doi: 10.3934/dcdsb.2020248

Finite-time cluster synchronization of coupled dynamical systems with impulsive effects

1. 

School of Mathematics, Southeast University, Nanjing 210096, China

2. 

Department of Mathematics, Luoyang Normal University, Luoyang 471934, China

3. 

Department of Applied Mathematics, Changsha University of Science and Technology, Changsha 410114, China

* Corresponding author: Jinde Cao

Received  December 2019 Revised  June 2020 Published  August 2020

In our paper, the finite-time cluster synchronization problem is investigated for the coupled dynamical systems in networks. Based on impulsive differential equation theory and differential inequality method, two novel Lyapunov-based finite-time stability results are proposed and be used to obtain the finite-time cluster synchronization criteria for the coupled dynamical systems with synchronization and desynchronization impulsive effects, respectively. The settling time with respect to the average impulsive interval is estimated according to the sufficient synchronization conditions. It is illustrated that the introduced settling time is not only dependent on the initial conditions, but also dependent on the impulsive effects. Compared with the results without stabilizing impulses, the attractive domain of the finite-time stability can be enlarged by adding impulsive control input. Conversely, the smaller attractive domain can be obtained when the original system is subject to the destabilizing impulses. By using our criteria, the continuous feedback control can always be designed to finite-time stabilize the unstable impulsive system. Several existed results are extended and improved in the literature. Finally, typical numerical examples involving the large-scale complex network are outlined to exemplify the availability of the impulsive control and continuous feedback control, respectively.

Citation: Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020248
References:
[1]

F. AmatoM. Ariola and C. Cosentino, Finite-time stability of linear time-varying systems: analysis and controller design, IEEE Trans. Automat. Control, 55 (2010), 1003-1008.  doi: 10.1109/TAC.2010.2041680.  Google Scholar

[2]

S. Arik, Stability analysis of delayed neural networks, IEEE Trans. Circuits Systems I Fund. Theory Appl., 47 (2000), 1089-1092.  doi: 10.1109/81.855465.  Google Scholar

[3]

K. L. Babcock and R. M. Westervelt, Dynamics of simple electronic neural networks, Physica D, 28 (1987), 305-316.  doi: 10.1016/0167-2789(87)90021-2.  Google Scholar

[4]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.  doi: 10.1126/science.286.5439.509.  Google Scholar

[5]

V. N. BelykhI. V. Belykh and M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Physica D, 195 (2004), 159-187.  doi: 10.1016/j.physd.2004.03.012.  Google Scholar

[6]

S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751-766.  doi: 10.1137/S0363012997321358.  Google Scholar

[7]

S. P. Bhat and D. S. Bernstein, Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Transactions on Automatic Control, 43 (1998), 678-682.  doi: 10.1109/9.668834.  Google Scholar

[8]

Y. CaoW. YuW. Ren and et. al, An overview of recent progress in the study of distributed Multi-Agent coordination, IEEE Transaction on Industrial Informations, 9 (2013), 427-438.  doi: 10.1109/TII.2012.2219061.  Google Scholar

[9]

W. Chen and L. C. Jiao, Finite-time stability theorem of stochastic nonlinear systems, Automatica J. IFAC, 46 (2010), 2105-2108.  doi: 10.1016/j.automatica.2010.08.009.  Google Scholar

[10]

D. Chen, W. Zhang, J. Cao, et. al, Fixed time synchronization of delayed quaternion-valued memristor-based neural networks, Adv. Difference Equ., 2020 (2020), Paper No. 92, 16 pp.. doi: 10.1186/s13662-020-02560-w.  Google Scholar

[11]

F. De Smet and D. Aeyels, Clustering in a network of non-identical and mutually interacting agents, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2019), 745-768.  doi: 10.1098/rspa.2008.0259.  Google Scholar

[12]

D. EfimovA. PolyakovE. Fridman and et. al, Comments on finite-time stability of time-delay systems, Automatica, 50 (2014), 1944-1947.  doi: 10.1016/j.automatica.2014.05.010.  Google Scholar

[13]

M. Galicki, Finite-time control of robotic manipulators, Automatica J. IFAC, 51 (2015), 49-54.  doi: 10.1016/j.automatica.2014.10.089.  Google Scholar

