Nonlinear DynDOI 10.1007/s11071-014-1375-4

ORIGINAL PAPER

Global Mittag-Leffler stability and synchronizationof impulsive fractional-order neural networks withtime-varying delays

Ivanka Stamova

Received: 6 November 2013 / Accepted: 15 March 2014© Springer Science+Business Media Dordrecht 2014

Abstract In this paper we consider a class of impul-sive Caputo fractional-order cellular neural networkswith time-varying delays. Applying the fractional Lya-punov method and Mittag-Leffler functions, we givesufficient conditions for global Mittag-Leffler stabilitywhich implies global asymptotic stability of the net-work equilibrium. Our results provide a design methodof impulsive control law which globally asymptoticallystabilizes the impulse free fractional-order neural net-work time-delay model. The synchronization of frac-tional chaotic networks via non-impulsive linear con-troller is also considered. Illustrative examples aregiven to demonstrate the effectiveness of the obtainedresults.

Keywords Global Mittag-Leffler stability ·Synchronization · Neural networks · Fractional-orderderivatives · Time-varying delays · Impulsive control ·Lyapunov method

1 Introduction

Neural networks have received increasing interest dueto their impressive applications in many areas of sci-ence and engineering [1,2]. Cellular Neural Networks

I. Stamova (B)Department of Mathematics, The University ofTexas at San Antonio, San Antonio,TX 78249, USAe-mail:ivanka.stamova@utsa.edu

(CNNs) were first introduced by Chua and Yang in 1988[3,4] as a novel class of information-processing sys-tems. Moreover, it is necessary to solve some adap-tive control theory, optimization, linear and nonlin-ear programming, associative memory, pattern recog-nition, and computer vision problems by using DelayedCellular Neural Networks (DCNNs) [5–7]. Such appli-cations heavily depend on the dynamic behavior ofnetworks; therefore, the qualitative analysis of thesedynamic behaviors is a necessary step for practicaldesign of neural networks.

On the other hand, many processes in DCNNs arecharacterized by the fact that at certain moments of timethey experience a change of state abruptly. The abruptchanges in the voltages produced by faulty circuit ele-ments are examples of impulsive phenomena that canaffect the transient behavior of the network. It is nowrecognized that real world phenomena that are subjectto short-term perturbations whose duration is negligiblein comparison with the duration of the process, are moreaccurately described using impulsive differential equa-tions. The study of impulsive integer-order DCNNs isof great importance and has gained considerable popu-larity in recent years. See, for example [8–14] and thereferences therein. Also, since impulsive control arisesnaturally in a wide variety of applications, differentimpulsive control approaches have been proposed inspatiotemporal chaos [15], a nonlinear oscillator witha non-symmetric potential [16], a quarter car modelforced by a road profile [17], as well as in DCNNsapplications [18–20].

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I. Stamova

In the past decade, a great progress in studyingfractional evolution models has been made. Indeed,fractional-order modeling has come to play a crucialrole in many fields such as physics, polymer rheology,regular variation in thermodynamics, biophysics, bloodflow phenomena, aerodynamics, electro-dynamics ofcomplex medium, viscoelasticity, capacitor theory,electrical circuits, electron-analytical chemistry, biol-ogy, control theory, fitting of experimental data, etc.[21–23]. Nevertheless, only relatively recently func-tional differential equations of fractional order havestarted to receive an increasing interest [24–26]. Inrelation to the mathematical simulation in chaos,fluid dynamics and many physical systems, recentlythe investigation of impulsive fractional differentialequations began [27–29]. Some stability results forfractional-order models, which are fundamental to allcontrol systems, are reported in [30–33]. Synchroniza-tion, has also received a great deal of interest amongscientists from various fields in the last few years, espe-cially in fractional-order chaotic systems [34–36].

