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Asymptotic Stability and Polynomial Stability of Impulsive Cohen–Grossberg Neural Networks with Multi-proportional Delays

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Abstract

This paper is concerned with the global asymptotic stability (GAS) and global polynomial stability (GPS) of impulsive Cohen–Grossberg neural networks (ICGNNs) with multi-proportional delays. The concept of GPS of the ICGNNs considered is first proposed and it is pointed out that the GPS is also one of the dynamics of recurrent neural networks with proportional delays. The GPS criteria depending on proportional delay factors are made by introducing tunable parameters, skillfully designing Lyapunov functionals and using inequality skills. The application scope of the ICGNNs considered parameters is extended by introducing tunable parameters. And the relationship of exponential stability, polynomial stability and asymptotic stability of the ICGNNs considered is revealed. These criteria proposed are checked by two numerical examples and simulations.

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References

  1. Zhang X, Han Q (2014) Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach. Neural Netw 54:57–69

    MATH  Google Scholar 

  2. Ali MS, Saravanan S, Rani ME, Elakkia S, Cao J, Alsaedi A, Hayat T (2017) Asymptotic stability of Cohen–Grossberg BAM neutral type neural networks with distributed time varying delays. Neural Process Lett 46(3):991–1007

    Google Scholar 

  3. Wang L, Ding X, Li M (2018) Global asymptotic stability of a class of generalized BAM neural networks with reaction–diffusion terms and mixed time delays. Neurocomputing 321:251–265

    Google Scholar 

  4. Wu B, Liu Y, Lu J (2012) New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays. Math Comput Model 55(3–4):837–843

    MathSciNet  MATH  Google Scholar 

  5. Wang X, Li C, Huang T, Duan S (2014) Global exponential stability of a class of memristive neural networks with time-varying delays. Neural Comput Appl 24(7–8):1707–1715

    Google Scholar 

  6. Liu X, Liu X, Tang M, Wang F (2017) Improved exponential stability criterion for neural networks with time-varying delay. Neurocomputing 234:154–163

    Google Scholar 

  7. Zhang G, Hu J, Jiang F (2018) Show more exponential stability criteria for delayed second-order memristive neural networks. Neurocomputing 315:439–446

    Google Scholar 

  8. Mao XR (1992) Almost sure polynomial stability for a class of stochastic differential equations. Quart J Math 43(3):339–348

    MathSciNet  MATH  Google Scholar 

  9. Santos ML, Rocha MPC, Gomes SC (2010) Polynomial stability of a coupled system of wave equations weakly dissipative. Appl Anal 86(10):1293–1302

    MathSciNet  MATH  Google Scholar 

  10. Liu Z, Rao B (2007) Frequency domain approach for the polynomial stability of a system of partially damped wave equations. J Math Anal Appl 335(2):860–881

    MathSciNet  MATH  Google Scholar 

  11. Es-Sarhir A, Renesse MKV, Stannat W (2012) Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow. Nonlinear Differ Equ Appl 19(6):663–675

    MathSciNet  MATH  Google Scholar 

  12. Milos̆ević M (2018) Convergence and almost sure polynomial stability of the backward and forward–backward Euler methods for highly nonlinear pantograph stochastic differential equations. Math Comput Simul 150:25–48

    MathSciNet  Google Scholar 

  13. Lan G, Xia F, Wang Q (2019) Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay. J Comput Appl Math 346:340–356

    MathSciNet  MATH  Google Scholar 

  14. Zhou L (2013) Delay-dependent exponential stability of cellular neural networks with multi-proportional delays. Neural Process Lett 38(3):321–346

    Google Scholar 

  15. Zhou L, Zhang Y (2016) Global exponential periodicity and stability of recurrent recurrent neural networks with multi-proportional delays. ISA Trans 60:89–95

