An Analysis of Stability of a Class of Neutral-Type Neural Networks with Discrete Time Delays

Abstract

The problem of existence, uniqueness, and global asymptotic stability is considered for the class of neutral-type neural network model with discrete time delays. By employing a suitable Lyapunov functional and using the homeomorphism mapping theorem, we derive some new delay-independent sufficient conditions for the existence, uniqueness, and global asymptotic stability of the equilibrium point for this class of neutral-type systems. The obtained conditions basically establish some norm and matrix inequalities involving the network parameters of the neural system. The main advantage of the proposed results is that they can be expressed in terms of network parameters only. Some comparative examples are also given to compare our results with the previous corresponding results and demonstrate the effectiveness of the results presented.

1. Introduction

In recent years, dynamical neural networks have been employed in solving many practical engineering problems such as signal and image processing, pattern recognition, associative memories, parallel computation, and optimization and control problems [1–10]. In such applications, it is important to know the dynamics of the designed neural networks. In addition, when using delayed neural networks, time delays might affect the transmission rate and cause instability. Therefore, the analysis of stability of neural networks with time delays is indispensable for solving engineering system problems. In the recent literature, many papers have studied the problem of global stability of different classes of neural networks by exploiting various analysis techniques and methods and presented some useful stability results for delayed neural networks. In practice, in order to precisely determine the equilibrium and stability properties of neural networks, the information about time derivatives of the past states must be introduced into the state equations of neural networks. A neural network of this model is called neutral-type neural networks. Some global stability results of various classes of neural networks with time delays have been reported in [1–33]. The goal of our paper is to present some new and alternative stability results of neutral-type neural networks with discrete time delays with respect to Lipschitz continuous activation functions.

Throughout this paper we will use these notations: for any matrix 𝑃=(𝑝_𝑖𝑗)_𝑛×𝑛, 𝑃>0 will denote that 𝑃 is symmetric and positive definite; 𝑃^𝑇, 𝑃⁻¹, 𝜆_𝑚(𝑃), and 𝜆_𝑀(𝑃) will denote the transpose of 𝑃, the inverse of 𝑃, the minimum eigenvalue of 𝑃, and the maximum eigenvalue of 𝑃, respectively. We will use the matrix norm ‖𝑃‖₂=[𝜆_𝑀(𝑃^𝑇𝑃)]^1/2. For any two positive definite matrices 𝑃=(𝑝_𝑖𝑗)_𝑛×𝑛 and 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛. If 𝑄>0, then 𝑃>𝑄 will imply that 𝑃>0. For 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇∈𝑅^𝑛, we will use the vector norms ‖𝑣‖₂=∑^𝑛_𝑖=1𝑣²_𝑖 and ‖𝑣‖₁=∑^𝑛_𝑖=1|𝑣_𝑖|.

2. Problem Statement

The class of neutral-type neural network model with discrete time delays is described by the following set of nonlinear differential equations: 𝑥̇_𝑖(𝑡)+^𝑛_𝑗=1𝑒_𝑖𝑗𝑥̇_𝑗𝑡−𝜏_𝑗=−𝑐_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑗+𝑢_𝑖,𝑖=1,…,𝑛, (1) where 𝑛 is the number of the neurons in the network, 𝑥_𝑖 denotes the state of the 𝑖th neuron, and the parameters 𝑐_𝑖 are some constants: the constants 𝑎_𝑖𝑗 denote the strengths of the neuron interconnections within the network; the constants 𝑏_𝑖𝑗 denote the strengths of the neuron interconnections with time delay parameters 𝜏_𝑗. 𝑒_𝑖𝑗 are coefficients of the time derivative of the delayed states, the functions 𝑓_𝑗(⋅) denote the neuron activations, and the constants 𝑢_𝑖 are some external inputs. In system (1), 𝜏_𝑗≥0 represents the delay parameter with 𝜏=max(𝜏_𝑗), 1≤𝑗≤𝑛. Accompanying the neutral system (1) is an initial condition of the form: 𝑥_𝑖(𝑡)=𝜙_𝑖(𝑡)∈𝐶([−𝜏,0],𝑅), where 𝐶([−𝜏,0],𝑅) denotes the set of all continuous functions from [−𝜏,0] to 𝑅.

We will assume that the activation functions 𝑓_𝑖(⋅), 𝑖=1,2,…,𝑛, are Lipschitz continuous; for example, there exist some constants ℓ_𝑖>0 such that 𝑓_𝑖(𝑥)−𝑓_𝑖𝑦≤ℓ_𝑖𝑥−𝑦,𝑖=1,2,…,𝑛,∀𝑥,𝑦∈𝑅,𝑥≠𝑦. (2) Neural network model (1) can be written in the vector-matrix form as follows: 𝑥̇(𝑡)+𝐸𝑥̇(𝑡−𝜏)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏))+𝑢, (3) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, 𝐸=(𝑒_𝑖𝑗)_𝑛×𝑛, 𝐶=diag(𝑐_𝑖>0), 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇, 𝑓(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),𝑓₂(𝑥₂(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇, and 𝑓(𝑥(𝑡−𝜏))=(𝑓₁(𝑥₁(𝑡−𝜏₁)),𝑓₂(𝑥₂(𝑡−𝜏₂)),…,𝑓_𝑛(𝑥_𝑛(𝑡−𝜏_𝑛)))^𝑇.

In order to obtain our main results, the following lemma will be needed.

Lemma 1 (see [23]). If a map 𝐻(𝑥)∈𝐶⁰ satisfies the following conditions:(i)𝐻(𝑥)≠𝐻(𝑦) for all 𝑥≠𝑦,(ii)‖𝐻(𝑥)‖→∞ as ‖𝑥‖→∞,then, 𝐻(𝑥) is homeomorphism of 𝑅^𝑛.

