Scholarly article on topic 'The pth moment exponential stability of stochastic cellular neural networks with impulses'

The pth moment exponential stability of stochastic cellular neural networks with impulses Academic research paper on "Mathematics"

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Academic research paper on topic "The pth moment exponential stability of stochastic cellular neural networks with impulses"

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Thepth moment exponential stability of stochastic cellular neural networks with impulses

Xiaoai Li1*, Jiezhong Zou1 and Enwen Zhu2*

Correspondence: xiaoaili@csu.edu.cn; engwenzhu@126.com

1SchoolofMathematicsand Statistics, Central South University, Changsha, Hunan 410083, China

2School ofMathematics and Computational Science, Changsha University of Science and Technology, Changsha, Hunan 410004, China

Abstract

This paper studies the pth moment exponential stability of stochastic cellular neural networks with time-varying delays under impulsive perturbations. Based on the Lyapunovfunction, Razumikhin theory, stochastic analysis and differential inequality technique, criteria on the pth moment exponential stability of this model are derived. These results generalize and improve some of the existing ones. A numerical example illustrates the effectiveness and improvements of our results.

Keywords: pth moment exponentially stable; stochastic cellular neural network; impulses; Razumikhin theory

ringer

1 Introduction

Since Chua and Yang [1, 2] introduced a cellular neural network in 1988, it has received great attention because of its various applications such as classification of patterns, associative memories and optimization, etc. It should be pointed out that time delays are commonly encountered in real systems, which are the source of oscillation and instability both in biological and artificial neural networks, hence it is necessary and important to discuss the delayed cellular neural networks models. Up to now, many results on the stability of delayed neural networks have been developed [3-5]. In fact, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Therefore, noise is unavoidable and should be taken into consideration in modeling. Some recent results of stochastic cellular neural networks with delays can be found in [6-15].

On the other hand, it is noteworthy that the state of electronic networks is often subjected to some phenomenon or other sudden noises. On that account, the electronic networks will experience some abrupt changes at certain instants that in turn affect dynamical behaviors of the systems. Therefore, it is necessary to take both stochastic effects and impulsive perturbations into account on dynamical behaviors of delayed neural networks. In recent years, the dynamic analysis of neural networks with impulsive and stochastic effects has been an attractive topic for many researchers, and a large number of stability criteria of these systems have been reported; see [3, 4,10-12,16,17].

© 2013 Li et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

In [8], Sun et al. investigated the following stochastic cellular neural networks model with time-varying delays:

dxi(t) =

-CiXi(t) + aijfj(xj(t)) + bijgj(xj(t - T/(t))) + Ii

;=i j=i

+ °ii(t,Xi(t),Xi(t - Ti(t))) dwi(t), (1.1)

where xi(t) denotes the potential (or voltage) of a cell i at time t; A = (1,2,...,«}, n corresponds to the number of units in a neural network; fj(■), gj(-) are activation functions; ci > 0 denotes the rate at which a cell i resets its potential to the resting state when isolated from other cells and inputs; aij and bij denote the strengths of connectivity between cells i and j, respectively; Ii denotes the external bias on the ith unit, Ti(t) satisfies 0 < Ti(t) < t and it is a transmission delay. a(t,x,y) = (ail(t,xi,yi))nxm e Rnxm is the diffusion coefficient matrix and ai(t, xi, yi) = (ai1(t, xi, yi),...,aim(t, xi, yi)) is the ith row vector of a (t, x, y). w(t) = (w1(t), w2(t),..., wm(t))T is an m-dimensional Brownian motion defined on a complete probability space F,P) with a natural filtration (Ft}t>0.

They investigated the pth moment exponential stability with the help of the method of variation parameter and inequality technique, where p > 2 denotes a positive constant. More precisely, they established the following fundamental assumptions:

(H) For each j = 1,2,..., n, Tj(t) is a differentiate function, namely, there exists Z such that

Tj(t) < z <1.

(HI) Functions fj (■) and gj(-) are Lipschitz-continuous on R with Lipschitz constants Li >0, Ni > 0. That is, for all x,y e R, i e A,

fi(x)-fi(y) \ < Li|x -y|, \gi(x)-gi(y) \ < Ni\x -y|.

(H2) There exist nonnegative constants li, ei, such that for all x,y,x',y' e R, i e A, [ai(t,x',y') - ai(t,x,y)][a^t,x',y') - ai(t,x,y)]T < ei \x' -x\2 + li\y' -y\2.

Huang et al. [6] studied (1.1) and obtained the pth moment exponential stability by using Dini-derivative and Halanay-type inequality without assumption (H). When k[ > k2, the equilibrium point of the system (1.1) is pth moment exponentially stable, where

k1 = min pa -(p -1) V (Lj\aij\ + Njlbj) - V — - V (P -2)(p-1) ej

1<i<n i—' i—' —i i—' 2

"" I j=1 j=1 11 j=1

- £ j - £ ^^ 4

j=1 —i j=1 2 J

k2 = max NiY — \bji \ + — (p - 1)li , (.2)

M —i —i J

where —i (i e A) are positive constants.

Very recently, Li [11] generalized (1.1); he considered a stochastic cellular neural network under impulsive perturbations. The condition k[ > k'2 is also needed to ensure exponential stability in mean square.

We have a question whether the condition k > /<2 in the theorems [6,11,12] is an essential condition or not for the equilibrium point of (1.1) to bepth moment exponentially stable.

In this paper, we solve this question and obtain the improved version of the pth moment exponential stability by applying Lyapunov functions, Razumikhin theory and inequality technique. An example is also provided to illustrate the effectiveness of the new results.

2 Preliminaries

In this paper, we study stochastic cellular neural networks with impulses described by the delayed differential equations

dxi(t) = [-CiXi(t) + E"=i afjj)) + E"=i bijgj(xj(t - jt))) + Ii] dt

+ £ m (t, Xi (t), Xi(t - Ti(t))) dwi (t), t = tk, (2.1)

Axi{tk) = pik(x(tk)) = Xi(tk) - Xi(t-), k e Z+, i e A,

where {tk} is the time sequence and satisfies 0 = to < t1 < t2 < ••• < tk < tk+1 ••• , limk^TO tk = to; xt (s) = x(t + s), s e [-t ,0]. For k = 1,2,..., pik (x(tk)) represents the abrupt change of the state xi(t) at the impulsive moments tk. System (2.1) is supplemented with the initial condition given by

xto(s) = f (s), s e [-t,0], (2.2)

where ^ (s) is J"0-measurable and continuous everywhere except at a finite number of points tk, at which ^(t+) and ^(t-) exist and ^(t+) = ^(tk).

Let PC1,2([tk, tk+1) x R+) denote the family of all nonnegative functions V(t,x) on [tk, tk+1) x which are continuous once differentiable in t and twice differentiable in x. If V(t,x) e PC1,2([tk, tk+1) x R+), define an operator LV associated with (2.1) as

LV(t,x) = Vt(t,x) + J2 Vx>(t,x)

-CiXi(t) + J2 aijfj{xj(t)) + J2 hij&j{xj{t - Tj(t))) + Ii j=1 j=1

+ 2 trace [a TVxx(t, x)o ], (.3)

d V (t, x) d V (t, x) (d V (t, x) Vt (t, x) = ———, Vxi (t, x) = —--, Vxx(t, x) = '

dt ' dxi V dxidxj / nxn

Throughout this paper, the following standard hypothesis is needed: (H3) pi(xi(tk)) = -fak(xi(tk) - x*), where x* is the equilibrium point of (2.1) with the initial condition (2.2), ¡¡ik satisfies |1 - ¡3ik| < dk, dk is a positive constant.

Let yi(t) =Xi(t) -x, then (2.1) can be written by

dyi(t) = [-Ciyi(t) + £;=1 afj)) + ^ bijgj(yj(t - Tj(t)))] dt

+ E 7=i ôu (t, yi(t), yi(t - Ti(t))) dwi (t), t = tk, (2.4)

^yi(tk) = pik(y(tk)), k e Z+, i e A,

fj (yj(t)) =fj(yj(t) + j -fj{x",), gj j - Tj(t))) = gjj - Tj(t^ + j - gj(x-),

ay(t,yi(t),yi(t - Ti(t^) = aij(t,yi(t) + xi,yi(t - Ti(t^ + x^ - j,x*,x*), pik(y(tk)) = pik(y(tk) +x") -pik(x").

In the following, for further study, we first give the following definitions and lemmas.

||x|| denotes a vector norm defined by

||x||p = V |xi(t)|p, ||r = sup (s)|.

i=1 se[-r,0]

Definition 2.1 (Mao [18]) The equilibrium point x" = (x1,x"2,...,x"n)T of the system (2.1) is said to be pth moment exponentially stable if there exist X >0 and M >0 such that

£||x(t) -x*||p < M|xo -x"|pe-X(t-t0), Vt > to, Vxo e Rn,x(to)=xo.

In such a case,

limsup1 ln£(|x(t)-x"|p) < -X. (2.5)

t^œ t

The right-hand side of (2.5) is commonly known as the pth moment Lyapunov exponent of this solution.

When p = 2, it is usually said to be exponentially stable in mean square. Lemma 2.2 Ifai (i = 1,2, ...,p) denotep nonnegative real numbers, then

p p p a\ + a2 + ••• + ap

a1a2 ■ •• a„ <-, (.6)

1 2 p p

where p > 1 denotes an integer. A particular form of (2.6), namely

p-1 _ (p - 1)ap aP2 f , 0 o

a1 a2 <-+ —, forp = 1,2,3,.

1 2 p p

3 Main result

Theorem 3.1 Assume that (H1)-(H3) hold; furthermore, let

k = min |pci -(p -1) V (Ljlaijl +

1<i<n I 1 i n j=1

" p(p -1) (p-1)(p-2)

- laji Li--2— ei--2-^ | > 0,

j=1 2 2

k2 = max WiV |bji| + (p -1)li ,

1 </< w I < J I

(i) there exist a >0, X >0 such that -k1 + < a - X;

(ii) p ln dk-1 < -(a + X)(tk - tk-1), k e N, then the equilibrium point of (2.1) ispth moment exponentially stable.

Proof We define a Lyapunov function V(t,y(t)) = £n=1 |yi(t)|p = \\y(t)\\p. Let t > t0 and t e [tk-1, tk), then we can get the operator LV(t,y) associated with the system (2.4) of the form as follows:

LV(t,y) = |yi(t)|p-1 sgn(yi(t))

-ciyi(t) + aifj{yj(t)) + E bijjyfc- Tj(t)))

p(p -1)

EN^rE a2(t, yi(t), y+t - Ti(t)))

i=1 nn

< -pEci|yi(t)|p + pEElaij|Lj|yi(t)Г1|yj(t)|

i=1 i=1 j=1 nn

+ lbij|Nj|yi(t)|p-1|yj(t - Tj(t)) |

i=1 j=1

+ p(p^^|yi(t)r-2(ei|yi(t)|2 + li|yi(t - Ti(t))|2)

< -pj^ci k(t) T + E E laj -1) |yi(t) |p + |yj(t) D

i=1 j=1

+ E E | (p -1) |yi(t) |p + |yj(t - Tj(t^ n

i=1 j=1

+ ¿ei|yi(t)r + (p-1) E^ - 2)|yi(t)|p + 2|yi(t- Ti(t))|p) i=1 i=1

n n ( 1 )

pci -(p -1) £ (Ljlaijl + Wj|bj|) - E lajilLi - ei

|yi(t)|p

li|yi(t)|p + £ i=1 i=1

< -k1 V(t,y(t)) + k2 sup V(s,y(s)),

w^ lbjil + (p -1)ii

|yi(t - Ti(t)) |p

k = min |pCi -(p -1) V {Lj|aij| + Nj\bij\)

1 i n j=1

" p(p -1) (p-1)(p-2), \ \aji\Li--5-ei--5-li ,

(Ni£|

k2 = max] \ bji | + (p -1)li [.

