ROBUST STABILITY FOR FUZZY STOCHASTIC HOPFIELD NEURAL NETWORKS WITH TIME-VARYING DELAYS

. This paper investigates delay-dependent robust stability problem of fuzzy stochastic Hop(cid:133)eld neural networks with random time-varying delays. Moreover, in this paper, the stochastic delay is assumed to satisfy a certain probability distribution. By introducing a stochastic variable with Bernoulli distribution, the neural networks with random time delays is trans- formed into one with deterministic delays and stochastic parameters. Based on a Lyapunov-Krasovskii functional and stochastic analysis approach, delay- probability-distribution-dependent stability criteria have been derived in terms of linear matrix inequalities (LMIs), which can be checked easily by the LMI control toolbox. Finally two numerical examples are given to illustrate the e⁄ectiveness of the theoretical results. 92B20, 37H30, 15A39, 93C43. time-varying delays


Introduction
In recent decades, Neural Networks (NNs) especially recurrent neural networks (RNNs) and Hop…eld neural networks (HNNs) have been successfully applied in various …elds such as pattern recognition, optimization problems, associative memories, signal processing, etc., see [1] - [18]. One of the best important works is to study the stability of the equilibrium point of NNs. Since time delays as a source of instability and poor performance always appear in many neural networks owing to the …nite speed of information processing, the stability analysis for the delayed neural network has received considerable attention [3][4][5][6][7]. 2020 Mathematics Subject Classi…cation. 92B20, 37H30, 15A39, 93C43. Keywords and phrases. Hop…eld neural networks, stochastic system, linear matrix inequality, time-varying delays Submitted via ICCSPAM 2020.
gopalakrishnan.n@srec.ac.in 0000-0002-2365-9305. On the other hand, the stability analysis of stochastic systems with time delays has been investigated by many researchers since stochastic modelling plays an important role in many …elds of science and engineering applications [8][9][10][11][12][13][14][15][16]. In a real system, time delay often exists in a random form, that is, some values of the time delay are very large. However the probability of the delay taking such large values is very small and it may lead to a more conservative result, only if the information of variation range of the time delay is considered. In addition, its probabilistic characteristic such as Bernoulli distribution and the Poisson distribution can also be obtained by statistical methods. Therefore, it is necessary and realizable to investigate the probability-distribution delay and therefore in recent years, the stability problems of NNs with probability-distribution delay have been widely investigated [17,18].
It is well known that fuzzy logic theory has shown to be an appealing and e¢ cient approach to dealing with the analysis and synthesis problems for complex nonlinear systems. The well-known Takagi-Sugeno (T-S) fuzzy model [19], is a popular and convenient tool to transform a complex nonlinear system to a set of linear submodels via some fuzzy models by de…ning a linear input/output relationship as its consequence of individual plant rule. Recently, a lot of research works have been produced on T-S fuzzy model in the existing available literature [20][21][22].
Based on the above discussion, we consider the problem of delay-dependent robust stability analysis for uncertain fuzzy stochastic Hop…eld neural networks with time-varying delays. Some su¢ cient condition for delay-probability-distributiondependent stability criteria of the addressed system have been derived in terms of linear matrix inequalities by constructing proper Lyapunov-Krasovskii functional and stochastic theory. Finally, numerical examples are provided to show the e¤ectiveness of the theoretical results.
Notations: Throughout this paper, R n and R n n denote, respectively, the ndimensional Euclidean space and the set of all n n real matrices. The superscript T denotes the transposition and the notation X Y (respectively, X > Y ), where X and Y are symmetric matrices, means that X Y is positive semi-de…nite (respectively, positive de…nite). I n is the n n identity matrix. k k is the Euclidean norm in R n . Moreover, let ; F; fF t g t 0 ; P be a complete probability space with a …ltration fF t g t 0 satisfying the usual conditions (i.e. the …ltration contains all P -null sets and is right continuous). Denoted by L p F0 [ ; 0]; R n the family of all where Ef:g stands for the mathematical expectation operator with respect to the given probability measure P .

Problem description and preliminaries
Consider the following uncertain stochastic HNNs with time-varying delays where x(t) 2 R n is the neural state vector, f (x(t)) = h f 1 (x 1 (t)); : : : ; f n (x n (t)) i T 2 R n is the neuron activation function with initial condition f (0) = 0. The timevarying delay (t) satis…es where and are constants. In (1), Further A = diagfa 1 ; a 2 ; : : : ; a n g has positive entries a i > 0, B, W , H 0 , H 1 are connection weight matrices with appropriate dimensions and A(t), B(t), W (t), H 0 (t) and H 1 (t) denote the time-varying and norm-bounded uncertainties.
Assumptions 2.2 Considering the information of probability distribution of the time delay (t), two sets and functions are de…ned by where 0 2 [0; ], 1 2 [0; 0 ) and 2 2 [ 0 ; ]. It is easy to know t 2 1 means the event (t) 2 [0; 0 ) occurs and t 2 2 means the event (t) 2 [ 0 ; ] occurs. Therefore, a stochastic variable (t) can be de…ned as 1 is a constant and Ef (t)g is the expectation of (t).

Remark 2.4 From Assumption 2.3, it is easy to know that
. By Assumption 2.2 and 2.3, the system (1) can be rewritten as which is equivalent to x(t) = (t); t 2 [ ; 0]: Remark 2.5 In this paper, the probability distribution of the delay taking values in some interval is assumed to be known in advance. Further, a new model of the SNNs (8) has been derived, which can be seen as an extension of the common SNNs (1). Specially, in the case of (t) 1, system (8) becomes system (1). Moreover, when the probability of time delay taking values is known a priori, the possible values that the delay takes may be larger than those previously obtained results based on the traditional methods, which will be illustrated via example later.
In this paper, we consider the following neural network with parameter uncertainties and stochastic perturbations which is represented by a T-S fuzzy model. The kth rule of the T-S fuzzy model is of the following form: Plant Rule k: IF 1 (t) is k 1 and : : : x(t) = (t); t 2 [ ; 0]; k = 1; 2; :::; r; where k i (i = 1; 2; :::; p) is the fuzzy set, (t) = h 1 (t); 2 (t); :::; p (t) i T is the premise variable vector and r is the number of IF-THEN rules. !(t) is a onedimensional Brownian motion de…ned on ; F t ; fF t g t 0 ; P . 2 L 2 F0 ([ ; 0]; R n ) is the initial value of (9). A k , B k , W k , H 0k and H 1k are constant known real matrices. A k (t), B k (t), W k (t), H 0k (t) and H 1k (t) denote the time-varying parameter uncertainties and we make the following assumption.
k and E H1 k are known real constant matrices with appropriate dimensions, and F (t) is the time-varying uncertain matrix which satis…es The defuzzi…ed output of the T-S fuzzy system (9) is represented as follows: in which k j ( j (t)) is the grade of membership of j (t) in k j . According to the theory of fuzzy sets, we have k ( (t)) 0, k = 1; 2; :::; r, r P k=1 k ( (t)) > 0 for all t. Therefore, it implies k ( (t)) 0, k = 1; 2; :::; r, r P k=1 k ( (t)) = 1 for all t. Let x(t; ) denotes the state trajectory of system (12) from the initial value It is easy to see that system (12) admits a trivial solution x(t; 0) 0.
The following de…nition and lemmas are used to prove our main result.
De…nition 2.6 [22] For system (9) and every 2 L 2 F0 ([ 1; 0]; R n ), the trivial solution is asymptotically stable in the mean square if Ejx(t; )j 2 = 0: I. Then (i) for any scalar > 0, Lemma 2.9 [25] Let M , P , Q be the given matrices such that Q > 0, then if and only if there exists a scalar " > 0 such that Lemma 2.11 [27] Assume that a( ) 2 R na , b( ) 2 R n b and N 2 R na n b are de…ned on the interval . Then for any matrices X 2 R na na , Y 2 R na n b and Z 2 R n b n b , the following holds
In the following part, we extend the above result to uncertain fuzzy stochastic Hop…eld neural network (UFSHNN) (12) and obtain the stability criteria as the following theorem by means of the feasibility of LMIs.

Remark 3.3
In [22], the authors dealt with the problem of delay-dependent robust stability for uncertain stochastic fuzzy Hop…eld neural networks with timevarying delays. However, the probability distribution delay was not taken into account in this model. In our paper, we study delay-dependent robust stability analysis for uncertain fuzzy stochastic Hop…eld neural networks with random timevarying delays. Thus, the results in this paper are lead to an improvement over the existing ones [22].

Remark 3.4
In the case of k = 1, the system (12) is reduced to same as in [18] and the stability criteria for the corresponding reduced system can be obtained by using Theorem 3.1. Moreover, the traditional assumption such as boundedness, monotonicity or di¤erentiability on the neuron activation functions [22] have been removed in this paper.

Numerical Examples
In this section, we will give two examples showing the e¤ectiveness of established theoretical results. Example 1 Consider the SNNs (14) without uncertain parameters de…ned as The activation function f (x(t)) = tanh(x(t)), the time-varying delays are chosen as 1 = 0:4 and 2 = 1. The derivative of time-varying delays _ 1 (t) 1 = 0:9, _ 2 (t) 2 = 0:9, 0 = 0:2, and using the Matlab LMI toolbox to solve the LMI in Theorem 3.1, we obtained the following matrices Therefore, it follows from Theorem 3.1, that the SNNs without uncertain parameters (14) is globally asymptotically stable in the mean square. The response of the state dynamics for the SNNs without uncertain parameters (14) which converges to zero asymptotically in the mean square are shown in Figures 1 and 2. Example 2 Consider the SNNs (12) with uncertain parameters de…ned as The activation function f (x(t)) = tanh(x(t)), the time-varying delays are chosen as 1 = 0:2, 2 = 1:2, The derivative of time-varying delays _ 1 (t) 1 = 0:8, _ 2 (t) 2 = 0:8, 0 = 0:1 and using the Matlab LMI toolbox to solve the LMI in Theorem 3.2, we obtained the following matrices  (12) is globally robustly asymptotically stable in the mean square. The response of the state dynamics for the UFSHNN (12) which converges to zero asymptotically in the mean square are shown in Figures 3 and 4.

Conclusion
The delay-dependent robust stability analysis for fuzzy stochastic Hop…eld neural networks with random time-varying delays has been investigated. By using the combination of Lyapunov stability theory and stochastic analysis approach, some delay-dependent criteria have been derived to guarantee that the global robust asymptotic stability of the system in the mean square. This criteria can be checked easily by the LMI control toolbox in Matlab. Finally, numerical examples have been provided to illustrate the advantages and usefulness of the proposed results.