Delay-Dependent Stability Analysis for Recurrent Neural Networks with Time-Varying Delays

This paper concerns the problem of delay-dependent stability criteria for recurrent neural networks with time varying delays. By taking more information of states and activation functions as augmented vectors, a new class of the Lyapunov functional is proposed. Then, some less conservative stability criteria are obtained in terms of linear matrix inequalities LMIs . Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.


Introduction
In the past few decades, the stability analysis for recurrent neural networks has been extensively investigated because of their successful applications in various scientific fields, such as pattern recognition, image processing, associative memories, and fixed-point computations.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, much more attention has been paid to recurrent delayed neural networks.Many interesting stability conditions, including delay-independent results 1, 2 and delay-dependent results 3-41 , have been obtained for neural networks with time delays.Generally speaking, the delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delay is small.For the delay-dependent case, some criteria have been derived by using Lyapunov-Krasovskii functional LKF .It is well known that the construction of an appropriate LKF is crucial for obtaining less conservative stability conditions.Thus, some new methods have been developed for reducing conservatism, such as free-weighting matrix method 4-8 , augmented LKF 9 , discretized LKF 10 , delay-partitioning method 12-18 , and delayslope-dependent method 19 .Some less conservative stability criteria were proposed in

Problem Formulation
According to the inequality 2.2 , one can obtain that Thus, under this assumption, the following inequality holds for any diagonal matrix Q 4 > 0, where Lemma 2.1 see 43 .For any constant matrix Z ∈ R n×n , Z Z T > 0, scalars h 2 > h 1 > 0, such that the following integrations are well defined, then x s ds.2.6

Main Results
In this section, a new Lyapunov functional is constructed and a less conservative delaydependent stability criterion is obtained.
Theorem 3.1.For given scalars h 0, u, diagonal matrices . ., k n , the system 2.3 is globally asymptotically stable if there exist symmetric positive matrices . ., λ n , and any matrices S i i 1, 2, . . ., 5 with appropriate dimensions, such that the following LMIs hold: where Proof.Construct a new Lyapunov functional as follow: where

3.6
Remark 3.2.Since the terms 2 n i 1 δ i z s ds are taken into account, it 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 2.4 is given by: V z t

3.11
By the use of Lemma

3.17
Furthermore, there exist positive diagonal matrices T 1 , T 2 , T 3 , such that the following inequalities hold based on 2.4 :

3.20
From 3.8 -3.20 , one can obtain that where z T s ds dθ .

3.22
If E A T RA < 0, then there exists a scalar ε > 0, such that Thus, according to 44 , system 2.1 is globally asymptotically stable.By Schur complement, E A T RA < 0 is equivalent to 3.3 , this completes the proof.
Remark 3.3.By taking the states t t−τ t f T z s ds, t−τ t t−h f T z s ds, as augmented variables, the stability condition in Theorem 3.1 utilizes more information about f z t on state variables, which may lead to less conservative results.Remark 3.4.Recently, the reciprocally convex optimization technique 42 is used to reduce the conservatism of stability criteria for systems with time-varying delays.Motivated by this work, the proposed method of 42 was utilized in 3.12 and 3.14 , which have potential to yield less conservative conditions.However, an augmented vector with z T s ds dθ was used, which is different from the method of 42 .
In many cases, u is unknown.Considering this situation, a rate-independent corollary for the delay τ t satisfying 0 ≤ τ t ≤ h is derived by setting X 0, Q 4 0 in the proof of Theorem 3.1.

is globally asymptotically stable if there exist symmetric positive matrices P
. ., λ n , and any matrices S i i 1, 2, . . ., 5 with appropriate dimensions, such that 3.1 , 3.2 and the following LMI hold:

3.25
The other E ij is defined in Theorem 3.1.

Numerical Examples
In this section, two numerical examples are given to demonstrate the effectiveness of the proposed method.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 the Table 2, this example is given to indicate significant improvements over some existing results.

Conclusions
In this paper, a new Lyapunov functional was proposed to investigate the stability of neural networks with time-varying delays.Some improved generalized delay-dependent stability criteria have been established.The obtained criteria are less conservative because a convex optimization approach is considered.Finally, two numerical examples have shown that these new stability criteria are less conservative than some existing ones in the literature.

Figure 1 :
Figure 1: The dynamical behavior of z t .

Table 1 :
Allowable upper bound of h for different u.E 11 E 12 E 13 E 14 E 15 −P 13 E 17 B T P 12 B T P 12 B T P 13 B T P 13 B T P 14 The upper bounds of h for different u are derived by Theorem 3.1 in our paper and the results in 13-16 are listed in Table1.According to Table1, this example shows that the stability condition in this paper gives much less conservative results than those in the literature.For h 2.7098, the global asymptotic stability with the initial state −0.2, 0.3, −0.4,0.2T is shown in Figure1.

Table 2 :
Allowable upper bound of h for different u.Example 4.2.Consider the system 2.3 with the following parameters: