July  2021, 26(7): 3595-3620. 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  July 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248
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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.

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

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[23]

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

[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.

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

[46]

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

[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.

[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.

[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.

[50]

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

[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.

[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.

[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.

[54]

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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
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