Scholarly article on topic 'Dynamic analysis of stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays'

Dynamic analysis of stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays Academic research paper on "Mathematics"

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Academic research paper on topic "Dynamic analysis of stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays"

Bao Advances in Difference Equations (2015) 2015:196 DOI 10.1186/s13662-015-0431-9

0 Advances in Difference Equations

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Dynamic analysis of stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays

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Hongmei Bao*

*Correspondence:

baohmmath@126.com

Faculty of Mathematics and Physics,

Huaiyin Institute of Technology,

Huai'an, Jiangsu 223003, P.R. China

Abstract

This paper is concerned with the problem of stochastic stability for a class of fuzzy Cohen-Grossberg neural networks, in which the interconnections and delays are time-varying. Based on a Lyapunov function and the Ito differential formula, a set of novel sufficient conditions on the pth moment exponential stability of the equilibrium of the system is derived. Moreover, an illustrative example is given to demonstrate the effectiveness of the results obtained.

Keywords: fuzzy Cohen-Grossberg neural networks; global pth moment exponential stability; time-varying delays; Ito differential formula

ringer

1 Introduction

Recently Cohen and Grossberg neural networks [1] have been extensively studied and applied in many different fields such as associative memory, signal processing, and some optimization problems. In such applications, it is of prime importance to ensure that the designed neural networks are stable [2]. In practice, due to the finite speeds of the switching and transmission of signals, time delays do exist in a working network and thus should be incorporated into the model equation [3-12]. In addition to the delay effects, studies have been intensively focused on stochastic models [13-18]. It has been realized that the synaptic transmission is a noisy process brought about by random fluctuations from the release of neurotransmitters and other probabilistic causes, and it is of great significance to consider stochastic effects on the stability of neural networks described by stochastic functional differential equations.

Stochastic effects constitute another source of disturbances or uncertainties in real systems. A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems or sudden environment switching. Therefore, stochastic perturbations should be taken into account when modeling neural networks. In recent years, the dynamic analysis of stochastic systems (including neural networks) with delays has been an attractive topic for many researchers, and a large number of stability criteria of these systems have been reported; see e.g. [19-26] and the references therein.

© 2015 Bao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly credited.

In this paper, I would like to integrate fuzzy operations into Cohen-Grossberg neural networks. Speaking of fuzzy operations, Yang and Yang [27] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have found that FCNNs are useful in image processing, and some results have been reported on the stability and periodicity of FCNNs [26-34].

However, to the best of my knowledge, few authors have considered the problem of the pth moment exponential stability and almost sure exponential stability of stochastic nonautonomous fuzzy Cohen-Grossberg neural networks. In fact, in the process of the electronic circuits' applications, assuring a constant connection matrix and delays is unrealistic. Therefore, in this sense, time-varying connection matrix and delays will be better candidates for modeling neural information processing.

Motivated by the above discussions, this paper is concerned with the following stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays:

dxi(t) = -at(xi(tj)

b,(x,(t)) -J2 cij(t)fj(xj(t)) - /\ aij(t)g(x(t - Tj(t)))

j=i j=i

-V ßj'(t)gj(xj(t - Tj(t)))+ Ii(t)

dt + Oij(xj(t)) doj(t). (1)

For i = 1,2,..., n, where n corresponds to the number of units in the neural networks, xi(t) corresponds to the state of the ith neuron. f (-), gj(•) are signal transmission functions. Tj(t) corresponds to the time delay required in processing and satisfies 0 < Tj(t) < t (t is a constant). ai(xi(t)) represents an amplification function at time t. bi(xi(t)) is an appropriately behaved function at time t such that the solutions of model (1) remain bounded; cij(t) represents the elements of the feedback template. Ii(t) = ~Ii(t) + /\jn=1 Tij(t)uj(t) + V;=1 Hij(t)uj(t). aij(t), pij(t), Tj(t), and Hij(t) are elements of the fuzzy feedback MIN template and the fuzzy feedback MAX template, fuzzy feed-forward MIN template, and fuzzy feed-forward MAX template, respectively; /\ and V denote the fuzzy AND and fuzzy OR operation, respectively; Uj(t) denotes the external input of the ith neurons. ~U(t) is the external bias of the ith unit. aij(•) is the diffusion coefficient, ai = (ai1,ai2,...,ain): a>(t) = (<y1(t),M2(t),...,Mn(t))T is an n-dimensional Brownian motion defined on a complete probability space (fi, F, {Ft}t>0, P) with a filtration {Ft}t>0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets).