Also, fractional calculus has been incorporated intoCNNs. The synchronization problem is studied in [37]for a class of fractional-order chaotic neural networks.By using the Mittag-Leffler function, M-matrix andlinear feedback control, a sufficient condition is devel-oped ensuring the synchronization of such neural mod-els with the Caputo fractional derivatives. In [38] afractional-order four-cell cellular neural network isproposed, and the complex dynamical behaviors ofsuch a network are investigated by means of numer-ical simulations. Stability and multi-stability, bifurca-tions and chaos of fractional-order neural networks ofHopfield type were investigated in [39]. In [40] theauthors investigated generalized chaotic synchroniza-tion of the general fractional-order weighted complexdynamical networks with nonidentical nodes. In [41]α-stability and α-synchronization for fractional-orderneural networks are investigated. The authors pointedout that fractional order recurrent neural networks areexpected to be very effective in applications such asparameter estimations due to the fact they are char-acterized by infinite memory. In [42] an algorithm ofnumerical solution is presented for fractional differen-tial equations. The study and experiment indicated thatthe chaos in fractional order neuron networks could becontrolled and synchronized. However, in spite of thegreat possibilities of applications, the investigations onthe fractional-order neural networks with delays is still

in the initial stage. Some stability results for fractional-order DCNN with constant delays have been presentedin [43] and [44]. Usually, constant fixed time delays inthe models of delayed feedback systems serve as goodapproximation in simple circuits having a small num-ber of cells. But, in most situations, neural networkshave a spatial extent due to presence of a multitudeof parallel pathways with a variety of axon sizes andlengths. Thus it is common to have time-varying delays.To the best of our knowledge, the problem for stabil-ity and synchronization of impulsive fractional-orderDCNNs with time-varying delays has not been studiedpreviously.

For extending the application of fractional calculusin nonlinear systems, Podlubny and his co-authors pro-posed in [45] the Mittag-Leffler stability and the frac-tional Lyapunov direct method with a view to enrichthe knowledge of both system theory and fractionalcalculus.

In the present paper, motivated by the above con-siderations, we study an impulsive DCNN of Caputofractional order with time-varying delays. We apply theMittag-Leffler stability concept, and we investigate theeffect of the impulses on the global asymptotic stabil-ity behavior of the equilibrium. The impulses are real-ized at fixed moments of time and can be considered asa control. Furthermore, the synchronization of impul-sive fractional chaotic networks using a linear non-impulsive controller is also proposed. The main resultsare obtained by using piecewise continuous Lyapunovfunctions and the Razumikhin technique [46,47]. Thisresearch serves as a first step to extend the global sta-bility results for fractional-order neural networks withdelays to the impulsive case. Examples are presentedto illustrate the theory.

2 Statement of the problem: preliminary notes anddefinitions

Let Rn denote the n-dimensional Euclidean space, andlet ||x || = ∑n

i=1 |xi | define the norm of x ∈ Rn . LetR+ = [0,∞) and t0 ∈ R+.

Definition 2.1 [22,23] For any t ≥ t0, the Caputo frac-tional derivative of order q, 0 < q < 1 with the lowerlimit t0 for a function l ∈ C1[[t0, b], R], b > t0, isdefined as

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Synchronization of impulsive fractional-order neural networks

Ct0 D

q

tl(t) = 1

�(1 − q)

t∫

t0

l ′(s)(t − s)q

ds.

Here and in what follows � denotes the Gamma func-tion.

We consider the following impulsive fractional-order CNNs with time-varying delays⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

Ct0 D

qt

xi (t) = −ci xi (t)+n∑

j=1

ai j f j(x j (t)

)

+n∑

j=1

bi j f j (x j (t − τ j (t)))+ Ii , t �= tk, t > t0,

�xi (tk)= xi (t+k )−xi (tk)= Pik(xi (tk)), k =1, 2,. . .,

(2.1)

i = 1, 2, . . . , n, where n corresponds to the numberof units in the neural network; xi (t) corresponds to thestate of the i th unit at time t ; f j (x j (t)) denotes theoutput of the j th unit at time t . Further, ai j , bi j , Ii , ci

are constants, ai j denotes the strength of the j th uniton the i th unit at time t , bi j denotes the strength of thej th unit on the i th unit at time t − τ j (t), Ii denotes theexternal bias on the i th unit, τ j (t) corresponds to thetransmission delay along the axon of the j th unit andsatisfies 0 ≤ τ j (t) ≤ τ (τ = constant), ci representsthe rate with which the i th unit will reset its potentialto the resting state in isolation when disconnected fromthe network and external inputs, tk , k = 1, 2, . . . are themoments of impulsive perturbations and satisfy t0 <t1 < t2 < . . . and lim

k→∞ tk = ∞. The numbers xi (tk) =xi (t

−k ) and xi (t

+k ) are, respectively, the states of the

i th unit before and after impulse perturbation at themoment tk , and the functions Pik represent the abruptchange of the state xi (t) at the impulsive moment tk .

