At present, the theory and application of fractional-order neural networks remain in the exploratory stage. We study the asymptotic stability of fractional-order neural networks with Riemann-Liouville (R-L) derivatives. For non-delayed and delayed systems, we propose an asymptotic stability criterion based on the combination of the Lyapunov method and linear matrix inequality (LMI) method. The highlights include the following: (1) for fractional-order neural networks with time delay, the existence and uniqueness of solutions are proven by using matrix analysis theory and contraction mapping theorem, and (2) based on the unique solution, a suitable Lyapunov functional is constructed. Based on the inequality theorem and LMI method, two sets of asymptotic stability criteria for fractional-order neural networks are proven, which avoids the difficulty of solving the fractional derivative by the Leibniz law. Finally, the results are verified using numerical simulations.INDEX TERMS fractional-order neural networks, time delay, asymptotic stability, Lyapunov, LMI

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