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Mittag-Leffler stability and synchronization for FOQVFNNs including proportional delay and Caputo derivative via fractional differential inequality approach

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Abstract

This article discusses the dynamics of Caputo fractional-order quaternion-valued fuzzy neural networks (FOQVFNNs) including proportional delay and derivative order interval \((0,1)\cup (1,2)\). By constructing a fractional-order differential equation according to the fractional differential inequality, a novel lemma is proposed in terms of the Laplace transform approach. Combining the Mittag-Leffler function, Lyapunov direct method, fuzzy inequality technique and the new lemma, several ML stability and synchronization results for FOQVFNNs under the different controller designs are established. The obtained criteria are concise and applicable in the form of algebraic inequalities. Three numerical simulation examples are given to further illustrate the theoretical results.

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References

  • Aadhithiyan S, Raja R, Zhu Q, Alzabut J, Niezabitowski M, Lim CP (2021) Modified projective synchronization of distributive fractional order complex dynamic networks with model uncertainty via adaptive control. Chaos Soliton Fract 147:110853

    Article  MathSciNet  MATH  Google Scholar 

  • Bao H, Park J, Cao J (2021) Adaptive synchronization of fractional-order output-coupling neural networks via quantized output control. IEEE Trans Neural Netw Learn Syst 32:3230–3239

    Article  MathSciNet  Google Scholar 

  • Chen J, Li C, Yang X (2018) Asymptotic stability of delayed fractional-order fuzzy neural networks with impulse effects. J Frankl Inst 355(15):7595–7608

    Article  MathSciNet  MATH  Google Scholar 

  • Chen J, Park J, Xu S (2021) Stability analysis for delayed neural networks via an improved negative-definiteness lemma. Inf Sci 576:756–768

    Article  MathSciNet  Google Scholar 

  • Diethelm K (2011) An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract Calc Appl Anal 14(3):475–490

    Article  MathSciNet  MATH  Google Scholar 

  • El Mfadel A, Melliani S, Elomari M (2024) On the initial value problem for fuzzy nonlinear fractional differential equations. Kragujevac J Math 48(4):547–554

    MATH  Google Scholar 

  • El Mfadel A, Melliani S, Elomari M (2021a) A note on the stability analysis of fuzzy nonlinear fractional differential equations involving the Caputo fractional derivative. Int J Math Math Sci. https://doi.org/10.1155/2021/7488524

  • El Mfadel A, Melliani S, Elomari M (2021b) On the existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. J Funct Space. https://doi.org/10.1155/2021/4099173

  • Garrappa R, Popolizio M (2022) A computationally efficient strategy for time-fractional diffusion-reaction equations. Comput Math with Appl 116:181–193

    Article  MathSciNet  MATH  Google Scholar 

  • Guan K, Wang Q (2018) Impulsive control for a class of cellular neural networks with proportional delay. Neural Process Lett 48:1459–1479

    Article  Google Scholar 

  • Jian J, Wu K, Wang B (2020) Global Mittag–Leffler boundedness of fractional-order fuzzy quaternion-valued neural networks with linear threshold neurons. IEEE Trans Fuzzy Syst 29(10):3154–3164

    Article  Google Scholar 

  • Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations. North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands

  • Lee T, Park M, Park J (2021) An improved stability criterion of neural networks with time-varying delays in the form of quadratic function using novel geometry-based conditions. Appl Math Comput 404(11):126226

    Article  MathSciNet  MATH  Google Scholar 

  • Li H, Hu C, Cao J, Jiang H, Alsaedi A (2019) Quasi-projective and complete synchronization of fractional-order complex-valued neural networks with time delays. Neural Netw 118:102–109

    Article  MATH  Google Scholar 

  • Liu M, Wu H, Zhao W (2020) Event-triggered stochastic synchronization in finite time for delayed semi-Markovian jump neural networks with discontinuous activations. Comput Appl Math 39:118

    Article  MathSciNet  MATH  Google Scholar 

  • Miron S, Bihan N, Mars J (2006) Quaternion-music for vector-sensorarray processing. IEEE Trans Signal Process 54(4):1218–1229

    Article  MATH  Google Scholar 

  • Odibat Z, Baleanu D (2020) Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives. Appl Numer Math 156:94–105

    Article  MathSciNet  MATH  Google Scholar 

  • Rakkiyappan R, Velmurugan G, Cao J (2014) Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays. Nonlinear Dyn 78:2823–2836

    Article  MathSciNet  MATH  Google Scholar 

  • Song Q, Chen Y, Zhao Z, Liu Y (2021) Robust stability of fractional-order quaternion-valued neural networks with neutral delays and parameter uncertainties. Neurocomputing 420:70–81

    Article  Google Scholar 

  • Soori Z, Aminataei A (2020) Correction to: Numerical solution of space fractional diffusion equation by spline method combined with Richardson extrapolation. Comp. Appl. Math 39:179

    Article  MathSciNet  MATH  Google Scholar 

  • Stamova I (2014) Global Mittag–Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dyn 77(4):1251–1260

    Article  MathSciNet  MATH  Google Scholar 

  • Syed Ali M, Hymavathi M (2021) Synchronization of fractional order neutral type fuzzy cellular neural networks with discrete and distributed delays via state feedback control. Neural Process Lett 53:929–957

    Article  Google Scholar 

  • Syed Ali M, Narayanan G, Shekher V, Alsulami H, Saeed T (2020) Dynamic stability analysis of stochastic fractional-order memristor fuzzy BAM neural networks with delay and leakage terms. Appl Math Comput 369:124896

