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Further analysis of stability of uncertain neural networks with multiple time delays

Sabri Arik

Author Affiliations

Department of Electrical and Electronics Engineering, Isik University, Sile, Istanbul, 34980, Turkey

Advances in Difference Equations 2014, 2014:41  doi:10.1186/1687-1847-2014-41

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2014/1/41


Received:11 July 2013
Accepted:8 January 2014
Published:27 January 2014

© 2014 Arik; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the robust stability of uncertain neural networks with multiple time delays with respect to the class of nondecreasing activation functions. By using the Lyapunov functional and homeomorphism mapping theorems, we derive a new delay-independent sufficient condition the existence, uniqueness, and global asymptotic stability of the equilibrium point for delayed neural networks with uncertain network parameters. The condition obtained for the robust stability establishes a matrix-norm relationship between the network parameters of the neural system, and therefore it can easily be verified. We also present some constructive numerical examples to compare the proposed result with results in the previously published corresponding literature. These comparative examples show that our new condition can be considered as an alternative result to the previous corresponding literature results as it defines a new set of network parameters ensuring the robust stability of delayed neural networks.

Keywords:
stability analysis; delayed neural networks; interval matrices; Lyapunov functionals

1 Introduction

Dynamical neural networks have recently received a great deal of attention due to their potential applications in image and signal processing, combinatorial optimization problems, pattern recognition, control engineering, and some other related areas. In the electronic implementation of analog neural networks, during the processing and transmission of signals in the network, due to the finite switching speed of amplifiers, some time delays occur which may change the dynamical behavior of the network from stable to unstable. Therefore, it is important to take into account the effects of the time delays in the dynamical analysis of neural networks. On the other hand, it is well known that unavoidably some disturbances are to be considered in the modeling and stability analysis neural networks. The major disturbances occur within the network, which is mainly due to the deviations in the values of the electronic components during the process of implementation. Therefore, in recent years, many papers have focused on studying the existence, uniqueness, and global robust asymptotic stability of the equilibrium point in the presence of time delays and parameter uncertainties for various classes of nonlinear neural networks, and one reported some robust stability results [1-43].

In the current paper, we aim to study the robust stability of a class of uncertain neural networks with multiple time delays. By using the Lyapunov functional and homeomorphism mapping theorems, a new delay-independent sufficient condition for global robust asymptotic stability of the equilibrium point for this class of neural networks is derived. Meanwhile, three numerical examples are presented to demonstrate the applicability of the condition and to show the advantages of our result over the previously published robust stability results.

We use the following notation. Throughout this paper, the superscript T represents the transpose. I stands for the identity matrix of appropriate dimension. For the vector <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M1">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M2">View MathML</a> will denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M3">View MathML</a>. For any real matrix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M5">View MathML</a> will denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M7">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M8">View MathML</a> will denote the minimum and maximum eigenvalues of Q, respectively. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4">View MathML</a> is a symmetric matrix, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M10">View MathML</a> will imply that Q is positive definite, i.e., Q has all eigenvalues real and positive. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M11">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4">View MathML</a> be two symmetric matrices. Then, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M13">View MathML</a> will imply that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M14">View MathML</a> for any real vector <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M15">View MathML</a>. A real matrix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M11">View MathML</a> is said to be a nonnegative matrix if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M17">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M18">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M11">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M4">View MathML</a> be two real matrices. Then, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M21">View MathML</a> will imply that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M22">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M18">View MathML</a>. We also recall the following vector and matrix norms:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M24">View MathML</a>

2 Preliminaries

The delayed neural network model we consider in this paper is described by the set of nonlinear differential equations of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M25">View MathML</a>

(2.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26">View MathML</a>, n is the number of the neurons, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M27">View MathML</a> denotes the state of the neuron i at time t, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M28">View MathML</a> denote activation functions, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M29">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M30">View MathML</a> denote the strengths of connectivity between neurons j and i at time t and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M31">View MathML</a>, respectively; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M32">View MathML</a> represents the time delay required in transmitting a signal from the neuron j to the neuron i, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M33">View MathML</a> is the constant input to the neuron i, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M34">View MathML</a> is the charging rate for the neuron i.

In order to accomplish the objectives of this paper in the sense of robust stability of dynamical neural networks, we will first define the class of the activation functions that we will employ in the neural network model (2.1) and the parametric uncertainties of the system matrices A, B, and C.

The activation functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M35">View MathML</a> are assumed to be nondecreasing and slope-bounded, that is, there exist some positive constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M36">View MathML</a> such that the following conditions hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M37">View MathML</a>

This class of functions will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M38">View MathML</a>.

We will set intervals for the system matrices <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M39">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M40">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M41">View MathML</a> in (2.1) as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M42">View MathML</a>

(2.2)

In what follows, we will give some basic definitions and lemmas that will play an important role in the proof of our robust stability results.

Definition 2.1 (See [28])

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M43">View MathML</a> be an equilibrium point of neural system (2.1). The neural network model (2.1) with the parameter ranges defined by (2.2) is globally asymptotically robust stable if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44">View MathML</a> is a unique and globally asymptotically stable equilibrium point of system (2.1) for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M45">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M46">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M47">View MathML</a>.

Lemma 2.1 (See [29])

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M48">View MathML</a>satisfies the conditions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M49">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M51">View MathML</a>as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M52">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M53">View MathML</a>is a homeomorphism of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54">View MathML</a>.

Lemma 2.2 (See [1])

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M55">View MathML</a>. If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M56">View MathML</a>

then, for any positive diagonal matrix P and a nonnegative diagonal matrix ϒ, the following inequality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M57">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M58">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M59">View MathML</a>.

3 Robust stability analysis

In this section, we will present a new sufficient condition that guarantees the global robust asymptotic stability of the equilibrium point of the neural network model (2.1), which is stated in the following theorem.

Theorem 3.1For the neural network model (2.1), assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M60">View MathML</a>and the network parameters satisfy (2.2). Then the neural network model (2.1) has a unique and globally robust asymptotically stable equilibrium point for eachu, if there exist a positive diagonal matrixPand a nonnegative diagonal matrix ϒ such that the following condition holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M61">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M62">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M63">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M64">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M65">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M66">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M67">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M68">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M69">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M70">View MathML</a>are positive constants such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M71">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M72">View MathML</a>.

Proof We will first prove the existence and uniqueness of the equilibrium point of system (2.1) by making use of the homeomorphism mapping theorem defined in Lemma 2.1. To this end, we define the mapping associated with system (2.1) as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M73">View MathML</a>

(3.1)

We point out here that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44">View MathML</a> is an equilibrium point of the neural network model (2.1), then, by definition, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44">View MathML</a> satisfies the following equilibrium equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M76">View MathML</a>

Therefore, every solution of the equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M77">View MathML</a> is an equilibrium point of system (2.1). Hence, if we shaw that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M78">View MathML</a> is homeomorphism of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54">View MathML</a>, then we will conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M80">View MathML</a> has a unique solution for each u. In order to prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M78">View MathML</a> is a homeomorphism of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54">View MathML</a>, we choose two real vectors <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M83">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M84">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50">View MathML</a>. In this case, we can write the following equation for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M78">View MathML</a> given by (3.1):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M87">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50">View MathML</a>. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M89">View MathML</a> when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50">View MathML</a>, then (3.2) takes the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M91">View MathML</a>

from which it follows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M92">View MathML</a> if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M93">View MathML</a> as C is a positive diagonal matrix. Now assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M94">View MathML</a> when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M93">View MathML</a>. In this case, if we multiply both sides of (3.2) by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M96">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M97">View MathML</a>

(3.3)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M98">View MathML</a> is a positive diagonal matrix.

In the light of Lemma 2.2, we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M99">View MathML</a>

(3.4)

where ϒ is a nonnegative diagonal matrix.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M38">View MathML</a> implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M101">View MathML</a>

(3.5)

We also note the following inequality:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M102">View MathML</a>

(3.6)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M69">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M70">View MathML</a> are positive constants such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M105">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M72">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M107">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M108">View MathML</a>.

