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Title: Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks
Author: Almeida, Ricardo
Agarwal, Ravi P.
Hristova, Snezhana
O’Regan, Donal
Keywords: Generalized Caputo proportional fractional derivative
Exponential stability
Mittag–Leffler stability
Quadratic Lyapunov functions
Hopfield neural networks
Issue Date: 2021
Publisher: MDPI
Abstract: A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.
Peer review: yes
DOI: 10.3390/axioms10040322
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Appears in Collections:CIDMA - Artigos
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