3. Existence and Uniqueness Analysis

This section deals with obtaining the sufficient conditions that ensure the existence and uniqueness of the equilibrium point for neutral-type neural network model (1). The main result is given in the following result.

Theorem 2. For the neutral-type neural network model (1), let ‖𝐸‖₂<1 and the activation functions satisfy (2). Then, the system (1) has unique equilibrium point for each 𝑢 if there exist positive diagonal matrices 𝐻 and 𝐷 and positive definite matrices 𝑃, 𝑄, and 𝑅 such that the following conditions hold: Υ₁=𝐶−𝑃−𝑄−𝐻−𝐶𝑅⁻¹Υ𝐶>0,₂=𝐶ℒ⁻²−𝐷−𝐴^𝑇𝑃⁻¹𝐴−𝐴^𝑇𝑅⁻¹Υ𝐴>0,₃=𝐷−𝐵^𝑇𝑄⁻¹𝐵−𝐵^𝑇𝑅⁻¹Υ𝐵>0,₄=𝐻−3𝐸^𝑇𝑅𝐸>0, (4) where ℒ=diag(ℓ₁,ℓ₂,…,ℓ_𝑛).

Proof. We will make use of the result of Lemma 1 for the proof of the existence and uniqueness of the equilibrium point for system (1). Let us define the following mapping associated with system (1): 𝐻(𝑥)+𝐸𝐻(𝑥)=−𝐶𝑥+𝐴𝑓(𝑥)+𝐵𝑓(𝑥)+𝑢. (5) If 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇 is an equilibrium point of (1), then 𝑥^∗ satisfies the equilibrium equation: 𝐻𝑥^∗+𝐸𝐻𝑥^∗=−𝐶𝑥^∗+𝐴𝑓𝑥^∗+𝐵𝑓𝑥^∗+𝑢=0. (6) Clearly, the solution of the equation 𝐻(𝑥)=0 is an equilibrium point of (1). Therefore, in the light of Lemma 1, we can conclude that, for the system defined by (1), there exists a unique equilibrium point for every input vector 𝑢 if 𝐻(𝑥) is homeomorphism of 𝑅^𝑛. We will now show that the conditions of Theorem 2 imply that 𝐻(𝑥) is a homeomorphism of 𝑅^𝑛. To this end, we choose any two vectors 𝑥∈𝑅^𝑛 and 𝑦∈𝑅^𝑛 such that 𝑥≠𝑦. When the activation functions satisfy (2), for 𝑥≠𝑦, we have two cases: first case is 𝑥≠𝑦 and 𝑓(𝑥)≠𝑓(𝑦), and the second case is 𝑥≠𝑦 and 𝑓(𝑥)=𝑓(𝑦). Let us carry out the existence and uniqueness analysis for the first case where 𝑥≠𝑦 and 𝑓(𝑥)≠𝑓(𝑦). In this case, for 𝐻(𝑥) defined by (5), we can write 𝐻(𝑥)𝑦𝐻(𝑥)𝑦=()𝐻(𝑥)𝑦𝑓(𝑥)𝑦𝑓(𝑥)𝑦.−𝐻+𝐸−𝐻𝐼+𝐸−𝐻=−𝐶𝑥−𝑦+𝐴−𝑓+𝐵−𝑓 (7) If we multiply both sides of (7) by the term 2(𝑥−𝑦)^𝑇(𝐼+𝐸)^𝑇, and then add the terms (𝑓(𝑥)−𝑓(𝑦))^𝑇𝐷(𝑓(𝑥)−𝑓(𝑦))−(𝑓(𝑥)−𝑓(𝑦))^𝑇𝐷(𝑓(𝑥)−𝑓(𝑦))=0 and (𝑥−𝑦)^𝑇𝐻(𝑥−𝑦)−(𝑥−𝑦)^𝑇𝐻(𝑥−𝑦)=0 to the right hand side of the resulting equation, we get 2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦=2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇×−𝐶𝑥−𝑦+𝐴𝑓(𝑥)−𝑓𝑦+𝐵𝑓(𝑥)−𝑓𝑦=2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇×𝑓(𝑥)𝑦𝑓(𝑥)𝑦+−𝐶𝑥−𝑦+𝐴−𝑓+𝐵−𝑓𝑥−𝑦^𝑇𝐻𝑥−𝑦−𝑥−𝑦^𝑇𝐻+𝑓(𝑥)𝑦𝑥−𝑦−𝑓^𝑇𝐷𝑓(𝑥)𝑦−𝑓(𝑥)𝑦−𝑓−𝑓^𝑇𝐷𝑓(𝑥)−𝑓𝑦=−2𝑥−𝑦^𝑇𝐶𝑥−𝑦+2𝑥−𝑦^𝑇𝐴𝑓(𝑥)−𝑓𝑦+2𝑥−𝑦^𝑇𝐵𝑓(𝑥)−𝑓𝑦−2𝑥−𝑦^𝑇𝐸^𝑇𝐶𝑥−𝑦+2𝑥−𝑦^𝑇𝐸^𝑇𝐴𝑓(𝑥)−𝑓𝑦+2𝑥−𝑦^𝑇𝐸^𝑇𝐵×𝑓(𝑥)−𝑓𝑦+𝑥−𝑦^𝑇𝐻𝑥−𝑦−𝑥−𝑦^𝑇×𝐻𝑥−𝑦+𝑓(𝑥)−𝑓𝑦^𝑇𝐷𝑓(𝑥)𝑦−𝑓(𝑥)𝑦−𝑓−𝑓^𝑇𝐷𝑓(𝑥)−𝑓𝑦. (8) We note the following inequalities: 2𝑥−𝑦^𝑇𝐴𝑓(𝑥)𝑦≤−𝑓𝑥−𝑦^𝑇𝑃𝑥−𝑦+𝑓(𝑥)−𝑓𝑦^𝑇×𝐴^𝑇𝑃⁻¹𝐴𝑓(𝑥)𝑦,2−𝑓𝑥−𝑦^𝑇𝐵𝑓(𝑥)𝑦≤−𝑓𝑥−𝑦^𝑇𝑄𝑥−𝑦+𝑓(𝑥)−𝑓𝑦^𝑇×𝐵^𝑇𝑄⁻¹𝐵𝑓(𝑥)−𝑓𝑦,−2𝑥−𝑦^𝑇𝐸^𝑇𝐶(𝑡)≤𝑥−𝑦𝑥−𝑦^𝑇𝐸^𝑇𝑅𝐸𝑥−𝑦+𝑥−𝑦^𝑇×𝐶^𝑇𝑅⁻¹𝐶,2𝑥−𝑦𝑥−𝑦^𝑇𝐸^𝑇𝐴𝑓(𝑥)𝑦≤−𝑓𝑥−𝑦^𝑇𝐸^𝑇𝑅𝐸𝑥−𝑦+𝑓(𝑥)−𝑓𝑦^𝑇×𝐴^𝑇𝑅⁻¹𝐴𝑓(𝑥)𝑦,2−𝑓𝑥−𝑦^𝑇𝐸^𝑇𝐵𝑓(𝑥)𝑦≤−𝑓𝑥−𝑦^𝑇𝐸^𝑇+𝑓(𝑥)𝑦𝑅𝐸𝑥−𝑦−𝑓^𝑇𝐵^𝑇𝑅⁻¹𝐵𝑓(𝑥)−𝑓𝑦. (9) Using (9) in (8) results in 2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦≤−𝑥−𝑦^𝑇𝐶𝑥−𝑦−𝑥−𝑦^𝑇𝐶+𝑥−𝑦𝑥−𝑦^𝑇𝑃𝑥−𝑦+𝑓(𝑥)−𝑓𝑦^𝑇𝐴^𝑇𝑃⁻¹×𝐴𝑓(𝑥)−𝑓𝑦+𝑥−𝑦^𝑇𝑄+𝑓(𝑥)𝑦𝑥−𝑦−𝑓^𝑇𝐵^𝑇𝑄⁻¹𝐵𝑓(𝑥)𝑦+−𝑓𝑥−𝑦^𝑇𝐸^𝑇𝑅𝐸𝑥−𝑦+𝑥−𝑦^𝑇𝐶^𝑇𝑅⁻¹×𝐶𝑥−𝑦+𝑥−𝑦^𝑇𝐸^𝑇+𝑓(𝑥)𝑦𝑅𝐸𝑥−𝑦−𝑓^𝑇𝐴^𝑇𝑅⁻¹𝐴𝑓(𝑥)−𝑓𝑦𝑥−𝑦^𝑇×𝐸^𝑇𝑅𝐸𝑥−𝑦+𝑓(𝑥)−𝑓𝑦^𝑇×𝐵^𝑇𝑅⁻¹𝐵𝑓(𝑥)−𝑓𝑦+𝑥−𝑦^𝑇𝐻−𝑥−𝑦𝑥−𝑦^𝑇𝐻𝑥−𝑦+𝑓(𝑥)−𝑓𝑦^𝑇×𝐷𝑓(𝑥)−𝑓𝑦−𝑓(𝑥)−𝑓𝑦^𝑇×𝐷𝑓(𝑥)−𝑓𝑦 (10) which is of the form 2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦≤−𝑥−𝑦^𝑇𝐶−𝑃−𝑄−𝐻−𝐶^𝑇𝑅⁻¹𝐶𝑥−𝑦^𝑇−𝑓(𝑥)−𝑓𝑦^𝑇𝐶ℒ⁻²−𝐷−𝐴^𝑇𝑃⁻¹𝐴−𝐴^𝑇𝑅⁻¹𝐴×𝑓(𝑥)−𝑓𝑦−𝑓(𝑥)−𝑓𝑦^𝑇×𝐷−𝐵^𝑇𝑄⁻¹𝐵−𝐵^𝑇𝑅⁻¹𝐵𝑓(𝑥)−𝑓𝑦−(𝑥−𝑦)^𝑇𝐻−3𝐸^𝑇𝑅𝐸𝑥−𝑦 (11) or equivalently 2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦≤−𝑥−𝑦^𝑇Υ₁−𝑓(𝑥)𝑦𝑥−𝑦−𝑓^𝑇Υ₂𝑓(𝑥)𝑦−𝑓(𝑥)𝑦−𝑓−𝑓^𝑇Υ₃𝑓(𝑥)𝑦−−𝑓𝑥−𝑦^𝑇Υ₄𝑥−𝑦. (12) Since 𝑥≠𝑦 and 𝑓(𝑥)≠𝑓(𝑦), Υ₁>0, Υ₂>0, Υ₃>0, and Υ₄>0 imply that 2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦<0 (13) implying that 𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦₂>0 (14) from which it follows that 𝑥−𝑦₂‖𝐼+𝐸‖²₂𝐻(𝑥)−𝐻𝑦₂>0. (15)‖𝐸‖₂<1 implies that ‖𝐼+𝐸‖²₂>0, and 𝑥≠𝑦 implies that ‖𝑥−𝑦‖₂>0. Therefore, it directly follows that ‖𝐻(𝑥)−𝐻(𝑦)‖₂>0, thus implying that ‖𝐻(𝑥)‖≠‖𝐻(𝑦)‖. Hence, we conclude that 𝐻(𝑥)≠𝐻(𝑦) for all 𝑥≠𝑦 and 𝑓(𝑥)≠𝑓(𝑦).
Now consider the case where 𝑥≠𝑦 and 𝑓(𝑥)=𝑓(𝑦). In this case, 𝐻(𝑥) defined by (5) satisfies 2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦≤−𝑥−𝑦^𝑇Υ₁𝑥−𝑦−𝑥−𝑦^𝑇Υ₄𝑧𝑥−𝑦, (16)𝑥≠𝑦; Υ₁>0 and Υ₄>0 imply that 2𝑥−𝑦^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)𝐻(𝑥)−𝐻𝑦<0. (17) Based on the analysis carried out for the previous case, we conclude that 𝐻(𝑥)≠𝐻(𝑦) for all 𝑥≠𝑦 for this case.
Now it is shown that the conditions of Theorem 2 imply that ‖𝐻(𝑥)‖→∞ as ‖𝑥‖→∞. For 𝑦=0, we can write 2𝑥^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)(𝐻(𝑥)−𝐻(0))≤−𝑥^𝑇Υ₁𝑥−𝑓(𝑥)−𝑓(0)^𝑇Υ₂𝑓(𝑥)(0)−𝑓(𝑥)(0−𝑓−𝑓)^𝑇Υ₃𝑓(𝑥)−𝑓(0)−𝑥^𝑇Υ₄𝑥. (18) Taking the absolute value of the both sides of the above inequality, we obtain 2𝑥^𝑇(𝐼+𝐸)^𝑇(𝐼+𝐸)(𝐻(𝑥)−𝐻(0))≥𝑥^𝑇Υ₁𝑥+𝑓(𝑥)−𝑓(0)^𝑇Υ₂𝑓(𝑥)(0)+𝑓(𝑥)(0)−𝑓−𝑓^𝑇Υ₃𝑓(𝑥)−𝑓(0)+𝑥^𝑇Υ₄𝑥 (19) from which it follows that ‖𝑥‖₂‖𝐼+𝐸‖²₂‖𝐻(𝑥)−𝐻(0)‖₂≥𝜆_𝑚Υ₁‖𝑥‖²₂+𝜆_𝑚Υ₂𝑓(𝑥)−𝑓(0)²₂+𝜆_𝑚Υ₃𝑓(𝑥)−𝑓(0)²₂+𝜆_𝑚Υ₄‖𝑥‖²₂≥𝜆_𝑚Υ₁‖𝑥‖²₂ (20) which yields ‖𝐻(𝑥)−𝐻(0)‖₂≥𝜆_𝑚Υ₁‖𝐼+𝐸‖²₂‖𝑥‖₂. (21) We note that ‖𝐻(𝑥)−𝐻(0)‖₂≤‖𝐻(𝑥)‖₂+‖𝐻(0)‖₂. Hence, from (21), it follows that ‖𝐻(𝑥)‖₂≥𝜆_𝑚Υ₁‖𝐼+𝐸‖²₂‖𝑥‖₂−‖𝐻(0)‖₂. (22) Since ‖𝐻(0)‖₂ is bounded, and ‖𝐼+𝐸‖²₂>0, then ‖𝐻(𝑥)‖₂→∞ as ‖𝑥‖₂→∞. Hence, the conditions of Theorem 2 ensure that 𝐻(𝑥) is homeomorphism of 𝑅^𝑛, proving that neutral system defined by (1) has unique equilibrium point for each 𝑢.
Choosing 𝐻, 𝐷, 𝑃, 𝑄, and 𝑅 in the conditions of Theorem 2 as 𝐻=ℎ𝐼, 𝐷=𝑑𝐼, 𝑃=𝑝𝐼, 𝑄=𝑞𝐼, and 𝑅=𝑟𝐼, we can express some special cases of Theorem 2 as follows.

Corollary 3. For the neutral-type neural network model (1), let ‖𝐸‖₂<1 and the activation functions satisfy (2). Then, the system (1) has unique equilibrium point for each 𝑢 if there exist some positive constants ℎ, 𝑑, 𝑝, 𝑞, and 𝑟 such that the following conditions hold: Υ^∗₁1=𝐶−𝑝+𝑞+ℎ𝐼−𝑟𝐶²Υ>0,_∗2=𝐶ℒ⁻²1−𝑑𝐼−𝑝𝐴^𝑇1𝐴−𝑟𝐴^𝑇Υ𝐴>0,_∗31=𝑑𝐼−𝑞𝐵^𝑇1𝐵−𝑟𝐵^𝑇Υ𝐵>0,_∗4=ℎ𝐼−3𝑟𝐸^𝑇𝐸>0, (23) where ℒ=diag(ℓ₁,ℓ₂,…,ℓ_𝑛).

A special case of Corollary 3 is the following result.

Corollary 4. For the neutral-type neural network model (1), let ‖𝐸‖₂<1 and the activation functions satisfy (2). Then, the system (1) has unique equilibrium point for each 𝑢 if there exist some positive constants ℎ, 𝑑, 𝑝, 𝑞, and 𝑟 such that the following conditions hold: 𝜌₁=𝑐_𝑚−−1𝑝+𝑞+ℎ𝑟𝑐²_𝑀𝜌>0,₂=𝑐_𝑚ℓ_𝑀⁻²1−𝑑−𝑝+1𝑟‖𝐴‖²₂𝜌>0,₃1=𝑑−𝑞+1𝑟‖𝐵‖²₂𝜌>0,₄=ℎ−3𝑟‖𝐸‖²₂>0, (24) where 𝑐_𝑚=min_{1≤𝑖≤𝑛}(𝑐_𝑖), 𝑐_𝑀=max_{1≤𝑖≤𝑛}(𝑐_𝑖), and ℓ_𝑀=max_{1≤𝑖≤𝑛}(ℓ_𝑖).

4. Stability Analysis

In this section, we will prove that the conditions obtained from Theorem 2 for the existence and uniqueness of the equilibrium point are also sufficient for the global stability of the equilibrium point of neutral system defined by (1). In order to simplify the proofs, we will first shift the equilibrium point 𝑥^∗=[𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛]^𝑇 of system (1) to the origin. By using the transformation 𝑧(𝑡)=𝑥(𝑡)−𝑥^∗, the neutral-type neural network model (1) can be put in the form: 𝑧̇_𝑖(𝑡)=−𝑐_𝑖𝑧_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑔_𝑗𝑧_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗𝑔_𝑗𝑧_𝑗𝑡−𝜏_𝑗+^𝑛_𝑗=1𝑒_𝑖𝑗𝑧̇_𝑗𝑡−𝜏_𝑗,𝑖=1,…,𝑛 (25) which can be written in vector-matrix form as follows: 𝑧̇(𝑡)=−𝐶𝑧(𝑡)+𝐴𝑔(𝑧(𝑡))+𝐵𝑔(𝑧(𝑡−𝜏))+𝐸𝑧̇(𝑡−𝜏), (26) where 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡))^𝑇∈𝑅^𝑛 is the state vector of transformed neural system, 𝑔(𝑧(𝑡))=(𝑔₁(𝑧₁(𝑡)),𝑔₂(𝑧₂(𝑡)),…,𝑔_𝑛(𝑧_𝑛(𝑡)))^𝑇 represents the new nonlinear activation, functions, and 𝑔(𝑧(𝑡−𝜏))=(𝑔₁(𝑧₁(𝑡−𝜏₁)),𝑔₂(𝑧₂(𝑡−𝜏₂)),…,𝑔_𝑛(𝑧_𝑛(𝑡−𝜏_𝑛)))^𝑇. The activation functions 𝑔_𝑖(𝑧_𝑖(𝑡)) in (25) satisfy 𝑔_𝑖𝑧_𝑖(𝑡)≤ℓ_𝑖𝑧_𝑖(𝑡),𝑖=1,2,…,𝑛. (27) We can now state the following stability result.

Theorem 5. For the neutral-type neural network model (25), let ‖𝐸‖₂<1 and the activation functions satisfy (27). Then, the origin of system (25) is globally asymptotically stable if there exist positive diagonal matrices 𝐻 and 𝐷 and positive definite matrices 𝑃, 𝑄, and 𝑅 such that the following conditions hold: Υ₁=𝐶−𝑃−𝑄−𝐻−𝐶^𝑇𝑅⁻¹Υ𝐶>0,₂=𝐶ℒ⁻²−𝐷−𝐴^𝑇𝑃⁻¹𝐴−𝐴^𝑇𝑅⁻¹Υ𝐴>0,₃=𝐷−𝐵^𝑇𝑄⁻¹𝐵−𝐵^𝑇𝑅⁻¹Υ𝐵>0,₄=𝐻−3𝐸^𝑇𝑅𝐸>0, (28) where ℒ=diag(ℓ₁,ℓ₂,…,ℓ_𝑛).

