STABILITY OF IMPULSIVE HOPFIELD NEURAL NETWORKS WITH MARKOVIAN SWITCHING AND TIME VARYING DELAYS

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1 Int. J. Appl. Math. Comput. Sci. 211 Vol. 21 No DOI: /v y STABILITY OF IMPULSIVE HOPFIELD NEURAL NETWORKS WITH MARKOVIAN SWITCHING AND TIME VARYING DELAYS RAMACHANDRAN RAJA RATHINASAMY SAKTHIVEL 1 SELVARAJ MARSHAL ANTHONI HYUNSOO KIM Department of Mathematics Periyar University Salem India antony.raja67@yahoo.com Department of Mathematics Sungyunwan University Suwon South Korea rsathivel@yahoo.com Department of Mathematics Anna University of Technology Coimbatore India smarshalanthoni@gmail.com The paper is concerned with stability analysis for a class of impulsive Hopfield neural networs with Marovian jumping parameters and time-varying delays. The jumping parameters considered here are generated from a continuous-time discrete-state homogenous Marov process. By employing a Lyapunov functional approach new delay-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities (LMIs). The proposed criteria can be easily checed by using some standard numerical pacages such as the Matlab LMI Toolbox. A numerical example is provided to show that the proposed results significantly improve the allowable upper bounds of delays over some results existing in the literature. Keywords: Hopfield neural networs Marovian jumping stochastic stability Lyapunov function impulses. 1. Introduction In recent years the study of stochastic Hopfield neural networs have been extensively intensified since it has been widely used to model many of the phenomena arising in areas such as signal processing pattern recognition static image processing associative memory especially for solving some difficult optimization problems (Cichoci and Unbehauen 1993; Hayin 1998). One of the important and interesting problems in the analysis of stochastic Hopfield neural networs is their stability. In the implementation of networs time delays exist due to the finite switching speed of amplifiers and transmission of signals in the networ community which may lead to oscillation chaos and instability. Consequently stability analysis of stochastic neural networs with time delays has attracted many researchers and some results related to this problem have been reported in the litera- 1 Author for correspondence. ture (Balasubramaniam and Raiyappan 29; Balasubramaniam et al. 29; Li et al. 28; Singh 27; Zhou and Wan 28). Marovian jump systems are a special class of hybrid systems with two different states. The first one refers to the mode which is described by a continuous-time finitestate Marovian process and the second one refers to the state which is represented by a system of differential equations. Jump or switching systems have the advantage of modeling dynamic systems to abrupt variation in their structures such as component failures or repairs sudden environmental disturbance changing subsystem interconnections operating at different points of a nonlinear plant. Neural networs with Marovian jumping parameters and time delay have received much attention (Mao 22; Shi et al. 23; Wang et al. 26; Yuan and Lygeros 25; Zhang and Wang 28). In the wor of Li et al. (28) the problem of delay-dependent robust stability of uncertain Hopfield neural networs with Maro-

2 128 R. Raja et al. vian jumping parameters delays is investigated. Sufficient conditions are derived by Lou and Cui (29) to guarantee the stochastic stability for a class of delayed neural networs of neutral type with Marovian jump parameters. Moreover many physical systems also undergo abrupt changes at certain moments due to instantaneous perturbations which leads to impulsive effects. Neural networs are often subject to impulsive perturbations that in turn affect dynamical behaviors of the systems. Therefore it is necessary to consider the impulsive effect when investigating the stability of neural networs. The stability of neural networs with impulses and time delays has received much attention (Li et al. 29; Raiyappan et al. 21; Song and Wang 28; Song and Zhang 28). However neural networs with Marovian jumping parameters and impulses have received little attention in spite of their practical importance (Dong et al. 29). To the best of our nowledge up to now the stability analysis problem of time varying delayed Hopfield neural networ with Marovian jumping parameters and impulses has not appeared in the literature and this motivates our present wor. The main aim of this paper is to study the stochastic stability for a class of time varying delayed Hopfield neural networs with Marovian jumping parameters and impulses by constructing a suitable Lyapunov Krasovsii functional. The stability conditions are formulated in terms of LMIs and can be easily solved by using the Matlab LMI Control Toolbox. Further a numerical example is given to show the stability criteria obtained in this paper are less conservative than some existing results. 2. Problem formulation Notation. The notation in this paper is of standard form. The superscript T stands for matrix transposition; R n denotes the n-dimensional Euclidean space; P > means that P is real symmetric and positive definite; I and represents the identity and zero matrices respectively. The symbol within a matrix represents the symmetric term of the matrix. Furthermore diag{ denotes a blocdiagonal matrix and E{ represents the mathematical expectation operator. Consider the following Hopfield neural networs with impulses and a time-varying delay: ẋ(t) =Ax(t)Bf(x(t h(t))) D tτ (t) f(x(s)) ds U t t x(t )=C x(t ) t = t (1) for t> and =1 2...where x(t) =[x 1 (t)x 2 (t)...x n (t)] T R n is the state vector associated with the n neurons at time t; f(x(t h(t))) [ = f 1 (x(th(t)))f 2 (x(th(t)))...f n (x(th(t))) denotes the activation function; U =[U 1 U 2...U n ] T is the constant external input vector; the matrix A = diag(a 1 a 2...a n ) has positive entries a i > ; h(t) and τ(t) denote time-varying delays; the matrices B = [b ij ] n n and D = [d ij ] n n represent the delayed connections weight matrix and the connection weight matrix respectively; x(t ) = C x(t ) is the impulse at moment t the fixed moment of time t satisfies t 1 <t 2 <...lim t = and x(t ) = lim s t x(s); C is a constant real matrix at moments of time t. Let PC([ρ ] R n ) denote the set of piecewise right continuous functions φ : [ρ ] R n with the sup-norm φ = sup ρ s φ(s). For given t and φ PC([ρ ] R n ) the initial condition of the system (1) is described as x(t t) =φ(t) for t [ρ ] φ PC([ρ ] R n ) ρ max{h τ. Throughout this paper we assume that the following conditions hold: (i) The neuron activation function f( ) is continuous and bounded on R and satisfies the following inequality: f q(s 1 ) f q (s 2 ) l q q =1 2...n s 1 s 2 s 1 s 2 R s 1 s 2. (ii) The time-varying delay h(t) satisfies r 1 h(t) r 2 ḣ(t) μ (2) where r 1 r 2 are constants. Furthermore the bounded function τ(t) represents the distributed delay of systems with τ(t) τ τ is a constant. Let x =(x 1x 2...x n) T R n be the equilibrium point of Eqn. (1). For simplicity we can shift the equilibrium x to the origin by letting y(t) =x(t) x and ψ(t) =x(t t) x. Then the system (1) can be transformed into the following one: where ẏ(t) =Ay(t)Bg(y(t h(t))) D tτ (t) g(y(s)) ds t t y(t )=C y(t ) t = t (3) y(t t) =ψ(t) t [ρ ] y(t) =(y 1 (t)y 2 (t)...y n (t)) T ] T

3 Stability of impulsive Hopfield neural networs with Marovian switching and time-varying delays 129 is the state vector of the transformed system. It follows from Assumption (i) that the transformed neuron activation function satisfies g j () = g q(y q ) l q y q y q q =1 2...n. (4) Now based on the model (3) we are in a position to introduce Hopfield neural networs with Marovian jumping parameters. Let {r(t)t be a rightcontinuous Marov process on the complete probability space (Ω F {F t t P) taing values in a finite state space S = {1 2...N with generator Π=(π ij ) N N given by p{r(t Δ)=j r(t) =i { π ij Δo(Δ) if i j = 1π ii Δo(Δ) if i = j where Δ > and lim Δ o(δ)/δ π ij is the transition rate from i to j if i j and π ii = j i π ij. In this paper we consider the following time varying delayed Hopfield neural networs with Marovian jumping parameters and impulses which are actually a modification of (3): ẏ(t) =A(r(t))y(t)B(r(t))g(y(t h(t))) D(r(t)) tτ (t) g(y(s)) ds t t y(t )=C (r(t))y(t ) t = t (5) y(t t) =ψ(t) t [ρ ] r() = r where r S is the mode of the continuous state. For simplicity we write r(t) =i while A(r(t))B(r(t)) and D(r(t)) are denoted as A i B i and D i respectively. Let us first give the following lemmas and definitions which will be used in the proofs of our main results. Lemma 1. (Gu et al. 23) Let a b R n P be a positive definite matrix. Then 2a T b<a T P 1 a b T Pb. Lemma 2. (Gu et al. 23) For any positive definite matrix W> two scalars b>a and a vector function ω :[a b] R n such that the integrations concerned are well defined the following inequality holds: ( b ) T ( b ) ω(s)ds W ω(s)ds a a < (b a) b a ω T (s)wω(s)ds. Definition 1. The system (5) is said to be stochastically stable when U = for any finite ψ(t) R n defined on [ρ ] and r() S the following condition is satisfied: { lim E y T (s)y(s)ds ψ r <. t Definition 2. (Zhang and Sun 25) The function V : [t ) R n R belongs to class v if (i) the function V is continuous on each of the sets [t 1 t ) R n and for all t t V (t ) ; (ii) V (t x) is locally Lipschitzian in x R n ; (iii) for each =1 2...there exist finite limits lim V (t q) =V (t (tq) (t x) x) lim V (t q) =V (t (tq) (t x) x) with V (t x)=v (t x). 3. Stochastic stability results In this section we will derive conditions for the stochastic stability of delayed Hopfield neural networs with Marovian jumping parameters and impulsive effects. Theorem 1. Consider the neural networ system (5) satisfying Assumptions (i) and (ii). Given scalars r 2 > r 1 μ and τ > the system (5) is stochastically stable if there exist positive definite matrices P i > Q 1 Q 2 Q 3 Q 4 > R 1 R 2 > S 1 S 2 > and diagonal matrices T j =diag{t 1j t 2j...t nj (j = 1 2) such that the following LMIs hold: C T ip j C i P i < (6) for i = s and = along with (7) where Φ 1 = P i A i A T i P i π ij P j Q 1 Q 2 Q 3 r1r 2 1 (r 2 r 1 ) 2 R 2 Φ 2 = Q 4 τ 2 S 1 2 τ 2 Di T S 2 D i 2T 2 Φ 3 = (1 μ)q 4 2T 1 Φ 4 = S 1 2Di T S 2D i. Proof. In order to prove the stability result we construct the following Lyapunov Krasovii functional V (t y(t)r(t) =i) =V 1 V 2 V 3 V 4 V 5 (8)

4 13 R. Raja et al. Φ 1 L T 2 T 2 P i B i P i D i (1 μ)q 1 L T 1 T 1 Q 2 Q 3 Υ i = Φ 2 < (7) Φ 3 R 1 R 2 Φ 4 where V 1 = y T (t)p i y(t) V 2 = th(t) y T (s)q 1 y(s)ds tr 1 y T (s)q 2 y(s)ds V 3 = r 1 th(t) V 4 =(r 2 r 1 ) V 5 = τ g T (y(s))q 4 g(y(s)) ds tr 1 [s (t r 1 )]y T (s)r 1 y(s)ds 2 τ r1 tr 2 τ tσ τ tr 2 y T (s)q 3 y(s)ds [s (t r 2 )]y T (s)r 2 y(s)ds (9) tσ When t = t wehave V (t yj) V (t yi) = y T (t )P j y(t ) y T (t )P iy(t ) t h(t ) g T (y(s))d T i S 2 D i g(y(s)) ds dσ. y T (s)q 1 y(s)ds t h(t ) y T (s)q 1 y(s)ds t r 1 y T (s)q 2 y(s)ds t r1 y T (s)q 2 y(s)ds t r 2 y T (s)q 3 y(s)ds t h(t ) g T (y(s))q 4 g(y(s)) ds t r2 y T (s)q 3 y(s)ds g T (y(s))q 4 g(y(s)) ds t h(t ) r 1 r 1 (r 2 r 1 ) (r 2 r 1 ) [ τ τ t r 1 [s (t r 1 )]y T (s)r 1 y(s)ds t r1 [s (t r 1 )]y T (s)r 1 y(s)ds r 1 t r 2 r1 τ t σ t r2 [s (t r 2 )]y T (s)r 2 y(s)ds [s (t r 2 )]y T (s)r 2 y(s)ds ] t σ [ 2 τ g T (y(s))dj T S 2 D j g(y(s)) ds dσ τ τ t σ ] g T (y(s))d T t σ i S 2 D i g(y(s)) ds dσ = y T (t )CT ip j C i y(t ) yt (t )P iy(t ) t h(t ) y T (s)q 1 y(s)ds t h(t ) y T (s)q 1 y(s)ds t r 1 y T (s)q 2 y(s)ds t r1 y T (s)q 2 y(s)ds t y T (s)q 3 y(s)ds t r2 y T (s)q 3 y(s)ds t t y T (s)q 1 y(s)ds y T (s)q 2 y(s)ds t r 2 y T (s)q 3 y(s)ds

5 Stability of impulsive Hopfield neural networs with Marovian switching and time-varying delays 131 t h(t ) t g T (y(s))q 4 g(y(s)) ds g T (y(s))q 4 g(y(s)) ds g T (y(s))q 4 g(y(s)) ds t h(t ) r 1 [s (t r 1 )]y T (s)r 1 y(s)ds t r 1 r 1 [s (t r 1 )]y T (s)r 1 y(s)ds t r 1 (r 2 r 1 ) (r 2 r 1 ) t r1 [s (t r 1 )]y T (s)r 1 y(s)ds r1 t r 2 r 1 t r1 [s (t r 2 )]y T (s)r 2 y(s)ds [s (t r 2 )]y T (s)r 2 y(s)ds r1 (r 2 r 1 ) [s (t r 2 )]y T (s)r 2 y(s)ds t r2 [ τ τ τ τ t σ t [ 2 τ τ τ ] t σ τ t σ t g T (y(s))d T j S 2 D j g(y(s)) ds dσ g T (y(s))d T j S 2D j g(y(s)) ds dσ ] g T (y(s))d T t σ i S 2D i g(y(s)) ds dσ = y T (t )(CT i P jc i P i )y(t ) t t y T (s)q 1 y(s)ds t y T (s)q 2 y(s)ds y T (s)q 3 y(s)ds g T (y(s))q 4 g(y(s)) ds t r 1 [s (t r 1 )]y T (s)r 1 y(s)ds t (r 2 r 1 ) τ 2 τ τ t τ r 1 t t r1 [s (t r 2 )]y T (s)r 2 y(s)ds g T (y(s))d T j S 2D j g(y(s)) ds dσ. Since C i are constant matrices the terms involving positive-definite constant matrices Q 1 Q 2 Q 3 Q 4 R 1 R 2 S 1 S 2 will be equal to zero and hence V (t yj) V (t yi) <. (1) Let F( ) be the wea infinitesimal generator of the process {y(t)r(t)t for the system (5) at the point {t y(t)r(t) given by F{V (t y(t)r(t)) = V t ẏt (t) V y r(t)=i π ij V (t y(t)ij). For t [t 1 t ) taing account of (8) FV can be derived as FV 1 (t) =2y T (t)p i ẏ(t)y T (t) π ij P j y(t) =2y T (t)p i [A i y(t)b i g(y(t h(t))) D i g(y(s)) ds] y T (t) tτ (t) π ij P j y(t) = y T (t)[p i A i A T i P i]y(t) 2y T (t)p i B i g(y(t h(t))) ( 2y T t ) (t)p i D i g(y(s)) ds y T (t) tτ (t) π ij P j y(t) (11) FV 2 (t) y T (t)q 1 y(t) (1 μ)y T (t h(t))q 1 y(t h(t)) y T (t)q 2 y(t) y T (t r 1 )Q 2 y(t r 1 ) y T (t)q 3 y(t) y T (t r 2 )Q 3 y(t r 2 ) g T (y(t))q 4 g(y(t)) (1 μ)g T (y(t h(t)))q 4 g(y(t h(t))) (12) FV 3 (t) =r1 2 yt (t)r 1 y(t) r 1 y T (s)r 1 y(s)ds tr 1 r1 2 yt (t)r 1 y(t) ( ) T ( ) y(s)ds R1 y(s)ds tr 1 tr 1 FV 4 (t) =(r 2 r 1 ) 2 y T (t)r 2 y(t) (r 2 r 1 ) r1 tr 2 y T (s)r 2 y(s)ds (13)

6 132 R. Raja et al. (r 2 r 1 ) 2 y T (t)r 2 y(t) ( r1 ) T ( r1 y(s)ds R2 tr 2 FV 5 (t) = τ 2 g T (y(t))s 1 g(y(t)) τ t τ tr 2 g T (y(s))s 1 g(y(s)) ds 2 τ 2 g T (y(t))d T i S 2 D i g(y(t)) 2 τ t τ ) y(s)ds (14) g T (y(s))d T i S 2D i g(y(s)) ds g T (y(t))[ τ 2 S 1 2 τ 2 Di T S 2 D i ]g(y(t)) ( T g(y(s)) ds) [S1 2Di T S 2D i ] t τ ( ) g(y(s)) ds. (15) t τ From Assumption (i) we can get the following inequalities: and 2g T (y(t h(t)))t 1 Ly(t h(t)) 2g T (y(t h(t)))t 1 g(y(t h(t))) (16) 2g T (y(t))t 2 Ly(t) 2g T (y(t))t 2 g(y(t)). (17) From (11) (15) we obtain FV (t) y T (t)[p i A i A T i P i π ij P j Q 1 Q 2 Q 3 r1r 2 1 (r 2 r 1 ) 2 R 2 ]y(t) y T (t)l T T 2 g(y(t)) 2y T (t)p i B i g(y(t)) 2y T (t)p i D i g(y(t h(t))) y T (t h(t))[(1 μ)q 1 ]y(t h(t)) y T (t h(t))l T T 1 g(y(t h(t))) y T (t r 1 )Q 2 y(t r 1 ) y T (t r 2 )Q 3 y(t r 2 ) g T (y(t))[q 4 τ 2 S 1 2 τ 2 Di T S 2D i 2T 2 ]g(y(t)) g T (y(t h(t)))[(1 μ)q 4 2T 1 ]g(y(t h(t))) ( ) T ( ) y(s)ds R1 y(s)ds tr 1 tr 1 ( r 1 ) T ( r 1 ) y(s)ds R2 y(s)ds tr 2 ( t τ tr 2 g(y(s)) ds) T [S1 2D T i S 2 D i ] ( ) g(y(s)) ds t τ = ζ T (t) Υ i ζ(t) (18) where Υ i is defined in (7) and [ ζ(t) = y T (t) y T (t h(t)) y T (t r 1 ) y T (t r 2 ) ( g T (y(t)) g T t ) T (y(t h(t))) y(s)ds tr 1 ( r1 ) T ( T ] T y(s)ds g(y(s)) ds). tr 2 t τ This implies that Υ i <. Setting δ 1 =min{λ min (Υ i )i S we get δ 1 >. Foranyt hwehave F[V (y(t)i)] δ 1 ζ T (t)ζ(t) δ 1 y T (t)y(t). By Dynin s formula we get E{V (y(t)i)e{v(y r ) { δ 1 E y T (s)y(s)ds and hence { E y T (s)y(s)ds 1 {V (ψ r ) E{V (y(t)i). (19) δ 1 On the other hand from the definitions of V i (y(t)i) (i = ) there exists a scalar δ 2 > such that for any t we have E{V (y(t)i) =E{V 1 (y(t)i) E{V 2 (y(t)i) E{V 3 (y(t)i) E{V 4 (y(t)i) E{V 5 (y(t)i) δ 2 E{y T (t)y(t) (2) where δ 2 =min{λ min (P i )i S >. From (19) and (2) it follows that { E y T (t)y(t) { β 1 E y T (s)y(s)ds β 2 V (y r ) where β 1 = δ 1 δ2 1 β 2 = δ2 1. Thus we have { E y T (s)y(s)ds ψ r β 1 1 β 2[1 exp(β 1 t)]v (y r ).

7 Stability of impulsive Hopfield neural networs with Marovian switching and time-varying delays As t there exists a scalar η> such that { lim E y T (s)y(s)ds ψ r t β 1 1 β 2V (y r ) η sup ψ(s) 2. ρ s Thus by Definition 2 the impulsive Hopfield neural networ with Marovian switching (5) is stochastically stable. The proof is thus complete. 4. Numerical example Example 1. Consider the stochastic Hopfield neural networs with Marovian jumping parameters and impulses (5): A 1 = Π= A 2 = B 1 = B 2 = D = D 2 = C = C 2 = L.3 1 =.3.4 L 2 =..6 By solving the LMIs in Theorem 1 for positive definite matrices P 1 P 2 Q 1 Q 2 Q 3 Q 4 R 1 R 2 S 1 S 2 and diagonal matrices T 1 T 2 it can be verified that the system (5) is stochastically stable and a set of feasible solutions can be obtained as follows: P 1 = P 2 = Q 1 = Q 2 = Q 3 = Q 4 = R 1 = R 2 = S 1 = S 2 = T 1 = T 2 = In the wor of Zhang and Sun (25) when μ = the maximum allowable bounds for r 2 and τ are obtained as r 2 =.3 and τ =.6. In the paper by Liu et al. (29) the stochastic stability of a delayed Hopfield neural networ with Marovian jumpings and constant delays is discussed but the upper bound of the delay is not taen into account. In this paper when μ =and r 1 =by using Theorem 1 we obtain the maximum allowable upper bounds r 2 = τ = Moreover it is obvious that the upper bound obtained in our paper is better than those found by Liu et al. (29) or Zhang and Sun (25). The result reveals the stability criteria obtained in this paper are less conservative than some existing results. If we tae the initial values of (5) as [y 1 (s)y 2 (s)] = [cos(s).3sin(s)] s [2 ]. Figure 1 depicts the time response of state variables y 1 and y 2 with and without impulsive effects. Acnowledgment The wor of R. Sathivel and H. Kim was supported by the Korean Research Foundation funded by the Korean government with the grant no. KRF The wor of the first author was supported by the UGC Rajiv Gandhi National Fellowship and the wor of the third author was supported by the CSIR New Delhi. References Balasubramaniam P. Lashmanan S. and Raiyappan R. (29). Delay-interval dependent robust stability criteria for stochastic neural networs with linear fractional uncertainties Neurocomputing 72(16 18): Balasubramaniam P. and Raiyappan R. (29). Delaydependent robust stability analysis of uncertain stochastic neural networs with discrete interval and distributed timevarying delays Neurocomputing 72(13 15): Cichoci A. and Unbehauen R. (1993). Neural Networs for Optimization and Signal Processing Wiley Chichester. Dong M. Zhang H. and Wang Y. (29). Dynamic analysis of impulsive stochastic Cohen Grossberg neural networs

8 134 R. Raja et al. y1y2 y1y y1 y taxis y1 y taxis Fig. 1. State response of variables y 1y 2 for Example 1 with and without impulses. with Marovian jumping and mixed time delays Neurocomputing 72(7 9): Gu K. Kharitonov V. and Chen J. (23). Stability of Time- Delay Systems Birhäuser Boston MA. Hayin S. (1998). Neural Networs: A Comprehensive Foundation Prentice Hall Upper Saddle River NJ. Li D. Yang D. Wang H. Zhang X. and Wang S. (29). Asymptotic stability of multidelayed cellular neural networs with impulsive effects Physica A 388(2 3): Li H. Chen B. Zhou Q. and Liz C. (28). Robust exponential stability for delayed uncertain hopfield neural networs with Marovian jumping parameters Physica A 372(3): Liu H. Zhao L. Zhang Z. and Ou Y. (29). Stochastic stability of Marovian jumping Hopfield neural networs with constant and distributed delays Neurocomputing 72(16 18): Lou X. and Cui B. (29). Stochastic stability analysis for delayed neural networs of neutral type with Marovian jump parameters Chaos Solitons & Fractals 39(5): Mao X. (22). Exponential stability of stochastic delay interval systems with Marovian switching IEEE Transactions on Automatic Control 47(1): Raiyappan R. Balasubramaniam P. and Cao J. (21). Global exponential stability results for neutral-type impulsive neural networs Nonlinear Analysis: Real World Applications 11(1): Shi P. Bouas E. and Shi Y. (23). On stochastic stabilization of discrete-time Marovian jump systems with delay in state Stochastic Analysis and Applications 21(1): Singh V. (27). On global robust stability of interval Hopfield neural networs with delay Chaos Solitons & Fractals 33(4): Song Q. and Wang Z. (28). Stability analysis of impulsive stochastic Cohen Grossberg neural networs with mixed time delays Physica A 387(13): Song Q. and Zhang J. (28). Global exponential stability of impulsive Cohen Grossberg neural networ with timevarying delays Nonlinear Analysis: Real World Applications 9(2): Wang Z. Liu Y. Yu L. and Liu X. (26). Exponential stability of delayed recurrent neural networs with Marovian jumping parameters Physics Letters A 356(4 5): Yuan C.G. and Lygeros J. (25). Stabilization of a class of stochastic differential equations with Marovian switching Systems and Control Letters 54(9): Zhang H. and Wang Y. (28). Stability analysis of Marovian jumping stochastic Cohen Grossberg neural networs with mixed time delays IEEE Transactions on Neural Networs 19(2): Zhang Y. and Sun J.T. (25). Stability of impulsive neural networs with time delays Physics Letters A 348(1 2): Zhou Q. and Wan L. (28). Exponential stability of stochastic delayed Hopfield neural networs Applied Mathematics and Computation 199(1): Ramachandran Raja received the M.Sc. and M.Phil. degrees in mathematics from Periyar University Salem India in 25 and 26 respectively. He is currently a Ph.D. candidate at the Mathematics Department at Periyar University India. His research interests include impulsive differential equations neural networs and stability analysis of dynamical systems. Rathinasamy Sathivel received the B.Sc. M.Sc. and Ph.D. degrees in mathematics from Bharathiar University Coimbatore India in and 1999 respectively. Soon after the completion of his Ph.D. degree he served as a lecturer at the Mathematics Department at the Sri Krishna College of Engineering and Technology India. From 21 to 23 he was a post-doctoral fellow at the Mathematics Department Inha University South Korea. He was a visiting fellow at Max Planc Institute Magdeburg Germany in 22. From 23 to 25 he was a JSPS (Japan Society for the Promotion of Science) fellow at the Department of Systems Innovation and Informatics Kyushu Institute of Technology Japan. After that he wored as a research professor at the Mathematics Department Yonsei University South Korea till 26. Then he was a post-doctoral fellow (Brain Pool Program) at the Department of Mechanical Engineering Pohang University of Science and Technology (POSTECH) Pohang Korea from 26 till 28. He is currently an assistant professor (since 28) at the Department of Mathematics Sungyunwan University South Korea. His research interests include

9 Stability of impulsive Hopfield neural networs with Marovian switching and time-varying delays 135 control theory robust control for nonlinear systems exact solutions of PDEs and neural networs. Selvaraj Marshal Anthoni received the M.Sc. and Ph.D. degrees in mathematics from Bharathiar University Coimbatore India in 1995 and 21 respectively. He served as a lecturer in mathematics at the Kumaraguru College of Technology Coimbatore India from 21 till 23. He was a post-doctoral fellow at the Department of Mathematics Yonsei University South Korea from 23 till 24. After that he wored as a lecturer in mathematics at Periyar University Salem India (24 28). He is currently an assistant professor of mathematics (since 28) at the Anna University of Technology Coimbatore India. His research interests include control theory neural networs and abstract differential equations. Hyunsoo Kim received the M.S and Ph.D. degrees in mathematics from Sungyunwan University Suwon South Korea in 1998 and 21 respectively. From 25 till 27 he was a post-doctoral fellow at the School of Information and Communication Engineering of the same university. He is currently a researcher (since 21) at the Department of Mathematics Sungyunwan University. His research interests include statistics nonlinear partial differential equations and stability of nonlinear systems. Received: 8 March 21 Revised: 7 September 21