Robust Stability of Cohen–Grossberg Neural Networks via State Transmission Matrix

This brief is concerned with the global robust exponential stability of a class of interval Cohen-Grossberg neural networks with both multiple time-varying delays and continuously distributed delays. Some new sufficient robust stability conditions are established in the form of state transmission matrix, which are different from the existing ones. Furthermore, a sufficient condition is also established to guarantee the global stability for this class of Cohen-Grossberg neural networks without uncertainties. Three examples are used to show the effectiveness of the obtained results.

I. Introduction

Cohen–Grossberg [9] proposed a neural network model in 1983 described by the following system of equations

$$\mathdot{u}_i (t) = - d_i (u_i (t))\left[a_i (u_i (t)) - \sum_{j = 1}^n {w_{ij}} \bar g_j (u_j (t))\right] \eqno{\hbox{(1)}}$$ where is a positive and bounded amplification function, is a well-defined function to guarantee the existence of the solution of system (1), is an activation function describing the effects of input on the output of neuron, and is the connection weight coefficient of the neural network, . System (1) includes a number of models from neurobiology, population biology, and evolution theory, as well as the Hopfield neural network model [11] as a special case. In electronic implementation of analog neural networks, delays always exist due to the transmission of signal and the finite switching speed of amplifiers [22]. On the other hand, it is desirable to introduce delays into neural networks when dealing with problems associated with motions [17], [18], [26], [27]. Therefore, model (1) and its delayed version have attracted the attention of many researchers and have been extensively investigated due to their potential applications such as associative content-addressable memories, pattern recognition, and optimization. Such applications rely on the qualitative stability properties of the network. Thus, the qualitative analysis of the networks dynamic behavior is a prerequisite step for the practical design and application of neural networks. Recently, some sufficient conditions for the global asymptotic/exponential stability of Cohen–Grossberg neural networks have been studied in the literatures; see, e.g., [6]–[8], [10], [12], [13], [20], [21], [30], [31], [33], and [34]. Among them, Ye et al. [33] introduced constant delays into (1), which yields the following form: $$\mathdot {u}_i (t) = - d_i (u_i (t))\left[a_i (u_i (t)) \vphantom{\sum_{k = 0}^N}- \sum_{k = 0}^N {\sum_{j = 1}^n w_{ij}^k} \bar g_j (u_j (t - \tau _k))\right]\eqno\hbox{(2)}$$ where are bounded constant delays, are the connection weight coefficients, and other notations are the same as those in system (1), .