A New Global Asymptotic Stability Result of Delayed Neural Networks via Nonsmooth Analysis

In the paper, we obtain new sufficient conditions ensuring existence, uniqueness, and asymptotic stability of the equilibrium point for delayed neural network via nonsmooth analysis, which makes use of the Lipschitz property of the functions. Based on this tool of nonsmooth analysis, we first obtain a couple of general re-sults concerning the existence and uniqueness of the equilibrium point. Then we drive some new sufficient conditions ensuring global asymptotic stability of the equilibrium point. Finally, there are the illustrative examples feasibility and effectiveness of our results. Throughout our paper, the activation function is a more general function which has a wide application.


Introduction
In recent years, the stability of a unique equilibrium point of delayed neural networks has extensively been discussed by many researchers [1][2][3][4][5].Several criteria ensuring the global asymptotic stability of the equilibrium point are given by using the comparison method, Lyapunov functional method, M-matrix, diagonal dominance technique and linear matrix inequality approach.In [1][2][3][4][5], some sufficient conditions are given for the global asymptotic stability of delayed neural networks by constructing Lyapunov functions.A new sufficient condition on the global asymptotic stability for delayed neural networks via nonsmoosh analysis is derived in this letter.The condition is independent of delay and imposes constraints on both the feedback matrix and delayed feedback matrix.Our results generalize and improve the preciously known works due to expending the activation function of the delay part.
Concerning the global stability of delayed neural networks described by the following differential equations with time delays where 1 ( ) ( ( ), , ( ))    is a constant input vector and  is the delay parameter.Furthermore, we assume that the function g , the activation function f and the activation function satisfy the following conditions: h A1) Each function : is a locally Lipschitz function and there exists such that ( ) for all at y   which i g is differentiable.A2) Each function : is a globally Lipschitz function with module , i.e., 1 2 1 ( ) ( ) is a globally Lipschitz function with module , i.e., The paper is organized as follows: Section 2 contains a short introduction to nonsmooth analysis for Lipschitz functions.In particular, the Lipschitzin Hadamard Theorem is explained and a homeomorphism theorem is obtained.Section 3 is to demonstrate how nonsmooth analysis can be carried out on (1) to derive sufficient conditions to ensure the existence and uniqueness of the equilibrium point of (1).In Section 4, we study new sufficient conditions with guarantee the GAS of the (1).In Section 5, we have an illustrative example and its simulations.We conclude in Section 6.
Notation: Let be the constants given in assumptions (A1), (A2) and (A3), define two diagonal matrices , , , 1,2, Let  denotes the Euclidean norm for vectors and the matrix norm for matrices for any vector

 
)  is the largest eigenvalue of the symmetric part of .All the mathematical facts concerning the eigenvalues of a matrix used in this paper can be found in the book [6].

Nonsmooth Analysis on Lipschitz Functions
We first review some concepts which are essential for conducting nonsmooth analysis on Lipschitz function.
Then we state the Lipschitzian Hadamard Theorem, which gives conditions for homeomorphism of Lipschitz functions.Finally, we give a sufficient condition which ensures the existence and uniqueness of the equilibrium point of ( ( ) x y where denotes the segmen onnecting t c x and y .
For any two locally Lipschitz functions : Now we are ready to state the Lipschitzian Hadamard Theroem which will lead to our homeomorphism result Theorem 1.
Lemma 1 [9] (Lipschitzian Hadamard Theorem): Suppo For ssions on a pplications, please refer to books [7,8] as w more discu the gener lized Jacobian and its various a ell as to the paper [10,11].Now, we analyze (1) from the viewpoint of nonsmooth analysis.We first recall that a state * n x   is called an equilibrium point of (1) if it satisfies To study the existence and uniqueness rium point for any input vector , we define the function of the equilib- Clearly, . By applying Theorem 1 to (4), we have the following.
2) Each element  W V is nonsin nd each activation function i f and i h is nondecreasing.
Then for each input vector Proof: We only p he result for the case (i).The se (2) be proved si as (1).
We prove it by a contradiction.Assume that there exists a sequence of matrices { } Then there exist two sequences of diagonal matrices We obse e matrix sequence This contradicts our assumption (6).Hence, th such that (5) holds.It then follows from Theorem at F is a Homeomorphism from n  onto itself.
The above homeomorphism result means that if each element in is nonsingular, then the neural network de V fined by (1) has a unique equilibrium point for any input vector n u   .This result is the starting point of the next two sections where we will consider what pracake the delay neural network stable.tical conditions m nd niqueness of the equilibrium point of (1).As consessumption e existing s

Existence and Uniqueness of the Equilibrium Point
In this section, based on Theorem 1 we present some new sufficient conditions which ensure the existence a u quences, we further show that the existence a on nnecessary in som equilibrium point is u ults for GAS.re Theorem 2: Suppose one of the following assumptions holds: 1 We have from the assumption i) of the theorem and the athe 2 , , ) m matical facts listed at the beginning of the proof that ) ) (( This means that the matrix For the remaining two cases, we need only to show that each element in is nonsingular.We note the fact efining ngular.Then the nonsingularity of W follows from the observation that the with positive diagonal, an a nonnegative diagonal m 2) The proof is trivial by noticing fact that for any three matrices  , , ) gular.We prove it by a contradiction.Assume at is singular, then there exists on both sides of (7), we have

Conditions for GAS
In this section, we present new conditions for the GAS of me that all the acti vation function are nondecreasing, i.e., the generalized nonsingular.This com-

New
the equilibrium point of (1).We assu -Jacobian of F at any point V Theorem 3: In addition to assump (A3), we assume that each activatio tions (A1), (A2) and n function is nondeeasing.Suppose cr h ique equilibrium in as a un po t which is GAS.
In particular, the DCNN where 1   k m has a unique equilibrium point for each input vector n u   and this equilibrium point is GAS if the following condition holds: ollows from Theorem 2 3) that(1)has a unique equilibrium Copyright © 2010 SciRes.IJCNS , is GAS.For simplicity, we shift to this origin through the transformation en can be equivale where the origin is GAS of (9).s ee point.Hence it remains to show that this equilibrium point, say * x Equation (1) th ntly written as the following system . Next, evaluating the time along the trajectories of ( 9), btain The Lebourg theorem for Lipshcitz functions means that (12) for some From the definition of , matrix is diago we denote D nal, and Putting those inequalit using (10), we have Rearranging terms in (19) and using above inequalities, Now we consider the following three cases.
ensures that is negative.

2)
)) We recall that , which for this case.
, that m point rding to [12] or [13] equivalently the equilibriu * f (1) is GAS.ffic 8) x o The su ient cond for DCNNs are direct nse of the general proved above.Next we prove the GAS of the equilibrium point, say onsequen from [9] that fo is nonsingular for the a and D .This fact in turn means that any element in V is nonsingular.Now the existence and x  , of formation as in the proof of 3, we con S of the system 9) at origin.Define a Lyapunov function as [10] follows: (1).Through the trans Theorem sider the GA ( the with , 0    ,being selected later on.It is easy to see that ( ( )) V z  is positive except at the origin the trajectory of ( 9and it is radially unbounded.Evaluating its time derivative along ), we obtain that Since the matrix in ( 16) is assumed to be nega e definite, we can prove that the origin is GAS of (9) following the very similar way of 1)-3) in the last part of the proof of Theorem 3.This accomplishes our proof.tiv by

Illustrative Examples
Example 1: Consider the following model: where g(x) = 0.3x, f(x) = 0.2x, h(x) = 0.1(x-sinx), obviously, satisfied the assumption ), (A2) and (A3), we obtain d the condition of Theorem 3. Using the Matlab, we have made graphics of the solution as the time in the system with initial conditions [-0.5 0.6 -0.8] and the delay  From the figure, we can easily see the system has a unique equilibrium point, and is GAS.
Example 2: In order to demonstrate the validity of our criterion of the Theorem 4, we consider a delayed neural network in (5) with parameters as obviously, (A1), (A2) and (A3) hold.

Conclusions
In un Our study is based on a thorough nonsmooth analysis on functions defining DNNs.The general Theorem 1 on existence and uniqueness of the equilibrium point is proved easy to apply.This general result, allows us to study sufficient conditions for GAS in which the spectral properties of the matrix  play an important role.Advantages of our results are illustrated by examples and also given a graph of the GAS.It would be very interesting to see how our approach can be used to study conditions which do not enjoy symmetric properties.

Figure 1 .
Figure 1.State trajectories of we present new conditions for the existence, iqueness, and GAS of the equilibrium point of DNNs.()A lA  1) for any input vector .