Improved Stability Criteria of Static Recurrent Neural Networks with a Time-Varying Delay

This paper investigates the stability of static recurrent neural networks (SRNNs) with a time-varying delay. Based on the complete delay-decomposing approach and quadratic separation framework, a novel Lyapunov-Krasovskii functional is constructed. By employing a reciprocally convex technique to consider the relationship between the time-varying delay and its varying interval, some improved delay-dependent stability conditions are presented in terms of linear matrix inequalities (LMIs). Finally, a numerical example is provided to show the merits and the effectiveness of the proposed methods.


Introduction
During the past decades, recurrent neural network (RNN) has been successfully applied in many fields, such as signal processing, pattern classification, associative memory design, and optimization. Therefore, the study of RNN has attracted considerable attention and various issues of neural networks have been investigated (see, e.g., [1][2][3][4] and the references therein). As the integration and communication delay is unavoidably encountered in implementation of RNN and is often the main source of instability and oscillations, much efforts have been expended on the problem of stability of RNNs with time delays (see, e.g., [5][6][7][8][9][10][11][12][13][14]).
RNNs can be classified as local field networks and static neural networks based on the difference of basic variables (local field states or neuron states) [15]. Recently, the stability of static recurrent neural networks (SRNNs) with timevarying delay was investigated in [16], where sufficient conditions were obtained guaranteeing the global asymptotic stability of the neural network. Nevertheless, some negative semi-definite terms were ignored in [16], which lead to the conservatism of the derived result. By retaining these terms and considering the low bound of the delay, some improved stability conditions were derived for SRNNs with interval time-varying delay in [17]. In [18], an input-output framework was proposed to investigate the stability of SRNNs with linear fractional uncertainties and delays. Based on the augmented Lyapunov-Krasovskii functional approach, some new conditions were derived to assure the stability of SRNNs in [19][20][21][22], but the results can be further improved.
In this paper, the problem of stability of SRNNs with timevarying delay is investigated based on the complete delaydecomposing approach [12]. By employing a reciprocally convex technique, some sufficient conditions are derived in the forms of linear matrix inequalities (LMIs). The effectiveness and the merit are illustrated by a numerical example.
Notations. Through this paper, and −1 stand for the transpose and the inverse of the matrix , respectively; > 0 ( ≥ 0) means that the matrix is symmetric and positive definite (semipositive definite); R denotes the -dimensional Euclidean space; diag{⋅ ⋅ ⋅ } denotes a blockdiagonal matrix; ‖ ‖ is the Euclidean norm of ; the symbol * within a matrix represents the symmetric terms of the matrix;

System Description
Consider the following delayed neural network: where ( ) = Furthermore, the neuron activation functions satisfy the following assumption.
Before presenting our main results, we first introduce two lemmas, which are useful in the stability analysis of the considered neural network.

Main Results
In the sequel, following the method proposed in [13], we decompose the delay interval The Scientific World Journal are to be determined, , and denotes the th row of matrix .

Remark 4.
Notice that a novel term 4 ( ) being continuous at ( ) = is included in the Lyapunov-Krasovskii functional (10), which plays an important role in reducing conservativeness of the derived result.
Next, we develop some new delay-dependent stability criteria for the delayed neural networks described by (5) and (6) with ( ) satisfying (2) and (3). By employing the Lyapunov-Krasovskii functional (10), the following theorem is obtained.

Remark 7.
In previous works such as [16,19], considerable attention has been paid to the case that the derivative of the time-varying delay( ) satisfies (3). However, in the case oḟ ( ) satifyinġ the treatment in [16,19] means that( ) in (27) is enlarged to( ) ≤ = max{ 1 , 2 , . . . , }, which may lead to conservativeness inevitably. By contrast, the case above can be taken fully into account by replacing with in Theorem 5.
For the case that the time-varying delay ( ) is nondifferentiable or( ) is unknown, setting Q = 0, = 1, 2, . . . , , in Theorem 5, a delay-dependent and rate-independent criterion is easily derived as follows.

Numerical Examples
In this section, we will provide a numerical example to show the effectiveness of the presented criteria.
This example has been discussed in [16][17][18][19][20][21][22]. By using Theorem 5 and Corollary 8 with = 2, for various , the upper bounds that guarantee the global asymptotic stability of neural network (1) are computed and listed in Table 1. It can be concluded that the upper bounds obtained by our method are much better than those in [16][17][18][19][20][21][22]. Obviously, the conditions proposed in this paper are an improvement over the existing ones.

Conclusions
This paper has studied the stability of SRNNs by constructing a complete delay-decomposing Lyapunov-Krasovskii functional. Some improved delay-dependent stability conditions have been derived by utilizing a reciprocally convex technique to consider the relationship between the time-varying delay and its varying interval, which are formulated in linear matrix inequalities (LMIs). Finally, a numerical example has been provided to show the effectiveness of the proposed methods.