[14]

L. V. Gambuzza and M. Frasca, A criterion for stability of cluster synchronization in networks with external equitable partitions, Automatica J. IFAC, 100 (2019), 212-218.  doi: 10.1016/j.automatica.2018.11.026.  Google Scholar

[15]

W. M. Haddad and A. L'Afflitto, Finite-time stabilization and optimal feedback control, IEEE Trans. Automat. Control, 61 (2016), 1069-1074.  doi: 10.1109/TAC.2015.2454891.  Google Scholar

[16]

J. HeP. ChengL. Shi and et. al, Time synchronzation in WSNS: A maximum-value-based consensus approach, IEEE Trans. Automat. Control, 59 (2014), 660-675.  doi: 10.1109/TAC.2013.2286893.  Google Scholar

[17]

Y. HongZ.-P. Jiang ZP and G. Feng, Finite-time input-to-state stability and applications to finite-time control design, SIAM J. Control Optim., 48 (2010), 4395-4418.  doi: 10.1137/070712043.  Google Scholar

[18]

Y. HongJ. Wang and D. Cheng, Adaptive finite-time control of nonlinear systems with parametric uncertainty, IEEE Trans. Automat. Control, 51 (2006), 858-862.  doi: 10.1109/TAC.2006.875006.  Google Scholar

[19]

B. HuZ.-H. GuanG. Chen and et. al, Multistability of delayed hybrid impulsive neural networks with application to associative memories, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 1537-1551.  doi: 10.1109/TNNLS.2018.2870553.  Google Scholar

[20]

C. HuJ. YuZ. Chen and et. al, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Networks, 89 (2017), 74-83.  doi: 10.1016/j.neunet.2017.02.001.  Google Scholar

[21]

C. HuJ. YuH. Jiang and et al, Exponential synchronization of complex networks with finite distributed delays coupling, IEEE Transactions on Neural Networks, 22 (2011), 1999-2010.   Google Scholar

[22]

J. HuangC. WenW. Wang and Y.-D. Song, Adaptive finite-time consensus control of a group of uncertain nonlinear mechanical systems, Automatica J. IFAC, 51 (2015), 292-301.  doi: 10.1016/j.automatica.2014.10.093.  Google Scholar

[23]

S. Jalan and R. E. Amritkar, Self-organized and driven phase synchronization in coupled maps, Physical Review Letters, 90 (2003), 014101. Google Scholar

[24]

S. Jalan, R. E. Amritkar and C. K. Hu, Synchronized clusters in coupled map networks. I. Numerical studies, Physical Review E, 72 (2005), 016212. Google Scholar

[25]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Prentice Hall, Upper Saddle River, 2002. Google Scholar

[26]

M. KumarD. P. Garg and V. Kumar, Segregation of heterogeneous units in a swarm of robotic agents, IEEE Trans. Automat. Control, 55 (2010), 743-748.  doi: 10.1109/TAC.2010.2040494.  Google Scholar

[27]

Z. LiZ. DuanG. Chen and et. al, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans. Circuits Syst. I. Regul. Pap., 57 (2010), 213-224.  doi: 10.1109/TCSI.2009.2023937.  Google Scholar

[28]

X. LiD. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.  Google Scholar

[29]

X. Liu, Adaptive finite time stability of delayed systems with applications to network synchronization, (2020), arXiv: 2002.00145. Google Scholar

[30]

X. Liu and T. Chen, Finite-time and fixed-time cluster synchronization with or without pinning control, IEEE Transactions on Cybernetics, 48 (2018), 240-252.  doi: 10.1109/TCYB.2016.2630703.  Google Scholar

[31]

Z. LiuW. S. Wong and H. Cheng, Cluster synchronization of coupled systems with nonidentical linear dynamics, Internat. J. Robust Nonlinear Control, 27 (2017), 1462-1479.   Google Scholar

[32]

J. LuD. W. C. Ho and J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica J. IFAC, 46 (2010), 1215-1221.  doi: 10.1016/j.automatica.2010.04.005.  Google Scholar

[33]

W. Lu, B. Liu and T. Chen, Cluster synchronization in networks of coupled nonidentical dynamical systems, phChaos, 20 (2010), 013120, 12 pp. doi: 10.1063/1.3329367.  Google Scholar

