Improved delay-dependent globally asymptotic stability of delayed uncertain recurrent neural networks with Markovian jumping parameters

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1 Improved delay-dependent globally asymptotic stability of delayed uncertain recurrent neural networks with Markovian jumping parameters Ji Yan( 籍艳 ) and Cui Bao-Tong( 崔宝同 ) School of Communication and Control Engineering, Jiangnan University, Wuxi , China (Received 17 May 2009; revised manuscript received 4 January 2010) In this paper, we have improved delay-dependent stability criteria for recurrent neural networks with a delay varying over a range and Markovian jumping parameters. The criteria improve over some previous ones in that they have fewer matrix variables yet less conservatism. In addition, a numerical example is provided to illustrate the applicability of the result using the linear matrix inequality toolbox in MATLAB. Keywords: recurrent neural networks, time-varying delays, linear matrix inequality, Lyapunov Krasovskii functional, Markovian jumping parameters PACC: 0545, Introduction In the past few decades, the mathematical properties of recurrent neural networks (RNNs), such as the stability, the attractivity and the oscillation, have been intensively investigated. RNNs have been successfully applied in many areas, including image processing, pattern recognition, associative memory, and optimization problems. In fact, the stability analysis issue for RNNs with time delays has been an attractive subject of research in the past few years, the time delays under consideration can be classified as constant delays, time-varying delays, and distributed delays, with various sufficient conditions, either delay-dependent or delay-independent. Since delay-independent criteria tend to be more conservative especially for small size delays, considerable attention has been devoted to delay-dependent ones, see for example Refs Recently, stability analysis issues for neural networks in the presence of parameter uncertainties have been proposed in some papers. Chen et al. 7 used free-weighting matrices to study the exponential stability of uncertain stochastic delayed neural networks in mean square. Rakkiyappan et al. 8 obtained robust stability results for uncertain stochastic neural networks with discrete interval and distributed timevarying delays by using LMI and free-weighting matrices. The free weighting matrix method, by contrast, can keep a balance between the conservatism and the computational effort; there is still some conservatism (see Refs. 9 11) and the criteria can be simplified. But this method needs many matrix variables; Shao 12 and Kwon 13 got delay-dependent stability criteria for systems with interval time-varying delay with fewer matrix variables than some recently reported ones. It is worth noting that the RNNs may have finite modes, and the modes may switch (or jump) from one to another at different