Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield Neural Networks with Unbounded Distributed Delays and Impulses R. Raja1, R.Sakthivel 2 and S.Marshal Anthoni 3 1Department of Mathematics, Periyar University, Salem-636 011, India 2Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea 3Department of Mathematics, Anna University of Technology, Coimbatore-641 047, India Email: antony.raja67@yahoo.com Abstract This paper is concerned with the stability analysis problem for a class of delayed stochastic uncertain Hopfield neural networks with unbounded distributed delays and impulses. A new Lyapunov-Krasovskii functional is constructed for the addressed system and several free- weighting matrices combined with the S-procedure are employed to derive the delay-dependent stability criterion. The criterion is derived and formulated in terms of linear matrix inequality (LMI). In addition to that, two illustrated examples with simulation results are given to show the effectiveness of the obtained theoretical results. Keywords Delay-dependent Stochastic Hopfield neural networks Distributed delays Lyapunov Krasovskii functional Linear matrix inequality Impulses. 1. Introduction During the past several years, the stability of a unique equilibrium point of Hopfield neural networks [2] with delays have received especially considerable attention due to their exten- sive applications in solving optimization problem, traveling salesman problem and many other subjects in recent years [4, 5, 6, 14, 15, 16, 17, 18, 23, 24]. Basically, the stability results of delayed Hopfield neural networks can be classified into two categories: delay dependent stability and delay independent stability. Delay-dependent stability results are generally less conservative than delay-independent stability when the delays are small. On the other hand, time-delays occurring in the interaction between neurons will affect the stability of a network by creating instability, oscillation and chaos phenomena. Recently, a number of global stability criteria of Hopfield networks with time-delays have been proposed (see [13, 15, 16, 17, 18]). The dynamical systems are often classified into two categories of either continuous-time or discrete-time systems. Apart from this two systems, yet there is a somewhat new category of dynamical systems, which is neither continuous-time nor purely discrete-time, these are called dynamical systems with impulses. A basic theory of impulsive differential equa- tions has been developed in [11]. The stability conditions in [19, 20, 22] were established by using the impulsive condition. In the real world, there are two common disturbances that affects the ISSN 1078-6236 International Institute for General Systems Studies, Inc. Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 93-109 network process, one is stochastic perturbations and the other one is uncertain parameters. Re- cently, there are some research papers about stochastic neural networks, has been investigated, see for example [3, 5, 6, 7, 9, 10, 16, 17, 18]. In [12] the stability problem for both discrete and dis- tributed delays were discussed. Yang [21] has been investigated the stability of neural networks with distributed delays. In practical, uncertainties often exist in most engineering and communication systems and may cause undesirable dynamic network behaviors such as oscillation, instability and chaos. More specifically, the connection weights of the neurons are inherent dependent on certain re- sistance and capacitance values that inevitably bring in uncertainties during the parameter iden- tification process. In the literature, uncertainties can possibly be described by norm bounded, polytopic or linear fractional uncertainties characterizations and have been widely employed in the field of robust control and performance analysis. Based on the above descriptions, this paper aims to develop the problem of asymptotic sta- bility for delayed stochastic uncertain Hopfield neural networks with unbounded distributed de- lays and impulses. By constructing an appropriate Lyapunov-Krasovskii functional, employing several free-weighting matrices and S-procedure, we obtain a delay-dependent stability crite- rion in terms of LMIs. Finally, two numerical examples with simulation results are provided to demonstrate the usefulness of the main results in this paper. 2. Network Model and Preliminaries The delayed stochastic Hopfield neural network model with unbounded distributed delays and impulses is defined by the following state equations: dxi(t) = [ − aixi(t) + n∑ j=1 bijfj(xj(t− τ)) + n∑ j=1 cij ∫ t −∞ kj(t− s)fj(xj(s))ds+ Ji ] dt + n∑ j=1 σij(t, xj(t))dwj(t), t 6= tk (1) xi(tk) = Ikx(t−k ), t = tk, k = 1, 2, ... where xi(t) is the state of the ith neuron at time t; ai > 0 denotes the passive decay rate; bij and cij are the synaptic connection strengths; fj denotes the neuron activation functions; Ji is the constant input from outside the system; τ represents the continuous delay and the delay kernel kj is a real valued continuous function defined on [0,+∞] and satisfies, for each i, ∫∞ 0 kj(s)ds = 1. The stochastic disturbance w(t) = (w1(t), w2(t), ..., wm(t))T is an m- dimensional Brownian motion; σij(·, ·) is locally Lipschitz continuous and satisfies the linear growth condition as well; xi(tk) = Ikx(t−k ) is the impulse at moment tk, the fixed moment of time tk satisfy t1 < t2 <, ..., limk→+∞ tk = +∞ and x(t−) = lims→t− x(s); Ik is a constant real matrix at the moments of time tk. Let PC([−τ, 0],Rn) denotes the set of piecewise right continuous functions φ : [−τ, 0]→ Rn with the sup-norm |φ| = sup−τ≤s≤0‖φ(s)‖. For given t0, and φ ∈ (PC[−τ, 0],Rn), the initial condition of system (1) is described as x(t0 + t) = φ(t), for t ∈ [−τ, 0], φ ∈ 94 Raja: Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield . . . . . . PC([−τ, 0],Rn). The following assumptions is utilized throughout this paper: (C1) The activation function f(x) is boundless and satisfies 0 ≤ fi(ξ1)− fi(ξ2) ξ1 − ξ2 ≤ li, for any