    Article  MathSciNet  MATH  Google Scholar 

  • Took C, Strbac G, Aihara K, Mandic D (2011) Quaternion-valued short-term joint forecasting of three-dimensional wind and atmospheric parameters. Renew Energy 36:1754–1760

    Article  Google Scholar 

  • Tyagi S, Martha S (2020) Finite-time stability for a class of fractional-order fuzzy neural networks with proportional delay. Fuzzy Set Syst 381:68–77

    Article  MathSciNet  MATH  Google Scholar 

  • Wang C, Zhang H, Stamova I, Cao J (2022) Global synchronization for BAM delayed reaction–diffusion neural networks with fractional partial differential operator. J Frankl Inst. https://doi.org/10.1016/j.jfranklin.2022.08.038

  • Wu A, Zeng Z (2016) Boundedness, Mittag–Leffler stability and asymptotical \(\omega \)-periodicity of fractional-order fuzzy neural networks. Neural Netw 74:73–84

    Article  MATH  Google Scholar 

  • Wu A, Zeng Z, Song X (2015) Global Mittag–Leffler stabilization of fractional-order bidirectional associative memory neural networks. Neurocomputing 177:489–496

    Article  Google Scholar 

  • Xiao J, Cheng J, Cao J, Zhong S, Wen S (2020) Novel methods to finite-time Mittag–Leffler synchronization problem of fractional-order quaternion-valued neural networks. Inf Sci 526:221–244

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao J, Wen S, Yang X, Zhong S (2020) New approach to global Mittag–Leffler synchronization problem of fractional-order quaternion-valued BAM neural networks based on a new inequality. Neural Netw 122:320–337

    Article  MATH  Google Scholar 

  • Xiao Q, Huang T, Zeng Z (2020) Synchronization of timescale-type nonautonomous neural networks with proportional delays. IEEE Trans Syst Man Cybern 52(4):2167–2173

    Article  Google Scholar 

  • Xu Y, Li Y, Li W (2020) Adaptive finite-time synchronization control for fractional-order complex-valued dynamical networks with multiple weights. Commun Nonlinear Sci Numer Simul 85:105239

    Article  MathSciNet  MATH  Google Scholar 

  • Yang S, Hu C, Yu J, Jiang H (2021) Projective synchronization in finite-time for fully quaternion-valued memristive networks with fractional-order. Chaos Soliton Fract 147:110911

    Article  MathSciNet  MATH  Google Scholar 

  • Yang Z, Zhang J, Hu J, Mei J (2021) New results on finite-time stability for fractional-order neural networks with proportional delay. Neurocomputing 442:327–336

    Article  Google Scholar 

  • Yao X, Zhong S (2021) EID-based robust stabilization for delayed fractional-order nonlinear uncertain system with application in memristive neural networks. Chaos Soliton Fract 144(10):110705

    Article  MathSciNet  Google Scholar 

  • You X, Song Q, Zhao Z (2020) Global Mittag–Leffler stability and synchronization of discrete-time fractional-order complex-valued neural networks with time delay. Neural Netw 122:382–394

    Article  MATH  Google Scholar 

  • Zeng D, Zhang R, Park J, Zhong S, Cheng J, Wu G (2021) Reliable stability and stabilizability for complex-valued memristive neural networks with actuator failures and aperiodic event-triggered sampled-data control. Nonlinear Anal-Hybri 39:100977

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang H, Ye M, Ye R, Cao J (2018) Synchronization stability of Riemann–Liouville fractional delay-coupled complex neural networks. Phys A 508:155–165

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang H, Ye R, Liu S, Cao J, Alsaedi A, Li X (2018) LMI-based approach to stability analysis for fractional-order neural networks with discrete and distributed delays. Int J Syst Sci 49:537–545

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang W, Capilnasiu A, Sommer G, Holzapfel G, Nordsletten D (2020) An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials. Comput Methods Appl Mech Eng 362:112834

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang H, Cheng J, Zhang HM, Zhang W, Cao J (2021) Quasi-uniform synchronization of Caputo type fractional neural networks with leakage and discrete delays. Chaos Soliton Fract 152:111432

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang H, Cheng Y, Zhang HM, Zhang W, Cao J (2022) Hybrid control design for Mittag–Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects. Math Comput Simul 197:341–354

    Article  MathSciNet  MATH  Google Scholar 

  • Zheng M, Li L, Peng H, Xiao J, Yang Y, Zhang Y, Zhao H (2018) Finite-time stability and synchronization of memristor-based fractional-order fuzzy cellular neural networks. Commun Nonlinear Sci Numer Simul 59:272–291

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to express our sincere thanks to Editors and anonymous Reviewers for their valuable suggestions and constructive comments to improve this paper. This work was supported by the Natural Science Foundation of Anhui Province of China (No. 1908085MA01).

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Authors and Affiliations

  1. School of Mathematics and Physics, Anqing Normal University, Anqing, 246133, People’s Republic of China

    Hai Zhang, Chen Wang, Weiwei Zhang & Hongmei Zhang

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  1. Hai Zhang
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  2. Chen Wang
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  3. Weiwei Zhang
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  4. Hongmei Zhang
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Correspondence to Hai Zhang.

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Communicated by Roberto Garrappa.

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Zhang, H., Wang, C., Zhang, W. et al. Mittag-Leffler stability and synchronization for FOQVFNNs including proportional delay and Caputo derivative via fractional differential inequality approach. Comp. Appl. Math. 41, 344 (2022). https://doi.org/10.1007/s40314-022-02062-3

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  • DOI: https://doi.org/10.1007/s40314-022-02062-3

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