times. Recently, it has been shown in Ref. 14 that the switching (or jumping) between different RNN modes can be governed by a Markovian chain. Hence, an RNN with such a jumping character may be modelled as a hybrid one; that is, the state space of the RNN contains both discrete and continuous states. For a specific mode, the dynamics of the RNN is continuous, but the parameter jumps among different modes may be seen as discrete events. When time delay control systems are subject to the influence of Markovian jumping parameters, the behaviour of the system becomes stochastic. Standard techniques for the design of robust controllers of deterministic systems mentioned above are no longer applicable. Therefore, RNNs with Markovian jumping parameters are of great significance in modelling Project supported by the National Natural Science Foundation of China (Grant No ) and the Jiangsu Provincial Natural Science Foundation of China (Grant No. BK ). Corresponding author. yji maths@yahoo.com.cn 2010 Chinese Physical Society and IOP Publishing Ltd

2 a class of neural networks with finite network modes. Note that the concept of Markovian neural networks or stochastic system has already been used in some papers, see e.g. Refs It should be pointed out that, up to now, the stability analysis problem for uncertain RNNs with Markovian switching has received little research attention, despite its practical importance. In this paper, we are interested in developing criteria for delaydependent stability of delayed recurrent neural networks with Markovian jumping parameters. By utilizing a Lyapunov Krasovskii functional and conducting the stochastic analysis, we recast the addressed stability analysis problem into a numerically solvable problem. A unified LMI approach is developed to establish sufficient conditions for the neural networks to be globally asymptotically stable. Note that LMIs can be easily solved by using the Matlab LMI toolbox, and no tuning of parameters is required. 20 The numerical example is provided to show the usefulness of the proposed global stability condition. Notations: The notations are quite standard. Throughout this paper, R n denotes the n-dimensional Euclidean space. The superscript T denotes matrix transposition and the notation X Y (respectively, X > Y ) X and Y are symmetric matrices means that X Y is positive-semidefinite (respectively, positive-definite). I n is the n n identity matrix. is the Euclidean norm in R n. If A is a matrix, denote by A its operator norm, i.e., A = sup{ Ax : x = 1} = λ max (A T A) λ max ( ) (respectively, λ min ( )) means the largest (respectively, smallest) eigenvalue of A. The symmetric terms in a symmetric matrix are denoted by. 2. Model descriptions and preliminaries Consider the following delayed recurrent neural networks with Markovian jumping described by dx(t) = d m (η t )x m (t) p a mn (η t )f n (x n (t)) q b mn (η t )f n (x n (t h(t))) p c mn f n (x n (s))ds I m dt, m, n = 1, 2,..., p, (1) x m (t) is the activation of the m-th neurons, f n (.) denotes the signal function of the n-th neurons; I m stands for the external inputs at time t; d m (η t ) is a positive number, and it denotes the charging time constant or passive decay rate of the m-th neurons; a mn (η t ), b mn (η t ), c mn (η t ) stand for the synaptic connection weights. η t, t 0 is a homogeneous, finitestate Markovian process with right continuous trajectories and takes values in a finite set S = {1, 2,..., N} with a given probability space (Ω, F, P) and the initial model η 0, and a finite-state Markovian process with right continuous trajectories with generator ℵ = π ij and transition probability from mode i at time t to mode j at time t δ, i, j S, p ij = P r(η tδ = j η t = i) π ij δ o(δ), i j, = (2) 1 π ij δ o(δ), i = j with transition probability rates π ij 0, for i, j S, i j and π ii = N π ij, δ > 0 and lim δ 0 o(δ)/δ = 0. To obtain our results, we make the following assumption. (i) The neuron activation functions in Eq. (1) satisfy f n (s 1 ) f n (s 2 ) L n s 1 s 2, s 1, s 2 R, s 1 s 2, L n (n = 1, 2,..., p) are positive constants. (ii) The neural activation function in Eq. (1) is bounded. (iii) The time delays h(t) satisfy 0 h 1 h(t) h 2, ḣ(t) h, 0 τ(t) τ, h 1, h 2, h are constants. It is clear that under the assumptions (i) and (ii), system (1) has at least one equilibrium. In order to simplify our proof, we shift the equilibrium point x = (x 1, x 2,..., x p) T of system (1) to the origin. This transformation g n (x n (t)) = f n (u n (t)x )f(x ), u m (t) = x m (t)x

3 puts system (1) into the following system (3) du m (t) = d m (η t )u m (t) p a mn (η t )g n (u n (t)) q b mn (η t )g n (u n (t h(t))) p c mn (η t ) g n (u n (s))ds dt, m, n = 1, 2,..., p (3) with the initial values given by u m (t) = φ m (t), t 2µ, 0, µ = max{h, τ}, φ m (t) is a continuous function g n (.) which satisfies the following properties. (H1) The neuron activation functions g n (.) (n = 1, 2,..., p) are bounded on R and Lipchitz continuous, that is, there exist constants L n such that g n (s 1 ) g n (s 2 ) L n s 1 s 2, for all s 1, s 2 R. Rewriting Eq. (3) into the vector form, we obtain du(t) = D(η t )u(t) A(η t )g(u(t)) B(η t )g(u(t h(t))) C(η t ) g(u(s))ds dt, (4) u(t) = (u 1 (t), u 2 (t),..., u p (t)) T, D(η t ) = diag(d 1 (η t ), d 2 (η t ),..., d p (η t )), A(η t ) = (a mn (η t )) p p, B(η t ) = (b mn (η t )) p p, C(η t ) = (c mn (η t )) p p. Some lemmas used in this correspondence are listed as follows. Lemma 1 (Schur complement) Given constant Ω 1, Ω 2, and Ω 3 with appropriate dimensions, Ω1 T = Ω 1 and Ω2 T = Ω 2 > 0, then if and only if Ω 1 Ω3 T Ω2 1 Ω 3 < 0, Ω 1 Ω T 3 Ω 2 < 0 or Ω 2 Ω 3 < 0. Ω 1 Lemma 2 16 For any constant matrix M > 0, any scalars a and b with a < b, and a vector function x(t) : a, b R n such that the integrals concerned are well defined, then the following inequality holds b T