Proof. Define the following positive definite Lyapunov functional: 𝑉(𝑧(𝑡))=(𝑧(𝑡)+𝐸𝑧(𝑡−𝜏))^𝑇(𝑧(𝑡)(++𝐸𝑧𝑡−𝜏))_𝑛𝑖=1ℎ_𝑖^𝑡_{𝑡−𝜏𝑖}𝑧²_𝑖(𝑠)𝑑𝑠+^𝑛_𝑖=1𝑑_𝑖^𝑡_{𝑡−𝜏𝑖}𝑔²_𝑖𝑧_𝑖(𝑠)𝑑𝑠, (29) where 𝑝_𝑖 and 𝑑_𝑖𝑖=1,2,…,𝑛 are some positive constants. The time derivative of 𝑉(𝑧(𝑡)) along the trajectories of the system (25) is obtained as follows: 𝑉̇(𝑧(𝑡))=2(𝑧(𝑡)+𝐸𝑧(𝑡−𝜏))^𝑇(𝑧̇(𝑡)𝑧̇(++𝐸𝑡−𝜏))_𝑛𝑖=1ℎ_𝑖𝑧²_𝑖(𝑡)−^𝑛_𝑖=1ℎ_𝑖𝑧²_𝑖𝑡−𝜏_𝑖+^𝑛_𝑖=1𝑑_𝑖𝑔²_𝑖𝑧_𝑖(𝑡)−^𝑛_𝑖=1𝑑_𝑖𝑔²_𝑖𝑧_𝑖𝑡−𝜏_𝑖=2(𝑧(𝑡)+𝐸𝑧(𝑡−𝜏))^𝑇(𝑧̇(𝑡)+𝐸𝑧̇(𝑡−𝜏))+𝑧^𝑇(𝑡)𝐻𝑧(𝑡)−𝑧^𝑇(𝑡−𝜏)𝐻𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡))𝐷𝑔(𝑧(𝑡))−𝑔^𝑇(𝑧(𝑡−𝜏))𝐷𝑔(𝑧(𝑡−𝜏)). (30) Since 𝑧(̇𝑡)+𝐸𝑧(̇𝑡−𝜏)=−𝐶𝑧(𝑡)+𝐴𝑔(𝑧(𝑡))+𝐵𝑔(𝑧(𝑡−𝜏)), we can write 𝑉̇(𝑧(𝑡))=2(𝑧(𝑡)+𝐸𝑧(𝑡−𝜏))^𝑇×−𝐶𝑧(𝑡)+𝐴𝑔(𝑧(𝑡))+𝐵𝑔(𝑧(𝑡−𝜏))+𝑧^𝑇(𝑡)𝐻𝑧(𝑡)−𝑧^𝑇(𝑡−𝜏)𝐻𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡))𝐷𝑔(𝑧(𝑡))−𝑔^𝑇(𝑧(𝑡−𝜏))𝐷𝑔(𝑧(𝑡−𝜏))=−2𝑧^𝑇(𝑡)𝐶𝑧(𝑡)+2𝑧^𝑇(𝑡)𝐴𝑔(𝑧(𝑡))+2𝑧^𝑇(𝑡)𝐵𝑔(𝑧(𝑡−𝜏))−2𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝐶𝑧(𝑡)+2𝑧^𝑇(𝑡−𝜏)×𝐸^𝑇𝐴𝑔(𝑧(𝑡))+2𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝐵𝑔(𝑧(𝑡−𝜏))+𝑧^𝑇(𝑡)𝐻𝑧(𝑡)−𝑧^𝑇(𝑡−𝜏)𝐻𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡))𝐷𝑔(𝑧(𝑡))−𝑔^𝑇(𝑧(𝑡−𝜏))𝐷𝑔(𝑧(𝑡−𝜏)). (31) We can write the following inequalities: 2𝑧^𝑇(𝑡)𝐴𝑔(𝑧(𝑡))≤𝑧^𝑇(𝑡)𝑃𝑧(𝑡)+𝑔^𝑇(𝑧(𝑡))𝐴^𝑇𝑃⁻¹𝐴𝑔(𝑧(𝑡)),2𝑧^𝑇(𝑡)𝐵𝑔(𝑧(𝑡−𝜏))≤𝑧^𝑇(𝑡)𝑄𝑧(𝑡)+𝑔^𝑇(𝑧(𝑡−𝜏))𝐵^𝑇𝑄⁻¹𝐵𝑔(𝑧(𝑡−𝜏)),−2𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝐶𝑧(𝑡)≤𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑧^𝑇(𝑡)𝐶^𝑇𝑅⁻¹𝐶𝑧(𝑡),2𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝐴𝑔(𝑧(𝑡))≤𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡))𝐴^𝑇𝑅⁻¹𝐴𝑔(𝑧(𝑡)),2𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝐵𝑔(𝑧(𝑡−𝜏))≤𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡−𝜏))𝐵^𝑇𝑅⁻¹𝐵𝑔(𝑧(𝑡−𝜏)), (32) where 𝑃, 𝑄, and 𝑅 are some positive definite matrices. Using (32) in (31) yields 𝑉̇(𝑧(𝑡))≤−𝑧^𝑇(𝑡)𝐶𝑧(𝑡)−𝑧^𝑇(𝑡)𝐶𝑧(𝑡)+𝑧^𝑇(𝑡)𝑃𝑧(𝑡)+𝑔^𝑇(𝑧(𝑡))𝐴^𝑇𝑃⁻¹𝐴𝑔(𝑧(𝑡))+𝑧^𝑇(𝑡)𝑄𝑧(𝑡)+𝑔^𝑇(𝑧(𝑡−𝜏))𝐵^𝑇𝑄⁻¹𝐵𝑔(𝑧(𝑡−𝜏))+𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑧^𝑇(𝑡)𝐶^𝑇𝑅⁻¹𝐶𝑧(𝑡)+𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡))𝐴^𝑇𝑅⁻¹×𝐴𝑔(𝑧(𝑡))+𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡−𝜏))𝐵^𝑇𝑅⁻¹𝐵𝑔(𝑧(𝑡−𝜏))+𝑧^𝑇(𝑡)×𝐻𝑧(𝑡)−𝑧^𝑇(𝑡−𝜏)𝐻𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡(𝑧(𝑡))×𝐷𝑔))−𝑔^𝑇(𝑧(𝑡−𝜏))𝐷𝑔(𝑧(𝑡−𝜏)). (33)
Equation (27) implies that 𝑧^𝑇(𝑡)𝐶𝑧(𝑡)≥𝑔^𝑇(𝑧(𝑡))𝐶ℒ⁻²𝑔(𝑧(𝑡)). (34) Hence, we have 𝑉̇(𝑧(𝑡))≤−𝑧^𝑇(𝑡)𝐶𝑧(𝑡)−𝑔^𝑇(𝑧(𝑡))𝐶ℒ⁻²𝑔(𝑧(𝑡))+𝑧^𝑇(𝑡)𝑃𝑧(𝑡)+𝑔^𝑇(𝑧(𝑡))𝐴^𝑇𝑃⁻¹𝐴𝑔(𝑧(𝑡))+𝑧^𝑇(𝑡)𝑄𝑧(𝑡)+𝑔^𝑇(𝑧(𝑡−𝜏))𝐵^𝑇𝑄⁻¹×(𝑧(𝐵𝑔𝑡−𝜏))+𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑧^𝑇(𝑡)𝐶^𝑇𝑅⁻¹𝐶𝑧(𝑡)+𝑧^𝑇(𝑡−𝜏)𝐸^𝑇×𝑅𝐸𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡))𝐴^𝑇𝑅⁻¹𝐴𝑔(𝑧(𝑡))+𝑧^𝑇(𝑡−𝜏)𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏)+𝑔^𝑇(𝑧(𝑡−𝜏))×𝐵^𝑇𝑅⁻¹𝐵𝑔(𝑧(𝑡−𝜏))+𝑧^𝑇(𝑡)𝐻𝑧(𝑡)−𝑧^𝑇()()𝑡−𝜏×𝐻𝑧𝑡−𝜏+𝑔^𝑇(𝑧(𝑡))𝐷𝑔(𝑧(𝑡))−𝑔^𝑇(𝑧(𝑡−𝜏))𝐷𝑔(𝑧(𝑡−𝜏)) (35) which can be written as 𝑉̇(𝑧(𝑡))≤−𝑧^𝑇(𝑡)𝐶−𝑃−𝑄−𝐻−𝐶^𝑇𝑅⁻¹𝐶𝑧(𝑡)−𝑔^𝑇(𝑧(𝑡×))𝐶ℒ⁻²−𝐷−𝐴^𝑇𝑃⁻¹𝐴−𝐴^𝑇𝑅⁻¹𝐴×𝑔(𝑧(𝑡))−𝑔^𝑇(𝑧(×𝑡−𝜏))𝐷−𝐵^𝑇𝑄⁻¹𝐵−𝐵^𝑇𝑅⁻¹𝐵×𝑔(𝑧(𝑡−𝜏))−𝑧^𝑇()×𝑡−𝜏𝐻−3𝐸^𝑇𝑅𝐸𝑧(𝑡−𝜏) (36) or equivalently 𝑉̇(𝑧(𝑡))≤−𝑧^𝑇(𝑡)Υ₁𝑧(𝑡)−𝑔^𝑇(𝑧(𝑡))Υ₂𝑔(𝑧(𝑡))−𝑔^𝑇(𝑧(𝑡−𝜏))Υ₃𝑔(𝑧(𝑡−𝜏))−𝑧^𝑇(𝑡−𝜏)Υ₄𝑧(𝑡−𝜏). (37) Clearly, Υ₁>0, Υ₂>0, Υ₃>0, and Υ₄>0 imply that 𝑉̇(𝑧(𝑡))<0 if any of the vectors 𝑧(𝑡), 𝑔(𝑧(𝑡−𝜏)), 𝑔^𝑇(𝑧(𝑡)), and 𝑧(𝑡−𝜏) is nonzero, thus implying that 𝑉̇(𝑧(𝑡))=0 if and only if 𝑧(𝑡)=𝑧(𝑡−𝜏)=𝑔(𝑧(𝑡−𝜏))=𝑧(𝑡−𝜏)=0 which is the origin of system (25). On the other hand, 𝑉(𝑧(𝑡))→∞ as ‖𝑧(𝑡)‖₂→∞, meaning that the Lyapunov functional used for the stability analysis is radially unbounded. Thus, it can be concluded from the standard Lyapunov theorems [34] that the origin of system (25) or equivalently the equilibrium point of system (1) is globally asymptotically stable.