Obviously, model (1) is quite general, and it includes several well-known neural networks models as its special cases such as Hopfield neural networks, cellular neural networks, and bidirectional association memory neural networks [12, 24]. There are at least three different types of stochastic stability to describe the limiting behaviors of stochastic differential equations: stability in probability, moment stability, and almost sure stability (see [23, 35]). When designing an associative memory neural network, we should make the convergence speed as high as possible to ensure the quick convergence of the network operation. Therefore, pth moment (p > 2) exponential stability and almost sure exponential stability are most useful concepts as they imply that the solutions will tend to the trivial solution exponentially fast. This motivates us to study the pth moment exponential stability for system (1).

The rest of this paper is organized as follows. In Section 2, the basic assumptions and preliminaries are introduced. In Section 3, the criterion for the pth moment (p > 2) ex-

ponential stability for system (1) is derived by using the Lyapunov function method and Ito differential inequality. An illustrative example is given in Section 4. Conclusions are drawn in Section 5.

2 Preliminaries and some assumptions

For convenience, we introduce several notations. Let C = C((-x, 0],Rn) be the Ba-nach space of continuous function which map into Rn with the topology of uniform convergence. For any x(t) = (x1(t),x2(t),...,cn(t))T e Rn, we define ||x(t)|| = ||x(t)||p =

(En=i Mt)ip)p (i<p < ().

The initial conditions for system (1) are x(s) = p(s), -t < s < 0, p e L]p0((-r,0],Rn), where LP ((-t,0],Rn) is Rn-valued stochastic process p(s), -t < s < 0, p(s) is F0 measurable, f_T £[|p(s)ip] ds < x. Throughout the paper, we make the following assumptions. (Al) There exist positive constants ai such that

0<a < ai(x) <ai, Vx e R,i = 1,2, ...,n. (2)

(A2) The signal transmission functionsf (-), gj(•) (j = 1,2,..., n) are Lipschitz

continuous on R with Lipschitz constants fj and Vj, namely, for any u, v e R,

fj(u)-fj(v)\< ju - v|, |gi(u) -gi(v)| < Viiu - v|, f(0) = g(0) = 0. (A3) bi(•) e C(R, R) and there exist positive constants hi such that bi(u)-bi(v)

> hi, Vu = v,i = 1,2,...,n.

(A4) a(x(t)) = (aij(xj(t)))nxn (i, j = 1,2,..., n), there exist nonnegative numbers si, i = 1,2,..., n, such that

trace[aT(x)a(x)] < ^six2. (3)

Remark 2.1 The activation functions are generally assumed to be continuous, differen-tiable, and monotonically increasing, such as the functions of sigmoid type. These restrictive conditions are no longer needed in this paper. Instead, only the Lipschitz condition is imposed in assumption (A2). Note that the type of activation functions in (A2) has already been used in numerous papers.

