Fin-TS and Fix-TS on fractional quaternion delayed neural networks with uncertainty via establishing a new Caputo derivative inequality approach

Qiong Wu; Zhimin Yao; Zhouping Yin; Hai Zhang; Qiong Wu; Zhimin Yao; Zhouping Yin; Hai Zhang

doi:10.3934/mbe.2022428

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 9: 9220-9243. doi: 10.3934/mbe.2022428

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Fin-TS and Fix-TS on fractional quaternion delayed neural networks with uncertainty via establishing a new Caputo derivative inequality approach

School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China

Academic Editor: Xiaodi Li

Received: 19 May 2022 Revised: 13 June 2022 Accepted: 21 June 2022 Published: 23 June 2022

This paper investigates the finite time synchronization (Fin-TS) and fixed time synchronization (Fix-TS) issues on Caputo quaternion delayed neural networks (QDNNs) with uncertainty. A new Caputo fractional differential inequality is constructed, then Fix-TS settling time of the positive definite function is estimated, which is very convenient to derive Fix-TS condition to Caputo QDNNs. By designing the appropriate self feedback and adaptive controllers, the algebraic discriminant conditions to achieve Fin-TS and Fix-TS on Caputo QDNNs are proposed based on quaternion direct method, Lyapunov stability theory, extended Cauchy Schwartz inequality, Jensen inequality. Finally, the correctness and validity of the presented results under the different orders are verified by two numerical examples.
- Fin-TS,
- Fix-TS,
- fractional differential inequality,
- quaternion direct method,
- distributed delay,
- Caputo derivative
Citation: Qiong Wu, Zhimin Yao, Zhouping Yin, Hai Zhang. Fin-TS and Fix-TS on fractional quaternion delayed neural networks with uncertainty via establishing a new Caputo derivative inequality approach[J]. Mathematical Biosciences and Engineering, 2022, 19(9): 9220-9243. doi: 10.3934/mbe.2022428

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Abstract

This paper investigates the finite time synchronization (Fin-TS) and fixed time synchronization (Fix-TS) issues on Caputo quaternion delayed neural networks (QDNNs) with uncertainty. A new Caputo fractional differential inequality is constructed, then Fix-TS settling time of the positive definite function is estimated, which is very convenient to derive Fix-TS condition to Caputo QDNNs. By designing the appropriate self feedback and adaptive controllers, the algebraic discriminant conditions to achieve Fin-TS and Fix-TS on Caputo QDNNs are proposed based on quaternion direct method, Lyapunov stability theory, extended Cauchy Schwartz inequality, Jensen inequality. Finally, the correctness and validity of the presented results under the different orders are verified by two numerical examples.