We can directly state the following corollaries.

Corollary 6. For the neutral-type neural network model (25), let ‖𝐸‖₂<1 and the activation functions satisfy (27). Then, the origin of system (25) is globally asymptotically stable if there exist some positive constants ℎ, 𝑑, 𝑝, 𝑞, and 𝑟 such that the following conditions hold: Υ^∗₁1=𝐶−𝑝+𝑞+ℎ𝐼−𝑟𝐶²Υ>0,_∗2=𝐶ℒ⁻²1−𝑑𝐼−𝑝𝐴^𝑇1𝐴−𝑟𝐴^𝑇Υ𝐴>0,_∗31=𝑑𝐼−𝑞𝐵^𝑇1𝐵−𝑟𝐵^𝑇Υ𝐵>0,_∗4=ℎ𝐼−3𝑟𝐸^𝑇𝐸>0, (38) where ℒ=diag(ℓ₁,ℓ₂,…,ℓ_𝑛).

Corollary 7. For the neutral-type neural network model (25), let ‖𝐸‖₂<1 and the activation functions satisfy (27). Then, the origin of system (25) is globally asymptotically stable if there exist some positive constants ℎ, 𝑑, 𝑝, 𝑞, and 𝑟 such that the following conditions hold: 𝜌₁=𝑐_𝑚−−1𝑝+𝑞+ℎ𝑟𝑐²_𝑀𝜌>0,₂=𝑐_𝑚ℓ_𝑀⁻²1−𝑑−𝑝+1𝑟‖𝐴‖²₂𝜌>0,₃1=𝑑−𝑞+1𝑟‖𝐵‖²₂𝜌>0,₄=ℎ−3𝑟‖𝐸‖²₂>0, (39) where 𝑐_𝑚=min_{1≤𝑖≤𝑛}(𝑐_𝑖), 𝑐_𝑀=max_{1≤𝑖≤𝑛}(𝑐_𝑖), and ℓ_𝑀=max_{1≤𝑖≤𝑛}(ℓ_𝑖).

5. A Comparative Example

In this section, we will give a numerical example to make a comparison between our results and some previous corresponding results derived in the literature. We should point our here that the stability results regarding the neutral-type neural networks involve complicated relationships between the network parameters and some positive definite matrices to be determined, which is a difficult task to achieve. Therefore, the example we give will show that, in a particular case, our results seem to be equivalent to the previous corresponding literature results. We now state some of the previous results.

