Stability properties of neural networks with non-instantaneous impulses

Ravi Agarwal; Snezhana Hristova; Donal O’Regan; Radoslava Terzieva; Ravi Agarwal; Snezhana Hristova; Donal O’Regan; Radoslava Terzieva

doi:10.3934/mbe.2019058

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 3: 1210-1227. doi: 10.3934/mbe.2019058

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Stability properties of neural networks with non-instantaneous impulses

1.
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
2.
Department of Applied Mathematics and Modeling, University of Plovdiv ”Paisii Hilendarski”, 4000 Plovdiv, Bulgaria
3.
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
4.
Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA

Received: 11 December 2018 Accepted: 14 January 2019 Published: 18 February 2019

In this paper, we consider neural networks in the case when the neurons are subject to a certain impulsive state displacement at fixed moments and the duration of this displacement is not negligible small (these are known as non-instantaneous impulses). We examine some stability properties of the equilibrium of the model. Several sufficient conditions for uniform Lipschitz stability of the equilibrium of neural networks with time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. These sufficient conditions are explicitly expressed in terms of the parameters of the system and hence they are easily verifiable. The case of non-Lipschitz activation functions is also studied. The theory is illustrated on particular nonlinear neural networks.
- nonlinear neural networks,
- non-instantaneous impulses,
- Lipschitz stability
Citation: Ravi Agarwal, Snezhana Hristova, Donal O’Regan, Radoslava Terzieva. Stability properties of neural networks with non-instantaneous impulses[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1210-1227. doi: 10.3934/mbe.2019058

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Abstract

In this paper, we consider neural networks in the case when the neurons are subject to a certain impulsive state displacement at fixed moments and the duration of this displacement is not negligible small (these are known as non-instantaneous impulses). We examine some stability properties of the equilibrium of the model. Several sufficient conditions for uniform Lipschitz stability of the equilibrium of neural networks with time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. These sufficient conditions are explicitly expressed in terms of the parameters of the system and hence they are easily verifiable. The case of non-Lipschitz activation functions is also studied. The theory is illustrated on particular nonlinear neural networks.

References

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