If V(t,x) e C2,1([-t, x) x Rn;R+), according to the Ito formula, we define an operator LV associated with (1) as

LV(t,x) = Vt(t,x) + J^ Vx;(t,x) j -aixi(t)

bi(xi(t)) (j(t)fj{xj(t))

y ____^ _ )) dt\

j=1 J J

- A aij (t)gj(xj(t - Tj (t))) -\J Pij(t)gj(xj(t - Tj(t))) + Ii(t) j=1 j=1

+ 22 trace[a TVxx(t, x)a],

v (tx)- dJVM V (tx)- dVM V <tx)J9v(t,x)

Vt (t,x) - -—-> Vxi(t,x) - ---> vxx(t,x) -

dt ' dxi V dxi dxj , nyn

Definition 2.1 The equilibrium x* of system (1) is said to be globalpth moment exponentially stable, if there exist positive constants M > 1, X > 0 such that

£(||x(t)-x*||p) < M||^ - x*\\pipe-x(t-t0\ t > to, Vxo e Rn, (4)

where x(t) - (x1(t), x2(t), ...,xn(t))T is any solution of model (1), p > 2 isa constant; when p - 2, it is usually said to be exponential stability in mean square.

Lemma 2.1 [34] Suppose x and y are two states of system (1), then we have

A aij(t)gj(x) - A aij(t)gj(y)

j-1 j-1

\Z Pij(t)gj(x)-\J Pij(t)gj(y)

j-i j-i

< J2\aij(t) \ \gj (x)-gj(y)\

\Pij(t)\\gj(x) -gj(y)\.

Lemma 2.2 Ifai > 0 (i = 1,2,..., m), denotep nonnegative real numbers, then

a, + a2 + ••• + am /

aa •••am <-, (5)

where p > 1 denotes an integer. A particular form of (5) is

p-1 ^(p - 1K ap

a1 a2 <-+ —.

3 Main results

In this section, we will consider the existence and the global pth moment exponential stability of system (1).

Theorem 3.1 Under condition (A1)-(A4), if there exist a positive diagonal matrix D = diag(di, d2,..., dn) and two constants 0 < N2, 0 < u <1, such that

0 < N2 < N2 (t) < uN (t), t > t0,

pahi ai(p -1) I cij(t) \Pj a \ cji(t) I Pj

j=1 j=1

-J2 ai(p-1)(\aij(i)\ + \Pij (t)|) Vj

" (p - 1)(p-2) "dj I -£-2-s -£ d{p-1)si ¡,

N2(t) = max v -jai(\cnj(t)\ + \Aj(t)\)Vj,

1<i<n di - - j=1 i

then x* = (x*, x2,...,x")t is a unique equilibrium which is globally pth moment exponentially stable, where p > 2 denotes a positive constant. When p = 2, the equilibrium x* of system (1) has exponential stability in mean square.

Proof The proof of the existence and uniqueness of the equilibrium for system is similar to that of [33]. So we omit it.

Suppose that x* = (x*,x*,...,x")t is the unique equilibrium of system (1). Set yi(t) = xi(t) - x*, aj(yj(t)) = aij(yj(t) + x*) - ajx*), then system (1) can be transformed into the following equation:

dyi(t) = -ai(yi(t)+x*

bi(yi(t) + x*) - bi(x*)

-£ Cij(t)tfj(?j(t)+xJ) -fjj)

A aij (t)gjjt - Tj(t)) + j -/\ cnj(t)gj(x*

j=1 j=1

- V Pji(t)gj{yj{t-Tj(t)) + x*) - \//3ji(t)gj(x*

+ £ aij{yj(t)) djt), t > t0, i = 1,2,..., n.

Consider the following Lyapunov function:

V(t,y) = £di \yi(t) \p = £ di \xi(t) -x* \p, p > 2.

Calculating the operator LV(t,y(t)), and using Lemma 2.2, associated with system (6), it has the form

LV(t,y(t)) = pj^di\yi(t)\P-1 sgn{yt(t)\ -ai(yi(i) + x*)

-£ cij (t)(fj(yj(t) +x*) ^./^(xjr)) j=1

- Aaij(t)gj(yj(t - Tj(t)) + xj") -/\ aij(t)gj(x

b^yi(t)+x*) - bi(x*)

