Existence and global stability of equilibrium point for delayed competitive neural networks with discontinuous activation functions

Abstract

In this article, the issue of global stability is discussed for competitive neural networks with time-varying delay and discontinuous activation functions. First, a sufficient criterion is presented towards the existence and global asymptotic stability of an equilibrium point (EP), by employing the Leray–Schauder alternative theorem, linear matrix inequality technique and generalised Lyapunov function method. In particular, for the case where the activation functions are monotonic increasing, the above criterion also ensures the global exponential stability of an EP. Second, based on the properties of M-matrix, topological degree theory of set-valued map and generalised Lyapunov function method, some sufficient conditions are derived for checking the existence and global exponential stability of an EP. In doing so, the viability problem is investigated and the obtained results are delay-dependent and independent of each other. Finally, two examples with simulations are provided to demonstrate the effectiveness of our results.

Keywords:

• competitive neural networks
• global exponential stability
• global asymptotic stability
• discontinuous activation functions

Acknowledgements

This work was jointly supported by the National Natural Science Foundation of China under Grant 60874088, the 333 Project of Jiangsu Province of China, and the Specialised Research Fund for the Doctoral Programme of Higher Education under Grant 20070286003.