References

[1]	K. S. Chiu, T. X. Li, New stability results for bidirectional associative memory neural networks model involving generalized piecewise constant delay, Math. Comput. Simul., 194 (2022), 719–743. https://doi.org/10.1016/j.matcom.2021.12.016 doi: 10.1016/j.matcom.2021.12.016
[2]	A. M. Alimi, C. Aouiti, E. A. Assali, Finite-time and fixed-time synchronization of a class of inertial neural networks with multi-proportional delays and its application to secure communication, Neurocomputing, 332 (2019), 29–43. https://doi.org/10.1016/j.neucom.2018.11.020 doi: 10.1016/j.neucom.2018.11.020
[3]	Z. Yang, D. Wang, X. Sun, J. Wu, Speed sensorless control of a bearingless induction motor with combined neural network and fractional sliding mode, Mechatronics, 82 (2022), 102721. https://doi.org/10.1016/j.mechatronics.2021.102721 doi: 10.1016/j.mechatronics.2021.102721
[4]	H. Zhang, R. Ye, S. Liu, J. Cao, A. Alsaedi, X. Li, LMI-based approach to stability analysis for fractional-order neural networks with discrete and distributed delays, Int. J. Syst. Sci., 49 (2018), 537–545. https://doi.org/10.1080/00207721.2017.1412534 doi: 10.1080/00207721.2017.1412534
[5]	F. Zhang, T. Huang, Q. Wu, Z. Zeng, Multistability of delayed fractional-order competitive neural networks, Neural Network, 140 (2021), 325–335. https://doi.org/10.1016/j.neunet.2021.03.036 doi: 10.1016/j.neunet.2021.03.036
[6]	X. D. Li, D. W.C. Ho, J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361–368. https://doi.org/10.1016/j.automatica.2018.10.024 doi: 10.1016/j.automatica.2018.10.024
[7]	X. D. Li, S. J. Song, J. H. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024–4034. https://doi.org/10.1109/TAC.2019.2905271 doi: 10.1109/TAC.2019.2905271
[8]	X. D. Li, D. X. Peng, J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908–4913. https://doi.org/10.1109/TAC.2020.2964558 doi: 10.1109/TAC.2020.2964558
[9]	L. M. Wang, Z. G. Zeng, M. F. Ge, A disturbance rejection framework for finite-time and fixed-time stabilization of delayed memristive neural networks, IEEE Trans. Syst. Man Cybern. Syst., 51 (2019), 905–915. https://doi.org/10.1109/TSMC.2018.2888867 doi: 10.1109/TSMC.2018.2888867
[10]	C. Hu, H. J. Jiang, Special functions-based fixed-time estimation and stabilization for dynamic systems, IEEE Trans. Syst. Man Cybern. Syst., 52 (2022), 3251–3262. https://doi.org/10.1109/TSMC.2021.3062206 doi: 10.1109/TSMC.2021.3062206
[11]	W. Yang, Y. W. Wang, I. C. Morarescu, X. K. Liu, Y. H. Huang, Fixed-time synchronization of competitive neural networks with multiple time scales, IEEE Trans. Neural Network Learn. Syst., 2021 (2021). https://doi.org/10.1109/TNNLS.2021.3052868
[12]	Y. Zhao, S. Ren, J. Kurths, Finite-time and fixed-time synchronization for a class of memristor-based competitive neural networks with different time scales, Chaos Solitons Fractals, 148 (2021), 111033. https://doi.org/10.1016/j.chaos.2021.111033 doi: 10.1016/j.chaos.2021.111033
[13]	C. J. Xu, Z. X. Liu, C. Aouiti, P. L. Li, L. Y. Yao, J. L. Yan, New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays, Cogn. Neurodyn., 2022 (2022). https://doi.org/10.1007/s11571-021-09763-1
[14]	T. Li, G. Wang, D. Yu, Q. Ding, Y. Jia, Synchronization mode transitions induced by chaos in modified MorrisLecar neural systems with weak coupling, Nonlinear Dyn., 108 (2022), 2611–2625. https://doi.org/10.1007/s11071-022-07318-5 doi: 10.1007/s11071-022-07318-5
[15]	S. Lin, X. Liu, Synchronization and control for directly coupled reaction-diffusion neural networks with multiple weights and hybrid coupling, Neurocomputing, 487 (2022), 144–156. https://doi.org/10.1016/j.neucom.2022.02.061 doi: 10.1016/j.neucom.2022.02.061
[16]	E. Viera-Martin, J. F. Gomez-Aguilar, J. E. Solis-Perez, J. A. Hernandez-Perez, R. F. Escobar-Jimenez, Artificial neural networks: a practical review of applications involving fractional calculus, Eur. Phys. J. Spec. Top., 2022 (2022). https://doi.org/10.1140/epjs/s11734-022-00455-3
[17]	X. He, Z. Hu, Optimization design of fractional-order Chebyshev lowpass filters based on genetic algorithm, Int. J. Circ. Theor. App., 50 (2022), 1420–1441. https://doi.org/10.1002/cta.3224 doi: 10.1002/cta.3224
[18]	X. Zhang, R. Li, J. Y. Hong, X. Zhou, N. Xin, Q. Li, Image-enhanced single-pixel imaging using fractional calculus, Opt. Express, 30 (2022), 81–91. https://doi.org/10.1364/OE.444739 doi: 10.1364/OE.444739
[19]	C. J. Xu, Z. X. Liu, C. Aouiti, et al, New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays, Cogn. Neurodyn., 2022 (2022). https://doi.org/10.1007/s11571-021-09763-1
