New Global Exponential Stability Criteria for Nonlinear Delay Differential Systems with Applications to BAM Neural Networks

TL;DR

This paper introduces new global exponential stability criteria for nonlinear delay differential systems, especially applied to BAM neural networks, using the M-matrix method for easily verifiable conditions.

Contribution

It provides novel stability conditions for nonlinear delay systems and their linear counterparts, utilizing the M-matrix approach for straightforward verification.

Findings

Stability conditions are based on explicit M-matrix criteria.

The criteria are easier to verify than previous methods.

Applications to BAM neural networks demonstrate the effectiveness.

Abstract

We consider a nonlinear non-autonomous system with time-varying delays $$ \dot{x_i}(t)=-a_i(t)x_{i}(h_i(t))+\sum_{j=1}^mF_{ij}(t,x_j(g_{ij}(t))) $$ which has a large number of applications in the theory of artificial neural networks. Via the M-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed by the system's parameters, is an $M$-matrix. Comparison with the tests obtained by K. Gopalsamy (2007) and B. Liu (2013) for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper.