Liu Xiaoyang, Cao Jinde
Department of Mathematics, Southeast University, Nanjing 210096, China.
Neural Netw. 2009 May;22(4):329-34. doi: 10.1016/j.neunet.2008.11.003. Epub 2008 Nov 27.
Discontinuous dynamical systems, especially neural networks with discontinuous activation functions, arise in a number of applications and have received considerable research attention in recent years. However, there still remain some fundamental issues to be investigated, for instance, how to define the solutions of such discontinuous systems and what conditions can guarantee the existence and stability of the solutions. In this paper, based on the concept of Filippov solution, the dynamics of a general class of neural networks with discontinuous activation functions is investigated. Sufficient conditions are obtained to ensure the existence and stability of the unique periodic solution for the neural networks by using the differential inclusions theory, the Lyapunov-Krasovskii functional method and linear matrix inequality (LMI) technique. Two numerical examples are given to illustrate the theoretical results.
非连续动力系统，特别是具有非连续激活函数的神经网络，出现在许多应用中，并且近年来受到了相当多的研究关注。然而，仍然存在一些基本问题有待研究，例如，如何定义此类非连续系统的解以及哪些条件可以保证解的存在性和稳定性。本文基于菲利波夫解的概念，研究了一类具有非连续激活函数的一般神经网络的动力学。利用微分包含理论、李雅普诺夫 - 克拉索夫斯基泛函方法和线性矩阵不等式（LMI）技术，得到了确保神经网络唯一周期解存在性和稳定性的充分条件。给出了两个数值例子来说明理论结果。