ξ1, ξ2 ∈ R, ξ1 6= ξ2, i = 1, 2, ..., n (C2) The function fi(xi(·)) = 0, i = 1, 2, ..., n satisfy 0 ≤ fi(xi(t)) xi(t) ≤ li, fi(0) = 0, ∀xi(t) 6= 0, i = 1, 2, ..., n where li, i = 1, 2, ..., n are positive constants. Remark 2.1 The above conditions ensures that the nonlinear resulting neuron activation func- tions should be non-monotonic and be more general than the usual sigmoid function as well as the commonly used Lipschitz condition. The equilibrium point y∗ = [y∗1, y ∗ 2, ..., y ∗ n] of system (1) will be shifted to the origin by the transformation y(·) = x(·)− x∗, transforms system (1) into the following form dy(t) = [−Ay(t) +Bg(y(t− τ)) + C ∫ t −∞ K(t− s)g(y(s))ds]dt+ σ(t, y(t))dw(t), t 6= tk (2) y(tk) = Iky(t−k ), t = tk, k = 1, 2, ... y(t0 + t) = ψ(t), t ∈ [−τ, 0] where y = [y1, y2, ..., yn]T , A = diag[a1, a2, ..., an], B = [bij ], C = [cij ], K(t − s) = diag[k1(t − s), k2(t − s), ..., kn(t − s)], g(y) = [g1(y1), g2(y2), ..., gn(yn)] with gj(yj(t)) = fj(yj(t) + x∗j )− fj(x∗j ). Note that since each function fj(·) satisfies the assumptions (C1) and (C2), hence each gj(·) satisfies g2j (ξj) ≤ L2 jξ 2, ξjgj(ξj) ≥ g2j (ξj) Lj ∀ξj ∈ R, gj(0) = 0 (C3) There exist a constant matrix D0 such that trace[σT (t, y(t))σ(t, y(t))] ≤ yT (t) D0 y(t) Lemma 2.2 [1] (S-Procedure) Let Ti ∈ Rn×n (i = 0, 1, ..., p) be symmetric matrices. The conditions on Ti, (i = 0, 1, ..., p) αTT0α > 0, ∀α 6= 0 s.t. αTTiα ≥ 0 (i = 0, 1, ..., p) hold, if there exist τi ≥ 0 (i = 0, 1, ..., p) such that T0 − p∑ i=1 τiTi > 0 Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 95 Lemma 2.3 Let U, V,W andM be real matrices of appropriate dimensions withM satisfying M = MT , then M + UVW +W TV TUT < 0, ∀ V TV ≤ I if and only if there exist a scalar ε > 0 such that M + ε−1UUT + εW TW < 0 Definition 2.4 [25] The function V : [t0,∞)× Rn → R+ belongs to class v0 if (1) the function V is continuous on each of the sets [tk−1, tk) × Rn and for all t ≥ t0, V (0, t) ≡ 0; (2) V (x, t) is locally Lipschitzian in x ∈ Rn; (3) for each k = 1, 2, ..., there exist finite limits lim (q,t)→(x,t−k ) V (q, t) = V (x, t−k ) lim (q,t)→(x,t+k ) V (q, t) = V (x, t+k ) with V (x, t+k ) = V (x, tk) satisfied. In the following section, we will develop delay-dependent condition for the given system such that the origin of the delayed stochastic Hopfield neural network (2) is asymptotically stable. 3. Asymptotic Stability Criterion Before discussing the stability analysis of the problem, we firstly introduce the Ito’s for- mula for a general stochastic system. Let V (y(t), t) : C([−τ, 0],Rn × R+ → R+) be a positive function which is continuously twice differentiable in y and once differentiable in t. Thus, an operator L acting on V (y(t), t), is defined by LV (y(t), t) = Vt(y(t), t) + Vy(y(t), t)[−Ay(t) +Bg(y(t− τ)) + C ∫ t −∞ K(t− s)g(y(s))ds] + 1 2 trace[σT (t, y(t))Vyy(y(t), t)σ(t, y(t))] (3) where Vt(y(t), t) = ∂V (y(t), t) ∂t , Vy(y(t), t) = (∂V (y(t), t) ∂y1 , ∂V (y(t), t) ∂y2 , ..., ∂V (y(t), t) ∂yn ) Vyy(y(t), t) = (∂2V (y(t), t) ∂yi∂yj ) n×n Now, the following theorem gives a new stability criterion for system (2) without uncertain parameters. 