ghosal and B.P.V. Kumar. Neural network based optimal routing algorithm for communication networks. Neural Netw., 15(10):1289–1298, 2002. <a href= "https://doi.org/10.1016/S0893-6080(02)00067-9" target="_blank" rel="noopener"> https://doi.org/10.1016/S0893-6080(02)00067-9</a>
Z. Wang, Z. Duan and J. Cao. Impulsive synchronization of coupled dynamical networks with nonidentical duffing oscillators and coupling delays. Chaos, 22(1):013140, 2012. <a href= "https://doi.org/10.1063/1.3692971" target="_blank" rel="noopener"> https://doi.org/10.1063/1.3692971</a>
D. West. Introduction to graph theory. Prentice Hall, Upper Saddle River, 1996.
J. Xiao, G. Hu and Z. Qu. Synchronization of spatiotemporal chaos and its application to multichannel spread-spectrum communication. Phys. Rev. Lett., 77(20):4162–4165, 1996. <a href= "https://doi.org/10.1103/PhysRevLett.77.4162" target="_blank" rel="noopener"> https://doi.org/10.1103/PhysRevLett.77.4162</a>
X. Yang and J. Cao. Stochastic synchronization of coupled neural networks with intermittent control. Phys. Lett. A, 373(36):3259–3272, 2009. <a href= "https://doi.org/10.1016/j.physleta.2009.07.013" target="_blank" rel="noopener"> https://doi.org/10.1016/j.physleta.2009.07.013</a>
J. Yu, C. Hu and H. Jiang. Exponential synchronization of Cohen-Grossberg neural networks via periodically intermittent control. Neurocomputing, 74(10):1776–1782, 2011. <a href= "https://doi.org/10.1016/j.neucom.2011.02.015" target="_blank" rel="noopener"> https://doi.org/10.1016/j.neucom.2011.02.015</a>
C. Zhang, W. Li and K. Wang. Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling. Appl. Math. Model., 37(7):5394–5402, 2013. <a href= "https://doi.org/10.1016/j.apm.2012.10.032" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2012.10.032</a>
C. Zhang, W. Li and K. Wang. A graph-theoretic approach to stability of neutral stochastic coupled oscillators network with time-varying delayed coupling. Math. Meth. Appl. Sci., 37(8):1179–1190, 2014. <a href= "https://doi.org/10.1002/mma.2879" target="_blank" rel="noopener"> https://doi.org/10.1002/mma.2879</a>
C. Zhang, W. Li and K. Wang. Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching. Nonlinear Anal.-Hybrid Syst., 15:37–51, 2015. <a href= "https://doi.org/10.1016/j.nahs.2014.07.003" target="_blank" rel="noopener"> https://doi.org/10.1016/j.nahs.2014.07.003</a>
G. Zhang and Y. Shen. Exponential stability of memristor-based chaotic neural networks with time-varying delays via intermittent control. IEEE Trans. Neural Netw. Learn. Syst., 26(7):1431–1441, 2015. <a href= "https://doi.org/10.1109/TNNLS.2014.2345125" target="_blank" rel="noopener"> https://doi.org/10.1109/TNNLS.2014.2345125</a>
J. Zhang and X. Guo. Stability and bifurcation analysis in the delay-coupled van der Pol oscillators. Appl. Math. Model., 34(9):2291–2299, 2010. <a href= "https://doi.org/10.1016/j.apm.2009.10.037" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2009.10.037</a>
W. Zhang, C. Li, T. Huang and J. Huang. Stability and synchronization of memristor-based coupling neural networks with time-varying delays via intermittent control. Neurocomputing, 173(Part 3):1066–1072, 2016. <a href= "https://doi.org/10.1016/j.neucom.2015.08.063" target="_blank" rel="noopener"> https://doi.org/10.1016/j.neucom.2015.08.063</a>
X. Zhang, W. Li and K. Wang. The existence and global exponential stability of periodic solution for a neutral coupled system on networks with delays. Appl. Math. Comput., 264:208–217, 2015. <a href= "https://doi.org/10.1016/j.amc.2015.04.109" target="_blank" rel="noopener"> https://doi.org/10.1016/j.amc.2015.04.109</a>
C. Zhou and T. Chen. Digital communication robust to transmission error via chaotic synchronization based on contraction maps. Phys. Rev. Lett., 56(2):1599–1604, 1997. <a href= "https://doi.org/10.1103/PhysRevE.56.1599" target="_blank" rel="noopener"> https://doi.org/10.1103/PhysRevE.56.1599</a>
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Guo, B., Xiao, Y., & Zhang, C. (2018). Graph-theoretic approach to exponential stability of delayed coupled systems on networks under periodically intermittent control. Mathematical Modelling and Analysis, 23(1), 44-63. https://doi.org/10.3846/mma.2018.004