A New Globally Exponential Stability Criterion for Neural Networks with Discrete and Distributed Delays

This paper concerns the problem of the globally exponential stability of neural networks with discrete and distributed delays. A novel criterion for the globally exponential stability of neural networks is derived by employing the Lyapunov stability theory, homomorphic mapping theory, and matrix theory. The proposed result improves the previously reported global stability results. Finally, two illustrative numerical examples are given to show the effectiveness of our results.


Introduction
The dynamics of neural networks has been widely studied in the past few decades, due to their practical importance and successful applications in many areas such as image processing, combinatorial optimization, signal processing, pattern recognition, and associative memories [1][2][3][4].In order to design an associative memory by using a neural network, we must choose the appropriate network parameters that allow the designed neural networks to have multiple equilibrium points for a particular input vector depending on the initial states of the neurons.Moreover, in order to solve some classes of optimization problems by employing neural networks, the equilibrium point of designed neural network must be unique and globally stable of the initial conditions.Therefore, the existence, uniqueness, and stability of the equilibrium point are some of the important dynamical properties from the application of the neural networks [1,[4][5][6].
It is well known that stability is one of the main properties of neural networks and stability is a crucial feature in the design of neural networks.However, time delays always occur in various neural networks and cause undesirable dynamic network behaviors such as oscillation and instability [3,[7][8][9].
A great deal of effort has been devoted to stability analysis of neural networks with various types of time delays such as constant delay, time-varying delay, and distributed delay [1][2][3][4].In [10], a model for neural networks with constant delay was investigated.Neural networks with time-varying delay have been considered in [2-4, 7-9, 11-24].Recently, there has been a growing interest in study of stability analysis for neural networks with discrete and distributed delays [2,3,[7][8][9]11].
The authors got a global stability condition about neural networks with discrete delays in [1,4], and distributed delay was not considered.In [2,3], neural networks with discrete and distributed delays were investigated by employing LMI methods.However, it is difficult to obtain the stability condition.In this paper, by using the Lyapunov stability theory, homomorphic mapping theory, and matrix theory, a novel delay-dependent sufficient condition for global exponential stability of neural networks with discrete and distributed delays is obtained.The process with which we get global stability is simple in this paper.The novel result improves conditions in [3,[7][8][9]11], and it is easy to be verified.Finally, some illustrative numerical examples are given to make a comparison between the proposed result and the previously corresponding results.
Lemma 2. For any real numbers  and , the following inequality holds: where  is any positive constant number.

Existence, Uniqueness, and Stability of Equilibrium Point
This section deals with the existence, uniqueness, and stability of the equilibrium point for system (2).The theorem is stated in the following. where Proof.The proof can be completed by two steps.Firstly, we prove the existence and uniqueness of the equilibruim point of system (2); it is equivalent to prove that the following mapping ( 12) is a homeomorphism on   .In the second step, the global asymptotical stability is proved.
Step 1.Consider the following mapping associated with system (2): To prove the existence and uniqueness of the equilibrium point, it is sufficient to show that () is a homeomorphism on   .We first prove that () is injective on   .For ∀ ̸ = , ,  ∈   , Multiplying both sides of ( 13) with ( − )  , From ( 14), we get the following inequalities: where Then, ( 16) can be written as follows: where Then, (18) can be written as follows: Then, for any  ̸ = , which implies that () ̸ = () for all  ̸ = .That is, () is injective on   .
Letting  = 0, we obtain it follows that and, therefore, has a unique equilibrium point.
Step 2. We show that the condition in Theorem 5 for the existence and uniqueness of the equilibrium point also implies the global asymptotical stability of neural network (2).From the above proof, the system has a unique equilibrium point where (()) = ( 1 ( Construct the following positive definite Lyapunov functional: where  1 (()),  2 (()), and  3 (()) are defined as follows: Taking the derivative of (()) along the trajectories of system (26) as follows: From (36), we get the following inequality: From the well-known Lyapunov theory, we can conclude that the system (2) is globally asymptotically stable.