[34]

E. Moulay and W. Perruquetti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appl., 323 (2006), 1430-1443.  doi: 10.1016/j.jmaa.2005.11.046.  Google Scholar

[35]

S. G. Nersesov and W. M. Haddad, Finite-time stabilization of nonlinear impulsive dynamical systems, Nonlinear Analysis: Hybrid Systems, 2 (2008), 832-845.  doi: 10.1016/j.nahs.2007.12.001.  Google Scholar

[36]

A. PratapR. RajaJ. Alzabut and et. al, Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field, Mathematical Methods in the Applied Sciences, 43 (2020), 6223-6253.  doi: 10.1002/mma.6367.  Google Scholar

[37]

M. T. Schaub, N. O'Clery N, Y. N. Billeh, et. al, Graph partitions and cluster synchronization in networks of oscillators, Chaos, 26 (2016), 094821, 14 pp. doi: 10.1063/1.4961065.  Google Scholar

[38]

Y. Shen and X. Xia, Semi-global finite-time observers for nonlinear systems, Automatica J. IFAC, 44 (2008), 3152-3156.  doi: 10.1016/j.automatica.2008.05.015.  Google Scholar

[39]

C. Song, S. Fei, Jinde Cao, et. al, Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control, Mathematics 7 (2019), 599. doi: 10.3390/math7070599.  Google Scholar

[40]

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, et. al, Complete characterization of the stability of cluster synchronization in complex dynamical networks, Science Advances, 2 (2016), e1501737. arXiv: 1507.04381v2. doi: 10.1126/sciadv.1501737.  Google Scholar

[41]

I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter GmbH & Co. KG, Berlin, 2009. doi: 10.1515/9783110221824.  Google Scholar

[42]

Y.-Z. Sun, S.-Y. Leng, Y.-C. Lai, et al, Closed-loop control of complex networks: a trade-off between time and energy, Phys. Rev. Lett., 119 (2017), 198301, 6 pp. doi: 10.1103/PhysRevLett.119.198301.  Google Scholar

[43]

Z.-Y. SunM.-M. Yun and T. Li, A new approach to fast global finite-time stabilization of high-order nonlinear system, Automatica J. IFAC, 81 (2017), 455-463.  doi: 10.1016/j.automatica.2017.04.024.  Google Scholar

[44]

Z. TangJ. H. Park and H. Shen, Finite-time cluster synchronization of Lur'e networks: A nonsmooth approach, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1213-1224.  doi: 10.1109/TSMC.2017.2657779.  Google Scholar

[45]

R. TangX. Yang and X. Wan, Finite-time cluster synchronization for a class of fuzzy cellular neural networks via non-chattering quantized controllers, Neural Networks, 113 (2019), 79-90.  doi: 10.1016/j.neunet.2018.11.010.  Google Scholar

[46]

Available from: http://link.aps.org/supplemental/10.1103/PhysRevLett.119.198301. Google Scholar

[47]

Y. Wang and J. Cao, Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems, Nonlinear Anal. Real World Appl., 14 (2013), 842-851.  doi: 10.1016/j.nonrwa.2012.08.005.  Google Scholar

[48]

X. YangJ. Cao and J. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Anal. Real World Appl., 12 (2011), 2252-2266.  doi: 10.1016/j.nonrwa.2011.01.007.  Google Scholar

[49]

X. YangD. W. C. HoJ. Lu and et. al, Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Transactions on Fuzzy Systems, 23 (2015), 2302-2316.  doi: 10.1109/TFUZZ.2015.2417973.  Google Scholar

[50]

T. Yang, Impulsive Control Theory, Springer-Verlag, Berlin, 2001.  Google Scholar

[51]

T. Yang and L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans. Circuits Systems I Fund. Theory Appl., 44 (1997), 976-988.  doi: 10.1109/81.633887.  Google Scholar

[52]

X. Yang and J. Lu, Finite-time synchronization of coupled networks with markovian topology and impulsive effects, IEEE Transactions on Automatic Control, 61 (2016), 2256-2261.  doi: 10.1109/TAC.2015.2484328.  Google Scholar

[53]

J. YinS. KhooZ. Man and et. al, Finite-time stability and instability of stochastic nonlinear systems, Automatica J. IFAC, 47 (2011), 2671-2677.  doi: 10.1016/j.automatica.2011.08.050.  Google Scholar

[54]

T. Yu and D. Cao, Stability analysis of impulsive neural networks with piecewise constant arguments, Neural Processing Letters, 47 (2018), 153-165.  doi: 10.1007/s11063-017-9638-y.  Google Scholar

[55]

T. Yu, D. Cao, S. Liu, et. al, Stability analysis of neural networks with periodic coefficients and piecewise constant arguments, J. Franklin Inst., 353 (2016), 409–425. doi: 10.1016/j.jfranklin.2015.11.010.  Google Scholar

[56]

T. YuH. WangM. Su and et. al, Distributed-delay-dependent exponential stability of impulsive neural networks with inertial term, Neurocomputing, 313 (2018), 220-228.  doi: 10.1016/j.neucom.2018.06.033.  Google Scholar

[57]

T. YuD. CaoY. Yang and et. al, Robust synchronization of impulsively coupled complex dynamical network with delayed nonidentical nodes, Chaos Solitons Fractals, 87 (2016), 92-101.  doi: 10.1016/j.chaos.2016.03.010.  Google Scholar

[58]

W. ZhangY. TangJ. Fang and et. al, Exponential cluster synchronization of impulsive delayed genetic oscillators with external disturbances, Chaos, 21 (2011), 6-12.  doi: 10.1063/1.3671609.  Google Scholar

[59]

C. ZhaoJ. HeP. Cheng P and et. al, Consensus-based energy management in a smart grid with transmission losses and directed communication, IEEE Transactions on Smart Grid, 8 (2016), 2019-2061.  doi: 10.1109/TSG.2015.2513772.  Google Scholar

[60]

Y. Zhou, X. Wan, C. Huang, et. al, Finite-time stochastic synchronization of dynamic networks with nonlinear coupling strength via quantized intermittent control, Appl. Math. Comput., 376 (2020), 125157, 14 pp. doi: 10.1016/j.amc.2020.125157.  Google Scholar

show all references

References:
[1]

F. AmatoM. Ariola and C. Cosentino, Finite-time stability of linear time-varying systems: analysis and controller design, IEEE Trans. Automat. Control, 55 (2010), 1003-1008.  doi: 10.1109/TAC.2010.2041680.  Google Scholar

[2]

S. Arik, Stability analysis of delayed neural networks, IEEE Trans. Circuits Systems I Fund. Theory Appl., 47 (2000), 1089-1092.  doi: 10.1109/81.855465.  Google Scholar

[3]

K. L. Babcock and R. M. Westervelt, Dynamics of simple electronic neural networks, Physica D, 28 (1987), 305-316.  doi: 10.1016/0167-2789(87)90021-2.  Google Scholar

[4]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.  doi: 10.1126/science.286.5439.509.  Google Scholar

[5]

V. N. BelykhI. V. Belykh and M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Physica D, 195 (2004), 159-187.  doi: 10.1016/j.physd.2004.03.012.  Google Scholar

[6]

S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751-766.  doi: 10.1137/S0363012997321358.  Google Scholar

[7]

S. P. Bhat and D. S. Bernstein, Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Transactions on Automatic Control, 43 (1998), 678-682.  doi: 10.1109/9.668834.  Google Scholar

[8]

Y. CaoW. YuW. Ren and et. al, An overview of recent progress in the study of distributed Multi-Agent coordination, IEEE Transaction on Industrial Informations, 9 (2013), 427-438.  doi: 10.1109/TII.2012.2219061.  Google Scholar

[9]

W. Chen and L. C. Jiao, Finite-time stability theorem of stochastic nonlinear systems, Automatica J. IFAC, 46 (2010), 2105-2108.  doi: 10.1016/j.automatica.2010.08.009.  Google Scholar

[10]

D. Chen, W. Zhang, J. Cao, et. al, Fixed time synchronization of delayed quaternion-valued memristor-based neural networks, Adv. Difference Equ., 2020 (2020), Paper No. 92, 16 pp.. doi: 10.1186/s13662-020-02560-w.  Google Scholar

[11]

F. De Smet and D. Aeyels, Clustering in a network of non-identical and mutually interacting agents, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2019), 745-768.  doi: 10.1098/rspa.2008.0259.  Google Scholar

[12]

D. EfimovA. PolyakovE. Fridman and et. al, Comments on finite-time stability of time-delay systems, Automatica, 50 (2014), 1944-1947.  doi: 10.1016/j.automatica.2014.05.010.  Google Scholar

[13]

M. Galicki, Finite-time control of robotic manipulators, Automatica J. IFAC, 51 (2015), 49-54.  doi: 10.1016/j.automatica.2014.10.089.  Google Scholar

[14]

L. V. Gambuzza and M. Frasca, A criterion for stability of cluster synchronization in networks with external equitable partitions, Automatica J. IFAC, 100 (2019), 212-218.  doi: 10.1016/j.automatica.2018.11.026.  Google Scholar

[15]

W. M. Haddad and A. L'Afflitto, Finite-time stabilization and optimal feedback control, IEEE Trans. Automat. Control, 61 (2016), 1069-1074.  doi: 10.1109/TAC.2015.2454891.  Google Scholar

[16]

J. HeP. ChengL. Shi and et. al, Time synchronzation in WSNS: A maximum-value-based consensus approach, IEEE Trans. Automat. Control, 59 (2014), 660-675.  doi: 10.1109/TAC.2013.2286893.  Google Scholar

[17]

Y. HongZ.-P. Jiang ZP and G. Feng, Finite-time input-to-state stability and applications to finite-time control design, SIAM J. Control Optim., 48 (2010), 4395-4418.  doi: 10.1137/070712043.  Google Scholar

[18]

Y. HongJ. Wang and D. Cheng, Adaptive finite-time control of nonlinear systems with parametric uncertainty, IEEE Trans. Automat. Control, 51 (2006), 858-862.  doi: 10.1109/TAC.2006.875006.  Google Scholar

[19]

B. HuZ.-H. GuanG. Chen and et. al, Multistability of delayed hybrid impulsive neural networks with application to associative memories, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 1537-1551.  doi: 10.1109/TNNLS.2018.2870553.  Google Scholar

[20]

C. HuJ. YuZ. Chen and et. al, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Networks, 89 (2017), 74-83.  doi: 10.1016/j.neunet.2017.02.001.  Google Scholar

[21]

C. HuJ. YuH. Jiang and et al, Exponential synchronization of complex networks with finite distributed delays coupling, IEEE Transactions on Neural Networks, 22 (2011), 1999-2010.   Google Scholar

[22]

J. HuangC. WenW. Wang and Y.-D. Song, Adaptive finite-time consensus control of a group of uncertain nonlinear mechanical systems, Automatica J. IFAC, 51 (2015), 292-301.  doi: 10.1016/j.automatica.2014.10.093.  Google Scholar

[23]

S. Jalan and R. E. Amritkar, Self-organized and driven phase synchronization in coupled maps, Physical Review Letters, 90 (2003), 014101. Google Scholar

[24]

S. Jalan, R. E. Amritkar and C. K. Hu, Synchronized clusters in coupled map networks. I. Numerical studies, Physical Review E, 72 (2005), 016212. Google Scholar

[25]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Prentice Hall, Upper Saddle River, 2002. Google Scholar

[26]

M. KumarD. P. Garg and V. Kumar, Segregation of heterogeneous units in a swarm of robotic agents, IEEE Trans. Automat. Control, 55 (2010), 743-748.  doi: 10.1109/TAC.2010.2040494.  Google Scholar

[27]

Z. LiZ. DuanG. Chen and et. al, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans. Circuits Syst. I. Regul. Pap., 57 (2010), 213-224.  doi: 10.1109/TCSI.2009.2023937.  Google Scholar

[28]

X. LiD. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.  Google Scholar

[29]

X. Liu, Adaptive finite time stability of delayed systems with applications to network synchronization, (2020), arXiv: 2002.00145. Google Scholar

[30]

X. Liu and T. Chen, Finite-time and fixed-time cluster synchronization with or without pinning control, IEEE Transactions on Cybernetics, 48 (2018), 240-252.  doi: 10.1109/TCYB.2016.2630703.  Google Scholar

[31]

Z. LiuW. S. Wong and H. Cheng, Cluster synchronization of coupled systems with nonidentical linear dynamics, Internat. J. Robust Nonlinear Control, 27 (2017), 1462-1479.   Google Scholar

[32]

J. LuD. W. C. Ho and J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica J. IFAC, 46 (2010), 1215-1221.  doi: 10.1016/j.automatica.2010.04.005.  Google Scholar

[33]

W. Lu, B. Liu and T. Chen, Cluster synchronization in networks of coupled nonidentical dynamical systems, phChaos, 20 (2010), 013120, 12 pp. doi: 10.1063/1.3329367.  Google Scholar

[34]

E. Moulay and W. Perruquetti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appl., 323 (2006), 1430-1443.  doi: 10.1016/j.jmaa.2005.11.046.  Google Scholar

[35]

S. G. Nersesov and W. M. Haddad, Finite-time stabilization of nonlinear impulsive dynamical systems, Nonlinear Analysis: Hybrid Systems, 2 (2008), 832-845.  doi: 10.1016/j.nahs.2007.12.001.  Google Scholar

[36]

A. PratapR. RajaJ. Alzabut and et. al, Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field, Mathematical Methods in the Applied Sciences, 43 (2020), 6223-6253.  doi: 10.1002/mma.6367.  Google Scholar

[37]

M. T. Schaub, N. O'Clery N, Y. N. Billeh, et. al, Graph partitions and cluster synchronization in networks of oscillators, Chaos, 26 (2016), 094821, 14 pp. doi: 10.1063/1.4961065.  Google Scholar

[38]

Y. Shen and X. Xia, Semi-global finite-time observers for nonlinear systems, Automatica J. IFAC, 44 (2008), 3152-3156.  doi: 10.1016/j.automatica.2008.05.015.  Google Scholar

[39]

C. Song, S. Fei, Jinde Cao, et. al, Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control, Mathematics 7 (2019), 599. doi: 10.3390/math7070599.  Google Scholar

[40]

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, et. al, Complete characterization of the stability of cluster synchronization in complex dynamical networks, Science Advances, 2 (2016), e1501737. arXiv: 1507.04381v2. doi: 10.1126/sciadv.1501737.  Google Scholar

[41]

I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter GmbH & Co. KG, Berlin, 2009. doi: 10.1515/9783110221824.  Google Scholar

[42]

Y.-Z. Sun, S.-Y. Leng, Y.-C. Lai, et al, Closed-loop control of complex networks: a trade-off between time and energy, Phys. Rev. Lett., 119 (2017), 198301, 6 pp. doi: 10.1103/PhysRevLett.119.198301.  Google Scholar

[43]

Z.-Y. SunM.-M. Yun and T. Li, A new approach to fast global finite-time stabilization of high-order nonlinear system, Automatica J. IFAC, 81 (2017), 455-463.  doi: 10.1016/j.automatica.2017.04.024.  Google Scholar

[44]

Z. TangJ. H. Park and H. Shen, Finite-time cluster synchronization of Lur'e networks: A nonsmooth approach, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1213-1224.  doi: 10.1109/TSMC.2017.2657779.  Google Scholar

[45]

R. TangX. Yang and X. Wan, Finite-time cluster synchronization for a class of fuzzy cellular neural networks via non-chattering quantized controllers, Neural Networks, 113 (2019), 79-90.  doi: 10.1016/j.neunet.2018.11.010.  Google Scholar

[46]

Available from: http://link.aps.org/supplemental/10.1103/PhysRevLett.119.198301. Google Scholar

[47]

Y. Wang and J. Cao, Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems, Nonlinear Anal. Real World Appl., 14 (2013), 842-851.  doi: 10.1016/j.nonrwa.2012.08.005.  Google Scholar

[48]

X. YangJ. Cao and J. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Anal. Real World Appl., 12 (2011), 2252-2266.  doi: 10.1016/j.nonrwa.2011.01.007.  Google Scholar

[49]