- V Pji(t)gj(yj{t - Tj(t)) + x*) - \/ Pji(t)gj{xj)

v=1 j=1

+^ l \yi(t)\p-2 t -Mt))

i=1 j=1

< -pJ2di\yi(t)r-1ai(y;(t) + xj)hiyi(t)sgn {yi(t)}

+ pY^di \yi(t)\p-1a^yi(t)+x^^ cij (t)fj(yj (t)) sgn{yi(t)}

i=1 I j=1

+ E\aij(t) \ kj - Tj(t)) + xj) -gj(xj) \ sgn{yi(t)}

E! j) \ Ls^CyjCt - +xj) - jx;) \ sg^{ yi(t)}

2 J2diypr2(t)J2rf sgn{yi(t)}

p(p -1) v^ p-2

i=1 j=1

/-l./wl . „¿jl. /aip-^

< -p^di\yi(t)\p- aihi\yi(t)\ + p^di^t)^- a^\cij(t)jjyj(t>j

+ pj^di \yi(t) ^^(\aij(t)\ + \Aj(t)DVj\yj(t)\

i=1 p(p -1)

^di\yi(t) sj)

= - dApOihi -J2ai(p -1) \cij(t) \p, -J2aj\ j(t) \p,

i=1 I j=1 j=1

- ¿a (p - 1)(\aij (t)\ + \Pij(t)\)vj - £ (p-1))(p-2) sj

-£ d(p -1si }\yi(t)r

+ E diJ2iMM^ + P (t)D vly^t - ti(t)) \p

i=1 j=1 di

< -N1(t)V(t,y(t)) + N2(t) sup V(s,y(s)),

N1 (t) = mi^ |pahi - ^ai(p -1) \cij(t) \pj - ^aj\ j(t) \Pj

- ¿afc - 1)(\aij(t) \ + \pii (t)\) Vj - £ p-1))(p-2) s,

-l d (p -1)si,

N2(t) = ma^V dat(\aij(t)\ + \Pij(t)\)v,.

1<i<n ^ u;

- - ,=1 i

j £ d

Applying the Ito formula, for t > t0,we obtain V(t + 5,y(t + 5)) - V(t,y(t))

pt+5 pt+5

= LV(s,y(s)) ds + / Vy(s,y(s))a(s,y(s)) do(s). (9)

Since E[Vx(s, y(s))a (s, y(s)) do(s)] = 0, taking expectations on both sides of the equality (9) and applying the inequality (8) yields

E(V(t + 5, y(t + 5)))- E(V(t, y(t)))

< i \-N1(t)E(V(s,y(s))) + N2(t)E( sup V(0,y(0)))! ds. (10)

Jt L s-T <0 <s

The Dini derivative D+ is

r+rtvt* u\W r E(V(t + 5,y(t + 5)))-E(V(t,y(t)))

D+EV i,y(t))) = lim sup---. (11)

5^0+ 5

Denote z(t) = E(V(t,y(t))), and (10) leads directly to

D+z(t) < -N1(t)z(t) +N2(t)|zt||p. (2)

Hence, from Lemma 3.2 of [22], we have

z(t) < ||z(t0)||pe-A(t-t0). Namely,

E[|x(t) -x*||P] < M\p -x* ||Pe-A(t-t0), t > to, where M = mXt'T^d-} > 1. X is the unique positive solution of the following equation: X = N1(t)-N2(t)eAT.

Therefore the equilibrium x* of system (1) is pth moment exponentially stable. The proof is completed. □

Specially, suppose that cij(t) = cij, aij(t) = aij, jt) = fy (i,j = 1,2,...,n); system (1) becomes the stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays,

dxi(t) = -ai(xi(t))

bi{xi(t)) - J2 cijfj{xj(t)) - /\ aijgjj - Tj(t))) j=1 j=1

-\/ Pjigj{xj{t - Tj (t))) + Ii(t)

dt + ^ aij(xj (t)) doj(t). (13)

For (13), we have the following corollary by Theorem 3.1.

