Improved Stability Analysis for Neural Networks with Time-Varying Delay

This paper concerned the problem of delay-dependent asymptotic stability for neural networks with time-varying delay. A new class of Lyapunov functional dividing the interval delay is constructed to derive some new delay-dependent stability criteria. The obtained criteria are less conservative because free-weighting matrices method, a convex optimization approach, and a mixed dividing delay interval approach are considered. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.


Introduction
In the past few decades, delayed neural networks have been investigated extensively because of their successful applications in various scientific areas, such as pattern recognition, image processing, associative memories, and parallel computation.It is well known that time delay is frequently encountered in neural networks, and it is often a major cause of instability and oscillation.Thus, the stability analysis of delayed neural networks has been widely considered by many research results, delay-independent ones 1-3 , and delay-dependent ones 4-37 .Generally speaking, the delay-dependent stability criteria are less conservative than delay-independent ones when the time delay is small.Therefore, much attention has been paid to develop delay-dependent derived in 6 by considering some useful terms and using the free-weighting matrices method.By the fact that the neuron activation functions are sector bounded and nondecreasing, 7 presents an improved method, named the delay-slope-dependent method, for stability analysis of neural networks with time-varying delays.The method includes more information on the slope of neuron activation functions Mathematical Problems in Engineering and fewer matrix variables in the constructing Lyapunov functionals.Then some new delay-dependent stability criteria with less conservatism are obtained.Recently, some new Lyapunov functionals based on the idea of decomposing the delay were introduced to investigate the stability of neural networks with time-invariant delay 10-12 and timevarying delay 13-16 , which significantly reduced the conservativeness of the derived stability criteria.In 13 , different from some previous results, the delay interval 0, d t is divided into some variable subintervals by employing weighting delays.Thus, some new delay-dependent stability criteria for neural networks with time varying delay are derived by applying the weighting-delay method, which are less conservative than the existing results.However, when the delay is time-varying, the information of subinterval is not considered sufficiently.For example, the time-varying delay τ t satisfies 0 τ t h.When the delay interval 0, τ t is divided into some subintervals, the delay interval 0, h is also divided into some subintervals, in essence.But in the construction of Lyapunov functional in 15 , this important information is ignored, which is a major source of conservativeness.Furthermore, the purpose of reducing conservatism is still limited due to the existence of multiple coefficients and the impact of subintervals with uniform size.Thus, it is still a quite difficult task to divide interval 0, τ t in a more reasonable manner, so that the functional with the augmented matrix can easily be constructed to obtain less conservative stability results, which motivates our present study.
In this paper, the problem of delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been considered.A new class of Lyapunov functional is constructed to derive some new delay-dependent stability criteria.The obtained criteria are less conservative because a mixed dividing delay interval approach is considered.Finally, the numerical examples are given to indicate significant improvements over some existing results.

Problem Formulation
Consider the following neural networks with time-varying delay:

Main Results
In this section, a new Lyapunov functional is constructed, and a new delay-dependent stability criterion is obtained.
Theorem 3.1.For given scalars K diag k 1 , k 2 , . . ., k n , h 0, u, and 0 < α < 1, the system 2.3 is globally asymptotically stable if there exist symmetric positive matrices . ., λ n , and any matrices P 1 , P 2 , N i , M i , L i , H i , Z i , S i , U i , V i , W i i 1, 2, 3 with appropriate dimensions, such that the following LMIs hold:

3.5
Proof.Construct a new class of Lyapunov functional candidate as follows: where is clear that the Lyapunov functional candidate in this paper is more general than that in 5, 6, 8, 9 .So the stability criteria in this paper may be more applicable.
The time derivative of V z t along the trajectory of system 2.3 is given by Vi z t , 3.9 where

3.10
Using Lemma 2.1, one can obtain that 1 For the case of 0 τ t αh, then it gets żT s R 3 ż s ds.

3.19
It is easy to see that żT s R 6 ż s ds dθ.

3.24
Using Lemma 2.1, it is easy to obtain that

3.25
Furthermore, there exist positive diagonal matrices T 1 , T 2 , such that the following inequalities hold based on 2.4 :

3.27
Mathematical Problems in Engineering From 3.10 -3.27 , one can obtain that where

3.29
Note that 0 τ t αh, αh 3 S T can be seen as the convex combination of NR −1 3 N T , LR −1 3 L T , and SR −1 3 S T on τ t .Therefore, Σ 1 < 0 holds if and only if Applying the Schur complement, the inequalities 3.30 and 3.31 are equivalent to the LMI 3.1 and 3.2 , respectively.żT s R 4 ż s ds.

3.33
It is easy to obtain that żT s R 4 ż s ds.

3.39
Applying the Schur complement, the inequalities 3.38 and 3.39 are equivalent to the LMI 3.3 and 3.4 , respectively.Therefore, if the LMIs 3.1 -3.4 are satisfied, then the system 2.3 is guaranteed to be asymptotically stable for 0 τ t h.

Mathematical Problems in Engineering
Remark 3.3.It is well known that the delay-dividing approach can reduce the conservatism notably.But some previous literature only uses single method to divide the delay interval 0, h .Unlike 10, 25 , the new Lyapunov functional in our paper which not only divides the delay interval 0, h into two ones 0, αh and αh, h but also divides the delay interval 0, h into three ones 0, ατ t , ατ t , τ t , and τ t , h is proposed.Each segment has a different positive matrix, which has the potential to yield less conservative results.
Remark 3.4.In this paper, by taking the states z t − αh , z t − τ t , z t − α 2 h , z t − h , and z t − ατ t as augmented variables, the stability in Theorem 3.1 utilizes more information on state variables.And in deriving upper bounds of integral terms in V4 z t , different freeweighting matrices are introduced in two different intervals 0 τ t αh and αh τ t h.These methods mentioned above may lead to obtain an improved feasible region for delaydependent stability criteria.
is evaluated by the LMIs 3.1 and 3.2 , which can help reduce much more conservatism than the results in 8 .Remark 3.6.In many cases, u is unknown.For this situation, a rate-independent criterion for a delay satisfying 0 ≤ τ t ≤ h is derived as follows by setting Q 1 0, Q 2 0, Q 6 0, and Q 0 in the proof of Theorem 3.1.

3.40
where Mathematical Problems in Engineering

3.41
The other E ij , Φ ij are defined in Theorem 3.1.

4.4
Therefore, it follows from Theorem 3.1 that the system 2.3 with given parameters is globally asymptotically stable.

Conclusions
In this paper, a new delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been proposed.A new class of Lyapunov functional has been introduced to derive some less conservative delay-dependent stability criteria by using the free-weighting matrices method and the technique of dealing with some integral terms.Finally, numerical examples have been given to illustrate the effectiveness of the proposed method.
where h and u are constants.In addition, it is assumed that each neuron activation function in 2.1 , g i • , i 1, 2, . . ., n, is bounded and satisfies the following condition: n is a constant input vector.A, B ∈ R n×n are the connection weight matrix and the delayed connection weight matrix, respectively.C diag C 1 , C 2 , . . ., C n with C i > 0, i 1, 2, . . ., n. τ t is a time-varying continuous function that satisfies 0 τ t h, τ t u, i , i 1, 2, . . ., n are positive constants.

Table 2
Our purpose is to estimate the allowable upper bounds delay h under different u such that the system 2.3 is globally asymptotically stable.According to Table1, this example shows that the stability criterion in this paper gives much less conservative results than those in the literature.By using the Matlab LMI toolbox, we solve LMIs 3.1 -3.4 for the case α 0.4, u 0.8, and h 2.9144 and obtain gives the comparison results on the maximum delay bound allowed via the methods in recent paper and our new study.According to Table2, this example shows that

Table 1 :
Allowable upper bound of h for different u Example 4.1 .

Table 2 :
Allowable upper bound of h for different u Example 4.2 .stability criterion in this paper can lead to less conservative results.By using the Matlab LMI toolbox, we solve LMIs 3.1 -3.4 for the case α 0.4, u 0.4, and h 4.7444 and obtain the