Improved Criteria on Delay-Dependent Stability for Discrete-Time Neural Networks with Interval Time-Varying Delays

and Applied Analysis 3 2. Problem Statements Consider the following discrete-time neural networks with interval time-varying delays: y k 1 Ay k W0g ( y k ) W1g ( y k − h k ) b, 2.1 where n denotes the number of neurons in a neural network, y k y1 k , . . . , yn k T ∈ R is the neuron state vector, g k g1 k , . . . , gn k T ∈ R denotes the neuron activation function vector, b b1, . . . , bn T ∈ R means a constant external input vector, A diag{a1, . . . , an} ∈ Rn×n 0 ≤ ai < 1 is the state feedback matrix, Wi ∈ Rn×n i 0, 1 are the connection weight matrices, and h k is interval time-varying delays satisfying 0 < hm ≤ h k ≤ hM, 2.2 where hm and hM are known positive integers. In this paper, it is assumed that the activation functions satisfy the following assumption. Assumption 2.1. The neurons activation functions, gi · , are continuous and bounded, and for any u, v ∈ R, u/ v, k− i ≤ gi u − gi v u − v ≤ k i , i 1, 2, . . . , n, 2.3 where k− i and k i are known constant scalars. As usual, a vector y∗ y∗ 1, . . . , y ∗ n T is said to be an equilibrium point of system 2.1 if it satisfies y∗ Ay∗ W0g y∗ W1g y∗ b. From 10 , under Assumption 2.1, it is not difficult to ensure the existence of equilibrium point of the system 2.1 by using Brouwer’s fixed-point theorem. In the sequel, we will establish a condition to ensure the equilibrium point y∗ of system 2.1 is globally exponentially stable. That is, there exist two constants α > 0 and 0 < β < 1 such that ‖y k − y∗‖ ≤ αβsup−hM≤s≤0‖y s − y∗‖. To confirm this, refer to 16 . For simplicity, in stability analysis of the network 2.1 , the equilibrium point y∗ y∗ 1, . . . , y ∗ n T is shifted to the origin by utilizing the transformation x k y k − y∗, which leads the network 2.1 to the following form: x k 1 Ax k W0f x k W1f x k − h k , 2.4 where x k x1 k , . . . , xn k T ∈ R is the state vector of the transformed network, and f x k f1 x1 k , . . . , fn xn k T ∈ R is the transformed neuron activation function vector with fi xi k gi xi k y∗ i − gi y∗ i and fi 0 0. From Assumption 2.1, it should be noted that the activation functions fi · i 1, . . . , n satisfy the following condition 10 : k− i ≤ fi u − fi v u − v ≤ k i , ∀u, v ∈ R, u / v, 2.5 4 Abstract and Applied Analysis which is equivalent to [ fi u − fi v − k− i u − v ][ fi u − fi v − k i u − v ] ≤ 0, 2.6 and if v 0, then the following inequality holds: [ fi u − k− i u ][ fi u − k i u ] ≤ 0. 2.7 Here, the aim of this paper is to investigate the delay-dependent stability analysis of the network 2.4 with interval time-varying delays. In order to do this, the following definition and lemmas are needed. Definition 2.2 see 16 . The discrete-time neural network 2.4 is said to be globally exponentially stable if there exist two constants α > 0 and 0 ≤ β ≤ 1 such that ‖x k ‖ ≤ αβ sup −hM≤s≤0 ‖x s ‖. 2.8 Lemma 2.3 Jensen inequality 21 . For any constant matrix 0 < M M ∈ Rn×n, integers hm and hM satisfying 1 ≤ hm ≤ hM, and vector function x k ∈ R, the following inequality holds: − hM − hm 1 hM ∑ k hm x k Mx k ≤ − ( hM ∑ k hm x k )T M ( hM ∑ k hm x k ) . 2.9 Lemma 2.4 Finsler’s lemma 22 . Let ζ ∈ R, Φ Φ ∈ Rn×n, and Γ ∈ Rm×n such that rank Γ < n. The following statements are equivalent: i ζΦζ < 0, ∀Γζ 0, ζ / 0, ii Γ⊥ΦΓ⊥ < 0, iii Φ XΥ ΥTXT < 0, ∀X ∈ Rn×m. 3. Main Results In this section, new stability criteria for the network 2.4 will be proposed. For the sake of simplicity on matrix representation, ei ∈ R10n×n i 1, . . . , 10 are defined as block entry matrices e.g., e2 0, I, 0, . . . , 0 } {{ } 8 T . The notations of several matrices are defined as Abstract and Applied Analysis 5 hd hM − hm,and Applied Analysis 5 hd hM − hm, ζ k [ x k , x k − hm , x k − h k , x k − hM ,Δx k ,Δx k − hm , Δx k − hM , f x k , f x k − h k , f x k 1 ]T , χ k [ x k , x k − hm , x k − hM , f x k ]T , ξ k [ x k ,Δx k ]T , Γ A − I , 0, 0, 0,−I, 0, 0,W0,W1, 0 , Π1 e1 e5, e2 e6, e4 e7, e10 , Π2 e1, e2, e4, e8 , Π3 e1, e5 , Π4 e2, e6 , Π5 e4, e7 , Π6 e2 − e3, e3 − e4 , Π7 e1, e8 , Π8 e3, e9 , Π9 e1 e5, e10 , Ξ1 Π1RΠT1 −Π2RΠ2 , Ξ2 Π3NΠ3 Π4 M −N ΠT4 −Π5MΠ5 , Ξ3 e5 ( hmQ1 ) e 5 e5 ( hdQ2 ) e 5 e1 hmP1 e T 1 − e2 hmP1 e 2 hd 3 ∑ i 2 ( eiPie T i − ei 1Pie i 1 ) , Ξ4 − e1 − e2 Q1 P1 e1 − e2 T −Π6 [ Q2 P2 S Q2 P3 ] Π6 , Ξ5 Π3 ( hmQ3 ) Π3 Π3 ( hdQ4 ) Π3 ,


Introduction
Neural networks have received increasing attention of researches from various fields of science and engineering such as moving image reconstructing, signal processing, pattern recognition, and fixed-point computation.In the hardware implementation of systems, there exists naturally time delay due to the finite information processing speed and the finite switching speed of amplifiers.It is well known that time delay often causes undesirable dynamic behaviors such as performance degradation, oscillation, or even instability of the systems.Since it is a prerequisite to ensure stability of neural networks before its application to various fields such as information science and biological systems, the problem of stability of neural networks with time delay has been a challenging issue 1-10 .Also, these days, most systems use digital computers usually microprocessor or microcontrollers with the necessary input/output hardware to implement the systems.The fundamental character of the digital computer is that it takes computed answers at discrete steps.Therefore, discretetime modeling with time delay plays an important role in many fields of science and engineering applications.With this regard, various approaches to delay-dependent stability criteria for discrete-time neural networks with time delay have been investigated in the literature 11-16 .In the field of delay-dependent stability analysis, one of the hot issues attracting the concern of the researchers is to increase the feasible region of stability criteria.The most utilized index to check the conservatism of stability criteria is to get maximum delay bounds for guaranteeing the globally exponential stability of the concerned networks.Thus, many researchers put time and efforts into some new approaches to enhance the feasible region of stability conditions.In this regard, Liu et al. 11 proposed a unified linear matrix inequality approach to establish sufficient conditions for the discrete-time neural networks to be globally exponentially stable by employing a Lyapunov-Krasovskii functional.In 12, 13 , the existence and stability of the periodic solution for discrete-time recurrent neural network with time-varying delays were studied under more general description on activation functions by utilizing free-weighting matrix method.Based on the idea of delay partitioning, a new stability criterion for discrete-time recurrent neural networks with time-varying delays was derived 14 .Recently, some novel delay-dependent sufficient conditions for guaranteeing stability of discrete-time stochastic recurrent neural networks with time-varying delays were presented in 15 by introducing the midpoint of the time delay's variational interval.Very recently, via a new Lyapunov functional, a novel stability criterion for discrete-time recurrent neural networks with time-varying delays was proposed in 16 and its improvement on the feasible region of stability criterion was shown through numerical examples.However, there are rooms for further improvement in delay-dependent stability criteria of discrete-time neural networks with time-varying delays.
Motivated by the above discussions, the problem of new delay-dependent stability criteria for discrete-time neural networks with time-varying delays is considered in this paper.It should be noted that the delay-dependent analysis has been paid more attention than delay-independent one because the sufficient conditions for delay-dependent analysis make use of the information on the size of time delay 17, 18 .That is, the former is generally less conservative than the latter.By construction of a suitable Lyapunov-Krasovskii functional and utilization of reciprocally convex approach 19 , a new stability criterion is derived in Theorem 3.1.Based on the results of Theorem 3.1 and motivated by the work of 20 , a further improved stability criterion will be introduced in Theorem 3.4 by applying zero equalities to the results of Theorem 3.1.Finally, two numerical examples are included to show the effectiveness of the proposed method.
Notation.R n is the n-dimensional Euclidean space, and R m×n denotes the set of all m × n real matrices.For symmetric matrices X and Y , X > Y resp., X ≥ Y means that the matrix X − Y is positive definite resp., nonnegative .X ⊥ denotes a basis for the null space of X.I denotes the identity matrix with appropriate dimensions.• refers to the Euclidean vector norm or the induced matrix norm.diag{• • • } denotes the block diagonal matrix.represents the elements below the main diagonal of a symmetric matrix.

Problem Statements
Consider the following discrete-time neural networks with interval time-varying delays: where n denotes the number of neurons in a neural network, y k y 1 k , . . ., y n k T ∈ R n is the neuron state vector, g k g 1 k , . . ., g n k T ∈ R n denotes the neuron activation function vector, b b 1 , . . ., b n T ∈ R n means a constant external input vector, A diag{a 1 , . . ., a n } ∈ R n×n 0 ≤ a i < 1 is the state feedback matrix, W i ∈ R n×n i 0, 1 are the connection weight matrices, and h k is interval time-varying delays satisfying where h m and h M are known positive integers.
In this paper, it is assumed that the activation functions satisfy the following assumption.
Assumption 2.1.The neurons activation functions, g i • , are continuous and bounded, and for any where k − i and k i are known constant scalars.
As usual, a vector y * y * 1 , . . ., y * n T is said to be an equilibrium point of system 2.1 if it satisfies y * Ay * W 0 g y * W 1 g y * b.From 10 , under Assumption 2.1, it is not difficult to ensure the existence of equilibrium point of the system 2.1 by using Brouwer's fixed-point theorem.In the sequel, we will establish a condition to ensure the equilibrium point y * of system 2.1 is globally exponentially stable.That is, there exist two constants α > 0 and 0 < β < 1 such that y k − y * ≤ αβ k sup −h M ≤s≤0 y s − y * .To confirm this, refer to 16 .For simplicity, in stability analysis of the network 2.1 , the equilibrium point y * y * 1 , . . ., y * n T is shifted to the origin by utilizing the transformation x k y k − y * , which leads the network 2.1 to the following form: where x k x 1 k , . . ., x n k T ∈ R n is the state vector of the transformed network, and i and f i 0 0. From Assumption 2.1, it should be noted that the activation functions f i • i 1, . . ., n satisfy the following condition 10 : which is equivalent to and if v 0, then the following inequality holds: Here, the aim of this paper is to investigate the delay-dependent stability analysis of the network 2.4 with interval time-varying delays.In order to do this, the following definition and lemmas are needed.Definition 2.2 see 16 .The discrete-time neural network 2.4 is said to be globally exponentially stable if there exist two constants α > 0 and 0 ≤ β ≤ 1 such that x s .

2.8
Lemma 2.3 Jensen inequality 21 .For any constant matrix 0 < M M T ∈ R n×n , integers h m and h M satisfying 1 ≤ h m ≤ h M , and vector function x k ∈ R n , the following inequality holds: and Γ ∈ R m×n such that rank Γ < n.The following statements are equivalent:

Main Results
In this section, new stability criteria for the network 2.4 will be proposed.For the sake of simplicity on matrix representation, e i ∈ R 10n×n i 1, . . ., 10 are defined as block entry matrices e.g., e 2 0, I, 0, . . ., 0

3.1
Now, the first main result is given by the following theorem.
3 , any symmetric matrices P i ∈ R n×n i 1, 2, 3 , and any matrix S ∈ R n×n satisfying the following LMIs: where Φ, Θ, and Γ are defined in 3.1 .
Proof.Define the forward difference of x k and V k as

3.5
Let us consider the following Lyapunov-Krasovskii functional candidate as where

3.7
The forward differences of V 1 k and V 2 k are calculated as

3.9
By calculating the forward differences of V 3 k and V 4 k , we get

3.11
For any matrix P , integers l 1 and l 2 satisfying l 1 < l 2 , and a vector function where k is the discrete time, the following equality holds: x T s 1 Px s 1 − x T s Px s .

3.12
It should be noted that Δx T s P Δx s 2x T s P Δx s .

3.13
From the equalities 3.12 and 3.13 , by choosing l 1 , l 2 as 0, h m , h m , h k and h k , h M , the following three zero equations hold with any symmetric matrices P 1 , P 2 , and P 3 : Δx T s P 2 Δx s 2x T s P 2 Δx s , Δx T s P 3 Δx s 2x T s P 3 Δx s .

3.16
By adding three zero equalities into the results of ΔV 3 k , we have

3.17
Abstract and Applied Analysis 9 where

3.19
By Lemma 2.3, when h m < h k < h M , the sum term Σ in 3.18 is bounded as where α k h M − h k /h d .By reciprocally convex approach 19 , if the inequality 3.3 holds, then the following inequality for any matrix S satisfies

It should be pointed out that when h k h m or h k h M , we have
respectively.Thus, the following inequality still holds:

3.24
Here, if the inequalities 3.4 hold, then ΔV 3 ΔV 4 is bounded as

3.26
Therefore, from 3.8 -3.16 and by application of the S-procedure 23 , ΔV has a new upper bound as where Φ and Θ are defined in 3.1 .Also, the system 2.4 with the augmented vector ζ k can be rewritten as where Γ is defined in 3.1 .Then, a delay-dependent stability condition for the system 2.4 is Finally, by utilizing Lemma 2.4, the condition 3.29 is equivalent to the following inequality From the inequality 3.30 , if the LMIs 3.2 -3.4 hold.From ii and iii of Lemma 2.4, if the stability condition 3.29 holds, then for any free maxrix X with appropriate dimension, the condition 3.29 is equivalent to Therefore, from 3.31 , there exists a sufficient small scalar ρ > 0 such that