Let J ⊂ R be an interval. Define the followingclasses of functions:

PC[J, Rn] = {σ : J → Rn : σ(t) is continuouseverywhere except at some points tk ∈ J at whichσ(t−k ) and σ(t+k ) exist and σ(t−k ) = σ(tk)

};

PC = PC[[−τ, 0], Rn];PC B[J, Rn] = {σ ∈ PC[J, Rn] : σ(t) is bounded

on J }.Let ϕ0 ∈ PC B[[−τ, 0], Rn]. Denote by x(t) =

x(t; t0, ϕ0), x ∈ Rn the solution of system (2.1) thatsatisfies the initial conditions:{

x(t; t0, ϕ0) = ϕ0(t − t0), t0 − τ ≤ t ≤ t0,x(t+0 ; t0, ϕ0) = ϕ0(0).

(2.2)

The solution x(t) = x(t; t0, ϕ0) = (x1(t; t0, ϕ0), . . . ,

xn(t; t0, ϕ0))T of problem (2.1), (2.2) is [28,29,32] a

piecewise continuous function with points of discon-tinuity of the first kind tk , k = 1, 2, . . ., where it iscontinuous from the left, i.e., the following relationsare valid

xi (t−k ) = xi (tk), k = 1, 2, . . . ,

xi (t+k ) = xi (tk)+ Pik(xi (tk)), tk > t0.

Especially, a constant point x∗ ∈ Rn , x∗ = (x∗1 , x∗

2 ,

. . . , x∗n )

T is called an equilibrium point of (2.1), if x∗ =x∗(t; t0, x∗) is a solution of (2.1).

We introduce the following conditions:H2.1. There exist constants Li > 0 such that

| fi (u)− fi (v)| ≤ Li |u − v|for all u, v ∈ R,u �= v, and fi (0) = 0, i = 1, 2, . . . , n.

H2.2. The functions Pik are continuous on R, i =1, 2, . . . , n, k = 1, 2, . . . .

H2.3. t0 < t1 < t2 < . . . < tk < tk+1 < . . . andtk → ∞ as k → ∞.

H2.4. There exists a unique equilibrium

x∗ = (x∗1 , x∗

2 , . . . , x∗n )

T

of the system (2.1) such that

ci x∗i =

n∑

j=1

ai j f j (x∗j )+

n∑

j=1

bi j f j

(x∗

j

)+ Ii ,

Pik(x∗i ) = 0, i = 1, 2, . . . , n, k = 1, 2, . . . .

Remark 2.1 The problems of existence and unique-ness of equilibrium states of fractional-order neuralnetworks without impulses have been investigated byseveral authors. For some efficient sufficient conditionswhich guarantee the existence and uniqueness of solu-tions for fractional DCNNs with constant delay see[43]. We will note that, if the system

Ct0 D

q

txi (t) = −ci xi (t)+

n∑

j=1

ai j f j(x j (t)

)

+n∑

j=1

bi j f j (x j (t − τ j (t)))+ Ii ,

has a unique solution x(t) = x(t; t0, ϕ0) with ϕ0 ∈C[[−τ, 0], Rn] on the interval [t0,∞), that means thatthe solution x(t) = x(t; t0, ϕ0) of the problem (2.1),(2.2) is defined on each of the intervals (tk−1, tk],k = 1, 2, . . .. From the hypotheses H2.2 nd H2.3 weconclude that it is continuable for t ≥ t0. For some

123

I. Stamova

recent existence results for fractional-order impulsivedelay systems see [28] and [29].

For ϕ ∈ PC , we equip the space PC with the norm||.||τ defined by ||ϕ||τ = sup

−τ≤s≤0||ϕ(s)||. In the case

τ = ∞ we have ||ϕ||τ = ||ϕ||∞ = sups∈(−∞,0]

||ϕ(s)||.We also introduce the following notations:

Gk = (tk−1, tk)× Rn, k = 1, 2, . . . ; G = ∪∞k=1Gk .

