Mittag-Leffler stability of fractional-order Hopfield neural networks

• An extended second method of Lyapunov is presented.
• A helpful Caputo fractional-order inequality is introduced.
• An appropriate Lyapunov function for fractional-order neural networks is chosen and proved.
• Mittag-Leffler stability of fractional-order Hopfield neural networks is analyzed.
• Synchronization for fractional-order Hopfield neural networks are discussed.

Fractional-order Hopfield neural networks are often used to model how interacting neurons process information. To show reliability of the processed information, it is needed to perform stability analysis of these systems. Here, we perform Mittag-Leffler stability analysis for them. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality that can be effectively used to this analysis. Importantly, these general results can help construct Lyapunov functions used to Mittag-Leffler stability analysis of fractional-order Hopfield neural networks. As a result, a set of sufficient conditions is derived to guarantee this stability. In addition, the general results can be easily used to the establishment of stability conditions for achieving complete and quasi synchronization in the coupling case of these networks with constant or time-dependent external inputs. Finally, two numerical examples are presented to show the effectiveness of our theoretical results.