Let y = infkeA -p—, there exist a >0,X >0 such that

-k1 + k2YeXT < -k1 + k2L_ < a - X (3.2)

ln y + Xt -(a + X)(tk - tk-1) > 0. (3.3)

Hence, we can choose M > 1 such that

e(a +X)(t1-t0) < M < yeXT. (.4)

For convenience, we denote that 0(s) = ty(s)- x* for s e [-t,0). It is obvious that

||<£||p < Mlpea(t1-t0) < M||<£||pe-X(t1-t0). (.5)

Now, we should prove

E||y(t)||p <M||^||pe-X(t-t0), Vt > t0. (.6)

Firstly, we prove when t e [to, t1),

EV(t,y(t)) = E||y(t)||p < M|mk.e-X(tl-to) < M||<£||pe-X(t-t0). (3.7)

If (3.7) is not true, there exists t e [t0, t1) such that

EV(t,y(t)) > Mmpe

pe-X(t1-10) > ||^Npea(t1-t0)

> ||0||p > EV(t0+ s,y(t0+ s)), s e [-t,0]. (3.8)

Since V(t,y(t)) is continuous on [t0, t1), which implies that there exists t e [t0, t) such that EV (t, y(t)) = M|mk.e-X(tl-to)

EV(t,y(t)) < EV(i,y(t)), Vt e [to - r, i), then there exists some t e [to, i) satisfying

ev (t, y(t)) = mp

EV (t, y(t)) > EV (t, y(t)), Vt e (t, i). Hence, for any s e [-r,0], t e (t, i),

EV (t + s, y(t + s)) < EV (i, y(t)) < y eXr 110 Hp?e-A(tl-to) < y eÀr ||milp < y eÀr EV (t, y(t)). By (3.l) and (3.2), we get

ELV(t,y) < (-/ci + k2y ekx)EV(t,y(t)) < (a - X)EV(t,y(t)), Vt e (t, i). Then

EV(i,y(t)) < EV(t,y(t))e(a-A)(i-i) < limilpea(tl-to) < MlimilPe-A(tl-to) = EV(i,y(t)),

which is a contradiction. Hence, (3.7) holds. Next, we will show

EV(t,y(t)) < Mlim ||pe-A(t-to), t e [t/-i, t/), k e A. (3.9)

Assuming (3.9) holds for k = l, 2,..., m, we shall show that it holds for k = m + l, i.e.,

EV(t,y(t)) = E\y(t)\p < MymiPe-A(t-to), Vt e [tm, tm+l). (3.lo)

Suppose (3.lo) is not true. Then we define t e [tm, tm+l) such that

EV (t, y(t)) > Mlimilpe-k(t-to). From (H3) we get

EV (tm, y(tm)) = £ E|yi (tm) + Pi(yi(tm) + **)|P = E^ jWim j^itmW

i=l i=l n

< Ej2dpm\yi(tm) \P = dPmEV(tm,y(tm)) < dmMlimilPe -A(tm- to)

= dmMlimilM^e-À(t-to) < dmMimiPPeÀ(tm+l-tm)e-À(t-to)

< MlimilPe-Art-to) < MlimilPe-A(tm-to),

which implies t e (tm, tm+1). Let

t = inf{t e (tm, tm+1) :EV(t,y(t)) = MUre^),

then for t e (tm - t, t), we can get EV(t, y(t)) < EV(t, y(t)). Hence, there exists t' e (tm, t) such that

EV (t, y(t')) = dpmM\\^\\pxe-X(i-t0)eX(tm+1-tm)

EV(t,y(t)) > EV(t,y(t*)), t e [t", t).