[20]	S. B. Rao, T. W. Zhang, L. J. Xu, Exponential stability and synchronization of fuzzy Mittag$-$Leffler discrete-time Cohen$-$Grossberg neural networks with time delays, Int. J. Syst. Sci., 2022 (2022), 1–23. https://doi.org/10.1080/00207721.2022.2051093 doi: 10.1080/00207721.2022.2051093
[21]	N. Huo, B. Li, Y. K. Li, Global exponential stability and existence of almost periodic solutions in distribution for Clifford-valued stochastic high-order Hopfield neural networks with time-varying delays, AIMS Math., 7 (2022), 3653–3679. https://doi.org/ 10.3934/math.2022202 doi: 10.3934/math.2022202
[22]	H. L. Li, C. Hu, L. Zhang, H. J. Jiang, J. Cao, Non-separation method-based robust finite-time synchronization of uncertain fractional-order quaternion-valued neural networks, Appl. Math. Comput., 409 (2021), 126377. https://doi.org/10.1016/j.amc.2021.126377 doi: 10.1016/j.amc.2021.126377
[23]	W. Zhang, H. Zhang, J. Cao, F. E. Alsaadi, D. Chen, Synchronization in uncertain fractional-order memristive complex-valued neural networks with multiple time delays, Neural Network, 110 (2019), 186–198. https://doi.org/10.1016/j.neunet.2018.12.004 doi: 10.1016/j.neunet.2018.12.004
[24]	J. Pan, Z. Y. Pan, Novel robust stability criteria for uncertain parameter quaternionic neural networks with mixed delays: Whole quaternionic method, Appl. Math. Comput., 407 (2021), 126326. https://doi.org/10.1016/j.amc.2021.126326 doi: 10.1016/j.amc.2021.126326
[25]	W. Wei, J. Yu, L. Wang, C. Hu, H. Jiang, Fixed/Preassigned-time synchronization of quaternion-valued neural networks via pure power-law control, Neural Network, 146 (2022), 341–349. https://doi.org/10.1016/j.neunet.2021.11.023 doi: 10.1016/j.neunet.2021.11.023
[26]	A. Kashkynbayev, A. Issakhanov, M. Otkel, J. Kurths, Finite-time and fixed-time synchronization analysis of shunting inhibitory memristive neural networks with time-varying delays, Chaos Solitons Fractals, 156 (2022), 111866. https://doi.org/10.1016/j.chaos.2022.111866 doi: 10.1016/j.chaos.2022.111866
[27]	R. Q. Tang, H. S. Su, Y. Zou, X. S. Yang, Finite-time synchronization of Markovian coupled neural networks with delays via intermittent quantized control: Linear programming approach, IEEE Trans. Neural Network Learn. Syst., 2021 (2021). https://doi.org/10.1109/TNNLS.2021.3069926
[28]	S. Yang, J. Yu, C. Hu, H. J. Jiang, Finite-time synchronization of memristive neural networks with fractional-order, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 3739–3750. https://doi.org/10.1109/TSMC.2019.2931046 doi: 10.1109/TSMC.2019.2931046
[29]	T. Q. Hou, J. Yu, C. Hu, H. J. Jiang, Finite-time synchronization of fractional-order complex-variable dynamic networks, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 4297–4307. https://doi.org/10.1109/TSMC.2019.2931339 doi: 10.1109/TSMC.2019.2931339
[30]	X. S. Yang, J. Lam, D. W.C. Ho, Z. G. Feng, Fixed-time synchronization of complex networks with impulsive effects via non-chattering control, IEEE Trans. Automat. Contr., 62 (2017), 5511–5521. https://doi.org/10.1109/TAC.2017.2691303 doi: 10.1109/TAC.2017.2691303
[31]	Y. Chen, X. Zhang, Y. Xue, Global exponential synchronization of high-order quaternion Hopfield neural networks with unbounded distributed delays and time-varying discrete delays, Math. Comput. Simul., 193 (2022), 173–189. https://doi.org/10.1016/j.matcom.2021.10.012 doi: 10.1016/j.matcom.2021.10.012
[32]	S. Rao, T. Zhang, L. Xu, Exponential stability and synchronization of fuzzy Mittag-Leffler discrete-time Cohen-Grossberg neural networks with time delays, Int. J. Syst. Sci., 2022 (2022). https://doi.org/10.1080/00207721.2022.2051093
[33]	D. Lin, X. Chen, G. Yu, Z. Li, Y. Xia, Global exponential synchronization via nonlinear feedback control for delayed inertial memristor-based quaternion-valued neural networks with impulses, Appl. Math. Comput., 401 (2021), 126093. https://doi.org/10.1016/j.amc.2021.126093 doi: 10.1016/j.amc.2021.126093
[34]	Y. Wang, Y. Tian, X. Li, Global exponential synchronization of interval neural networks with mixed delays via delayed impulsive control, Neurocomputing, 420 (2021), 290–298. https://doi.org/10.1016/j.neucom.2020.09.010 doi: 10.1016/j.neucom.2020.09.010
[35]	A. Kumar, S. Das, V. K. Yadav, Global exponential synchronization of complex-valued recurrent neural networks in presence of uncertainty along with time-varying bounded and unbounded delay terms, Int. J. Dyn. Control, 2021 (2021), 1–15. https://doi.org/10.1007/s40435-021-00838-9 doi: 10.1007/s40435-021-00838-9
