Finite-time and global Mittag-Leffler stability of fractional-order neural networks with S-type distributed delays

Wei Liu; Qinghua Zuo; Chen Xu; Wei Liu; Qinghua Zuo; Chen Xu

doi:10.3934/math.2024405

AIMS Mathematics

2024, Volume 9, Issue 4: 8339-8352. doi: 10.3934/math.2024405

Previous Article Next Article

Research article

Finite-time and global Mittag-Leffler stability of fractional-order neural networks with S-type distributed delays

1.
Department of Mathematics, Shaoxing University Yuanpei College, Qunxian Road 2799#, Shaoxing, 312000, Zhejiang, China
2.
Department of Mathematics, Shaoxing University, Chengnan Ave 990#, Shaoxing, 312000, Zhejiang, China

Received: 29 December 2023 Revised: 20 February 2024 Accepted: 21 February 2024 Published: 27 February 2024
MSC : 92B20, 34K20

This paper was mainly concerned with the stability analysis of a class of fractional-order neural networks with S-type distributed delays. By using the properties of Riemann-Liouville fractional-order derivatives and integrals, along with the additivity of integration intervals and initial conditions, fractional-order integrals of the state function with S-type distributed delays were transformed into fractional-order integrals of the state function without S-type distributed delays. By virtue of the theory of contractive mapping and the Bellman-Gronwall inequality, the sufficient conditions for finite-time stability and global Mittag-Leffler stability were obtained when certain conditions were satisfied. Moreover, the correctness and realizability of the conclusion were verified through the presentation of two illustrative numerical simulation examples.
- fractional-order neural networks,
- S-type distributed delays,
- finite-time stability,
- global Mittag-Leffler stability
Citation: Wei Liu, Qinghua Zuo, Chen Xu. Finite-time and global Mittag-Leffler stability of fractional-order neural networks with S-type distributed delays[J]. AIMS Mathematics, 2024, 9(4): 8339-8352. doi: 10.3934/math.2024405

Related Papers:

Abstract

This paper was mainly concerned with the stability analysis of a class of fractional-order neural networks with S-type distributed delays. By using the properties of Riemann-Liouville fractional-order derivatives and integrals, along with the additivity of integration intervals and initial conditions, fractional-order integrals of the state function with S-type distributed delays were transformed into fractional-order integrals of the state function without S-type distributed delays. By virtue of the theory of contractive mapping and the Bellman-Gronwall inequality, the sufficient conditions for finite-time stability and global Mittag-Leffler stability were obtained when certain conditions were satisfied. Moreover, the correctness and realizability of the conclusion were verified through the presentation of two illustrative numerical simulation examples.

References

[1]	Z. Zhang, Y. Wang, J. Zhang, Z. Ai, F. Y. Cheng, F. Liu, Novel fractional-order decentralized control for nonlinear fractional-order composite systems with time delays, ISA T., 128 (2022), 230–242. https://doi.org/10.1016/j.isatra.2021.11.044 doi: 10.1016/j.isatra.2021.11.044
[2]	S. Ha, L. Chen, H. Liu, Adaptive fuzzy variable structure control of fractional-order nonlinear systems with input nonlinearities, Int. J. Fuzzy Syst., 23 (2021), 2309–2323. https://doi.org/10.1007/s40815-021-01105-x doi: 10.1007/s40815-021-01105-x
[3]	G. F. Anaya, O. M. Fuentes, A. J. M. Vázquez, J. D. S. Torres, L. A. Q. Téllez, F. M. Vázquez, Passive decomposition and gradient control of fractional-order nonlinear systems, Nonlinear Dynam., 109 (2022), 1705–1722. https://doi.org/10.1007/s11071-022-07531-2 doi: 10.1007/s11071-022-07531-2
[4]	S. Liu, H. Wang, T. Li, Adaptive composite dynamic surface neural control for nonlinear fractional-order systems subject to delayed input, ISA T., 134 (2023), 122–133. https://doi.org/10.1016/j.isatra.2022.07.027 doi: 10.1016/j.isatra.2022.07.027
[5]	H. Qiu, H. Liu, X. Zhang, Historical data-driven composite learning adaptive fuzzy control of fractional-order nonlinear systems, Int. J. Fuzzy Syst., 25 (2022), 1156–1170. https://doi.org/10.1007/s40815-022-01430-9 doi: 10.1007/s40815-022-01430-9
[6]	M. Cui, S. Tong, Event-triggered predefined-time output feedback control for fractional-order nonlinear systems with input saturation, IEEE T. Fuzzy Syst., 31 (2023), 4397–4409. https://doi.org/10.1109/TFUZZ.2023.3283783 doi: 10.1109/TFUZZ.2023.3283783
[7]	C. Xu, M. Liao, P. Li, Y. Guo, Z. Liu, Bifurcation properties for fractional order delayed BAM neural networks, Cogn. Comput., 13 (2021), 322–356. https://doi.org/10.1007/s12559-020-09782-w doi: 10.1007/s12559-020-09782-w
[8]	L. Si, M. Xiao, G. Jiang, Z. Cheng, Q. Song, J. Cao, Dynamics of fractional-order neural networks with discrete and distributed delays, IEEE Access, 8 (2019), 46071–46080. https://doi.org/10.1109/ACCESS.2019.2946790 doi: 10.1109/ACCESS.2019.2946790
[9]	I. Stamova, Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays, Nonlinear Dynam., 77 (2014), 1251–1260. https://doi.org/10.1007/s11071-014-1375-4 doi: 10.1007/s11071-014-1375-4
[10]	B. Zheng, Z. Wang, Mittag-Leffler synchronization of fractional-order coupled neural networks with mixed delays, Appl. Math. Comput., 430 (2022), 127303. https://doi.org/10.1016/j.amc.2022.127303 doi: 10.1016/j.amc.2022.127303
[11]	Z. Zhang, J. Cao, Novel finite-time synchronization criteria for inertial neural networks with time delays via integral inequality method, IEEE T. Neur. Net. Lear., 30 (2019), 1476–1485. https://doi.org/10.1109/TNNLS.2018.2868800 doi: 10.1109/TNNLS.2018.2868800
[12]	F. Du, J. Lu, New results on finite-time stability of fractional-order Cohen-Grossberg neural networks with time delays, Asian J. Control, 24 (2022), 2328–2337. https://doi.org/10.1002/asjc.2641 doi: 10.1002/asjc.2641
[13]	F. Du, J. Lu, New approach to finite-time stability for fractional-order BAM neural networks with discrete and distributed delays, Chaos Soliton. Fract., 151 (2021), 111225. https://doi.org/10.1016/j.chaos.2021.111225 doi: 10.1016/j.chaos.2021.111225
[14]	Z. Zhang, J. Cao, Finite-time synchronization for fuzzy inertial neural networks by maximum value approach, IEEE T. Fuzzy Syst., 30 (2022), 1436–1446. https://doi.org/10.1109/TFUZZ.2021.3059953 doi: 10.1109/TFUZZ.2021.3059953
[15]	Z. Yang, J. Zhang, Z. Zhang, J. Mei, An improved criterion on finite-time stability for fractional-order fuzzy cellular neural networks involving leakage and discrete delays, Math. Comput. Simulat., 203 (2023), 910–925. https://doi.org/10.1016/j.matcom.2022.07.028 doi: 10.1016/j.matcom.2022.07.028
[16]	B. He, H. Zhou, Asymptotic stability and synchronization of fractional order Hopfield neural networks with unbounded delay, Math. Method. Appl. Sci., 46 (2023), 3157–3175. https://doi.org/10.1002/mma.8000 doi: 10.1002/mma.8000
[17]	Z. Yao, Z. Yang, Y. Fu, J. Li, Asymptotical stability for fractional-order Hopfield neural networks with multiple time delays, Math. Method. Appl. Sci., 45 (2022), 10052–10069. https://doi.org/10.1002/mma.8355 doi: 10.1002/mma.8355
[18]	F. Wang, J. Zhang, Y. Shu, X. G. Liu, Stability analysis for fractional-order neural networks with time-varying delay, Asian J. Control, 25 (2023), 1488–1498. https://doi.org/10.1002/asjc.2944 doi: 10.1002/asjc.2944
