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Axioms 2018, 7(2), 30; https://doi.org/10.3390/axioms7020030

Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays

1
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
2
Florida Institute of Technology, Melbourne, FL 32901, USA
3
Department of Applied Mathematics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
4
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 CF50 Galway, Ireland
Distinguished University Professor of Mathematics.
*
Author to whom correspondence should be addressed.
Received: 29 March 2018 / Revised: 3 May 2018 / Accepted: 5 May 2018 / Published: 9 May 2018
(This article belongs to the Special Issue Fractional Differential Equations)
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Abstract

One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks. View Full-Text
Keywords: nonlinear Caputo fractional neural networks; delays; Lyapunov functions; stability; fractional derivative of Lyapunov functions nonlinear Caputo fractional neural networks; delays; Lyapunov functions; stability; fractional derivative of Lyapunov functions
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Agarwal, R.; Hristova, S.; O’Regan, D. Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays. Axioms 2018, 7, 30.

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