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Robust stability of dynamical neural networks with multiple time delays: a review and new results

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Abstract

Robust stability properties of continuous-time dynamical neural networks involving time delay parameters have been extensively studied, and many sufficient criteria for robust stability of various classes of delayed dynamical neural networks have been obtained in the past literature. The class of activation functions and the types of delay terms involved in the mathematical models of dynamical neural networks are two main parameters in the determination of stability conditions for these neural network models. In this article, we will analyse a neural network model of relatively having a more complicated mathematical form where the neural system has the multiple time delay terms and the activation functions satisfy the Lipschitz conditions. By deriving a new and alternative upper bound value for the \(l_2\)-norm of uncertain intervalised matrices and constructing some various forms of the same type of a Lyapunov functional, this paper will first propose new results on global robust stability of dynamical Hopfield neural networks having multiple time delay terms in the presence of the Lipschitz activation functions. Then, we show that some simple modified changes in robust stability conditions proposed for multiple delayed Hopfield neural network model directly yield robust stability conditions of multiple delayed Cohen-Grossberg neural network model. We will also make a very detailed review of the previously published robust stability research results, which are basically in the nonsingular M-matrix or various algebraic inequalities forms. In particular, the robust stability results proposed in this paper are proved to generalize almost all previously reported robust stability conditions for multiple delayed neural network models. Some concluding remarks and future works regarding robust stability analysis of dynamical neural systems are addressed.

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Author notes

  1. Ezgi Aktas and Sabri Arik have contributed equally to this work.

Authors and Affiliations

  1. Department of Computer Engineering, Faculty of Engineering and Architecture, Istanbul Nisantasi University, Istanbul, Turkey

    Ezgi Aktas

  2. Department of Mathematic, Faculty of Science, Istanbul University, Vezneciler, 34134, Istanbul, Turkey

    Ozlem Faydasicok

  3. Department of Computer Engineering, Faculty of Engineering, Istanbul University-Cerrahpasa, Avcilar, 34320, Istanbul, Turkey

    Sabri Arik

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  1. Ezgi Aktas
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  2. Ozlem Faydasicok
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Correspondence to Ozlem Faydasicok.

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Aktas, E., Faydasicok, O. & Arik, S. Robust stability of dynamical neural networks with multiple time delays: a review and new results. Artif Intell Rev 56 (Suppl 2), 1647–1684 (2023). https://doi.org/10.1007/s10462-023-10552-x

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