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Exponential Synchronization for Variable-order Fractional Complex Dynamical Networks via Dynamic Event-triggered Control Strategy

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Abstract

This paper is concerned with the global exponential synchronization issues for variable-order fractional complex dynamic networks (FCDNs). Firstly, a new derivative operator, which is called as the generalized Caputo variable-order fractional derivative, is developed, and some properties and lemmas are rigorous proved. Secondly, a new dynamic event-triggered control mechanism is designed to realize the synchronization objective, where the generalized Caputo variable-order fractional derivative is applied to characterize the evolution state of internal dynamic variable. And the exclusion for Zeno behavior is verified by contradiction analysis method. Thirdly, a class of functions, which is an extension of the Lipschitz function, is introduced to model the nonlinear dynamics for the considered system. With the aid of fractional Lyapunov functional method, some auxiliary functions and advanced mathematical analysis techniques, the global exponential synchronization conditions are established in terms of linear matrix inequalities (LMIs). Finally, the correctness of the theoretical results and the feasibility of the designed controller in this paper are confirmed by applying a numerical simulation example.

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Abbreviations

R :

Real numbers set

\(R^{n}\) :

n-dimensional real vector

\(R^{n\times m} \) :

\(n\times m\) real matrices set

\(I_{n}\) :

n-dimensional identify matrix

\(A>0\) \((A<0)\) :

Positive (negative) definite matrix

||x||:

\((\sum _{i=1}^{n})|x_{i}|^{2})^{\frac{1}{2}}\), Euclidean norm of x

\(||x||_{1}\) :

1-Norm of x

\(\otimes \) :

Kronecker product

\(A^{T}\) :

Transpose of matrix A

\(A^{-1}\) :

Inverse of matrix A

\(\lambda _{\max }(A)\)(\(\lambda _{\min }(A)\)):

The maximum(minimum) eigenvalue of A

\(A^{s}\) :

\(\frac{1}{2}(A^{T}+A)\)

References

  1. Huberman BA, Adamic LA (1999) Growth dynamics of the world-wide web. Nature 401:23–25

    Article  Google Scholar 

  2. Pastorsatorras R, Smith E, Sole RV (2003) Evolving protein interaction networks through gene duplication. J Theor Biol 222:199–210

    Article  MathSciNet  MATH  Google Scholar 

  3. Strogatz SH (2001) Exploring complex networks. Nature 410:268–276

    Article  MATH  Google Scholar 

  4. Tan GQ, Wang ZS, Shi Z (2021) Proportional-integral state estimator for quaternion-valued neural networks with time-varying delays. IEEE Trans Neural Netw Learn Syst. 10.1109TNNLS.2021.3103979

  5. Li XN, Wu HQ, Cao JD (2023) Prescribed-time synchronization in networks of piecewise smooth systems via a nonlinear dynamic event-triggered control strategy. Math Comput Simul 203:647–668

    Article  MathSciNet  MATH  Google Scholar 

  6. Tan GQ, Wang ZS (2022) Stability analysis of systems with time-varying delay via a delay-product-type integral inequality. Math Meth Appl Sci. https://doi.org/10.1002/MMA.8186

    Article  MathSciNet  Google Scholar 

  7. Zhang ZQ, Wu HQ (2022) Cluster synchronization in finite/fixed time for semi-Markovian switching T-S fuzzy complex dynamical networks with discontinuous dynamic nodes. AIMS Math 7:11942–11971

    Article  MathSciNet  Google Scholar 

  8. Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50:15–67

    Article  Google Scholar 

  9. Fleury V, Schwartz L (1999) Diffusion limited aggregation from shear stress as a simple model of vasculogenesis. Fractals 7:33–39

    Article  Google Scholar 

  10. Rossikhin YA, Shitikova MV (2001) A new method for solving dynamic problems of fractional derivative viscoelasticity. Int J Eng Sci 39:149–176