Corollary 3.1 If assumptions (A1)-(A4) hold, and there are constants Ni > 0 (i = 1,2), 0 < u <1 such that

0 < N2 < uN1,

N1 = min I pafii - y at(p - 1)\ctj\pj - V* aj\cM

1<i<n I L—' L—'

"" I j=1 j=1

- ¿«i(p -1)(\aj \ + \jvj - j (p-1))(p-2) j - itd (p-1)s^,

j=1 j=1 2 j=1 di i

N2 = max V"ai(\aj\ + \hij\)vj,

1< i< n di

- - j=1 i

then the unique equilibrium x* = (x* , X2,..., Xn )T of system (13) is globally pth moment exponentially stable.

Remark 3.1 The delay functions T-(t) considered in this paper only need to be bounded and can be nondifferential. This generalized some published results in [20]. It should be noted that the stability of system (1) is dependent on the magnitude of noise, therefore, stochastic noise fluctuation is one of the very important aspects in designing a stable network and should be considered adequately.

Remark 3.2 Compared with [20, 21], the method in this paper does not resort to the semimartingale convergence theorem. Since system (1) does not require the delays to be constants, furthermore, the model is nonautonomous and includes fuzzy operation, it is clear that the results obtained in [12,14, 20-23] cannot be applicable to system (1). This implies that the results of this paper are essentially new and complement some corresponding ones already known.

4 An example

Example 4.1 Consider the following impulsive stochastic fuzzy neural networks with time-varying delays and distributed delays:

dXi(t) = -(3 + COSXx(t))[11Xx(t) - cn(t)fi(Xi(t)) - c12(tf2(X2(t))

- A'=1 av(t)gj(Xj(t - Tj(t)) + I1(t) + Aj=1 T-(t)uj(t)

- V'=1 hj(t)gj(Xj(t - T-(t)))) + V?=1 Hj(t)u-(t)] dt + on(X1(t)) d«1 + 012 (X2 (t)) d«2,

dX2(t) = -(2 + SinX2(t))[17X2(t) - c21(t)f1(X1(t)) - c22(t)f2(X2(t))

- Aj=1 a2j(t)gj(Xj(t - Tj(t))) + I2 (t) + Aj=1 T2j(t)uj(t)

- V2=1 h2j(t)gj(Xj(t - Tj(t))) + V|=1 H2j(t)uj(t)] dt + ff21(X1(t)) d«1 + CT22(X2(t)) d«2,

Tj(t) = 0.3| sin 11 +0.1, i, j = 1,2,

a21(x) = 0.1x, a22(x) = 0.2x, Ii (t) = 2 + 3t (i, j = 1,2).

Obviously, system (14) satisfies assumptions (A1)-(A3) with

a =2, a1 = 4, a2 = 1, a2 = 3, ¿1=11, Ä2=17, ß = Vi = 1 (i = 1,2).

It can easily be checked that the assumption (A4) is satisfied with s1 = 0.05, s2 = 0.13. Let p = 2. It is easy to compute N1 = 19.97, N2 = 10. There exists a positive number 0 < u = 0.6 < 1 such that 0 < N2 = 10 < uN1 = 0.6 x 19.97 = 11.98. Clearly, all conditions of Corollary 3.1 are satisfied. Thus system (14) has a unique equilibrium point x* which is globally mean square exponential stable.

5 Conclusions

In this paper, we have studied the existence, uniqueness, and pth moment exponential stability of the equilibrium point for stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays. Some sufficient conditions set up here are easily verified and these conditions are correlated with parameters and the magnitude of noise the system (1). The obtained criteria can be applied to design globally mean square exponentially stable fuzzy Cohen-Grossberg neural networks.

Competing interests

The author declares that she has no competing interests. Author's contributions

The author performed all tasks of this research: drafting, the design of the study, and writing and revision of paper. Acknowledgements

The author would like to thank the editor and anonymous reviewers for their helpful comments and valuable suggestions, which have greatly improved the quality of this paper.

Received: 30 September 2014 Accepted: 2 March 2015 Published online: 26 June 2015 References

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