[36]	H. Zhang, Y. Cheng, H. M. Zhang, W. Zhang, J. Cao, Hybrid control design for Mittag-Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects, Math. Comput. Simul., 197 (2022), 341–354. https://doi.org/10.1016/j.matcom.2022.02.022 doi: 10.1016/j.matcom.2022.02.022
[37]	Z. Li, Y. Zhang, The boundedness and the global Mittag-Leffler synchronization of fractional-Order inertial Cohen-Grossberg neural networks with time delays, Neural Process. Lett., 54 (2022), 597–611. https://doi.org/10.1007/s11063-021-10648-x doi: 10.1007/s11063-021-10648-x
[38]	S. Aadhithiyan, R. Raja, J. Alzabut, Q. Zhu, M. Niezabitowski, Robust nonfragile Mittag$-$Leffler synchronization of fractional order nonlinear complex dynamical networks with constant and infinite distributed delays, Math. Methods Appl. Sci., 45 (2022), 2166–2189. https://doi.org/10.1002/mma.7915 doi: 10.1002/mma.7915
[39]	Y. Zhang, Q. Li, Finite-time projective synchronization of neural networks with uncertainties via adaptive fault-tolerant control, in 2021 IEEE 7th International Conference on Control Science and Systems Engineering (ICCSSE), (2021), 29–33. https://doi.org/10.1109/ICCSSE52761.2021.9545093
[40]	W. Chen, Y. Yu, X. Hai, G. Ren, Adaptive quasi-synchronization control of heterogeneous fractional-order coupled neural networks with reaction-diffusion, Appl. Math. Comput., 427 (2022), 127145. https://doi.org/10.1016/j.amc.2022.127145 doi: 10.1016/j.amc.2022.127145
[41]	F. Ma, X. Gao, Synchronization and quasi-synchronization of delayed fractional coupled memristive neural networks, Neural Process. Lett., 2022 (2022). https://doi.org/10.1007/s11063-021-10698-1
[42]	L. M. Wang, H. B. He, Z. G. Zeng, Global synchronization of fuzzy memristive neural networks with discrete and distributed delays, IEEE Trans. Fuzzy Syst., 28 (2020), 2022–2034. https://doi.org/10.1109/TFUZZ.2019.2930032 doi: 10.1109/TFUZZ.2019.2930032
[43]	C. Hu, H. B. He, H. J. Jiang, Edge-based adaptive distributed method for synchronization of intermittently coupled spatiotemporal networks, IEEE Trans. Automat. Control, 67 (2022), 2597–2604. https://doi.org/10.1109/TAC.2021.3088805 doi: 10.1109/TAC.2021.3088805
[44]	J. Xiao, J. Cheng, K. Shi, R. Zhang, A general approach to fixed-time synchronization problem for fractional-order multi-dimension-valued fuzzy neural networks based on memristor, IEEE Trans. Fuzzy Syst., 30 (2022), 968–977. https://doi.org/ 10.1109/TFUZZ.2021.3051308 doi: 10.1109/TFUZZ.2021.3051308
[45]	M. Dutta, B. K. Roy, A new memductance-based fractional-order chaotic system and its fixed-time synchronization, Chaos Solitons Fractals, 145 (2021), 110782. https://doi.org/10.1016/j.chaos.2021.110782 doi: 10.1016/j.chaos.2021.110782
[46]	W. Zhang, H. Zhang, J. Cao, H. M. Zhang, F. E. Alsaadi, A. Alsaedi, Global projective synchronization in fractional-order quaternion valued neural networks. Asian J. Control, 24 (2022), 227–236. https://doi.org/10.1002/asjc.2485
[47]	H. Zhang, J. S. Cheng, H. M. Zhang, W. W. Zhang, J. Cao, Quasi-uniform synchronization of Caputo type fractional neural networks with leakage and discrete delays, Chaos Solitons Fractals, 152 (2021), 111432. https://doi.org/10.1016/j.chaos.2021.111432 doi: 10.1016/j.chaos.2021.111432
[48]	M. S. Ali, M. Hymavathi, S. Senan, V. Shekher, S. Arik, Global asymptotic synchronization of impulsive fractional-order complex-valued memristor-based neural networks with time varying delays. Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104869. https://doi.org/10.1016/j.cnsns.2019.104869
[49]	H. Li, J. Cao, H. Jiang, A. Alsaedi, Graph theory-based finite-time synchronization of fractional-order complex dynamical networks, J. Franklin Inst., 355 (2018), 5771–5789. https://doi.org/10.1016/j.jfranklin.2018.05.039 doi: 10.1016/j.jfranklin.2018.05.039
[50]	H. Zhang, M. Ye, R. Ye, J. Cao, Synchronization stability of Riemann-Liouville fractional delay-coupled complex neural networks, Phys. A, 508 (2018), 155–165. https://doi.org/10.1016/j.physa.2018.05.060 doi: 10.1016/j.physa.2018.05.060
[51]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[52]	W. Chen, H. G. Yu, X. D. Hai, G. J. Ren, Adaptive quasi-synchronization control of heterogeneous fractional-order coupled neural networks with reaction-diffusion, Appl. Math. Comput., 427 (2022), 127145. https://doi.org/10.1016/j.amc.2022.127145 doi: 10.1016/j.amc.2022.127145

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