Mittag-Leffler stability of impulsive fractional-order bi-directional associative memory neural networks with time-varying delays

Abstract

In the paper, a class of impulsive Caputo fractional-order bi-directional associative memory neural networks with time-varying delays is considered. Applying the fractional Lyapunov method, a sufficient condition for Mittag-Leffler stability of the equilibrium point of the system under consideration is derived. Some earlier results are extended and improved. Our results provide an impulsive control law which stabilizes the impulse-free fractional-order neural network time-delay model. An example is provided to demonstrate the effectiveness of the proposed results.

Keywords

Neural networks Mittag-Leffler stability fractional-order derivatives time-varying delays impulsive control Lyapunov method

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References

Acevedo-Mosqueda

Yáñez-Marquez

Acevedo-Mosqueda

(2013) Bidirectional associative memories: Different approaches. AACM Computing Surveys 45(2): 1–30.

Agilandeeswari

Ganesan

(2016) A bi-directional associative memory based multiple image watermarking on cover video. Multimedia Tools and Applications 75: 7211–7256.

Aguila-Camacho

Duarte-Mermoud

Gallegos

(2014) Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation 19: 2951–2957.

Cao

Bai

(2014) Finite-time stability of fractional-order BAM neural networks with distributed delay. Abstract and Applied Analysis 2014: 1–8.

Chen

Cao

Huang

(2004) Exponential stability of BAM neural networks with transmission delays. Neurocomputing 57: 435–454.

Chen

(2015) Global asymptotical ω-periodicity of a fractional-order non-autonomous neural networks. Neural Networks 68: 78–88.

Chen

Yang

(2017) Stability of a class of nonlinear fractional order impulsive switched systems. Transactions of the Institute of Measurement and Control 39(5): 781–790.

Chen

Zeng

Jiang

(2014) Global Mittag–Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Networks 51: 1–8.

Chen

Cao

et al . (2015) Stability and synchronization of memristor-based fractional-order delayed neural networks. Neural Networks 71: 37–44.

10.

Diethelm

(2010) The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Berlin: Springer.

11.

Huang

Cao

Xiao

et al . (2017) Bifurcations in a delayed fractional complex-valued neural network. Applied Mathematics and Computation 292: 210–227.

12.

Huang

Zhao

Wang

et al . (2012) Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94: 13–21.

13.

Kaslik

Sivasundaram

(2012) Nonlinear dynamics and chaos in fractional order neural networks. Neural Networks 32: 245–256.

14.

Kilbas

Srivastava

Trujillo

(2006) Theory and Applications of Fractional Differential Equations. New York, NY: Elsevier.

15.

Kosko

(1987) Adaptive bidirectional associative memories. Applied Optics 26: 4947–4960.

16.

Kosko

(1988) Bi-directional associative memories. IEEE Transactions, Systems, Man, and Cybernetics 18: 49–60.

17.

Kosko

(1992) Neural Networks and Fuzzy Systems – A Dynamical Systems Approach to Machine Intelligence. Englewood Cliffs, NJ: Prentice-Hall.

18.

Lakshmikantham

Bainov

Simeonov

(1989) Theory of Impulsive Differential Equations. Singapore: World Scientific.

19.

Lakshmikantham

Leela

Vasundhara

(2009) Theory of Fractional Dynamic Systems. Cambridge: Cambridge Scientific Publishers.

20.

Chen

Podlubny

(2010) Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers & Mathematics with Applications 59: 1810–1821.

21.

Yang

(2006) Global exponential stability analysis on impulsive BAM neural networks with distributed delay. Journal of Mathematical Analysis and Applications 324: 1125–1139.

22.

Lundstrom

Higgs

Spain

et al . (2008) Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11: 1335–1342.

23.

Naghshtabrizi

Hespanha

Teel

(2010) Stability of delay impulsive systems with application to networked control systems. Transactions of the Institute of Measurement and Control 32(5): 511–528.

24.

Podlubny

(1999) Fractional Differential Equations. San Diego, CA: Academic Press.

25.

Rakkiyappan

Cao

Velmurugan

(2015) Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Transactions on Neural Networks and Learning Systems 26(1): 84–97.

26.

Razumikhin

(1988) Stability of Systems with Retardation. (In Russian.) Moscow: Nauka.

27.

Song

Cao

(2007) Global exponential stability of bidirectional associative memory neural networks with distributed delays. Journal of Computational and Applied Mathematics 202: 266–279.

28.

Stamova

(2009) Stability Analysis of Impulsive Functional Differential Equations. Berlin: Walter de Gruyter.

29.

Stamova

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

30.

Stamova

Henderson

(2016) Practical stability analysis of fractional-order impulsive control systems. ISA Transactions 64: 77–85.

31.

Stamova

Stamov

(2011) Impulsive control on global asymptotic stability for a class of impulsive bidirectional associative memory neural networks with distributed delays. Mathematical and Computer Modelling 53: 824–831.

32.

Stamova

Stamov

(2014) Stability analysis of impulsive functional systems of fractional order. Communications in Nonlinear Science and Numerical Simulation 19: 702–709.

33.

Stamov

(2012) Almost Periodic Solutions of Impulsive Differential Equations. Berlin: Springer.

34.

Tarasov

(2011) Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. New York, NY: Springer.

35.

Wang

Yang

(2015a) Asymptotic stability of delayed fractional-order neural networks with impulsive effects. Neurocomputing 154: 239–244.

36.

Wang

Yang

et al . (2017) Global asymptotic stability of impulsive fractional-order BAM neural networks with time delay. Neural Computing and Applications 28(2): 345–352.

37.

Wang

(2012) Existence results for fractional functional differential equations with impulses. Journal of Applied Mathematics and Computing 38: 85–101.

38.

Wang

Wen

et al . (2015b) Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154: 15–23.

39.

Wang

Zhou

Fečkan

(2012) On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Computers and Mathematics with Applications 64(10): 3008–3020.

40.

Zeng

Song

(2016a) Global Mittag-Leffler stabilization of fractional-order bidirectional associative memory neural networks. Neurocomputing 177: 489–496.

41.

Zhang

Xue

et al . (2016b) LMI conditions to global Mittag-Leffler stability of fractional-order neural networks with impulses. Neurocomputing 193: 148–154.

42.

Fečkan

(2015) Stability analysis of impulsive fractional-order systems by vector comparison principle. Nonlinear Dynamics 82: 2007–2019.

43.

Hei

Chen

(2013) Finite-time stability of fractional-order neural networks with delay. Communication in Theoretical Physics 60: 189–193.

44.

Yang

Song

Liu

et al . (2014) Uniform stability analysis of fractional-order BAM neural networks with delays in the leakage terms. Abstract and Applied Analysis 2014: 1–16.

45.

Yan

Shen

(1999) Impulsive stabilization of impulsive functional differential equations by Lyapunov-Razumikhin functions. Nonlinear Analysis 37: 245–255.

46.

Jiang

(2012) α-stability and α-synchronization for fractional-order neural networks. Neural Networks 35: 82–87.

47.

Zaslavsky

(2005) Hamiltonian Chaos and Fractional Dynamics. Oxford: Oxford University Press.

48.

Zhao

(2002) Global stability of bidirectional associative memory neural networks with distributed delays. Physics Letters A 297: 182–192.

49.

Zhong

Bao

et al . (2008) Impulsive control for fractional-order chaotic systems. Chinese Physics Letters 25: 2812–2815.

50.

Zhou

Wan

(2009) Impulsive effects on stability of Cohen-Grossberg-type bidirectional associative memory neural networks with delays. Nonlinear Analysis: Real World Applications 10: 2531–2540.