Stability of delay Hopfield neural networks with generalized proportional Riemann-Liouville fractional derivative

Ravi P. Agarwal; Snezhana Hristova; Ravi P. Agarwal; Snezhana Hristova

doi:10.3934/math.20231372

AIMS Mathematics

2023, Volume 8, Issue 11: 26801-26820. doi: 10.3934/math.20231372

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Stability of delay Hopfield neural networks with generalized proportional Riemann-Liouville fractional derivative

Ravi P. Agarwal ¹,
Snezhana Hristova ^{2
,
,}

1.
Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 78363, USA
2.
Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv 4000, Bulgaria

Received: 04 July 2023 Revised: 19 July 2023 Accepted: 17 August 2023 Published: 20 September 2023
MSC : 34A08, 34A34, 34D20

The general delay Hopfield neural network is studied. It is considered the case of time-varying delay, continuously distributed delays, time varying coefficients and a special type of a Riemann-Liouville fractional derivative (GPRLFD) with an exponential kernel. The presence of delays and GPRLFD in the model require two special types of initial conditions. The applied GPRLFD also required a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. We use Razumikhin method and Lyapunov functions to study stability properties of the equilibrium of the model. We apply Lyapunov functions defined by absolute values as well as quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GPRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included.
- Hopfield neural networks,
- delays,
- Riemann-Liouville type fractional derivative,
- Lyapunov functions,
- Razumikhin method
Citation: Ravi P. Agarwal, Snezhana Hristova. Stability of delay Hopfield neural networks with generalized proportional Riemann-Liouville fractional derivative[J]. AIMS Mathematics, 2023, 8(11): 26801-26820. doi: 10.3934/math.20231372

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Abstract

The general delay Hopfield neural network is studied. It is considered the case of time-varying delay, continuously distributed delays, time varying coefficients and a special type of a Riemann-Liouville fractional derivative (GPRLFD) with an exponential kernel. The presence of delays and GPRLFD in the model require two special types of initial conditions. The applied GPRLFD also required a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. We use Razumikhin method and Lyapunov functions to study stability properties of the equilibrium of the model. We apply Lyapunov functions defined by absolute values as well as quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GPRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included.

References

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