X. YangD. W. C. HoJ. Lu and et. al, Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Transactions on Fuzzy Systems, 23 (2015), 2302-2316.  doi: 10.1109/TFUZZ.2015.2417973.  Google Scholar

[50]

T. Yang, Impulsive Control Theory, Springer-Verlag, Berlin, 2001.  Google Scholar

[51]

T. Yang and L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans. Circuits Systems I Fund. Theory Appl., 44 (1997), 976-988.  doi: 10.1109/81.633887.  Google Scholar

[52]

X. Yang and J. Lu, Finite-time synchronization of coupled networks with markovian topology and impulsive effects, IEEE Transactions on Automatic Control, 61 (2016), 2256-2261.  doi: 10.1109/TAC.2015.2484328.  Google Scholar

[53]

J. YinS. KhooZ. Man and et. al, Finite-time stability and instability of stochastic nonlinear systems, Automatica J. IFAC, 47 (2011), 2671-2677.  doi: 10.1016/j.automatica.2011.08.050.  Google Scholar

[54]

T. Yu and D. Cao, Stability analysis of impulsive neural networks with piecewise constant arguments, Neural Processing Letters, 47 (2018), 153-165.  doi: 10.1007/s11063-017-9638-y.  Google Scholar

[55]

T. Yu, D. Cao, S. Liu, et. al, Stability analysis of neural networks with periodic coefficients and piecewise constant arguments, J. Franklin Inst., 353 (2016), 409–425. doi: 10.1016/j.jfranklin.2015.11.010.  Google Scholar

[56]

T. YuH. WangM. Su and et. al, Distributed-delay-dependent exponential stability of impulsive neural networks with inertial term, Neurocomputing, 313 (2018), 220-228.  doi: 10.1016/j.neucom.2018.06.033.  Google Scholar

[57]

T. YuD. CaoY. Yang and et. al, Robust synchronization of impulsively coupled complex dynamical network with delayed nonidentical nodes, Chaos Solitons Fractals, 87 (2016), 92-101.  doi: 10.1016/j.chaos.2016.03.010.  Google Scholar

[58]

W. ZhangY. TangJ. Fang and et. al, Exponential cluster synchronization of impulsive delayed genetic oscillators with external disturbances, Chaos, 21 (2011), 6-12.  doi: 10.1063/1.3671609.  Google Scholar

[59]

C. ZhaoJ. HeP. Cheng P and et. al, Consensus-based energy management in a smart grid with transmission losses and directed communication, IEEE Transactions on Smart Grid, 8 (2016), 2019-2061.  doi: 10.1109/TSG.2015.2513772.  Google Scholar

[60]

Y. Zhou, X. Wan, C. Huang, et. al, Finite-time stochastic synchronization of dynamic networks with nonlinear coupling strength via quantized intermittent control, Appl. Math. Comput., 376 (2020), 125157, 14 pp. doi: 10.1016/j.amc.2020.125157.  Google Scholar

Figure 1.  Phase plots of (a). the system (61) and (b). the system (62) in Example 1
Figure 2.  Time histories of (a). the coupled system without control input, (b-d). the variables $ x_{i1} $, $ x_{i2} $ and $ x_{i3} $ of the coupled system with synchronization impulsive effect in Example 1
Figure 3.  Under control input, time histories of (a). the error function $ E(t) $ in Eq. (63), (b-d). the variables $ e_{i1} $, $ e_{i2} $ and $ e_{i3} $ of the synchronization error system in Example 1
Figure 4.  With desynchronization impulses, time histories of (a-c). the variables $ x_{i1} $, $ x_{i2} $ and $ x_{i3} $ of the coupled system with nonidentical nodes (61) and (62), (d-f). the variables $ e_{i1} $, $ e_{i2} $ and $ e_{i3} $ of the synchronization error system in Example 1
Figure 5.  Time histories of (a-c). the variables $ x_{i1} $, $ x_{i2} $ and $ x_{i3} $ of the complex networks, (d-f). the variables $ e_{i1} $, $ e_{i2} $ and $ e_{i3} $ of the synchronization error system in Example 2
[1]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[2]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[3]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[4]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[5]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[6]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[7]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[8]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[9]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[10]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[11]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[13]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[14]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[15]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[16]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[17]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[18]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[19]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[20]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]