Definition 2.2 The equilibrium x∗ =(x∗1 , x∗

2 ,. . ., x∗n )

T

of system (2.1) is said to be:

(a) stable, if

(∀t0 ∈ R+)(∀ε > 0)(∃δ = δ(t0, ε) > 0)

(∀ϕ0 ∈ PC : ||ϕ0 − x∗||τ < δ)

(∀t ≥ t0) : ||x(t; t0, ϕ0)− x∗|| < ε;(b) globally attractive, if

limt→∞ x(t; t0, ϕ0) = x∗;

(c) globally asymptotically stable, if it is stable andglobally attractive.

We shall use the following definition of global Mittag-Leffler stability of the equilibrium of (2.1) which isanalogous to the definition given in [45].

Definition 2.3 The equilibrium x∗ =(x∗1 , x∗

2 ,. . ., x∗n )

T

of system (2.1) is said to be globally Mittag-Lefflerstable, if for ϕ0 ∈ PC there exist constants c > 0 andd > 0 such that

||x(t; t0, ϕ0)− x∗|| ≤ {m(ϕ0 − x∗)Eq(−c(t − t0)q)}d ,

t ≥ t0,

where Eq is the corresponding Mittag-Leffler function,m(0) = 0, m(ϕ) ≥ 0, and m(ϕ) is Lipschitzian withrespect to ϕ ∈ PC .

Remark 2.2 Global Mittag-Leffler stability impliesglobal asymptotic stability; see [45].

If x∗ = (x∗1 , . . . , x∗

n )T is an equilibrium of (2.1), one

can derive from (2.1) that the error e ∈ Rn , e = (e1,e2, . . ., en)

T , ei (t) = xi (t)− x∗i satisfies

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

Ct0 D

qtei (t) = −ci ei (t)+

n∑

j=1

ai j

(f j

(x∗

j + e j (t))

− f j (x∗j )

)

+n∑

j=1

bi j

(f j

(x∗

j +e j (t −τ j (t)))− f j (x

∗j )

), t �= tk , t> t0,

�ei (tk) = Qik(ei (tk)), k = 1, 2, . . . ,

(2.3)

where Qik(ei (tk)) = Pik(ei (tk)+ x∗i ), i = 1, 2, . . . , n,

k = 1, 2, . . ., and

e(s) = ϕ0(s)− x∗, s ∈ [−τ, 0], e(t+0 ) = ϕ0(0)− x∗.

Further on we shall use piecewise continuous Lyapunovfunctions V : [t0,∞)× Rn → R+.

Definition 2.4 We say that the function V : [t0,∞)×Rn → R+, belongs to the class V0 if the followingconditions are fulfilled:

1. The function V is continuous in ∪∞k=1Gk and

V (t, 0) = 0 for t ∈ [t0,∞).2. The function V satisfies locally the Lipschitz con-

dition with respect to e on each of the sets Gk .3. For each k = 1, 2, . . . and e ∈ Rn there exist the

finite limits

V (t−k , e) = limt→tkt<tk

V (t, e), V (t+k , e) = limt→tkt>tk

V (t, e).

4. For each k = 1, 2, . . . and e ∈ Rn the followingequalities are valid

V (t−k , e) = V (tk, e).

For a function V ∈ V0 we define the following frac-tional order derivative (Dini-like derivative) in Caputo’ssense.

Definition 2.5 Given a function V ∈ V0. For t ∈ [tk−1,tk), k = 1, 2, . . . and ϕ ∈ PC the upper right-handderivative of V in Caputo’s sense of order q, 0 < q < 1with respect to system (2.3) is defined by

C Dq+V (t, ϕ(0)) = lim

h→0+ sup1

hq[V (t, ϕ(0))

−V (t − h, ϕ(0)− hq F(t, ϕ))],where

F(t, ϕ) = (F1, F2, . . . , Fn)T ,

and

Fi (t, ϕ) = −ciϕi (t)+n∑

j=1

ai j

(f j

(x∗

j +ϕ j (t))− f j (x

∗j )

)

+n∑

j=1

bi j

(f j

(x∗

j + ϕ j (t − τ j (t)))

− f j (x∗j )

), i = 1, 2, . . . , n.

We shall use the following comparison result from [32]:

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Synchronization of impulsive fractional-order neural networks

Theorem 2.1 [32] Assume that:

1. The function g : [t0,∞)× R+ → R is continuousin each of the sets (tk−1, tk] × R+, k = 1, 2, . . ..

2. Bk ∈ C[R+, R+] and ψk(u) = u + Bk(u) ≥0, k = 1, 2, . . . are non-decreasing with respectto u.

3. The maximal solution u+(t; t0, u0) of the scalarproblem⎧⎨

⎩

Ct0 D

qtu(t) = g(t, u(t)), t �= tk,

u(t0) = u0 ≥ 0,�u(tk) = Bk(u(tk)), tk > t0, k = 1, 2, . . .

is defined in the interval [t0,∞).4. The function V ∈ V0 is such that for t ∈ [t0,∞),ϕ ∈ PC,

V (t+, ϕ(0)+�(ϕ)) ≤ ψk(V (t, ϕ(0))),

t = tk, k = 1, 2, . . . ,

and the inequalityC Dq

+V (t, ϕ(0)) ≤ g(t, V (t, ϕ(0))),

t �= tk, k = 1, 2, . . .

is valid whenever

V (t + s, ϕ(s)) ≤ V (t, ϕ(0)), −τ ≤ s ≤ 0. (2.4)

Then sup−τ≤s≤0V (t0 + s, ϕ0(s)) ≤ u0 implies

V (t, e(t)) ≤ u+(t; t0, u0), t ∈ [t0,∞),

where e(t) = e(t; t0, ϕ0 − x∗) is any solution of(2.3) existing on [t0,∞).

Remark 2.3 The condition (2.4) is called the Razu-mikhin condition, and the corresponding technique isknown as Razumikhin technique [11,18,19,32,46,47].

In the case when g(t, u) = Mu for (t, u) ∈ [t0,∞)×R+, where M ∈ R is a constant, andψk(u) = u for u ∈R+, k = 1, 2, . . ., we deduce the following corollaryfrom Theorem 2.1.

Corollary 2.1 Assume that the function V ∈ V0 is suchthat for t ∈ [t0,∞), ϕ ∈ PC,

V (t+, ϕ(0)+�(ϕ)) ≤ V (t, ϕ(0)),

t = tk, k = 1, 2, . . . ,

and the inequalityC Dq

+V (t, ϕ(0)) ≤ MV (t, ϕ(0)), t �= tk, k = 1, 2, . . .

is valid whenever V (t + s, ϕ(s)) ≤ V (t, ϕ(0)), for−τ ≤ s ≤ 0. Then sup−τ≤s≤0V (t0 + s, ϕ0(s)) ≤ u0

implies

V (t, e(t))≤V (t+0 , e(t+0 ))Eq(M(t − t0)q), t ∈[t0,∞).

3 Global Mittag-Leffler stability

In this section, a sufficient condition for global Mittag-Leffler stability which implies global asymptotic sta-bility of the equilibrium of impulsive fractional-orderDCNN (2.1) is derived.

Theorem 3.1 Assume that:

1. Conditions H2.1-H2.4 are fulfilled.2. The system parameters ai j , bi j , ci (i, j = 1, 2, . . . , n)

satisfy

min1≤i≤n

⎛

⎝ci −Li

n∑

j=1

|a ji |⎞

⎠> max1≤i≤n

⎛

⎝Li

n∑

j=1

|b ji |⎞

⎠>0.

3. The functions Pik are such that

Pik(xi (tk)) = −σik(xi (tk)− x∗i ), 0 < σik < 2,

i = 1, 2, . . . , n, k = 1, 2, . . . .

Then the equilibrium x∗ of (2.1) is globally Mittag-Leffler stable.

Proof We define a Lyapunov function

V (t, e) =n∑

i=1

|ei (t)|.

