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 ,