[19]	Z. Zhang, Z. Yang, Asymptotic stability for quaternion-valued fuzzy BAM neural networks via integral inequality approach, Chaos Soliton. Fract., 169 (2023), 113227. https://doi.org/10.1016/j.chaos.2023.113227 doi: 10.1016/j.chaos.2023.113227
[20]	J. Zhou, X. Ma, Z. Yan, S. Arik, Non-fragile output-feedback control for time-delay neural networks with persistent dwell time switching: A system mode and time scheduler dual-dependent design, Neural Networks, 169 (2024), 733–743. https://doi.org/10.1016/j.neunet.2023.11.007 doi: 10.1016/j.neunet.2023.11.007
[21]	Z. Yan, D. Zuo, T. Guo, J. Zhou, Quantized $\mathcal{H}_\infty$ stabilization for delayed memristive neural, Neural Comput. Appl., 35 (2023), 16473–16486. https://doi.org/10.1007/s00521-023-08510-3 doi: 10.1007/s00521-023-08510-3
[22]	F. Zhang, Z. Zeng, Multistability of fractional-order neural networks with unbounded time-varying delays, IEEE T. Neur. Net. Lear., 32 (2020), 177–187. https://doi.org/10.1109/TNNLS.2020.2977994 doi: 10.1109/TNNLS.2020.2977994
[23]	F. Du, J. Lu, Improved quasi-uniform stability criterion of fractional-order neural networks with discrete and distributed delays, Asian J. Control, 25 (2023), 229–240. https://doi.org/10.1002/asjc.2758 doi: 10.1002/asjc.2758
[24]	R. Guo, S. Xu, J. Guo, Sliding-mode synchronization control of complex-valued inertial neural networks with Leakage delay and time-varying delays, IEEE T. Syst. Man Cy.-S., 53 (2023), 1095–1103. https://doi.org/10.1109/TSMC.2022.3193306 doi: 10.1109/TSMC.2022.3193306
[25]	X. Mao, X. Wang, Y. Lu, H. Qin, Synchronizations control of fractional-order multidimension-valued memristive neural networks with delays, Neurocomputing, 563 (2024), 126942. https://doi.org/10.1016/j.neucom.2023.126942 doi: 10.1016/j.neucom.2023.126942
[26]	L. Wang, D. Xu, Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays, Int. J. Syst. Sci., 33 (2002), 869–877. https://doi.org/10.1080/00207720210161777 doi: 10.1080/00207720210161777
[27]	Z. Huang, X. Li, S. Mohamad, Z. Lu, Robust stability analysis of static neural network with S-type distributed delays, Appl. Math. Model., 33 (2009), 760–769. https://doi.org/10.1016/j.apm.2007.12.006 doi: 10.1016/j.apm.2007.12.006
[28]	W. Han, Y. Kao, L. Wang, Global exponential robust stability of static interval neural networks with S-type distributed delays, J. Franklin I., 348 (2011), 2072–2081. https://doi.org/10.1016/j.jfranklin.2011.05.023 doi: 10.1016/j.jfranklin.2011.05.023
[29]	H. Zheng, B. Wu, T. Wei, L. Wang, Y. Wang, Global exponential robust stability of high-order Hopfield neural networks with S-type distributed time delays, J. Appl. Math., 2014 (2014), 1–8. https://doi.org/10.1155/2014/705496 doi: 10.1155/2014/705496
[30]	C. Ma, F. Zhou, Global exponential stability of high-order BAM neural networks with S-type distributed delays and reaction diffusion terms, WSEAS T. Math., 10 (2011), 333–345.
[31]	Y. Wang, C. Lu, G. Ji, L. Wang, Global exponential stability of high-order Hopfield-type neural networks with S-type distributed time delays, Commun. Nonlinear Sci., 16 (2011), 3319–3325. https://doi.org/10.1016/j.cnsns.2010.11.005 doi: 10.1016/j.cnsns.2010.11.005
[32]	Q. Yao, L. Wang, Y. Wang, Existence-uniqueness and stability of reaction-diffusion stochastic Hopfield neural networks with S-type distributed time delays, Neurocomputing, 275 (2018), 470–477. https://doi.org/10.1016/j.neucom.2017.08.060 doi: 10.1016/j.neucom.2017.08.060
[33]	Q. Yao, Y. F. Wang, L. S. Wang, Periodic solutions to stochastic reaction-diffusion neural networks with S-type distributed delays, IEEE Access, 7 (2019), 110905–110911. https://doi.org/10.1109/ACCESS.2019.2911962 doi: 10.1109/ACCESS.2019.2911962
[34]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1999.
[35]	L. Ke, Mittag-Leffler stability and asymptotic $\omega$-periodicity of fractional-order inertial neural networks with time-delays, Neurocomputing, 465 (2021), 53–62. https://doi.org/10.1016/j.neucom.2021.08.121 doi: 10.1016/j.neucom.2021.08.121
[36]	J. Slotine, W. Li, Applied nonlinear control, Englewood Cliffs: Prentice Hall, 1991.
[37]	Y. Ke, C. Miao, Stability analysis of fractional-order Cohen-Grossberg neural networks with time delay, Int. J. Comput. Math., 92 (2015), 1102–1113. https://doi.org/10.1080/00207160.2014.935734 doi: 10.1080/00207160.2014.935734

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)