New Results on Passivity for Discrete-Time Stochastic Neural Networks with Time-Varying Delays

The problem of passivity analysis for discrete-time stochastic neural networks with time-varying delays is investigated in this paper. New delay-dependent passivity conditions are obtained in terms of linear matrix inequalities. Less conservative conditions are obtained by using integral inequalities to aid in the achievement of criteria ensuring the positiveness of the Lyapunov-Krasovskii functional. At last, numerical examples are given to show the effectiveness of the proposed method.


Introduction
Neural networks have been greatly applied in many areas in the past few decades, such as static processing, pattern recognition, and combinatorial optimization [1][2][3].In practice, time-delays are frequently encountered in neural networks.As the finite signal propagation time and the finite speed of information processing, the existence of the delays may cause oscillation, instability, and divergence in neural networks.Moreover, stochastic perturbations and parameter uncertainties are two main resources which could reduce the performances of delayed neural networks.Due to the importance in both theory and practice, the problem of stability for stochastic delayed neural networks with parameter uncertainties is one of hot issues.Therefore, there have been lots of important and interesting results in this field [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].
It should be noticed that most neural networks are focused on continuous-time case [3,[7][8][9][10][11][12].However, discrete-time systems play crucial roles in today's information society.Particulary, when implementing the delayed continuous-time neural networks for computer simulation, it needs to formulate discrete-time system.Thus, it is necessary to research the dynamics of discrete-time neural networks.In recent years, a lot of important results have been published in the literatures [13][14][15][16][17]. Kwon et al. [14] discussed the stability criteria for the discrete-time system with time varying delays.Wang et al. [16] researched the exponential stability of discrete-time neural networks with distributed delays by means of Lyapunov-Krasovskill functional theory and linear matrix inequalities technology.In [17], the authors are concerned with the robust state estimation for discrete neural networks with successive packet dropouts, linear fractional uncertainties, and mixed time-delays.
On the other hand, passivity is a significant concept that represents input-output feature of dynamic systems, which can offer a powerful tool for analyzing mechanical systems, nonlinear systems, and electrical circuits [18].The passivity theory was firstly presented in the circuit analysis [19].During the past several decades, the passivity theory has found successful applications in various areas such as complexity, signal processing, stability, chaos control, and fuzzy control.Thus, the problem of passivity for time-delay neural networks has received much attention and lots of effective approaches have been proposed in this research area [20][21][22][23][24][25][26][27].The authors [21,22] discussed the problem of passivity for neural networks with time-delays.Recently, Lee et al. [23] further studied the problem of dissipative analysis for neural networks with times-delays by using reciprocally convex approach and 2 Mathematical Problems in Engineering linear matrix inequality technology.Very recently, in [24], the problem of passivity criterion of discrete-time stochastic bidirectional associative memory neural networks with timevarying delays has been developed.In [25], some delaydependent sufficient passivity conditions have been obtained for stochastic discrete-time neural networks with timevarying delays in terms of linear matrix inequalities technology and free-weighting matrices approach.A less conservative passivity criterion for discrete-time stochastic neural networks with time-varying delays was derived in [26].However, there is still a room for decreasing the conservatism.
Motivated by the above discussion, the problem of passivity for discrete-time stochastic neural networks with timevarying delays is studied.The major contribution of this paper lies in that, first of all, different from the traditional way, a new inequality is introduced to deal with terms This method can effectively reduce the conservatism.Secondly, we do not need all the symmetric matrices in the Lyapunov functional to be positive definite and take advantage of the relationships of () −   and   − ().New passivity conditions are presented in terms of matrix inequalities.Finally, numerical examples are given to indicate the effectiveness of the proposed method.
Notations.Throughout this paper, the superscripts "−1" and "" stand for the inverse and transpose of a matrix, respectively;  > 0 ( ⩾ 0,  < 0,  ⩽ 0) means that the matrix  is symmetric positive definite (positive semidefinite, negative definite, and negative semidefinite); {⋅} stands for the mathematical expectation operator with respect to the given probability measure; ‖⋅‖ refers to the Euclidean vector norm; (Ω, F, P) denotes a complete probability space with a filtration containing all -null sets and is right conditions; [, ] denotes the discrete interval given [, ] = {,  + 1, . . .,  − 1, };   denotes -dimensional Euclidean space;  × is the set of  ×  real matrices; * denotes the symmetric block in symmetric matrix;  max () and  min () denote, respectively, the maximal and minimal eigenvalue of matrix .
The following lemmas and definition will be used in proof of main results.
the following inequality holds: Mathematical Problems in Engineering 3 Proof.In fact, we have ) .
Thus, one can easily obtain The proof is completed.
Remark 5.The new inequality was proposed in [5,6] for continuous-time systems; it is worth noting that we firstly extend this method to study discrete-time neural networks in this paper.
Remark 11.It should be pointed out that the new inequality is introduced to deal with ∑ −1 =−    () 1 () and ∑ −  −1 =−    () 1 (), which is immensely different from traditional ways.This method can effectively reduce the conservatism of the results.
Remark 12.In this paper, not all the matrices in the Lyapunov functional need to be positive definite.In fact, the conditions in (15) assure the positive definiteness of the Lyapunov functional; this is greatly different from traditional ways for passivity researches of discrete-time neural network, because the traditional methods always need Lyapunov matrices to be positive definite.1) is passive, if there exist scalars  > 0,  > 0, matrices  > 0,  1 > 0,   16) and ( 17 where Δ(), Δ 1 (), and Δ 2 () denote the parameter uncertainties that are assumed to be of the form where ,   ( = 1, 2, 3) are known constant matrices, and () is the unknown matrix valued function subject to  ()   () ≤ .

Numerical Examples
In this section, some numerical examples are proposed to show the effectiveness of the results obtained in this paper.
The activation functions are taken as It can be verified that In this example, if   = 3 and   = 12, the optimal passivity performance obtained is  = 12.2439 by the method in [25] and  = 5.4007 by the method in [26], while by Theorem 10 in this paper, the optimal passivity performance  = 3.9546.
The comparisons of  are listed in Table 1, when   = 3,   = 8, 9, 10, 11, 12.Then, when we assume   = 13, the optimal passivity performance  obtained by Theorem 10 for different   can be found in Table 2.It can be seen that our results are less conservative than the ones in [25,26].(47) For this example, when  = 3 and   = 1, by Theorem 10, we can get that the upper bound of the time-varying delay is   = 8.When  = 4 and   = 1, by Theorem 10, we can get   = 10; we can obtain upper bound of   for different  and   , which are summarized in Table 3.It can be found from Table 3 that, for the same   , a larger passivity performance  corresponds to a larger   ; with the same   , a smaller passivity performance  corresponds to a larger   .

Conclusions
In this paper, the problem of passivity analysis for discretetime stochastic neural networks with time-varying delays has been investigated.The presented sufficient conditions are based on the Lyapunov-Krasovskii functional, a new inequality and linear matrix inequality approach.Numerical examples are given to demonstrate the usefulness and effectiveness of the proposed results.Finally, it should be worth noticing that the proposed method in this paper may be extensively applicable in many other areas, such as Markov jump neural networks, Markov jump neural networks with incomplete transition descriptions, and switched neural networks, which deserves further investigation.

Example 1 .
Consider the system (1) with the following parameters:

Table 1 :
Optimal passivity performance  for different   .

Table 2 :
Optimal passivity performance  for different   .

Table 3 :
Allowable upper bounds of   for different  and   .