New LMI-Based Conditions on Neural Networks of Neutral Type with Discrete Interval Delays and General Activation Functions

and Applied Analysis 3 where yi t is the state of the ith neuron at timet, ci > 0 denotes the passive decay rate, wij1, wij2, aij , and bij are the interconnection matrices representing the weight coefficients of the neurons, fj · , gj · , and vj · are activation functions, and Ii is an external constant input. The delay kj is a real valued continuous nonnegative function defined on 0, ∞ , which is assumed to satisfy ∫∞ 0 kj s ds 1, j 1, . . . , n. For system 2.1 , the following assumptions are given. Assumption 2.1. For i ∈ {1, 2, . . . , n}, the neuron activation functions in 2.1 satisfy l̃− i ≤ fi x1 − fi x2 x1 − x2 ≤ l̃ i , i 1, 2, . . . , n, x1, x2 ∈ , x1 / x2, l̂− i ≤ gi x1 − gi x2 x1 − x2 ≤ l̂ i , i 1, 2, . . . , n, x1, x2 ∈ , x1 / x2, l − i ≤ vi x1 − vi x2 x1 − x2 ≤ l i , i 1, 2, . . . , n, x1, x2 ∈ , x1 / x2, 2.2 where l̃− i , l̃ i , l̂ − i , l̂ i , l − i , and l i are some constants. Assumption 2.2. The time-varying delays τ t and h t satisfy 0 ≤ τ1 ≤ τ t ≤ τ2, τ̇ t ≤ τd < 1, 0 < h t ≤ h, ḣ t ≤ hd < 1, 2.3 where τ1, τ2, τd, h, and hd are constants. Assume y∗ y∗ 1, y ∗ 2, . . . , y ∗ n T is an equilibrium point of 2.1 . Through xi yi − y∗ i , system 2.1 can be transformed into the following system: ẋ t −Cx t W1f x t W2g x t − τ t A ∫ t −∞ K t − s v x s ds Bẋ t − h t , 2.4 where x t x1 t , . . . , xn t T ∈ n is the neural state vector, f x t f1 x1 t , . . . , fn xn t T ∈ n is the neuron activation function vector with f 0 0, g x t g1 x1 t , . . . , gn xn t T ∈ n is the neuron activation function vector with g 0 0, v x t v1 x1 t , . . . , vn xn t T ∈ n is the neuron activation function vector with v 0 0. C diag{c1, . . . , cn} > 0, and W1 ∈ n×n, W2 ∈ n×n, A ∈ n×n, and B ∈ n×n are the connection weight matrices. 4 Abstract and Applied Analysis Note that since functions fi · , gi · , and vi · satisfy Assumption 2.1, fi · , gi · , and vi · also satisfy l̃− i ≤ fi x1 − fi x2 x1 − x2 ≤ l̃ i , i 1, 2, . . . , n, x1, x2 ∈ , x1 / x2, l̂− i ≤ gi x1 − gi x2 x1 − x2 ≤ l̂ i , i 1, 2, . . . , n, x1, x2 ∈ , x1 / x2, l − i ≤ vi x1 − vi x2 x1 − x2 ≤ l i , i 1, 2, . . . , n, x1, x2 ∈ , x1 / x2, 2.5 where l̃− i , l̃ i , l̂ − i , l̂ i , l − i , and l i are some constants. 3. Stability Analysis In order to obtain the main results of stability analysis, the following lemma is introduced. Lemma 3.1. For any constant matrix M > 0, any scalars a and b such that a < b, and a vector function x t : a, b → n such that the integrals concerned are well defined, the following holds: