IEEE Trans Cybern. 2018 May;48(5):1660-1671. doi: 10.1109/TCYB.2017.2776283.
This paper is concerned with global asymptotic stability of delayed neural networks. Notice that a Bessel-Legendre inequality plays a key role in deriving less conservative stability criteria for delayed neural networks. However, this inequality is in the form of Legendre polynomials and the integral interval is fixed on . As a result, the application scope of the Bessel-Legendre inequality is limited. This paper aims to develop the Bessel-Legendre inequality method so that less conservative stability criteria are expected. First, by introducing a canonical orthogonal polynomial sequel, a canonical Bessel-Legendre inequality and its affine version are established, which are not explicitly in the form of Legendre polynomials. Moreover, the integral interval is shifted to a general one . Second, by introducing a proper augmented Lyapunov-Krasovskii functional, which is tailored for the canonical Bessel-Legendre inequality, some sufficient conditions on global asymptotic stability are formulated for neural networks with constant delays and neural networks with time-varying delays, respectively. These conditions are proven to have a hierarchical feature: the higher level of hierarchy, the less conservatism of the stability criterion. Finally, three numerical examples are given to illustrate the efficiency of the proposed stability criteria.
本文研究了时滞神经网络的全局渐近稳定性。注意，贝塞尔-勒让德不等式在推导时滞神经网络更保守稳定性准则方面起着关键作用。然而，这个不等式是勒让德多项式的形式，积分区间固定在[0,1]上。因此，贝塞尔-勒让德不等式的应用范围有限。本文旨在开发贝塞尔-勒让德不等式方法，以期望得到更不保守的稳定性准则。首先，通过引入一个典型的正交多项式序列，建立了典型的贝塞尔-勒让德不等式及其仿射版本，它们不是显式的勒让德多项式形式。此外，积分区间被转移到一个更一般的区间[0,τ]上。其次，通过引入一个适当的扩充李雅普诺夫-克拉索夫斯基泛函，专门针对典型的贝塞尔-勒让德不等式，分别为具有常数时滞和时变时滞的神经网络制定了全局渐近稳定性的充分条件。这些条件被证明具有层次特征：层次越高，稳定性准则的保守性越低。最后，给出了三个数值实例来说明所提出的稳定性准则的有效性。