b x(s)ds M x(s)ds a (b a) b a a x T (s)mx(s)ds. Lemma 3 5 Let A, D and N be real constant matrices with appropriate dimensions; matrix F (t) satisfies F T (t)f (t) I. Then we have: For any ϵ 1 > 0, DF (t)n N T F T (t)d T ϵ 1 DD T ϵn T N. Lemma 4 13 For any scalar h(t) > 0, and any constant matrix Q R n n, Q = Q T > 0, the following inequality holds: ẋ T (s)qẋ(s)ds τ(t)ξ T (t)x Q 1 ξ(t) 2ξ T (t)x(t) x(t τ(t)), ξ T (t) = x T (t), x T (t τ(t)), x T (t h 1 ), x T (t h 2 ), ẋ T (t), z T (t), G T (x(t)), G T (x()) and X is a free-weighting matrix with appropriate dimensions. In what follows, for simplicity, while η t = i, the matrices D(η t ), A(η t ), B(η t ), C(η t ) are represented by D i, A i, B i, C i. 3. Main results In this section, some sufficient conditions of global stability for systems (3) or (4) are obtained. Theorem 1 For system (4) satisfying assumptions (i) (iii), the equilibrium solution of neural network (4) is globally asymptotically stable in the mean square if there exist matrices Q i > 0, P k > 0 (k = 1, 2, 3), S 1 > 0, S 2 > 0, P > 0, and any matrices X 1, X 2, Y 1, Y 2, Z 1, Z 2 as well as positive scalars b 1, b 2 satisfying the following linear matrix inequalities Σ 1i R R T h 1 X h 2 Y h 12 Z S Ω i = < 0, (5) S 2 0 S 2 Σ 1i = (Σ mni ) 8 8 i, s Σ 11i = π ij Q i P 1 P 2 P 3 b 1 L T L W i D i D i W T i, Σ 1ki = 0 (k = 2,..., 4), Σ 15i = W i D T i W i Q i,

4 Σ 16i = W i A i, Σ 17i = W i B i, Σ 18i = W i C i, Σ 22i = (1 h)p 3 b 2 L T L, Σ 2ki = 0 (k = 3,..., 8), Σ 33i = P 1, Σ 3ki = 0 (k = 4,..., 8) Σ 44i = P 2, Σ 4ki = 0 (k = 5, 6, 7, 8), Σ 55 = h 2 1S 1 h 2 12S 2 W i W T i, Σ 56i = W i A i, Σ 57i = W i B i, Σ 58i = W i C i, Σ 66i = τ 2 P b 1 I, Σ 6ki (k = 7, 8) = 0, Σ 77i = b 2 I, Σ 78i = 0, Σ 88i = P, X = h 1 X T 1, 0, X T 2, 0, 0, 0, 0, 0 T, Y = h 2 0, Y T 1, 0, Y T 2, 0, 0, 0, 0 T, Z = h 12 0, Z T 1, Z T 2, 0, 0, 0, 0, 0 T, R = X, X Z, Z Y, Y, 0, 0, 0, 0, h 12 = h 2 h 1. Proof Construct the following Lyapunov Krasovskii functional: 3 V (u(t), t, i) = V j (u(t), t, i), (6) V 1 (u(t), t, i) = u T (t)q i u(t) V 2 (u(t), t, i) = V 3 (u(t), t, i) = τ 2 i=1 0 h 1 tθ h1 h 2 tθ 0 τ th(t) th i u T (s)p i u(s)ds, tθ u T (s)p 3 u(s)ds h 2 u T (s)s 1 u(s)dsdθ h 12 u T (s)s 2 u(s)dsdθ, g T (u(s))p g(u(s))dsdθ. Then it can be obtained by Ito s formula that L (V 1 (u(t), t, i)) = 2u T (t)q i u(t) ( s u T (t) π ij Q i )u(t) 3 u T (t)p k u(t) k=1 (1 h)u T (t h(t)))p 3 u(t h(t)) 2 u(t h k ) T P k u(t h k ), (7) k=1 L (V 2 (u(t), t, i)) = u T (t)(h 2 1S 1 h 2 12S 2 ) u(t) th 1 h 1 u T (s)s 1 u(s)ds h1 th 2 h 12 u T (s)s 2 u(s)ds, (8) L (V 3 (u(t), t, i)) = τ 2 f T (u(t))p f(u(t)) τ By Lemma 4, we have tτ f T (u(s))p f(u(s))ds. (9) th 1 h 1 u T (s)s 1 u(s)ds h 2 1ξ T (t)x S 1 1 X T ξ(t) 2h 1 ξ T (t)x (u(t) u(t h 1 )), (10) = h1 th 2 h(t) th 2 h1 th(t) h 12 u T (s)s 2 u(s)ds (h 2 h(t)) u T (θ)s 2 u(θ)dθ (h(t) h 1 ) u T (θ)s 2 u(θ)dθ (h 2 h(t)) 2 Y ξ(t) T S 1 2 Y T ξ(t) 2(h 2 h(t))y (u(t h(t)) u(t h 2 )) (h(t) h 1 ) 2 Z ξ T (t)s 1 2 Z T ξ(t) 2(h(t) h 1 )Z (u(t h 1 ) u(t h(t)) (11) with ξ T (t) = u T (t), u T (th(t)), u T (th 1 ), u T (th 2 ), u T (t), g T (u(t)), g T (u(t h(t))), ( g(u(s)))t ds. From assumption (i), it is obvious that b 1 f T (x(t))f(x(t)) b 1 x T (t)l T Lx(t) 0, b 2 f T (x(t h(t)))f(x(t h(t))) b 2 x T (t h(t))l T Lx(t h(t)) 0. (12) The first equation in Eq. (4) ensures ( 2 (u T (t) u )W T i u(t) D i u(t) A i g(u(t)) B i g(u(t h(t))) C i ) g(u(s))ds = 0. (13) Then, combining expressions (7) (13), it follows that with L V (u(t), t, i) ξ T (t)ω 1 ξ(t) (14) Ω 1 = Σ 1i R R T h 2 1ξ T (t)x S 1 1 X T ξ(t) (h 2 τ(t)) 2 Y ξ(t) T S 1 2 Y T ξ(t) (τ(t) h 1 ) 2 Z ξ T (t)s 1 2 Z T ξ(t) Σ 1 R R T h 2 1ξ T (t)x S 1 1 X T ξ(t) h 2 2Y ξ(t) T S 1 2 Y T ξ(t) h 2 12Z ξ T (t)s 1 2 Z T ξ(t). (15) By using the Schur complement (Lemma 1), we get Ω 1 < 0. Taking the mathematical expectation, we have dev (u(t), t, i) Eξ T (t)ω 1 ξ(t) λe u(t) 2, dt

5 λ is the eigenvalue of Ω 1 ; this implies that the equilibrium point x of system (4) is globally asymptotically stable. This completes the proof. Remark 1 Theorem 1 solves the stability problem for time-varying delay over a range and presents a delay-range-dependent criterion. Further, for h 1 = 0 it implies that 0 h(t) h 2, which was investigated in Refs. 7 and 16. Now we consider the robust stability of the uncertain system du(t) = (D i D i (t))(u(t) (A i A i (t))g(u(t)) (B i B i (t))g(u(t h(t))) (C i C i ) g(u(s))ds dt, (16) A i, B i, C i, D i, for any i S, are known constant matrices of appropriate dimensions, A i, B i, C i and D i for i S, are unknown matrices that represent time-varying parameter uncertainties and are assumed to be of the form: D i, A i, B i, C i = H i F i (t)e 1i, E 2i, E 3i, E 4i, (17) H i, E 1i, E 2i, E 3i, E 4i are known real constant matrices and F i (t), for any i S are unknown timevarying matrix functions satisfying F T i (t)f i (t) I, i S. Definition 1 The equilibrium solution of the neural networks (16) is said to be globally robustly asymptotically stable for all admissible uncertainties satisfying Eq. (17) in the mean square if the following condition holds: E x(t) 2 = 0. The system (16) can be written as du(t) = D i (t)(u(t) A i (t)g(u(t)) B i (t)g(u(t h(t)) C i (t) g(u(s))ds dt, (18) Theorem 2 For system (18) satisfying assumptions (i) (iii), the equilibrium solution of neural network (18) is globally asymptotically stable in the mean square if there exist matrices Q i > 0, P k > 0 (k = 1, 2, 3), S 1 > 0, S 2 > 0, P > 0, and any matrices X 1, X 2, Y 1, Y 2, Z 1, Z 2 as well as positive scalars ε i, b 1, b 2 that satisfy the following linear matrix inequalities Γ 1i R 1 R1 T h 1 X h 2 Y h 12 Z S Ω 1 = S 2 0 < 0, (19) S 2 Γ 1i = (Γ mni ) 9 9 i, Γ 11i = Σ 11i ε i E T 1iE 1i, Γ 1ki = 0 (k = 2,..., 4), Γ 15i = Σ 15i, Γ 16i = Σ 16i ε i E T 1iE 2i, Γ 17i = Σ 17i ε i E T 1iE 3i, Γ 18i = Σ 18i ε i E T 1iE 4i, Γ 19i = W i H i, Γ 22i = Σ 22i, Γ 2ki = 0 (k = 3,..., 9), Γ 33i = P 1, Γ 3ki = 0 (k = 4,..., 9), Γ 44i = P 2, Γ 4ki = 0 (k = 5, 6, 7, 9), Γ 55i = Σ 55i, Γ 56i = Σ 56i, Γ 57i Σ 57i, Γ 58i = Σ 58i, Γ 59i = W i H, Γ 66i = Σ 66i εe T 2iE 2i, Γ 67i = ε i E T 2iE 3i, Γ 68i = ε i E T 2iE 4i, Γ 69i = 0, Γ 77i = b 2 I ε i E T 3iE 3i, Γ 78i = ε i E T 3iE 4i, Γ 79i = 0, Γ 88i = P ε i E T 4iE 4i, Γ 89i = 0, Γ 99i = ε i I, X = h 1 X T 1, 0, X T 2, 0, 0, 0, 0, 0, 0 T, Y = h 2 0, Y T 1, 0, Y T 2, 0, 0, 0, 0, 0 T, Z = h 12 0, Z T 1, Z T 2, 0, 0, 0, 0, 0, 0 T, R 1 = X, X Z, Z Y, Y, 0, 0, 0, 0, 0, h 12 = h 2 h 1. Proof Replacing D i, A i, B i, C i in Eq. (4) with D i D i (t), A i A i (t), B i B i (t) and C i C i (t), we get Γ 1i = Σ 1i E 1i, 0, 0, 0, 0, E 2i, E 3i, E 4i T F T i (t) H T i W T i, 0, 0, 0, 0, H T i W T i, 0, 0 H T i W T i, 0, 0, 0, 0, H T i W T i, 0, 0 T F i (t) E 1i, 0, 0, 0, 0, E 2i, E 3i, E 4i. (20) From Lemma 1, we know that Γ 1i is equivalent to Γ 2i = Σ 1i ε 1 i Hi T Wi T, 0, 0, 0, 0, Hi T Wi T, 0, 0 T H T i W T i, 0, 0, 0, 0, H T i W T i, 0, 0 ε i E 1i, 0, 0, 0, 0, E 2i, E 3i, E 4i T E 1i, 0, 0, 0, 0, E 2i, E 3i, E 4i. (21)

6 Using the Schur complement, we have that Γ 2i is equivalent to Ω 1i. This completes the proof. When h 1 = 0, we can get the following result. Corollary 1 For system (18) satisfying assumptions (i) (iii), the equilibrium solution of neural network (18) is globally asymptotically stable in the mean square if there exist matrices Q i > 0, P k > 0 (k = 2, 3), S 1 > 0, S 2 > 0, P > 0, and any matrices Y 1, Y 2, Z 1, as well as positive scalars ε i, b 1, b 2 that satisfy the following linear matrix inequalities Ω = Σ R R T h 2 Y h 2 Z S 2 0 S 2 Σ i = (Σ mni ) 8 8 i, s Σ 11i = π ij Q i P 2 P 3 b 1 L T L W i D i D i W T i ε i E T 1iE 1i, < 0, (22) Σ 1ki =0 (k = 2,..., 4), Σ 15i = W i D T i W i Q i, Σ 16i = W i A i ε i E T 1iE 2i, Σ 17i = W i B i ε i E T 1iE 3i, Σ 18i = W i C i ε i E T 1iE 4i, Σ 19i = W i H i, Σ 22i = (1 h)p 3 b 2 L T L, Σ 2ki = 0 (k = 3,..., 8), Σ 33i = P 2, Σ 3ki = 0 (k = 5, 6, 7, 8), Σ 44 = h 2 2S 2 W i W T i, Σ 45i = W i A i, Σ 46i = W i B i, Σ 47i = W i C i, Σ 48i = W i H, Σ 55i = τ 2 P b 1 I εe T 2iE 2i, Σ 56i = ε i E T 2iE 3i, Σ 57i = ε i E T 2iE 4i, Σ 58i = 0, Σ 66i = b 2 I ε i E T 3iE 3i, Σ 67i = ε i E T 3iE 4i, Σ 68i = 0, Σ 77i = P ε i E T 4iE 4i, Σ 78i = 0, Σ 88i = ε i I, Y = h 2 0, Y T 1, Y T 2, 0, 0, 0, 0, 0 T, Z = h 2 0, Z T 1, 0, 0, 0, 0, 0, 0 T, R = 0, Z, Y, 0, 0, 0, 0, 0. When the information of the time derivative of delay is unknown, by setting P 3 = 0 and P = 0, we have the following results from Theorem 2. Corollary 2 For system (18) satisfying assumptions (i) (iii) the equilibrium solution of neural network (18) is exponentially stable in the mean square if there exist matrices Q i > 0, P k > 0 (k = 1, 2), S 1 > 0, S 2 > 0 and any matrices X 1, X 2, Y 1, Y 2, Z 1, Z 2 as well as positive scalars ε i, b 1, b 2 that satisfy the following linear matrix inequalities Σ 1i R R T τ 1 X τ 2 Y τ 12 Z S Ω = S 2 0 < 0, (23) S 2 Σ 1i = (Σ mni ) 9 9 i, s Σ 11i = π ij Q i P 1 P 2 b 1 L T L W i D i D i W T i ε i E T 1iE 1i, Σ 1ki =0 (k = 2,..., 4), Σ 15i = W i D T i W i Q i, Σ 16i = W i A i ε i E T 1iE 2i, Σ 17i = W i B i ε i E T 1iE 3i, Σ 18i = W i C i ε i E T 1iE 4i, Σ 19i = W i H i, Σ 22i = b 2 L T L, Σ 2ki = 0 (k = 3,..., 9), Σ 33i = P 1, Σ 3ki = 0 (k = 4,..., 9), Σ 44i = P 2, Σ 4ki = 0 (k = 5, 6, 7, 9), Σ 55 = h 2 1S 1 h 2 12S 2 W i W T i, Σ 56i = W i A i, Σ 57i = W i B i, Σ 58i = W i C i, Σ 59i = W i H, Σ 66i = τ 2 P b 1 I ε i E T 2 E 2, Σ 67i = ε i E T 2 E 3, Σ 68i = ε i E T 2 E 4, Σ 69i = 0, Σ 77i = b 2 I ε i E T 3 E 3, Σ 78i = ε i E T 3 E 4, Σ 79i = 0, Σ 88i = P ε i E T 4 E 4, Σ 89i = 0, Σ 99i = ε i I, X = h 1 X T 1, 0, X T 2, 0, 0, 0, 0, 0, 0 T, Y = h 2 0, Y T 1, 0, Y T 2, 0, 0, 0, 0, 0 T, Z = h 12 0, Z T 1, Z T 2, 0, 0, 0, 0, 0, 0 T, R = X, X Z, Z Y, Y, 0, 0, 0, 0, 0, h 12 = h 2 h 1. Remark 2 Theorem 2 can be applied to both discrete and distributed time-varying delays only if h is known. But when h(t) is not differentiable or h is unknown, Theorem 2 of this paper and Theorem 1 of Ref. 16 fail to work, while Corollary 2 can be turned to for checking the stability of system (18). 4. Illustrative example In this section, we present an example to illustrate the effectiveness of our results. Example Consider the delayed uncertain recurrent neural networks with Markovian jumping param

7 eters as follows: u(t) = Chin. Phys. B Vol. 19, No. 6 (2010) D i (t)u(t) A i (t)g(u(t)) B i (t)g(u(t h(t))) C i (t) g(u(s))ds, (24) D 1 = A 1 = B 1 = C 1 = H 1 = ℵ = D 2 = A 2 = B 2 = C 1 = H 2 = E 1i = E 2i = E 3i = E 4i (i = 1, 2) = L = I P 2 = P 3 = Q = S 1 = S 2 = Q 1 = W = W 1 = b 1 = , b 2 = Therefore, it follows from Theorem 2 that the Markovian jumping delayed neural network (24) is globally asymptotically stable in the mean square. with g(x) = 1 2 ( u1 u1 ). Obviously, assumption (i) is satisfied with L = diag{1, 1}. Applying Matlab Toolbox, by Theorem 2, for h 2 = , h 1 = 0, r = , h = 0, we can get the solution P = P 1 = Conclusions In this paper, delay-dependent stability of delayed recurrent neural networks with Markovian jumping parameters is considered. With a different Lyapunov functional proposed for the derivative of it, simplified delay-dependent criteria are derived. It is established that the criteria, though involved with fewer matrix variables, are less conservative than some existing ones. An example is given to verify the theoretical results. References 1 Li D, Wang H, Yang D, Zhang X H and Wang S L 2008 Chin. Phys. B Huang H and Feng G 2007 Physica A Zhang Q, Ma R N, Wang C and Xu J 2003 Chin. Phys Zhang Q 2008 Chin. Phys. B Chen B and Wang J 2004 Phys. Lett. A Yu J J, Zhang K J and Fei S M 2009 Commun. Nonlinear Sci. Numer. Simulat Chen W and Lu X 2008 Phys. Lett. A Rakkiyappan R, Balasubramaniam P and Lakshmanan S 2008 Phys. Lett. A

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