Equation (3.6) can be written in the following form:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M109">View MathML</a>

(3.7)

where α and β are some positive constants. Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M110">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M111">View MathML</a> in (3.7) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M112">View MathML</a>

(3.8)

Using (3.4), (3.5), and (3.8) in (3.3) results in

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M113">View MathML</a>

which can be written in the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M114">View MathML</a>

(3.9)

For the activations functions belonging to the class , it has been shown in [27] that for the inequality in the form of (3.9), if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M92">View MathML</a>, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M50">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M119">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M120">View MathML</a>. Hence, we have proved that the map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M121">View MathML</a> is a homomorphism of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M54">View MathML</a>, meaning that the condition of Theorem 3.1 implies the existence and uniqueness of the equilibrium point for neural network model (2.1).

It will be now shown that the condition obtained for the existence and uniqueness of the equilibrium point of neural network model (2.1) in Theorem 3.1 also implies the global asymptotic stability of the equilibrium point. To this end, we will shift the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44">View MathML</a> of system (2.1) to the origin. The transformation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M124">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26">View MathML</a>, puts the network model (2.1) into the following form:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M126">View MathML</a>

(3.10)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M127">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26">View MathML</a>, satisfies the following property:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M129">View MathML</a>

Note that equilibrium and stability properties of the neural network models are identical. Therefore, proving the asymptotic stability of the origin of system (3.10) will directly imply the asymptotic stability of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M44">View MathML</a>. Now, consider the following positive definite Lyapunov functional for system (3.10):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M131">View MathML</a>

where the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M132">View MathML</a>, α, β, γ, and ε are positive constants to be determined later. The time derivative of the functional along the trajectories of system (3.10) is obtained as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M133">View MathML</a>

(3.11)

We note that the following inequalities hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M134">View MathML</a>

(3.12)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M135">View MathML</a>

(3.13)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M136">View MathML</a>

(3.14)

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M137">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M138">View MathML</a>

(3.15)

In the light of Lemma 2.2, we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M139">View MathML</a>

(3.16)

Using (3.12)-(3.16) in (3.11) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M140">View MathML</a>

(3.17)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M141">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M142">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M143">View MathML</a>. We now note the following inequalities:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M144">View MathML</a>

(3.18)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M145">View MathML</a>

(3.19)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M146">View MathML</a>

(3.20)

Using (3.18)-(3.20) in (3.17) leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M147">View MathML</a>

(3.21)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M148">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M149">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M150">View MathML</a>. Then (3.21) takes the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M151">View MathML</a>

(3.22)

It has been shown in [27] that, for the Lyapunov functional defined above, if its time derivative is in the form of (3.22) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M152">View MathML</a> with Π being positive definite, then the origin system (3.10), or equivalently the equilibrium point of system (2.1) is globally asymptotically stable. Hence, we have shown that the condition of Theorem 3.1 implies the global robust asymptotic stability of system (2.1). □

4 Comparison and examples

In this section, we present some constructive numerical materials to demonstrate the effectiveness and applicability of the proposed conditions and to show the advantages of our results over the previous corresponding robust stability result derived in the literature. In order to make a precise comparison between the results, we will first restate results from the previous literature.

Theorem 4.1 (See [27])

For the neural network model (2.1), assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M153">View MathML</a>and the network parameters satisfy (2.2). Then the neural network model (2.1) is globally asymptotically robust stable if there exists a positive diagonal matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M154">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M155">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M156">View MathML</a>is a positive diagonal matrix, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M157">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M158">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M159">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M160">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M107">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M66">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M67">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M164">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M165">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M166">View MathML</a>are positive constants such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M167">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M168">View MathML</a>.

Theorem 4.2 (See [27])

For the neural network model (2.1), assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M153">View MathML</a>and the network parameters satisfy (2.2). Then the neural network model (2.1) is globally asymptotically robust stable, if there exists a positive diagonal matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M154">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M171">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M156">View MathML</a>is a positive diagonal matrix, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M107">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M66">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M67">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M164">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M165">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M166">View MathML</a>are positive constants such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M167">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M168">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M181">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M182">View MathML</a>.

