Further analysis of global robust stability of neural networks with multiple time delays

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Abstract

This paper deals with the problem of the global robust asymptotic stability of the class of dynamical neural networks with multiple time delays. We propose a new alternative sufficient condition for the existence, uniqueness and global asymptotic stability of the equilibrium point under parameter uncertainties of the neural system. We first prove the existence and uniqueness of the equilibrium point by using the Homomorphic mapping theorem. Then, by employing a new Lyapunov functional, the Lyapunov stability theorem is used to establish the sufficient condition for the asymptotic stability of the equilibrium point. The obtained condition is independent of time delays and relies on the network parameters of the neural system only. Therefore, the equilibrium and stability properties of the delayed neural network can be easily checked. We also make a detailed comparison between our result and the previous corresponding results derived in the previous literature. This comparison proves that our result is new and improves some of the previously reported robust stability results. Some illustrative numerical examples are given to show the applicability and advantages of our result.

Section snippets

System description and preliminaries

In recent years, neural networks have been widely used in solving various classes of engineering problems such as signal processing, optimization, image processing, associative memory design and control systems. In such applications, it is important to know the global stability properties of the designed neural network, which makes the analysis of dynamical behavior of neural networks one of the key factors in the design and applications of neural networks. Therefore, there has been a great

Global robust stability analysis

In this section, we present a theorem which states the conditions that guarantee the existence, uniqueness and global asymptotic stability of the equilibrium point of system (1) under the parameter uncertainties given by Eq. (2). Before we proceed any further, in order to simplify the proofs of the stability part of the theorem, we will shift the equilibrium point x of system (1) to the origin by using the transformation zi(·)=xi(·)xi, i=1,2,,n. This transformation puts the system (1) in

Comparison and examples

In this section, by giving some constructive numerical examples, we will compare our results with the previous robust stability result derived in the literature. In order to make a precise comparison, we first restate the previous literature results obtained for the robust stability of neural network model (1):

Theorem 2

Ozcan and Arik [2]

Let fL. Then, the neural network model (1) is globally asymptotically robust stable, if the following condition holds: Θ=cmM(A2+A2)12M(B^1+B^)>0where M=max(i) and cm=min

Conclusions

The main contribution of this paper is the result that ensures the existence, uniqueness and global robust stability of equilibrium point for neural networks with multiple time delays with respect to the Lipschitz activation functions. The obtained condition has established a relationship between the network parameters of neural system and is independently of the delay parameters. A comparison between the result of this paper and the corresponding robust stability results derived in the

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