Stability of inertial BAM neural network with time-varying delay via impulsive control

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Stability of inertial BAM neural network with time-varying delay via impulsive control Jiangtao Qi, Chuandong Li, Tingwen Huang

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Received date: 28 September 2014 Revised date: 8 January 2015 Accepted date: 16 February 2015 Cite this article as: Jiangtao Qi, Chuandong Li, Tingwen Huang, Stability of inertial BAM neural network with time-varying delay via impulsive control, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.02.052 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Stability of inertial BAM neural network with time-varying delay via impulsive control Jiangtao Qi1 , Chuandong Li1 , Tingwen Huang2 1

College of Computer Science, Chongqing University, Chongqing 400044, PR China 2

Department of Mathematics, Texas A & M University at Qatar, Qatar

Abstract In this paper, a class of inertial BAM neural networks with time-varying delays is considered. By choosing proper variable transformation, the inertial BAM neural networks can be rewritten as first-order differential equations. Based on the Lyapunov functional method and the comparison principle, we derive some sufficient conditions guaranteeing the exponential stability of the neural networks under impulsive control. For different equilibrium point, the different impulsive controller can be obtained. Moreover, less conservatism impulsive controller will be designed to stabilize the zero equilibrium point. Simulation results finally demonstrate the effectiveness of the theoretical results. Keywords: Global exponential, Inertial BAM neural network, impulsive effects,time-varying delays. 1. Introduction Recurrent neural networks (RNNs), especially Hopfield neural networks [1], cellular neural networks (CNNs) [2], [3], Cohen-Grossberg neural networks (CGNNs) [4], bidirectional associative memory (BAM) neural netwroks [5] have been extensively investigated in recent years because of their poenetial application in the ares of signal and image processing, financial industry, parallel computation, and optimization problems and so on. In the design of RNNs, the dynamical properties of networks, such as the stability of the networks, play important roles. And there have been many literatures to investigate the stability of equilibrium points for neural networks [6]-[12]. Noticing that lots of previous studies mainly focused on neural networks with only first derivative of the states, whereas it is also of significant importance to introduce an inertial term, or Preprint submitted to Elsevier

February 20, 2015

equivalently, the influence of inductance, into the artificial neural networks, because the inertial terms are considered to a critical tool to generate complicated bifurcation behavior and chaos [13]-[14]. Up to now, there are only few literatures to study the dynamical behavior of the equilibrium point of the inertial neural networks [15]-[17]. [15] studied the stability of inertial Cohen-Grossberg-type neural networks with time delays. In [16], the martix measure strategies were used to analyze the stability and synchronization of inertial BAM neural network with time delays. The authors in [17] presented the convergence analysis for second-order interval CohenGrossberg neural networks. All those literatures presented sufficient conditions to ensure the stability of the inertial neural networks. However, there is many neural networks are unstable in nature. In this case, controller has to be added to the neural networks to ensure the stability. There are many control strategies such as feedback control[18], intermittent control [19]-[21] and impulsive control method to stabilize the RNNs. To be noted that the impulsive control method is effective and robust in the investigation of stability analysis, since it needs small control gains and acts only at discrete times; thus control cost and the amount of transmitted information can be reduced drastically. Due to these advantages, there are many results on stability analysis of neural networks using impulsive control [22]-[27]. However, there is few literatures focusing on designing an impulsive controller to stabilize the inertial neural network. Motivated by the aforementioned discussions, this paper investigates the problem of global exponential stability of inertial BAM neural networks under the impulsive effects. By introducing a variable transformation, the inertial BAM neural networks can be rewritten as first-order differential equations in which the parameters are dependent on the variable. For different choosing of the variable, the different dynamical behavior of the first-order differential equations can be obtain which is related to the dynamical behavior of different equilibrium point of the inertial BAM neural networks. By mathematical analysis, we find that to stabilize the differen equilibrium point of the inertial network, different impulsive controller will be designed. Therefore, we can design a corresponding impulsive controller to stabilize the state trajectory to different equilibrium. For some special conditions, when we consider the stability of the zero equilibrium point of the inertial neural networks, we can choose arbitrary variable transformation. Therefore, how to design a less conservative impulsive controller will be given in the following. The contributions of this paper include the following: 1) a variable transformation is given to simplify the stability analysis of the inertial BAM neural networks ; 2) by using Lyapunov functional method and the comparison principle, different impulsive controllers are designed to 2

stabilize the state trajectory of networks to different equilibrium, which are more effective in comparison with only one impulsive controller for all equilibrium; 3) when considering the zero equilibrium point of the networks, we use some equation method to get a more effective results. To this end, the theoretical results will be illustrated by numerical simulations. The structure of this paper is outlined as follows: In Section 2, we will interpret the inertial neural networks with impulsive effects model and the variable transformation. Some stability conditions are presented in Section 3. An illustrative example is given to demonstrate the effectiveness of the proposed approach in Section 4. Finally, Section 5 concludes the article. Notations:

The standard notations will be used in this paper. λmax (·)/λmin(·) is used to

denote the maximum/minimum eigenvalue of a real matrix. R+ and Rn denote, respectively, the set of nonnegative real numbers and the n-dimensional Euclidean space. Rn×n are n × n real matrices.  ·  is the Euclidean norm in Rn . A = (aij )n×n is a n × n matrix. Let N+ = {1, 2, ...}. The superscript T denotes matrix or vector transposition. In is the n × n identity matrix. 2. Problem formulation and preliminaries In this paper, we consider the following inertial BAM neural networks with time delays [16]:   dui (t) d2 ui (t) − b = −a u (t) + c g (u (t)) + dij gj (uj (t − τ (t))) + Ii i i i ij j j dt2 dt j=1 j=1 n

n

(1)

for i = 1, 2, ..., n, where the second derivative is called an inertial term of system (1), ui (t) denotes the states variable of the ith neuron at time t, ai > 0, bi > 0 are constants, cij , dij are constants and denotes the connection weights of the neural networks, gi denotes the activation function of the jth neuron at time t and is assumed to be bounded in this paper, τ (t) is a time varying delay that satisfies 0 ≤ τ (t) ≤ τ . For the nonlinear function gi (·), we have the following assumption. Assumption 1: Assume that gi (·)i = 1, 2, ..., n are globally Lipchitz continuous functions, i.e., there exist constants li > 0(i = 1, 2, ..., n) such that |gi (x1 ) − gi (x2 )| ≤ li |x1 − x2 |,

(2)

hold for any x1 , x2 ∈ R. Denote L = diag{l1 , l2 , ..., ln } for convenience. Let variable transformation: vi (t) =

dui (t) + ξi ui (t), i = 1, 2, ..., n, dt 3

(3)

then of (1) can be rewritten as ⎧ dui (t) ⎪ ⎪ = −ξi ui (t) + vi (t), ⎪ ⎪ ⎨ dt dvi (t) = −[bi + ξi (ξi − ai )]ui (t) − (ai − ξi )vi (t) dt ⎪ ⎪ ⎪   ⎪ ⎩ + nj=1 cij gj (uj (t)) + nj=1 dij gj (uj (t − τ (t))) + Ii

(4)

for i = 1, 2, ..., n. Denote u(t) = (u1 (t), u2(t), ..., un (t))T , v(t) = (v1 (t), v2 (t), ..., vn (t))T , the system (4) can be written as ⎧ ⎪ ⎨ du(t) = −Λu(t) + v(t), dt

⎪ ⎩ dv(t) = −Au(t) − Bv(t) + Cg(u(t)) + Dg(u(t − τ (t))) + I, dt

(5)

where A = diag{b1 +ξ1 (ξ1 −a1 ), ..., bn +ξn (ξn −an )}, B = diag{a1 −ξ1 , ..., an −ξn }, C = (cij )n×n , D = (dij )n×n , Λ = diag{ξ1, ..., ξn }, I = diag{I1 , ..., In }. Remark 1: For simplicity, here, we only used one variable to express the transformation. In order to get less conservative results, we can use two variable to express the transformation, i.e., vi (t) = ζi dudti (t) + ξi ui (t), i = 1, 2, ..., n. Definition 1: The point (u∗ , v ∗ ) with u∗ = (u1 , ..., un )T , v ∗ = (v1 , ..., vn )T is called an equilibrium point of system (5) if ⎧ ⎪ ⎨−Λu∗ + v ∗ = 0, ⎪ ⎩−Au∗ − Bv ∗ + Cf (u∗) + Df (u∗) + I = 0,

(6)

Let (u∗ , v ∗ ) be the equilibrium point of (5). For the purpose of simplicity, we can shift the intended equilibrium (u∗ , v ∗) to be original by letting x = u − u∗ , y = v − v ∗ , then the system (5) can be transformed into ⎧ ⎪ ⎨ dx(t) = −Λx(t) + y(t), dt

⎪ ⎩ dy(t) = −Ax(t) − By(t) + Cf (x(t)) + Df (x(t − τ (t))), dt

(7)

where x(t) = [x1 (t), ..., xn (t)]T , y(t) = [y1 (t), ..., yn (t)]T is the state vector of the transformed system. It follows from (2) that the function f (x) = g(x + u∗ ) − g(u∗) satisfies |f (x1 ) − f (x2 )| ≤ L|x1 − x2 |, for any x1 , x2 ∈ Rn . 4

(8)

By considering the impulse effects, the impulsive inertial neural networks can be obtained in the following form: ⎧ dx(t) ⎪ ⎪ = −Λx(t) + y(t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ dy(t) = −Ax(t) − By(t) + Cf (x(t)) + Df (x(t − τ (t))), t = tk dt ⎪ − ⎪ x(t+ ⎪ k ) = αk x(tk ), ⎪ ⎪ ⎪ ⎪ ⎩y(t+ ) = β y(t− ), k ∈ N , k + k k

(9)

where {t1 , t2 , ...} is a sequence of strictly increasing impulsive moments, αk , βk ∈ R represents the strength of impulses. We assume that x(t) is right continuous at t = tk , i.e., x(tk ) = x(t+ k ). Hence, the solution of (8) are piecewise right-hand continuous functions with discontinuities at t = tk for k ∈ N+ . The initial condition of (8) is given by x(t) = φ(t) and y(t) = ψ(t) + Λφ(t). Before giving the main results, we also need the following definition and lemmas. Definition 2: The impulsive inertial neural network (8) is said to be globally exponential stable if there exist constant M > 0 and λ > 0 such that for any initial values |x(t)|2 ≤ Me−λt

(10)

hold for all t ≥ T0 . Lemma 1 [25]: For any vectors x, y ∈ Rn , Q is a diagonal positive definite matrix with appropriate dimensional, then the following inequality holds xT y + y T x ≤ xT Qx + y T Q−1 y.

(11)

Lemma 2 [23]: If P ∈ Rn×n is a symmetric positive definite matrix and Q ∈ Rn×n is a symmetric matrix. then λmin (P −1 Q)xT P x ≤ xT Qx ≤ λmax (P −1 Q)xT P x, x ∈ Rn×n .

(12)

Lemma 3 [16]: If the activation function gi is bounded, i.e, |gi (x)| ≤ M, for each i = 1, 2, ..., n, then for every given input I = diag{I1 , ..., In }, there exists an equilibrium for (1). Lemma 4: Let 0 ≤ τ (t) ≤ τ . F (t, u, u1 , ..., um) : R+ × R · · × R −→ R be nondecreasing  × ·

m+1

in ui for each fixed (t, u, u1 , ..., ui−1 , ui+1 , ..., um ) , and Ik (u) : R → R be nondecreasing in u . Suppose that u(t), v(t) satisfy 5

⎧ ⎪ ⎨D + u(t) ≤ F (t, u(t), u(t − τ1 (t)), ..., u(t − τm (t))), t ≥ 0, ⎪ ⎩u(tk ) ≤ Ik (u(t− )), k ∈ N+ k and ⎧ ⎪ ⎨D + v(t) > F (t, v(t), v(t − τ1 (t)), ..., v(t − τm (t))), t ≥ 0, ⎪ ⎩v(tk ) ≥ Ik (v(t− )), k ∈ N+ k where the upper-right Dini derivative D+ y(t) = limh→0+ (y(t + h) − y(t))/h, where h → 0+ means that h approaches zero from the right-hand side. Then, u(t) ≤ v(t), for −τ ≤ t ≤ 0, implies that u(t) ≤ v(t), for t ≥ 0. 3. Main results Theorem 1: Under Assumption 1, if there exists matrix Λ = {ξ1 , ..., ξn } such that the following inequality holds (p +

ln b )∗b+q < 0 d

(13)

where, p = max{λmax (−2Λ + 2I + LT L), λmax (I + AAT − 2B + CC T + DD T )}, q = λmax (LT L), 0 < max{αk2 , βk2 } < b < 1, 0 ≤ tk+1 − tk ≤ d, then the equilibrium (u∗ , v ∗ ) of the inertial neural networks will be globally exponential stable under the impulsive control. Proof: Consider the following Lyapunov function: V (t) = xT (t)x(t) + y T (t)y(t)

(14)

Then take the derivative of V (t) along the trajectories of the system (9) can be obtained as

6

follows: ˙ + 2y T (t)y(t) ˙ D + V (t) = 2xT (t)x(t) = 2xT (t)(−Λx(t) + y(t)) +2y T (t)(−Ax(t) − By(t) + Cf (x(t)) + Df (x(t − τ (t)))) = −2xT (t)Λx(t) + 2xT (t)y(t) − 2y T (t)Ax(t) − 2y T (t)By(t)

(15)

+2y T (t)Cf (x(t)) + 2y T (t)Df (x(t − τ (t)))) ≤ −2xT (t)Λx(t) + xT (t)x(t) + y T (t)y(t) + y T (t)AAT y(t) + xT (t)x(t) −2y T (t)By(t) + 2y T (t)Cf (x(t)) + 2y T (t)Df (x(t − τ (t)))) By Lemma 1 and Assumption 1, we can get

2y T (t)Cf (x(t)) ≤ y T (t)CC T y(t) + f T (x(t))f (x(t))

(16)

≤ y T (t)CC T y(t) + xT (t)LT Lx(t)

2y T (t)Df (x(t − τ (t)))) ≤ y T (t)DD T y(t) + f T (x(t − τ (t)))f (x(t − τ (t)))

(17)

≤ y T (t)DD T y(t) + xT (t − τ (t))LT Lx(t − τ (t)) From (15)-(17), we can get D + V (t) = xT (t)(−2Λ + 2I + LT L)x(t) + λmax (LT L)xT (t − τ (t))x(t − τ (t)) +y T (t)(I + AAT − 2B + CC T + DD T )y(t)

(18)

≤ pV (t) + qV (t − τ (t)) For t = tk , from the third and fourth equations of (9), we get T + + T + + V (t+ k ) = x (tk )x(tk ) + y (tk )y(tk ) − 2 T − − ≤ αk2 xT (t− k )x(tk ) + βk y (tk )y(tk )

≤ bV (t− k) 7

(19)

For any constant > 0, let υ(t) be a unique solution of the following delayed impulsive system: ⎧ ⎪ ⎪ υ(t) ˙ = pυ(t) + qυ(t − τ (t)), t = tk ⎪ ⎪ ⎨ (20) υ(t+ ) = bυ(t− k ), t = tk , k ∈ N+ , ⎪ ⎪ ⎪ ⎪ ⎩υ(t) = V (t), − τ ≤ s ≤ 0. Note that υ(s) ≥ V (s) ≥ 0 for −τ ≤ s ≤ 0. Then it follows from (18), (19) and Lemma 4 that 0 ≤ V (t) ≤ υ(t), t ≥ 0

(21)

By the formula for the variation of parameters, one obtains from (20) that t υ(t) = P (t, 0)υ(0) + P (t, s)[qυ(s − τ (s)) + ]ds 0

(22)

where P (t, s), t, s ≥ 0, is the Cauchy matrix of linear system ⎧ ⎪ ⎨y(t) ˙ = py(t), t = tk ⎪ ⎩y(t) = by(t− ), t = tk , k ∈ N+ . k According to the representation of the Cauchy matrix, we get the following estimation:

P (t, s) = ep(t−s)

b

s
≤ ep(t−s) b

t−s −1 d

≤ b−1 ep(t−s) e(t−s)

ln b d

≤ b−1 e−π(t−s) , where π = −(p +

ln b ), d

(23)

and we have used the conditions 0 < b < 1 and d = maxk∈N {tk+1 − tk }.

Let ξ = supτ ≤s≤0 V (t)τ . It follows from (22) and (23) that

−πt

υ(t) ≤ ξe

+

0

t

b−1 e−π(t−s) [qυ(s − τ (s)) + ]ds.

(24)

Now define w(ν) = ν − π + b−1 qeντ .

8

(25)

It follows from (13) that w(0) < 0. Since w(+∞) = +∞ and w(0) ˙ > 0, there exist a unique θ > 0 such that θ − π + b−1 qeθτ = 0.

(26)

Moreover, it can be derived from (13) that πb − q > 0. Hence υ(t) = V (0)τ ≤ ξ < ξe−θt +

, − τ ≤ t ≤ 0. πb − q

(27)

Next, we shall prove the following inequality: υ(t) < ξe−θt +

, t ≥ 0. πb − q

(28)

If the inequality (28) does not hold, there exists a t∗ such that

πb − q

(29)

, t < t∗ . πb − q

(30)

∗

υ(t∗) ≥ ξe−θt + and υ(t) < ξe−θt + It follows from (24), (25), (30) that −πt∗

∗

< < = <

t∗

∗

+ b−1 e−π(t −s) [qυ(s − τ (s)) + ]ds     0   t∗



∗ −πt −πs −θ(s−τ (s)) + + ds ξ+ q ξe e + e πb − q πb − q 0   t∗ t∗ π

−πt∗ −1 θτ (π−θ)s πs e ds + e ds e ξ+ + b qξe πb − q πb − q 0 0   1



−πt∗ −1 θτ (π−θ)t∗ πt∗ − 1) + e + b qξe (e (e − 1) ξ+ πb − q π−θ πb − q  



1 −1 θt −θt∗ 1 −1 θτ −πt∗ b qe ξe +e b qe ) − + ξ(1 − π−θ πb − q π−θ πb − q

∗ ξe−θt + πb − q

υ(t ) ≤ ξe <



(31)

Thus,

∗

υ(t∗ ) < ξe−θt + 9

πβ − q

(32)

which contradicts (29), and so (28) holds. Letting → 0, one derives form that

V (t) ≤ υ(t) ≤ ξe−θt = sup V (t)τ e−θt . τ ≤s≤0

(33)

Remark 1: In Theorem 1, the parameter Λ and A play an important role in designing the impulsive controller which are corresponding to the choose of ξi . Meanwhile, from (5), we conclude that the equilibrium point are also related to ξi . So for different ξi , depending on Theorem 1, we can design different impulsive controller to stabilize the state trajectories of the neural network to different equilibrium point. Remark 2: However, when the equilibrium point are assumed to be (0, 0), from (5), the only need is that Cf (0) + Df (0) + I = 0. So we can choose arbitrary ξi to design the impulsive controller. Corollary 1: Under Assumption 1, if the following inequality holds (p +

ln b )∗b+q < 0 d

(34)

where, p = max{λmax (−2Λ + 2I + LT L), λmax (I + AAT − 2B + CC T + DD T )}, q = λmax (LT L), 0 < max{αk2 , βk2} < b < 1, 0 ≤ tk+1 − tl ≤ d, ξi = arg min{p}, Cf (0) + Df (0) + I = 0, then the equilibrium (0, 0) of the inertial neural networks will be globally exponential stable under the impulsive control. Remark 3: For then given impulse strength b, the smaller the p is, the larger the impulse interval d is. Then we can choose ξi properly such that p is small which corresponding to the less conservativeness impulsive controller. For ξi = arg min{p}, by using some equation method, we can get the feasible solution. 4. Numerical examples In order to verify the effectiveness of the theoretical results, we will present one numerical example. Example 1: we consider the following delayed inertial BAM neural network with a1 = 1, a2 = 1, b1 = 1.5, b2 = 1.5, f1 (u1 (t)) = sin(u1 (t)), f2 (u2(t)) = cos(u2 (t)), τ (t) = et /(1 + et ), τ = 1,

10

I = 0. ⎧ d2 ui (t) ⎪ ⎪ = − dudti (t) − 1.5ui (t) + 0.3 sin(u1(t)) + 0.1 cos(u2 (t)) ⎪ 2 ⎪ ⎨ dt + 0.4 sin(u1 (t − τ (t))) + 0.2 cos(u1 (t − τ (t))), ⎪ ⎪ ⎪ ⎪ d2 ui (t) ⎩ = − dudti (t) − 1.5ui (t) + 0.2 sin(u1(t)) + 0.2 cos(u2 (t)) + 0.2 cos(u1 (t − τ (t))), dt2

(35)

choose ξ1 = ξ2 = 2 and αk = βk = 0.8. Write the above system in form (5), we can get the corresponding matrix ⎞ ⎞ ⎛ ⎛ 0.4 0.2 0.3 0.1 ⎠ ⎠, D = ⎝ C=⎝ 0 0.2 0.2 0.2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ξ1 0 0 2 0 b + ξ1 (ξ1 − a1 ) 3.5 0 ⎠=⎝ ⎠, A = ⎝ 1 ⎠=⎝ ⎠ Λ=⎝ 0 ξ2 0 b2 + ξ2 (ξ2 − a2 ) 0 2 0 3.5 ⎞ ⎛ ⎞ ⎛ 0 −1 0 a1 − ξ1 ⎠=⎝ ⎠ B=⎝ 0 a2 − ξ2 0 −1 Assume that μik = 0.64, based on Theorem 1, we can get that when tk − tk−1 < 0.0299, the neural network with impulsive effect can be globally exponential stable. Then we choose tk − tk−1 = 0.025. The state trajectories inertial neural networks without and with impulses are simulated in Fig. 1 and Fig. 2. Remark 4: It should be noted that most of the existing results such as [15]-[17] on stability problem of delayed inertial neural networks without impulses cannot be applied to equation (40) since it is originally unstable. But the stability can be effectively guaranteed via the impulsive control strategies in this paper. Example 2: we consider the following delayed inertial BAM neural network with a1 = 1, a2 = 1, b1 = 1.5, b2 = 1.5, f1 (u1 (t)) = sin(u1 (t)), f2 (u2(t)) = cos(u2 (t)), τ (t) = et /(1 + et ), τ = 1, I = 0, αk = βk = 0.8.. ⎧ d2 ui (t) ⎪ ⎪ = − dudti (t) − 1.5ui (t) + 0.3 sin(u1 (t)) + 0.1 cos(u2 (t)) ⎪ dt2 ⎪ ⎪ ⎪ ⎪ ⎨ + 0.4 sin(u1 (t − τ (t))) − 0.1 cos(u1 (t − τ (t))), ⎪ d2 ui (t) ⎪ = − dudti (t) − 1.5ui (t) + 2 sin(u1 (t)) − 0.2 cos(u2 (t)) ⎪ dt2 ⎪ ⎪ ⎪ ⎪ ⎩ + 1.5 sin(u1 (t − τ (t))) + 0.2 cos(u1 (t − τ (t))), 11

(36)

Based on Corollary 1, we can get when ξ1 = ξ2 = −0.8905, the result are less conservativeness. Write the above system in form (5), we can get the corresponding matrix

⎛ Λ=⎝

ξ1 0

⎛ ⎞ ⎛ ⎞ 0.3 0.1 0.4 −0.1 ⎠D = ⎝ ⎠ C=⎝ 2 −0.2 1.5 0.2 ⎞ ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ 3.1835 0 0 0 −0.8905 0 b + ξ1 (ξ1 − a1 ) ⎠=⎝ ⎠ ⎠=⎝ ⎠A = ⎝ 1 0 3.1835 ξ2 0 b2 + ξ2 (ξ2 − a2 ) 0 −0.8905 ⎞ ⎛ ⎞ ⎛ 0 1.8905 0 a1 − ξ1 ⎠=⎝ ⎠ B=⎝ 0 a2 − ξ2 0 1.8905

Assume that μik = 0.64, based on Corollary 1, we can get that when tk − tk−1 < 0.0523, the neural network with impulsive effect can be globally exponential stable. Then we choose tk − tk−1 = 0.05. The state trajectories inertial neural networks without and with impulses are simulated in Fig. 3 and Fig. 4. 5. Conclusions In this paper, we have investigated the issue of global exponential stability of inertial BAM neural networks with impulsive effects. Based on the Lyapunov functional method and the comparison principle, some new sufficient conditions are establish. Based on the results, the different impulsive controller can be designed to stabilize the different equilibrium point. In addition, a less conservatism impulsive controller can be designed when stabilize the zero equilibrium point. Acknowledgment

This publication was made possible by NPRP grant  NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. This work was also supported by Natural Science Foundation of China (grant no: 61374078, 61403313)

12

100 x1

80

x2 dx /dt

60

1

dx2/dt xi(t), dxi(t)/dt,i=1,2

40 20 0 −20 −40 −60 −80 −100

0

2

4

6

8

10

t

Figure 1: State trajectories of neural network (40) without impulses. 60 x x 40

1 2

dx1/dt dx /dt

xi(t), dxi(t)/dt,i=1,2

2

20

0

−20

−40

−60

0

0.5

1

1.5 t

2

2.5

3

Figure 2: State trajectories of neural network (40) with impulses.

References [1] J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, inProc. Nat. Academy Sci. United States Amer., vol. 81, no. 10, pp. 3088¨C3092, May. 1984. [2] L. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Syst., vol. 35, no. 10, pp. 1273¨C1290, Oct. 1988.

13

60 x1 x2

40

dx /dt 1

xi(t), dxi(t)/dt,i=1,2

dx2/dt 20

0

−20

−40

−60

0

1

2

3

4

5

t

Figure 3: State trajectories of neural network (41) without impulses. 60 x x 40

1 2

dx1/dt dx /dt

xi(t), dxi(t)/dt,i=1,2

2

20

0

−20

−40

−60

0

1

2

3

4

5

t

Figure 4: State trajectories of neural network (41) with impulses. [3] L. Chua and L. Yang, Cellular networks: Theory, IEEE Trans. Circuits Syst., vol. 35, no. 10, pp. 1257¨C1272, Oct. 1988. [4] M. Cohen and S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst., Man, Cybern., vol. 13, no. 5, pp. 815¨C826, Feb. 1983.. [5] B. Kosko, Bi-directional associative memories, IEEE Trans. Syst., Man, Cybern., vol. 18, no. 1, pp. 49¨C60, Jan.¨CFeb. 1988.

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