96 Raja: Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield . . . . . . Theorem 3.1 Suppose that the assumption (C1)− (C3) is satisfied. If there exist matrices P = [ P11 P12 P T12 P22 ] ≥ 0 with P11 > 0, R = [ R11 R12 RT12 R22 ] ≥ 0 Q1 > 0, Q2 > 0, H1, H2, H3, H4, diagonal matrices E > 0, S > 0 and a positive scalar ρ > 0 such that the following inequalities hold: P < ρI (4) ITk P11Ik + 2ITk P12Ik + ITk P22Ik − P11 − 2P12 − P22 < 0 (5) Ω =  Ξ11 Ξ12 Ξ13 Ξ14 −τATP T12 + τP T22 −τH1 −τATR22 ∗ Ξ22 Ξ23 −HT 4 −τP T22 −τH2 0 ∗ ∗ Ξ33 0 τBTP12 −τH3 −τBTR22 ∗ ∗ ∗ −E τCTP12 −τH4 −τCTR22 ∗ ∗ ∗ ∗ −τR11 −τR12 0 ∗ ∗ ∗ ∗ ∗ −τR22 0 ∗ ∗ ∗ ∗ ∗ ∗ −τR22  < 0 (6) where Ξ11 = −P11A−ATP T11 + P12 + P T12 + ρD0 +Q1 + LEL+ τR11 +H1 +HT 1 − τR12A Ξ12 = P12 −H1 +HT 2 , Ξ13 = P11B +HT 3 + τR12B Ξ14 = P11C +HT 4 + τR12C, Ξ22 = −Q1 −H2 −HT 2 , Ξ23 = LS −HT 3 , Ξ33 = −Q2 − 2S, L = {l1, l2, ..., ln}. Then the origin of system (2) is the unique equilibrium point and it is globally asymptotically stable. Proof. Define new state variables g1(t) = −Ay(t) +Bg(y(t− τ)) + C ∫ t −∞ K(t− s)g(y(s))ds (7) g2(t) = σ(t, y(t)) (8) To prove the asymptotic stability result, let us consider the following Lyapunov functional can- didate for system (2) as V1 = δT1 (t)Pδ1(t), V2 = ∫ t t−τ yT (s)Q1y(s)ds, V3 = ∫ t t−τ gT (y(s))Q2g(y(s))ds V4 = ∫ 0 −τ ∫ t t−τ δT2 (s)Rδ2(s)dsdσ, V5 = n∑ j=1 ej ∫ ∞ 0 kj(ξ) ∫ t t−ξ g2j (yj(γ))dγdξ (9) Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 97 where P = [ P11 P12 P T12 P22 ] ≥ 0 with P11 > 0, R = [ R11 R12 RT12 R22 ] ≥ 0 and δ1(t) = [ y(t)∫ t t−τ y(s)ds ] , δ2(t) = [ y(t) g1(t) ] , By Newton-Leibnitz formula, the following equation is true for any matrices Hi (i = 1, 2, 3, 4) with appropriate dimensions: 2 [ yT (t)H1 + yT (t− τ)H2 + gT (y(t− τ))H3 + (∫ t −∞ K(t− s)g(y(s))ds ) H4 ] × [ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] = 0 (10) when t 6= tk the derivative of V can be calculated by using Ito’s differential formula. Then the trajectories of the system (2) is given as: LV1 = 2δT1 (t)P δ̇1(t) = 2  y(t)∫ t t−τ y(s)ds T P11 P12 P T12 P22 −Ay(t) +Bg(y(t− τ)) + C ∫ t −∞K(t− s)g(y(s))ds y(t)− y(t− τ)  = −2yT (t)P11Ay(t) + 2yT (t)P11Bg(y(t− τ)) + 2yT (t)P11C (∫ t −∞ K(t− s)g(y(s))ds ) + 2yT (t)P12y(t)− 2yT (t)P12y(t− τ)− 2 ∫ t t−τ yT (s)P T12Ay(t)ds+ ∫ t t−τ yT (s)P T12B × g(y(t− τ))ds+ 2 ∫ t t−τ yT (s)P22y(t)ds− 2 ∫ t t−τ yT (s)P22y(t− τ)ds+ 2 (∫ t t−τ yT (s)ds ) × P T12C (∫ t −∞ K(t− s)g(y(s))ds ) + trace[σT (t, y(t))Pσ(t, y(t))] (11) LV2 = yT (t)Q1y(t)− yT (t− τ)Q1y(t− τ) (12) LV3 = gT (y(t))Q2g(y(t))− gT (y(t− τ))Q2g(y(t− τ)) (13) LV4 = τδT2 (t)Rδ2(t)− ∫ t t−τ δT2 (s)Rδ2(s)ds (14) LV5 = n∑ j=1 ej ∫ ∞ 0 kj(ξ)g 2 j (yj(t))dξ − n∑ j=1 ej ∫ ∞ 0 kj(ξ)g 2 j (yj(t− ξ))dξ = gT (y(t))Eg(y(t))− n∑ j=1 ej ∫ ∞ 0 kj(ξ)dξ ∫ ∞ 0 kj(ξ)g 2 j (yj(t− ξ))dξ ≤ yT (t)LELy(t)− n∑ j=1 ej (∫ ∞ 0 kj(ξ)g 2 j (yj(t− ξ))dξ )2 = yT (t)LELy(t)− (∫ t −∞ K(t− s)g(y(s))ds ) E (∫ t −∞ K(t− s)g(y(s))ds ) (15) 98 Raja: Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield . . . . . . It is noted from (C2) that, gi(yi(t− τ))[gi(yi(t− τ))− liyi(t− τ)] ≤ 0, i = 1, 2, ..., n (16) Now, by applying the S-procedure, we find that system (2) is asymptotically stable, if there exist S = diag{s1, s2, ..., sn} such that LV = LV1 + LV2 + LV3 + LV4 + LV5 + 2 [ yT (t)H1 + yT (t− τ)H2 + gT (y(t− τ))H3 + (∫ t −∞ K(t− s)g(y(s))ds ) H4 ][ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] ≤ LV1 + LV2 + LV3 + LV4 + LV5 + 2 [ yT (t)H1 + yT (t− τ)H2 + gT (y(t− τ))H3 + (∫ t −∞ K(t− s)g(y(s))ds ) H4 ][ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] −2 n∑ i=1 sigi(yi(t− τ))(gi(yi(t− τ))− liyi(t− τ)) ≤ LV1 + LV2 + LV3 + LV4 + LV5 + 2 [ yT (t)H1 + yT (t− τ)H2 + gT (y(t− τ))H3 + (∫ t −∞ K(t− s)g(y(s))ds ) H4 ][ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] −2gT (y(t− τ(t)))Sg(y(t− τ(t))) + 2gT (y(t− τ(t)))Sg(y(t− τ(t))) ≤ yT (t)[−P11A−ATP T11 + P12 + P T12 + ρD0 +Q1 + LEL+ τR11 +H1 +HT 1 − τR12A] ×y(t) + yT (t)[P12 −H1 +HT 2 ]y(t− τ(t)) + yT (t)[P11B +HT 3 + τR12B]g(y(t− τ)) +yT (t)[P11C +HT 4 + τR12C] (∫ t −∞ K(t− s)g(y(s))ds ) + yT (t)[−τATP T12B + τP T22 × (∫ t t−τ y(s)ds ) + yT (t)[−τH1] (∫ t t−τ g1(s)ds ) + yT (t− τ)[−Q1 −H2 −HT 2 ] ×y(t− τ) + yT (t− τ)[−HT 3 + LS]g(y(t− τ)) + yT (t− τ)[−τP T22] (∫ t t−τ y(s)ds ) +yT (t− τ)[−τH2] (∫ t t−τ g1(s)ds ) + gT (y(t− τ))[−Q2 − 2S]g(y(t− τ)) + yT (t− τ)) ×[−HT 4 ] (∫ t −∞ K(t− s)g(y(s))ds ) + gT (y(t− τ)[τBTP12] (∫ t t−τ y(s)ds ) +gT (y(t− τ)[−τH3] (∫ t t−τ g1(s)ds ) + (∫ t −∞ K(t− s)g(y(s))ds )T [−E] Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 99 × (∫ t −∞ K(t− s)g(y(s))ds ) + (∫ t −∞ K(t− s)g(y(s))ds )T [τCTP12] (∫ t t−τ y(s)ds ) + (∫ t −∞ K(t− s)g(y(s))ds )T [−τH4] (∫ t t−τ g1(s)ds ) + (∫ t t−τ y(s)ds )T [−τR11] × (∫ t t−τ y(s)ds ) + (∫ t t−τ y(s)ds )T [−τR12] (∫ t t−τ g1(s)ds ) + (∫ t t−τ g1(s)ds )T [−τR22] × (∫ t t−τ g1(s)ds ) (17) It is easy to see y(t)− y(t− τ)− ∫ t t−τ g1(s)ds = ∫ t t−τ(t) g1(s)ds = 1 τ ∫ t t−τ(t) τ(τ−1τ(t)g1(s))ds (18) This together with (16), implies LV = 1 τ ∫ t t−τ(t) βT (t, s)Ωβ(t, s)ds where βT (t, s) = [yT (t) yT (t− τ) gT (y(t− τ)) (∫ t −∞ K(t− s)g(y(s))ds )T yT (s) gT1 (s)] This implies that LV (y(t), t) < 0. When t = tk, we obtain the following result: V (y(tk), (tk))− V (y(tk), t − k ) = δT1 (tk) P11 P12 P T12 P22  δ1(tk)− δT1 (t−k ) P11 P12 P T12 P22  δ1(t−k ) = yT (t−k ) { ITk P11 P12 P T12 P22  Ik − P11 P12 P T12 P22 }y(t−k ) = yT (t−k )ITk P11Iky(t−k ) + yT (t−k )ITk P12Iky(t−k ) + yT (t−k )ITk P T 12Iky(t−k ) + yT (t−k )ITk P22Iky(t−k )− yT (t−k )P11y(t−k )− yT (t−k )P12y(t−k ) − yT (t−k )P T12y(t−k )− yT (t−k )P22y(t−k ) = yT (t−k ) [ ITk P11Ik + 2ITk P12Ik + ITk P22Ik − P11 − 2P12 − P22 ] y(t−k ) Based on the Lyapunov stability theorem, it follows that the delayed stochastic Hopfield neural network (2) is globally asymptotically stable in the mean square. The proof of the theorem is completed. � 100 Raja: Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield . . . . . . 4. Robust Asymptotic Stability Criterion Consider the system (2) with norm bounded parameter uncertainties that is dy(t) = [−(A+ ∆A(t))y(t) + (B + ∆B(t))g(y(t− τ)) + (C + ∆C(t)) ∫ t −∞ K(t− s) g(y(s))ds]dt+σ(t, y(t))dw(t), t 6= tk (19) y(tk) = Iky(t−k ), t = tk, k = 1, 2, ... where A+ ∆A(t), B + ∆B(t) and C + ∆C(t) are of the following structure: [∆A(t) ∆B(t) ∆C(t)] = MF (t)[N1 N2 N3] where M,N1, N2, N3 are known constant matrices with appropriate dimensions and bounded which satisfies F T (t)F (t) ≤ I, t ≥ 0 Theorem 3.2 Suppose that the assumption (C1)− (C3) is satisfied. If there exist matrices P = [ P11 P12 P T12 P22 ] ≥ 0 with P11 > 0, R = [ R11 R12 RT12 R22 ] ≥ 0 Q1 > 0, Q2 > 0, H1, H2, H3, H4, diagonal matrices E > 0, S > 0 and four positive scalars ρ > 0, ε1 > 0, ε2 > 0, ε3 > 0 such that the following inequalities hold: P < ρI (20) ITk P11Ik + 2ITk P12Ik + ITk P22Ik − P11 − 2P12 − P22 < 0 (21) Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 101 Ω1 = Θ11 Θ12 Θ13 Θ14 −τATP T12 + τP T22 −τH1 P11M 0 −τATR22 0 ∗ Ξ22 Ξ23 −HT 4 −τP T22 −τH2 0 0 0 0 ∗ ∗ Θ33 0 τBTP12 −τH3 0 0 −τBTR22 0 ∗ ∗ ∗ Θ44 τCTP12 −τH4 0 0 −τCTR22 0 ∗ ∗ ∗ ∗ −τR11 −τR12 0 τR12M 0 0 ∗ ∗ ∗ ∗ ∗ −τR22 P22M 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ −ε1I 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε2I 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −τR22 τR22M ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε3I  < 0 (22) where Θ11 = −P11A−ATP T11 + P12 + P T12 + ρD0 +Q1 + LEL+ τR11 +H1 +HT 1 − τR12A + ε1N T 1 N1 + τε2N T 1 N1 + τε3N T 1 N1 Θ13 = P11B +HT 3 + τR12B − ε1NT 1 N2 − τε2NT 1 N2 − τε3NT 1 N2, Θ14 = P11C +HT 4 + τR12C − ε1NT 1 N3 − τε2NT 1 N3 − τε3NT 1 N3, Θ33 = −Q2 − 2S + ε1N T 2 N2 + τε2N T 2 N2 + τε3N T 2 N2, Θ44 = −E + ε1N T 3 N3 + τε2N T 3 N3 + τε3N T 3 N3 and Ξ22,Ξ23, L are stated as in Theorem 3.1. Then the origin of system (2) is the unique equilibrium point and it is globally asymptotically stable. Proof. In order to prove the robust asymptotic stability, we use the same Lyapunov-Krasovskii functional as defined in (9). By replacing A,B and C in (2) with A+ ∆A(t), B + ∆B(t) and C + ∆C(t), respectively and by Ito’s differential formula, we can calculate the trajectories of 102 Raja: Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield . . . . . . the system (19), then we have Ω1 = Ω + ε−11  P11M 0 0 0 0 P22M  × [ MTP11 0 0 0 0 MTP22 ] + ε1  −N1 0 N2 N3 0 0  × [ −NT 1 0 NT 2 NT 3 0 0 ] +τε−12  0 0 0 0 R12M 0  × [ 0 0 0 0 MTR12 0 ] + τε2  −N1 0 N2 N3 0 0  × [ −NT 1 0 NT 2 NT 3 0 0 ] +τε−13  0 0 0 0 0 R22M  × [ 0 0 0 0 0 MTR22 ] + τε3  −N1 0 N2 N3 0 0  × [ −NT 1 0 NT 2 NT 3 0 0 ] Therefore, under condition (20)-(22), system (19) is robustly globally asymptotically stable with respect to the uncertain parameters ∆A(t), ∆B(t) and ∆C(t). This completes the proof of the theorem. � Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 103 If we neglect the impulsive term and stochastic perturbations in (2), then it reduces to ẏ(t) = −Ay(t) +Bg(y(t− τ)) + C ∫ t −∞ K(t− s)g(y(s))ds (23) Corollary 3.3 Suppose that the assumption (C1)− (C3) is satisfied. If there exist matrices P = [ P11 P12 P T12 P22 ] ≥ 0 with L11 > 0, R = [ R11 R12 RT12 R22 ] ≥ 0 Q1 > 0, Q2 > 0, H1, H2, H3, H4 and diagonal matrices E > 0, S > 0 such that the following inequalities hold: Ω =  Ξ11 Ξ12 Ξ13 Ξ14 −τATP T12 + τP T22 −τH1 −τATR22 ∗ Ξ22 Ξ23 −HT 4 −τP T22 −τH2 0 ∗ ∗ Ξ33 0 τBTP12 −τH3 −τBTR22 ∗ ∗ ∗ −E τCTP12 −τH4 −τCTR22 ∗ ∗ ∗ ∗ −τR11 −τR12 0 ∗ ∗ ∗ ∗ ∗ −τR22 0 ∗ ∗ ∗ ∗ ∗ ∗ −τR22  < 0 (24) where Ξ11 = −P11A−ATP T11 + P12 + P T12 +Q1 + LEL+ τR11 +H1 +HT 1 − τR12A Ξ12 = P12 −H1 +HT 2 , Ξ13 = P11B +HT 3 + τR12B Ξ14 = P11C +HT 4 + τR12C, Ξ22 = −Q1 −H2 −HT 2 , Ξ23 = LS −HT 3 , Ξ33 = −Q2 − 2S, L = diag{l1, l2, ..., ln}. Then the origin of system (2) is the unique equilibrium point and it is globally asymptotically stable. Proof. By arguing similar to the proof of Theorem 1, we can show that the equilibrium point of system (2) is globally asymptotically stable in the mean square. This completes the proof of the theorem. � Remark 3.4 The authors Chen and Cao, [2] discussed the global asymptotic stability of de- layed Hopfield neural networks. Wan et al. investigated the mean square exponential stability of stochastic delayed Hopfield neural networks. In [18], Wang et al. proposed the robust stability for stochastic Hopfield neural networks with time delays and Zhang et al., [24, 26] obtained the global stability results for delayed Hopfield neural network. As a result, in all the above men- tioned references, impulsive effect has not been taken into account. However, in our paper, we 104 Raja: Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield . . . . . . derived the delay-dependent stability results for stochastic Hopfield neural networks with im- pulsive effects. Therefore, our main result is new, quite effective and leads to less conservative results when compared with some existing works [2, 8, 19, 20]. Remark 3.5 It is noteworthy that in our paper, we employed several free weighting matrices and S-procedure to derive the delay-dependent stability criterion. The derived criterion is ob- tained in LMI forms whose feasibility can be readily checked by using the Matlab LMI toolbox. Different from the conventional stability criteria that depend on the M-matrix computation, no tuning of parameters will be needed when employing our LMI-based stability criteria. More- over, two numerical examples with simulation results will show the effectiveness of the stability conditions in this paper. 5. Illustrated Examples In this section, we provide two numerical examples to demonstrate the effectiveness of the main results presented in this paper. Example 4.1 Consider a delayed stochastic Hopfield neural network (2) with parameters as: A = [ 1.7679 0 0 1.8860 ] , B = [ −0.2376 −0.4769 −0.6707 −0.7654 ] , C = [ −0.1052 −0.5069 −0.0257 −0.2808 ] , L = [ 0.5219 0 0 1.8993 ] , D0 = [ 0.33 0 0 0.25 ] , Ik = I = [ 0.2 0 0 0.2 ] It can be checked that system (2) satisfies the