    Google Scholar 

  16. Zhou L, Zhang Y (2016) Global exponential stability of a class of impulsive recurrent neural networks with proportional delays via fixed point theory. J Frankl Inst 353(2):561–57

    MathSciNet  MATH  Google Scholar 

  17. Zhou L, Liu X (2017) Mean-square exponential input-to-state stability of stochastic recurrent neural networks with multi-proportional delays. Neurocomputing 219:396–403

    Google Scholar 

  18. Tan C, Zhang Y (2009) New sufficient conditions for global asymptotic stability of Cohen–Grossberg neural networks with time-varying delays. Nonlinear Anal RWA 10:2139–2145

    MathSciNet  MATH  Google Scholar 

  19. Chen Z, Liu B (2013) Convergence behavior of Cohen–Grossberg neural networks with time-varying delay in the leakage terms. Neurocomputing 120:518–523

    Google Scholar 

  20. Jian J, Jiang W (2015) Lagrange exponential stability for fuzzy Cohen–Grossberg neural networks with time-varying delays. Fuzzy Set Syst 277:65–80

    MathSciNet  MATH  Google Scholar 

  21. Zheng C, Gong C, Wang Z (2012) Stability criteria for Cohen–Grossberg neural networks with mixed delays and inverse Lipschitz neuron activations. J Frankl Inst 349(9):2903–2924

    MathSciNet  MATH  Google Scholar 

  22. Wang D, Huang L (2018) Robust synchronization of discontinuous Cohen–Grossberg neural networks: pinning control approach. J Frankl Inst 355(13):5866–5892

    MathSciNet  MATH  Google Scholar 

  23. Maharajan C, Raja R, Cao J, Rajchakit G, Alsaedif A (2018) Impulsive Cohen–Grossberg BAM neural networks with mixed time-delays: an exponential stability analysis issue. Neurocomputing 275:2588–2602

    Google Scholar 

  24. Yang W, Yu W, Cao J, Alsaadi FE, Hayat T (2018) Global exponential stability and lag synchronization for delayed memristive fuzzy Cohen–Grossberg BAM neural networks with impulses. Neural Netw 98:122–153

    MATH  Google Scholar 

  25. Zhou L (2014) Global asymptotic stability of cellular neural networks with proportional delays. Nonlinear Dyn 77(1–2):41–47

    MathSciNet  MATH  Google Scholar 

  26. Zheng C, Li N, Cao J (2015) Matrix measure based stability criteria for high-order neural networks with proportional delay. Neurcomputing 149:1149–1154

    Google Scholar 

  27. Hien LV, Son DT (2015) Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. App Math Comput 14:14–23

    MathSciNet  MATH  Google Scholar 

  28. Yu Y (2016) Global exponential convergence for a class of HCNNs with neutral time-proportional delays. Appl Math Comput 285:1–7

    MathSciNet  MATH  Google Scholar 

  29. Liu B (2017) Global exponential convergence of non-autonomous SICNNs with multi-proportional delays. Neural Comput Appl 28(7):1927–1931

    Google Scholar 

  30. Yu Y (2017) Finite-time stability on a class of non-autonomous SICNNs with multi-proportional delays. Asian J Control 19(S1):87–94

    MathSciNet  MATH  Google Scholar 

  31. Zhou L (2018) Delay-dependent and delay-independent passivity of a class of recurrent neural networks with impulse and multi-proportional delay. Neurocomputing 308:235–244

    Google Scholar 

  32. Huang Z, Bin H, Cao J, Wang B (2018) Synchronizing neural networks with proportional delays based on a class of \(q-\)type allowable time scales. IEEE Trans Neural Netw Learn Syst 29(8):3418–3428

    MathSciNet  Google Scholar 

  33. Li N, Cao J (2018) Global dissipativity analysis of quaternion-valued memristor-based neural networks with proportional delay. Neurocomputing 321:103–113

