Dynamical Analysis for High-Order Delayed Hopfield Neural Networks with Impulses

and Applied Analysis 3 the synaptic connection weight of the unit j on the unit i at time t; constant bij denotes the synaptic connection weight of the unit j on the unit i at time t − τ t ; Ii is the input of the unit i; τ t is the transmission delay such that 0 < τ t ≤ τ and τ̇ t ≤ ρ < 1, t ≥ t0; τ , ρ are constants. The initial conditions associated with system 2.1 are of the form x s φ s , s ∈ t0 − τ, t0 , 2.2 where x s x1 s , x2 s , . . . , xn s T , φ s φ1 s , φ2 s , . . . , φn s T ∈ PC −τ, 0 ,R , PC −τ, 0 ,R {ψ : −τ, 0 → R is continuous everywhere except at finite number of points tk, at which ψ t k and ψ t − k exist and ψ t k ψ tk }. For ψ ∈ PC −τ, 0 ,R , the norm of ψ is defined by ||ψ||τ sup−τ≤θ≤0|ψ θ |. For any t0 ≥ 0, let PCδ t0 {ψ ∈ PC −τ, 0 ,R : ||ψ|| < δ}. Assume that x∗ x∗ 1, x ∗ 2, . . . , x ∗ n T is an equilibrium point of system 2.1 . Impulsive operator is viewed as perturbation of the equilibrium point x∗ of such system without impulsive effects. We assume that Δxi|t tk xi tk − xi ( tk ) d i k ( xi ( tk ) − x∗ i ) , d i k ∈ R, i ∈ Λ, k ∈ Z . 2.3 Since x∗ is an equilibrium point of system 2.1 , one can derive from system 2.1 2.2 that the transformation yi xi − x∗ i , i ∈ Λ transforms such system into the following system for more details, please see papers 12, 13 : y′ t − Cy t AFy t ) BGy t − τ t ) ΓT G ( y t − τ t , t / tk, t ≥ t0, y tk Dky ( tk ) , k ∈ Z , y t0 θ φ θ , θ ∈ −τ, 0 , 2.4 where φ θ x t0 θ − x∗, y t ( y1 t , y2 t , . . . , yn t )T , y t − τ t y1 t − τ t , y2 t − τ t , . . . , yn t − τ t )T , F ( y t ) [ F1 ( y1 t ) , F2 ( y2 t ) , . . . , Fn ( yn t )]T , G ( y t − τ t ) G1 ( y1 t − τ t ) , G2 ( y2 t − τ t ) , . . . , Gn ( yn t − τ t )]T , Fj ( yj t ) fj ( x∗ j yj t ) − fj ( x∗ j ) , Gj ( yj t − τ t ) gj ( x∗ j yj t − τ t ) − gj ( x∗ j ) , C diag c1, c2, . . . , cn , A ( aij ) n×n, B ( bij ) n×n, Ti ( Tijl ) n×n, T ( T1 T 1 , T2 T T 2 , . . . , Tn T T n )T , 4 Abstract and Applied Analysis Γ diag ς, ς, . . . , ς , ς ς1, ς2, . . . , ςn T , Dk diag [ 1 d 1 k , 1 d 2 k , . . . , 1 d n k ] , 2.5 in which ςl is a real value between gl xl t − τ t and gl x∗ l , l ∈ Λ. Remark 2.1. Obviously, 0, 0, . . . , 0 T is an equilibrium point of 2.4 . Therefore, there exists at least one equilibrium point of system 2.1 . So, the stability analysis of the equilibrium point x∗ of 2.1 can now be transformed to the stability analysis of the trivial solution y 0 of 2.4 . In the following, the notations X and X−1 mean the transpose of and the inverse of a square matrix X. We will use the notation X > 0 or X < 0, X ≥ 0, X ≤ 0 to denote that the matrix X is a symmetric and positive definite negative definite, positive semidefinite, negative semidefinite matrix. Let λmax X , λmin X , respectively, denote the largest and smallest eigenvalue of matrix X. Throughout this paper, we assume that there exist constants χi > 0,M,N ≥ 0 such that |gi xi | ≤ χi, i ∈ Λ, F y F y ≤ Myy, G y G y ≤ Nyy. We introduce some definitions as follows. Definition 2.2 see 5 . Leting V : R × R → R , for any t, x ∈ tk−1, tk × R, the upper right-hand Dini derivative of V t, x along the solution of 2.4 is defined by D V t, x lim sup h→ 0 1 h { V [ t h, x h ( − Cy t AFy t ) BGy t − τ t ) ΓT G ( y t − τ t ) )] − V t, x } . 2.6 Definition 2.3 see 25 . Assume y t y t0, φ t is the solution of 2.4 through t0, φ . Then the zero solution of 2.4 is said to be uniformly stable, if, for any ε > 0 and t0 ≥ 0, there exists some δ δ ε > 0 such that φ ∈ PCδ t0 implies ||y t || < ε, t ≥ t0. Definition 2.4 see 5 . The equilibrium point x∗ of the system 2.1 is globally exponentially stable, if there exists constant μ > 0,M ≥ 1 such that, for any initial value φ, ∥x ( t0, φ ) t − x∗∥ < M∥φ − x∗∥τe−μ t−t0 , t ≥ t0. 2.7 Next, in order to obtain our results, we need to establish the following lemma. Lemma 2.5 see 13 . For any vectors a, b ∈ R, the inequality ±2aTb ≤ aXa bTX−1b 2.8 holds, in which X is any n × n matrix with X > 0. Abstract and Applied Analysis 5 Lemma 2.6 see 31 . Let X ∈ Rn×n, then λmin X aa ≤ aXa ≤ λmax X aa 2.9and Applied Analysis 5 Lemma 2.6 see 31 . Let X ∈ Rn×n, then λmin X aa ≤ aXa ≤ λmax X aa 2.9 for any a ∈ R if X is a symmetric matrix. 3. Main Results In this section, some sufficient delay-dependent conditions of global exponential stability and uniform stability for system 2.1 are obtained. Theorem 3.1. Assume that there exist constants ε > 0, δ ∈ 0, ε and n×n symmetric and positive definite matrices P , Q1, Q2 such that i ε P − PC − CP PAQ−1 1 AP λmax Q1 ME Nλmax ( Q2 T T ) 1 − ρ E e PBQ−1 2 B P e ∥χ ∥2P 2 ≤ 0, 3.1 where χ χ1, χ2, . . . , χn T , ii there exists constant W ≥ 0 such that