On the other hand, for any t e [t*, t), s e [-t, 0], either t + s e [tm - t, tm)or t + s e [tm, t]. If t + s e [tm - t, tm), we can obtain

EV(t + 5,y(t + s)) <M||<£||me-À(t+s-to) <M||<£||meÀTe-À(t-to)eA(tm+1-tm) = —EV(t,y(t")).

If t + s e [tm, t], we can get

_ _ e—

EV(t + s,y(t + s)) < EV(i,y(i)) <-mEV(t,y(f )).

Then for any s e [-r ,0], we get

gkx gkx

EV(t + s, y(t + s)) < ~fEV{t, y(t)) < —pEV(t, y(t)), t e [t',"t). Hence,

/ k2eXr \

LV (t, y) < i-k1 + kL—\EV(t, y(i) ) < (a - X)EV(t, y(t)). Then

EV(t,y(t)) < EV(f ,y(t*))e(a-À)(t-t*) = d^My^ypme-Art-to)eA(tm+1-tm)e(a-A)(i-t*). From the condition (ii), it is obvious that

dm ex(tm+1-tm) e(a -A)(t-t*) < 1.

Hence,

EV(t,y(t)) <My^ype-A(t-to) = EV(t,y(t)), which is a contradiction. Hence, (3.10) holds.

By induction, we can obtain that (3.9) holds for any k e A, i.e.,

EV(t,y(t)) < M\mPpe-x(t-to), Vt > t0,

which implies that the equilibrium point of the impulsive system (2.1) is pth moment exponentially stable. This completes the proof of the theorem. □

Theorem 3.2 Assume that (H1)-(H3) hold, fii(i e A)> 0,

(i) if there exist a >0, k >0 such that -k1 + k2 dpkl < a - k;

(ii) pdk-1 < -(a + k)(tk - tk-1), k e N, where

k = min Ipci -(p -1) £ {Lj\atj\ + Nj\btj\)

" I j=i

sr^H, ,, P(P -1) (P-1)(P-2), _

~ aL--~-e--~-h \ >0,

j=i !M 2 2 J

k2 = max Ni}— \ bji \ + (p -1)/; ,

then the equilibrium point of the system (2.1) is pth moment exponentially stable.

Proof Let V(t,y(t)) = 22n=1 lxi \yi (t) \p, the proof of the theorem is similar to that of Theorem 3.1 hence it is omitted. □

Corollary 3.3 Assume that (H1)-(H3) hold, /xi(i e A)> 0,

In n 1

2ci - V (Lj \ aij \ + Nj \ bij \ ) - V ^ \ aji \ Lt - eA >0,

M M ^ i

k2=mg{Ni t % b\+l],

(i) if there exist a >0, k >0 such that -k1 + — a - k;

(ii) 2lndk-1 < -(a + k)(tk - tk-1), k e N,

then the equilibrium point of the system (2.1) is exponentially stable in mean square.

Remark 3.4 In many stability results for stochastic cellular neural networks, ELV — 0 is an important condition for their conclusions [13-15], which means that the origin systems without impulses need to be stable. However, by constructing the impulses, we do not need this condition to ensure the equilibrium point of the impulsive system (2.1) is pth moment exponentially stable. Our results show that impulses play an important role in the pth moment exponential stability for the stochastic cellular neural network with time delay, even if the corresponding systems may be unstable themselves. It should be mentioned that our results develop an effective impulse control strategy to stabilize underlying retarded cellular neural networks. And it is particularly meaningful for some practical applications.

Remark 3.5 It is important to emphasize that, in contrast to some existing exponential stability results, see [6,11,12,19], the condition k1 > k2 is needed to ensure the equilibrium point of the system (2.1) is pth moment exponentially stable, while in our paper we omit it and obtain the results.

4 Illustrative example

In the following, we will give an example to illustrate the advantages of our results. Example 1 Consider the following model:

dxi(t) = [-cixi(t) + £j=1 aijfj(xj(t)) + Xj=1 bijgj(xj(t - Tj(t)))] dt

+ ai(t,xi(t),xi(t - Ti(t))) dw(t), t = tk, (4.1)

xi(tk) = (1 - ¡¡ik)xi(tk)k e Z+, i e A = (1,2),

wheref(x) =gi(x) = tanh(x), 0 < Ti(t) < t = 0.5, tk - tk-1 = 0.1, x1(tk) = ^x^, x2(tk) = X2|k-).

C2x1 = (2), KO2*2 = (°;3 0°^), (bij)2x2 = ( _002L 0°^).

Obviously, Li = Ni = 1, ei = 0, l, = 1 (i = 1,2), ¡¡1k = 2/3, ¡¡2k = 3/4.

Let dk = 2/3, a = 3.4, X = 0.2, ¡1 = ¡2. Then forp = 2, we can get k1 = min(1.4,1.2) = 1.2 > 0, k2 = max(1.3,1.7) = 1.7, -k1 + k2 dp = -1.2 + 1.7 x = 3.027 < a - X = 3.4 - 0.2 = 3.2, 2lndk-1 = 2ln(2/3) = -0.811 < -(a + X)(tk - th) = -3.6 x 0.1 = -0.36.

All conditions of Corollary 3.3 are satisfied, then the equilibrium point is exponentially stable in mean square.

Remark 4.1 If ¡1 = ¡2, we have computed k1 = 1.2, k2 = 1.7. If ¡1 = ¡2, set ¡2 = a, then

k1 = min{1.7 - 0.3a,2 - 0.8/a}, k2 = max{1.2 + 0.1a, 1.4 + 0.3/a}. If ¡¡1 < ¡2, then a > 1, we can compute k1 < 1.4 and k2 > 1.4; if ¡1 > ¡2, then 0 < a < 1, k1 < 1.2 and k2 > 1.4. Hence, in either case, we always have k1 < k2, so the exponential stability in mean square of the system (4.1) cannot be derived by applying the corresponding exponential stability result for cellular neural networks given in the literature [6,11,12,19], since k1 > k2 is not satisfied.

Remark 4.2 Since p [C^MM^C+MM2K+NN1 + WW2)] = 33.635, where p [C-1(MM1K + MM2K + NN1 + NN2)] was defined in [8], the condition p[C-1(MM1K + MM2K + NN1 + NN2)] < 1 is not satisfied. Hence, the results in [8] are useless to judge the exponential stability of the system (4.1).

Competing interests

The authors declare that they have no competing interests. Authors' contributions

XL completed the proof and wrote the initialdraft. JZ gave some suggestions on the amendment. XL then finalized the manuscript. Correspondence was mainly done by EZ. Allauthors read and approved the finalmanuscript.

Acknowledgements

The authors would like to thank the editor and the anonymous referees for their detailed comments and valuable suggestions which considerably improved the presentation of this paper. This work was supported in part by the NationalNaturalScience Foundation of China under Grant No. 11101054, the Hunan ProvincialNaturalScience Foundation of China under Grant No. 12jj4005, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grant No. 11FEFM11 and the Scientific Research Funds of Hunan ProvincialScience and Technology Department of China under Grant No. 2012SK3096.

Received: 7 September 2012 Accepted: 18 December 2012 Published: 9 January 2013

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doi:10.1186/1687-1847-2013-6

Cite this article as: LI et al.: The pth moment exponential stability of stochastic cellular neural networks with impulses. Advances in Difference Equations 2013 2013:6.

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