Finite-time synchronization for fuzzy shunting inhibitory cellular neural networks

Zhangir Nuriyev; Alfarabi Issakhanov; Jürgen Kurths; Ardak Kashkynbayev; Zhangir Nuriyev; Alfarabi Issakhanov; Jürgen Kurths; Ardak Kashkynbayev

doi:10.3934/math.2024623

AIMS Mathematics

2024, Volume 9, Issue 5: 12751-12777. doi: 10.3934/math.2024623

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Research article

Finite-time synchronization for fuzzy shunting inhibitory cellular neural networks

1.
Department of Mathematics, Nazarbayev University, Astana 010000, Kazakhstan
2.
Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany
3.
Institute of Physics, Humboldt University Berlin, Berlin 10099, Germany

Received: 25 January 2024 Revised: 10 March 2024 Accepted: 18 March 2024 Published: 08 April 2024
MSC : 00A69, 34A07, 34D06, 92B20, 92C42

Finite-time synchronization is a critical problem in the study of neural networks. The primary objective of this study was to construct feedback controllers for various models based on fuzzy shunting inhibitory cellular neural networks (FSICNNs) and find out the sufficient conditions for the solutions of those systems to reach synchronization in finite time. In particular, by imposing global assumptions of Lipschitz continuous and bounded activation functions, we prove the existence of finite-time synchronization for three basic FSICNN models that have not been studied before. Moreover, we suggest both controllers and Lyapunov functions that would yield a feasible convergence time between solutions that takes into account the chosen initial conditions. In general, we consecutively explore models of regular delayed FSICNNs and then consider them in the presence of either inertial or diffusion terms. Using criteria derived by means of the maximum-value approach in its different forms, we give an upper bound of the time up to which synchronization is guaranteed to occur in all three FSICNN models. These results are supported by 2D and 3D computer simulations and two respective numerical examples for $ 2\times 2 $ and $ 2\times 3 $ cases, which show the behavior of the solutions and errors under different initial conditions of FSICNNs in the presence and absence of designed controllers.
- finite-time synchronization,
- cellular neural networks,
- nonlinear partial differential equations,
- feedback controllers,
- Lyapunov function
Citation: Zhangir Nuriyev, Alfarabi Issakhanov, Jürgen Kurths, Ardak Kashkynbayev. Finite-time synchronization for fuzzy shunting inhibitory cellular neural networks[J]. AIMS Mathematics, 2024, 9(5): 12751-12777. doi: 10.3934/math.2024623

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Abstract

Finite-time synchronization is a critical problem in the study of neural networks. The primary objective of this study was to construct feedback controllers for various models based on fuzzy shunting inhibitory cellular neural networks (FSICNNs) and find out the sufficient conditions for the solutions of those systems to reach synchronization in finite time. In particular, by imposing global assumptions of Lipschitz continuous and bounded activation functions, we prove the existence of finite-time synchronization for three basic FSICNN models that have not been studied before. Moreover, we suggest both controllers and Lyapunov functions that would yield a feasible convergence time between solutions that takes into account the chosen initial conditions. In general, we consecutively explore models of regular delayed FSICNNs and then consider them in the presence of either inertial or diffusion terms. Using criteria derived by means of the maximum-value approach in its different forms, we give an upper bound of the time up to which synchronization is guaranteed to occur in all three FSICNN models. These results are supported by 2D and 3D computer simulations and two respective numerical examples for $ 2\times 2 $ and $ 2\times 3 $ cases, which show the behavior of the solutions and errors under different initial conditions of FSICNNs in the presence and absence of designed controllers.

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