3.32
By using the similar method of 11, 12 , the system 2.4 is globally exponentially stable for any time-varying delay h m ≤ h k ≤ h M from Definition 2.2.This completes our proof.
Remark 3.2.In Theorem 3.1, the stability condition is derived by utilizing a new augmented vector ζ k including f x k 1 .This state vector f x k 1 which may give more information on dynamic behavior of the system 2.4 has not been utilized as an element of augmented vector ζ k in any other literature.Correspondingly, the state vector f x k 1 is also included in 3.26 .
Remark 3.3.As mentioned in 10 , the activation functions of transformed system 2.4 also satisfy the condition 2.6 .In Theorem 3.4, by choosing u, v in 2.6 as x k , x k − h k and x k − h k , f x k 1 , more information on cross-terms among the states f x k , f x k − h k , f x k 1 , x k , and x k − h k will be utilized, which may lead to less conservative stability criteria.In stability analysis for discrete-time neural networks with time-varying delays, this consideration has not been proposed in any other literature.Through two numerical examples, it will be shown that the newly proposed activation condition may enhance the feasible region of stability criterion by comparing maximum delay bounds with the results obtained by Theorem 3.1.
As mentioned in Remark 3.3, from 2.6 , we add the following new inequality with any positive diagonal matrices H i diag{h i1 , . . ., h in } i 4, 5, 6 to be chosen as 1, 2 , positive diagonal matrices H i ∈ R n×n i 1, . . ., 6 , any symmetric matrices P i ∈ R n×n i 1, 2, 3 , and any matrix S ∈ R n×n satisfying the following LMIs: where Φ, Γ, and Ω are defined in 3.1 and Θ is in 3.33 .
Proof.With the same Lyapunov-Krasovskii functional candidate in 3.6 , by using the similar method in 3.8 -3.16 , and considering inequality 3.36 , the procedure of deriving the condition 3.34 -3.36 is straightforward from the proof of Theorem 3.1, so it is omitted.

Numerical Examples
In this section, we provide two numerical examples to illustrate the effectiveness of the proposed criteria in this paper.
The activation functions satisfy Assumption 2.1 with

4.3
When a 0.9, for different values of h m , maximum delay bounds obtained by 12-14, 16 and our Theorems are listed in Table 2. From Table 2, it can be confirmed that all the results of Theorems 3.1 and 3.4 provide larger delay bounds than those of 12-14 .Also, our results are better than or equal to the results of 16 .For the case of a 0.7, another comparison of our results with those of 15, 16 is conducted in Table 3, which shows all the results obtained by Theorems 3.1 and 3.4 give larger delay bounds than those of 15, 16 .

Conclusions
In this paper, improved delay-dependent stability criteria were proposed for discrete-time neural networks with time-varying delays.In Theorem 3.1, by constructing the suitable Lyapunov-Krasovskii's functional and utilizing some recent results introduced in 19, 20 , the sufficient condition for guaranteeing the global exponential stability of discrete-time neural network having interval time-varying delays has been derived.Based on the results of Theorem 3.1, by constructing new inequalities of activation functions, the further improved stability criterion was presented in Theorem 3.4.Via two numerical examples, the improvement of the proposed stability criteria has been successfully verified.

Example 4 . 1 .
Consider the discrete-time neural networks 2 5 , e 2 e 6 , e 4 e 7 , e 10 , Π 2 e 1 , e 2 , e 4 , e 8 , − e 3 , e 8 − e 9 , Π 11 e 3 − e 1 − e 5 , e 9 − e 10 , and Π 12 e 5 , e 10 − e 8 .We will add this inequality 3.33 in Theorem 3.4.Now, we have the following theorem.For given positive integers h m and h M , diagonal matrices K m diag{k − 1 , . . ., k − n } and K p diag{k 1 , . . ., k n }, the network 2.4 is globally exponentially stable for h m For various h m , the comparison of maximum delay bounds h M obtained by Theorems 3.1 and 3.4 with those of 12, 16 is conducted in Table1.From Table1, it can be confirmed that the results of Theorem 3.1 give a larger delay bound than those of 12 and are equal to the results of 16 .However, the results obtained by Theorem 3.4 are better than the results of 16 and Theorem 3.1, which supports the effectiveness of the proposed idea mentioned in Remark 3.3.

Table 1 :
Maximum bounds h M with different h m Example 4.1 .

Table 2 :
Maximum bounds h M with different h m and a 0.9 Example 4.2 .

Table 3 :
Maximum bounds h M with different h m and a 0.7 Example 4.2 .