Theorem 4.3 (See [30])

For the neural network model (2.1), assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M153">View MathML</a>and the network parameters satisfy (2.2). Then the neural network model (2.1) is globally asymptotically robust stable if there exist positive constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M184">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M26">View MathML</a>, such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M186">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M187">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M188">View MathML</a>.

We will now consider the following examples.

Example 4.1 Consider the neural system (2.1) with the following network parameters:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M189">View MathML</a>

from which we obtain the following matrices:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M190">View MathML</a>

We note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M191">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M192">View MathML</a>. We consider a special case of Theorem 4.1 where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193">View MathML</a>, in which case the matrix S in Theorem 4.1 is of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M194">View MathML</a>

Then, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M195">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M196">View MathML</a>, Λ in Theorem 4.1 is obtained as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M197">View MathML</a>

Note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M198">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M199">View MathML</a>. Hence, the robust stability condition imposed by Theorem 4.1 is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M199">View MathML</a>. For the same network parameters of this example, we will obtain the robust stability condition imposed by Theorem 3.1. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M202">View MathML</a>

Then we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M203">View MathML</a>

Π in Theorem 3.1 takes the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M204">View MathML</a>

It can be calculated that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M206">View MathML</a>. Therefore, Theorem 3.1 imposes a less restrictive stability condition on the network parameters than Theorem 4.1 does.

Example 4.2 Consider the neural system (2.1) with the following network parameters:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M207">View MathML</a>

From the above matrices, we can obtain the following matrices:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M208">View MathML</a>

from which we can calculate the norms <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M191">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M210">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M212">View MathML</a>

Then we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M213">View MathML</a>

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M195">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M196">View MathML</a>, the matrix Π in Theorem 3.1 is in the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M216">View MathML</a>

The condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116">View MathML</a> is satisfied if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M218">View MathML</a>. For the parameters of this example, Γ in Theorem 4.2 is in the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M219">View MathML</a>

The choice <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M220">View MathML</a> implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M221">View MathML</a>, which ensures the global robust stability of neural system (2.1). Hence, for the network parameters of this example, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M222">View MathML</a>, then the result of Theorems 4.2 does not hold, whereas the result of Theorem 3.1 is still applicable.

Example 4.3 Assume that the network parameters of neural system (2.1) are given as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M223">View MathML</a>

For the matrices given above, we can obtain the following matrices:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M224">View MathML</a>

from which we calculate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M225">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M226">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M193">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M228">View MathML</a>

We have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M229">View MathML</a>

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M195">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M196">View MathML</a>, the matrix Π in Theorem 3.1 is of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M232">View MathML</a>

Clearly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M116">View MathML</a> holds if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M234">View MathML</a>. Therefore, for this example, Theorem 3.1 ensures the global robust stability of neural system (2.1) under the condition that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M234">View MathML</a>.

When checking the condition of Theorem 4.3 for the same network parameters of this example, we search for the existence of the positive constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M236">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M237">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M238">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M239">View MathML</a> such that the following conditions hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M240">View MathML</a>

which can be written in form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M241">View MathML</a>

From the properties of the nonsingular M-matrices [44], in order to ensure the existence of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M236">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M237">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M238">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M239">View MathML</a> the symmetric matrix in the above inequality must be positive definite, which holds if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M246">View MathML</a>. Obviously, for the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/41/mathml/M247">View MathML</a>, our condition obtained in Theorem 3.1 is satisfied, but the result of Theorem 4.3 does not hold.

5 Conclusions

In this paper, we have focused on the existence, uniqueness, and global robust stability of an equilibrium point for neural networks with multiple time delays with respect to the class of nondecreasing activation functions. We have employed a suitable Lyapunov functional and made use of the homeomorphism mapping theorem to derive a new time-independent robust stability condition for dynamical neural networks with multiple time delays. The obtained condition basically establishes a relationship between the network parameters of the neural system and the number of neurons. We have also presented some numerical examples, which enabled us to show the advantages of our result over previously reported robust stability results. We should point here that in the neural network model we have considered, the delay parameters are constant and the stability condition we obtain is delay independent. However, it is possible to derive some delay-dependent stability conditions for the same neural network model by employing different classes of Lyapunov functionals.

Competing interests

The author declares that he has no competing interests.

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