Stability Analysis of Two Kinds of Fractional-Order Neural Networks Based on Lyapunov Method

Abstract

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

Lyapunov stability analysis for nonlinear nabla tempered fractional order systems

“…Remark Theorem 3 can be regarded the generalization of Zhang et al [44, Theorem 1], Zhang et al [45, Theorem 1] and Zhang [43, Theorem 3.4.1]. This work removes the condition $$$ \underset{k\to +\infty }{\lim }h(k)=0 $$$, which is the refinement.…”

“…Remark Compared with Zhang [43, Theorem 3.3.1], the continuous time case is extended to the discrete time case, the classical case is extended to the tempered case, and the zero initial instant is extended to the nonzero case. The condition $$$ \lambda <0 $$$ is adopted here.…”

“…Compared with Zhang et al [44, Theorem 1], [45, Theorem 1] and Zhang [43, Theorem 3.4.1], a new variable $$$ \gamma $$$ is introduced here, which could reduce the conservativeness and enhance the applicability. For example, letting $$$ h(k):= \frac{\left(k-a\right)\overline{{}^{-\alpha }}}{\Gamma \left(1-\alpha \right)} $$$, $$$ \gamma \in \left(0,\alpha \right) $$$, one has $$$$ {\sum}_{j=a+1}^{+\infty}\mid h(j)\mid ={\left[\frac{\left(j-a\right)\overline{{}^{1-\alpha }}}{\Gamma \left(2-\alpha \right)}\right]}_{j=+\infty }=+\infty, $$$$ …”