Theorem 8 (see [23]). For the neutral-type neural network model (1), let ‖𝐸‖₂<1 and the activation functions satisfy (2). Then, system (1) is globally asymptotically stable if there exist some positive constants 𝑘, 𝑝, 𝑞, and 𝑟 such that the following conditions hold: 𝛿₁=(1−𝑘)𝛾²−11+𝑝+1𝑞‖𝐴‖²₂𝛿>0,₂=𝑘𝛾²−11+𝑝+𝑟‖𝐵‖²₂𝛿>0,₃=1−1+𝑞+𝑟‖𝐸‖²₂>0, (40) where 𝛾=min_{1≤𝑖≤𝑛}(𝑐_𝑖/ℓ_𝑖).

Theorem 9 (see [22]). For the neutral-type neural network model (1), let ‖𝐸‖₂<1 and the activation functions satisfy (2). Then, system (1) is globally asymptotically stable if there exist positive constants 𝑝, 𝑝, 𝑞, and 𝑞 such that the following conditions hold: 𝜖=(2−𝑟)𝑐_𝑚−𝑝+𝑞−2𝑐_𝑚‖𝐸‖₂−𝑝+𝑞‖𝐸‖²₂>0,Φ=𝑟𝑐_𝑚ℓ_𝑀⁻²−1𝑝+1𝑝‖𝐴‖²₂−1𝑞+1𝑞‖𝐵‖²₂≥0, (41) where 𝑐_𝑚=min_{1≤𝑖≤𝑛}(𝑐_𝑖), 𝑐_𝑀=max_{1≤𝑖≤𝑛}(𝑐_𝑖), and ℓ_𝑀=max_{1≤𝑖≤𝑛}(ℓ_𝑖).

Theorem 10 (see [24]). For the neutral-type neural network model (1), let ‖𝐸‖₂<1 and the activation functions satisfy (2). Then, system (1) is globally asymptotically stable if the following condition holds: 𝛿=𝑐_𝑚−ℓ_𝑀‖𝐴‖₂1+‖𝐸‖₂−ℓ_𝑀‖𝐵‖₂1+‖𝐸‖₂−𝑐_𝑚‖𝐸‖₂>0, (42) where 𝑐_𝑚=min_{1≤𝑖≤𝑛}(𝑐_𝑖), ℓ_𝑀=max_{1≤𝑖≤𝑛}(ℓ_𝑖).

We now consider the following example.

Example 11. Assume that the network parameters of neutral-type neural system (1) are given as follows: 𝐴=𝐵=𝑎𝑎𝑎𝑎−𝑎−𝑎𝑎𝑎𝑎−𝑎𝑎−𝑎−𝑎𝑎𝑎−𝑎, (43) where 𝑎>0 is real number. Assume that 𝑐₁=𝑐₂=𝑐₃=𝑐₄=1 and ℓ₁=ℓ₂=ℓ₃=ℓ₄=1. We have ‖𝐴‖₂=‖𝐵‖₂=2𝑎.
For the sufficiently small values of ‖𝐸‖₂ and ℎ and sufficiently large value of 𝑟, 𝑑=1/2, and 𝑝=𝑞, the conditions of Corollary 7 can be approximately stated as follows: 𝜌₁𝜌≅1−2𝑝>0,₂≅12−1𝑝4𝑎²𝜌>0,₃≅12−1𝑝4𝑎²𝜌>0,₄≅ℎ−3𝑟‖𝐸‖²₂>0. (44) The two required conditions for stability are 𝑝<1/2 and 𝑎²<𝑝/8, implying that 𝑎<1/4.
In the case of Theorem 8, for the sufficiently small value of ‖𝐸‖₂ and sufficiently large values of 𝑟 and 𝑞, 𝑘=1/2, and 𝑝=1, the conditions of Theorem 8 can be approximately stated as follows: 𝛿₁≅12−8𝑎²𝛿>0,₂≅12−8𝑎²𝛿>0,₃≅1−1+𝑞+𝑟‖𝐸‖²₂>0. (45) The required condition for stability is 𝑎<1/4.
In the case of Theorem 9, for the sufficiently small value of ‖𝐸‖₂ and sufficiently large values of 𝑝 and 𝑞, 𝑟=1, and 𝑝=𝑞, the conditions of Theorem 9 can be approximately stated as follows: 2𝜖≅1−2𝑝>0,Φ≅1−𝑝4𝑎²≥0. (46) The two required conditions for stability are 𝑝<1/2 and 𝑎²<𝑝/8, implying that 𝑎<1/4.
In the case of Theorem 10, for a sufficiently small value of ‖𝐸‖₂, the condition of Theorem 10 can be approximately stated as follows: 𝛿≅1−4𝑎>0. (47) The required condition for stability is 𝑎<1/4.

6. Conclusions

In this paper, we have obtained some sufficient conditions for the existence, uniqueness, and global asymptotic stability of the equilibrium point for the class of neutral-type systems with discrete time delays. The results we obtained establish various relationships between the network parameters of the system. We have also given an example to show the applicability of our results and make a comparison between our results and some previous corresponding results derived in the literature. Most of the literature results express the stability conditions in terms of LMIs (linear matrix inequalities), which are then solved by some software tools. Such results may give less conservative results; however, the computational burden of this method can be high. Our results establish less complex relationships between the network parameters of the system. We should also point out that the delay-independent conditions may be more conservative than delay-dependent ones. In our paper, our stability conditions are independent of the time delays; this is due to the Lyapunov functional that we have employed in the analysis of our network model. In order to apply our techniques to obtain some delay-dependent conditions, one needs to modify the Lyapunov functional that we have used to include time delays in the conditions, which probably could be the subject of another study.