Delay-Dependent Stability Criterion of Caputo Fractional Neural Networks with Distributed Delay

This paper is concerned with the finite-time stability of Caputo fractional neural networks with distributed delay. The factors of such systems including Caputo’s fractional derivative and distributed delay are taken into account synchronously. For the Caputo fractional neural network model, a finite-time stability criterion is established by using the theory of fractional calculus and generalized Gronwall-Bellman inequality approach. Both the proposed criterion and an illustrative example show that the stability performance of Caputo fractional distributed delay neural networks is dependent on the time delay and the order of Caputo’s fractional derivative over a finite time. 1. Introduction It is well known that the fractional calculus is a generalization and extension of the traditional integer-order differential and integral calculus. The fractional calculus has gained importance in both theoretical and engineering applications of several branches of science and technology. It draws a great application in nonlinear oscillations of earthquakes and many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic models. Many practical systems in interdisciplinary fields can be described through fractional derivative formulation. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [1], Podlubny [2], Diethelm [3], and Kilbas et al. [4]. In the last few years, there has been a surge in the study of the theory of fractional dynamical systems. Some recent works the theory of fractional differential systems can be seen in [5–10] and references therein. In particular, for the first time, Lazarevi？ [7] investigated the finite-time stability of fractional time-delay systems. In [8], Lazarevi？ and Spasi？ further introduced the Gronwall’s approach to discuss the finite-time stability of fractional-order dynamic systems. Compared with the classical integer-order derivatives, fractional-order derivatives provide an excellent approach for the description of memory and hereditary properties of various processes. Therefore, it may be more accurate to model by fractional-order derivatives than integer-order ones. In [11–13], fractional operators were introduced into artificial neural network, and the fractional-order formulations of artificial neural network models were also proposed in research works about biological neurons. Recently, there has been an increasing interest in the investigation of the fractional-order neural networks, and some important and interesting results were obtained [13–19], due