assumptions (C1) − (C3). For the delay bound τ = 0.957, we have obtained the following feasible solutions to the LMIs (4) - (6) in Theorem 1 P11 = [ 25.7039 −1.6632 −1.6632 39.0949 ] , P12 = [ 4.3286 −0.9483 −0.9483 3.8977 ] , P22 = [ 8.8576 −1.6157 −1.6157 7.3501 ] Q1 = [ 18.9787 −6.6658 −6.6658 16.4166 ] , Q2 = [ 29.9330 20.6811 20.6811 51.6907 ] , R11 = [ 21.3444 −6.6192 −6.6192 19.6772 ] R12 = [ 6.4801 −1.9881 −1.9881 4.5859 ] , R22 = [ 7.2495 −2.6422 −2.6422 3.5791 ] , H1 = [ −1.5463 0.9452 −1.2486 1.3717 ] H2 = [ 1.6552 0.8485 −0.6090 1.9241 ] , H3 = [ 1.8527 1.2408 2.4549 1.7661 ] , H4 = [ −1.0116 2.4402 −1.5775 1.8661 ] Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 105 S = [ 18.3098 0 0 8.9915 ] , E = [ 22.2513 0 0 16.6959 ] , ρ = 1.8663× 103 In order to show the significant improvement of our results, we summerize the comparisons between the previous works and the obtained result. For this example, the delay-dependent sta- bility analysis in [3, 26, 27], cannot be satisfied for any τ > 0. Table 1 shows the maximum upper bound of the previous works [8, 19, 20] as 0.4121, 1.7484 and 1.7644, respectively. However, by theorem 1, we have that the origin of delayed stochastic Hopfield neural networks with im- pulsive effect is globally asymptotically stable for any constant allowable upper bound τ > 0. Hence, it is clear that the proposed method shows the less conservativeness than the existing works [2, 8, 19, 20]. Fig. 1 State trajectories of y1, y2 for Example 1 Example 4.2 Consider a delayed stochastic uncertain Hopfield neural network (2) with the following parameters: A = [ 0.7679 0 0 0.8860 ] , B = [ −0.1746 −0.8642 −0.2892 −0.7300 ] , C = [ −0.8252 −0.4912 −0.4732 −0.8858 ] , M = [ 0.07051 0 0 0.0342 ] , N1 = [ 0.3526 −0.1904 0.3322 −0.1564 ] , N2 = [ 0.2446 0.3674 −0.1753 0.2956 ] , 106 Raja: Delay-dependent Stability Criteria of Stochastic Uncertain Hopfield . . . . . . Table 1: Maximum allowable bound of the delay Method Maximum upper bound of τ In Ref [2, 26, 27] - In Ref [8] 0.4121 In Ref [19] 1.7484 In Ref [20] 1.7644 In this paper for any large finite τ > 0 N3 = [ 0.1981 −0.1313 0.1185 0.1645 ] , L = [ 0.07051 0 0 0.0342 ] , D0 = [ 0.85 0 0 0.85 ] , For τ = 0.957 and by solving the LMIs (20)-(22) in Theorem 3.2, we get the following feasible solution as follows: P11 = [ 133.1390 −39.2261 −39.2261 103.1396 ] , P12 = [ 5.2825 −5.7032 −5.7032 10.4098 ] , P22 = [ 18.1026 −10.6945 −10.6945 25.6053 ] Q1 = [ 30.8319 −27.8887 −27.8887 47.6789 ] , Q2 = [ 90.1163 28.7709 28.7709 147.2412 ] , R11 = [ 39.5332 −37.1370 −37.1370 66.5761 ] R12 = [ 21.2050 −14.1850 −14.1850 21.1791 ] , R22 = [ 31.5030 −17.8344 −17.8344 21.7477 ] , H1 = [ −13.2142 4.3068 −3.1365 −1.7307 ] H2 = [ 13.9993 −1.7353 −3.6838 8.4480 ] , H3 = [ −9.5150 11.0562 −15.8051 17.3187 ] , H4 = [ 12.1911 −1.3902 −7.3453 12.0990 ] S = [ 72.1101 0 0 263.5610 ] , E = [ 491.5915 0 0 339.6219 ] , ρ = 7.1793× 103 ε1 = 26.1631, ε2 = 15.5658, ε3 = 16.3759. 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