    Google Scholar 

  34. Li H, Li C, Zhang W, Xu J (2018) Global dissipativity of inertial neural networks with proportional delay via new generalized Halanay 1nequalities. Neural Process Lett 48(3):1543–1561

    Google Scholar 

  35. Yang G, Wang W (2019) New results on convergence of CNNs with neutral type proportional delays and D operator. Neural Process Lett 49(1):321–330

    Google Scholar 

  36. Xu C, Chen L, Li P (2019) Effect of proportional delays and continuously distributed leakage delays on global exponential convergence of CNNs. Asian J Control 21(5):2476–2483

    MathSciNet  MATH  Google Scholar 

  37. Xu Y, Zhong J (2019) Convergence of neutral type proportional-delayed HCNNs with D operators. Int J Biomath 12(1):1950016

    MathSciNet  MATH  Google Scholar 

  38. Shen W, Zhang X, Wang Y (2020) Stability analysis of high order neural networks with proportional delays. Neurocomputing 372:33–39

    Google Scholar 

  39. Xu C, Li P (2018) Global exponential convergence of fuzzy cellular neural networks with leakage delays, distributed delays and proportional delays. Circuit Syst Signal Process 37(1):163–177

    MathSciNet  MATH  Google Scholar 

  40. Xu C, Li P (2017) Global exponential convergence of neutral-type Hopfield neural networks with multi-proportional delays and leakage delays. Chaos Soliton Fract 96:139–144

    MathSciNet  MATH  Google Scholar 

  41. Huang C (2019) Exponential stability of inertial neural networks involving proportional delays and non-reduced order method. J Exp Theor Artif Intell. https://doi.org/10.1080/0952813X.2019.1635654

    Article  MATH  Google Scholar 

  42. Huang C, Wen S, Huang L (2019) Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays. Neurocomputing 357:47–52

    Google Scholar 

  43. Huang C, Zhang H (2019) Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method. Int J Biomath 12(2):1950016

    MathSciNet  MATH  Google Scholar 

  44. Li CD, Li CJ, Liu C (2009) Destabilizing effects of impulse in delayed BAM neural networks. Mod Phys Lett B 23(29):3503–3513

    MATH  Google Scholar 

  45. Zhu Q, Cao J (2012) Stability of Markovian jump neural networks with impulse control and time varying delays. Nonlinear Anal RWA 13(5):2259–2270

    MathSciNet  MATH  Google Scholar 

  46. Tan J, Li C (2017) Finite-time stability of neural networks with impulse effects and time-varying delay. Neural Process Lett 46(1):29–39

    Google Scholar 

  47. Guo Y (2018) Globally robust stability analysis for stochastic Cohen–Grossberg neural networks with impulse control and time-varying delays. Ukrainian Math J 69(8):1220–1233

    MathSciNet  MATH  Google Scholar 

  48. Guan K (2019) Global power stability of neural networks with impulses and proportional Delays. B Malays Math Sci Soc 42(5):2237–2264

    MathSciNet  MATH  Google Scholar 

  49. Zou Y, Yang X, Tang R, Cheng Z (2019) Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects. J Frankl Inst. https://doi.org/10.1016/j.jfranklin.2019.05.017

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editor and the anonymous reviewers for their valuable comments and constructive suggestions. This work is supported by the National Science Foundation of Tianjin (No.18JCYBJC85800), the National Natural Science Foundation of China (No.11901433) and the innovative talents cultivation of young middle aged backbone teachers of Tianjin (No.135205GC38).

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  1. School of Mathematics Science, Tianjin Normal University, Tianjin, 300387, China

    Liqun Zhou & Zhixue Zhao

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  1. Liqun Zhou
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Correspondence to Liqun Zhou.

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Zhou, L., Zhao, Z. Asymptotic Stability and Polynomial Stability of Impulsive Cohen–Grossberg Neural Networks with Multi-proportional Delays. Neural Process Lett 51, 2607–2627 (2020). https://doi.org/10.1007/s11063-020-10209-8

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