    Article  Google Scholar 

  11. Li RH, Wu HQ, Cao JD (2022) Impulsive exponential synchronization of fractional-order complex dynamical networks with derivative couplings via feedback control based on discrete time state observations. Acta Math Sci 42:737–754

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu J, Wu HQ, Cao JD (2020) Event-triggered synchronization in fixed time for semi-Markov switching dynamical complex networks with multiple weights and discontinuous nonlinearity. Commun Nonlinear Sci Numer Simul 90:105400

    Article  MathSciNet  MATH  Google Scholar 

  13. Bai J, Wu HQ, Cao JD (2022) Secure synchronization and identification for fractional complex networks with multiple weight couplings under DoS attacks. Comput Appl Math 41:187

    Article  MathSciNet  MATH  Google Scholar 

  14. Onoda J, Minesugi K (1996) Semiactive vibration suppression by variable-damping members. Aiaa J 34:355–361

    Article  MATH  Google Scholar 

  15. Atangana A, Botha JF (2013) A generalized groundwater flow equation using the concept of variable-order derivative. Bound Value Probl. https://doi.org/10.1186/1687-2770-2013-53

    Article  MathSciNet  MATH  Google Scholar 

  16. Patnaik S, Hollkamp JP, Semperlotti F (2020) Applications of variable-order fractional operators: a review. Proc R Soc A-Math Phys Eng Sci 476:20190498

    Article  MathSciNet  MATH  Google Scholar 

  17. Samko SG, Ross B (1993) Integration and differentiation to a variable fractional order. Integr Transf Spec F 1:277–300

    Article  MathSciNet  MATH  Google Scholar 

  18. Sun HG, Chen W, Chen YQ (2009) Variable-order fractional differential operators in anomalous diffusion modeling. Phys A 388:4586–4592

    Article  Google Scholar 

  19. Ostalczyk P (2010) Stability analysis of a discrete-time system with a variable-fractional order controller. Bull Pol Acad Sci Tech Sci 58:613–619

    MATH  Google Scholar 

  20. Jiang JF, Chen HT, Guirao JLG, Cao DQ (2019) Existence of the solution and stability for a class of variable fractional order differential systems. Chaos Soliton Fract 128:269–274

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiang JF, Cao DQ, Chen HT (2020) Sliding mode control for a class of variable-order fractional chaotic systems. J Frankl Inst 357:10127–10158

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang L, Yu CL, Liu T (2016) Control of finite-time anti-synchronization for variable-order fractional chaotic systems with unknown parameters. Nonlinear Dyn 86:1–14

    Article  MATH  Google Scholar 

  23. Årzén K (1999) A simple event-based PID controller, 14th IFAC World Congress : 423-428

  24. Åström K, Bernhardsson B (1999) Comparison of periodic and event based sampling for first-order stochastic systems, 14th IFAC World Congress: 301-306

  25. Xie WJ, Zhu QX (2020) Self-triggered state-feedback control for stochastic nonlinear systems with Markovian switching. IEEE Trans Syst Man Cybern Syst 50:3200–3209

    Article  Google Scholar 

  26. Zhu QX, Huang TW (2021) \(H_{\infty }\) control of stochastic networked control systems with time-varying delays: the event-triggered sampling case. Int J Robust Nonlinear Control 31:9767–9781

    Article  MathSciNet  Google Scholar 

  27. Zhang Y, Wu H, Cao J (2021) Global Mittag-Leffler consensus for fractional singularly perturbed multi-agent systems with discontinuous inherent dynamics via event-triggered control strategy. J Frankl Inst 358:2086–2114

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang XT, Wang H, Zhu QX (2022) Event-triggered predictive control of nonlinear stochastic systems with output delay. Automatica 140:110230

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhu QX (2019) Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. IEEE Trans Autom Control 64:3764–3771

    Article  MathSciNet  MATH  Google Scholar 

  30. Girard A (2013) Dynamic triggering mechanisms for event-triggered control. IEEE Trans Autom Control 60:1992–1997

    Article  MathSciNet  MATH  Google Scholar 

  31. Hu WF, Yang CH, Huang TW, Gui WH (2020) A distributed dynamic event-triggered control approach to consensus of linear multiagent systems with directed networks. IEEE Trans Cybern 50:869–874

    Article  Google Scholar 

  32. Yi XL, Liu K, Dimarogonas DV, Johansson KH (2019) Dynamic event-triggered and self-triggered control for multi-agent systems. IEEE Trans Autom Control 64:3300–3307

    Article  MathSciNet  MATH  Google Scholar 

  33. Amini A, Asif A, Mohammadi A (2022) A unified optimization for resilient dynamic event-triggering consensus under denial of service. IEEE Trans Cybern 52:2872–2874

    Article  Google Scholar 

  34. Ding D, Tang Z, Park JH, Wang Y, Ji ZC (2022) Dynamic self-triggered impulsive synchronization of complex networks with mismatched parameters and distributed delay. IEEE Trans Cybern. https://doi.org/10.1109/ZCYB.2022.3168854

    Article  Google Scholar 

  35. Zhao GL, Hua CC (2021) A hybrid dynamic event-triggered approach to consensus of multi-agent systems with external disturbances. IEEE Trans Autom Control 66:3213–3220

    Article  MATH  Google Scholar 

  36. Lorenzo CF, Hartley TT (2002) Variable order and distributed order fractional operators. Nonlinear Dyn 29:57–98

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang ZL, Yang DS, Ma TD, Sun N (2014) Stability analysis for nonlinear fractional-order systems based on comparison principle. Nonlinear Dyn 75:387–402

    Article  MathSciNet  MATH  Google Scholar 

  38. Aguila-Camacho N, Duarte-Mermoud MA, Gallegos JA (2014) Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 19:2951–2957

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang S, Hu C, Yu J, Jiang HJ (2020) Exponential stability of fractional-order impulsive control systems with applications in synchronization. IEEE Trans Cybern 50:3157–3168

    Article  Google Scholar 

  40. Corduneanu C (1971) Principles of differential and integral equations. Allyn and Bacon, Boston

    MATH  Google Scholar 

  41. Wan P, Jian JG (2019) Impulsive stabilization and synchronization of fractional-order complex-valued neural networks. Neural Process Lett 20:2201–2218

    Article  Google Scholar 

  42. DeLellis P, Bernardo MD, Russo G (2011) On QUAD, Lipschitz, and contracting vector fields for consensus and synchronization of networks. IEEE Trans Circuits Syst I-Regul Pap 58:576–583

    Article  MathSciNet  MATH  Google Scholar 

  43. Yu K, Lu JQ, Qiu JL, Jurgen K (2019) Exponential synchronization of time-varying delayed complex-valued neural networks under hybrid impulsive controllers. Neural Netw 114:157–163

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by the Natural Science Foundation of China (No.12171416) and Key Project of Natural Science Foundation of China (No. 61833005). Furthermore, the authors would like to thank Editors and Reviewers for their insightful and constructive comments, which help to enrich the content and improve the presentation of the results in this paper.

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Authors and Affiliations

  1. School of Science, Yanshan University, Qinhuangdao, 066001, China

    Ruihong Li & Huaiqin Wu

  2. School of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  3. Yonsei Frontier Lab, Yonsei University, Seoul, 03722, Korea

    Jinde Cao

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  1. Ruihong Li
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  2. Huaiqin Wu
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  3. Jinde Cao
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Correspondence to Huaiqin Wu.

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Li, R., Wu, H. & Cao, J. Exponential Synchronization for Variable-order Fractional Complex Dynamical Networks via Dynamic Event-triggered Control Strategy. Neural Process Lett 55, 8569–8588 (2023). https://doi.org/10.1007/s11063-023-11169-5

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