Then, for t ≥ t0 and t = tk , from condition 3 of thetheorem we obtain

V (t+k , e(t+k )) =n∑

i=1

|ei (tk)+ Qik(ei (tk))|

=n∑

i=1

|xi (tk)− x∗i − σik(xi (tk)− x∗

i )|

=n∑

i=1

|1 − σik ||xi (tk)− x∗i |

<

n∑

i=1

|xi (tk)− x∗i |=V (tk, e(tk)), k = 1, 2, . . . .

(3.1)

Let t ≥ t0 and t ∈ [tk−1, tk). If ei (t) = 0, i =1, 2, . . . , n, then C Dq

+V (t, e(t)) = 0. If ei (t) > 0,i = 1, 2, . . . , n, then

Ct0 D

q

t|ei (t)| = 1

�(1 − q)

t∫

t0

|ei (s)|′(t − s)q

ds

= 1

�(1 − q)

t∫

t0

e′i (s)

(t − s)qds = C

t0 Dq

tei (t).

123

I. Stamova

If ei (t) < 0, i = 1, 2, . . . , n, then

Ct0 D

q

t|ei (t)| = 1

�(1 − q)

t∫

t0

|ei (s)|′(t − s)q

ds

= − 1

�(1−q)

t∫

t0

e′i (s)

(t−s)qds =−C

t0 Dq

tei (t).

Therefore,

Ct0 D

q

t|ei (t)| = sgn(ei (t))

Ct0 D

q

tei (t).

Then for t ≥ t0 and t ∈ [tk−1, tk) for the upper rightderivative C Dq

+V (t, e(t)) along the solution of system(2.3), we get

C Dq+V (t, e(t))

≤n∑

i=1

⎡

⎣−ci |ei (t)| +n∑

j=1

L j |ai j ||e j (t)|

+n∑

j=1

L j |bi j ||e j (t − τ j (t))|⎤

⎦

= −n∑

i=1

⎡

⎣ci − Li

n∑

j=1

|a ji |⎤

⎦ |ei (t)|

+n∑

j=1

n∑

i=1

L j |bi j ||e j (t − τ j (t))|

≤ − min1≤i≤n

⎛

⎝ci − Li

n∑

j=1

|a ji |⎞

⎠n∑

i=1

|ei (t)|

+ max1≤i≤n

⎛

⎝Li

n∑

j=1

|b ji |⎞

⎠n∑

i=1

|e j (t − τ j (t))|

≤ −k1V (t, e(t))+ k2 supt−τ≤s≤t

V (s, e(s)),

where

k1 = min1≤i≤n

⎛

⎝ci − Li

n∑

j=1

|a ji |⎞

⎠ > 0,

k2 = max1≤i≤n

⎛

⎝Li

n∑

j=1

|b ji |⎞

⎠ > 0.

From the above estimate, for any solution e(t) of (2.3)which satisfies the Razumikhin condition

V (s, e(s)) ≤ V (t, e(t)), t − τ ≤ s ≤ t,

we have

C Dq+V (t, e(t)) ≤ −(k1 − k2)V (t, e(t)).

By virtue of condition 2 of Theorem 3.1, there exits areal number α > 0 such that

k1 − k2 ≥ α,

and it follows that

C Dq+V (t, e(t)) ≤ −αV (t, e(t)), (3.2)

t �= tk , t > t0.Then using (3.1), (3.2) and Corollary 2.1, we get

V (t,e(t))≤V (t+0 , e(t+0 ))Eq(−α(t−t0)q), t ∈[t0,∞).

So,

||x(t)− x∗||=

n∑

i=1

∣∣xi (t)− x∗

i

∣∣ ≤ Eq(−α(t − t0)

q)

×n∑

i=1

∣∣xi (t

+0 )− x∗

i

∣∣

= ||ϕ0(0)− x∗||Eq(−α(t − t0)q)

≤ ||ϕ0 − x∗||τ Eq(−α(t − t0)q), t > t0.

Let m = ||ϕ0 − x∗||τ . Then

||x(t)− x∗|| ≤ m Eq(−α(t − t0)q), t > t0,

where m ≥ 0 and m = 0 holds only if ϕ0(s) = x∗ fors ∈ [−τ, 0], which implies that the equilibrium x∗ of(2.1) is globally Mittag-Leffler stable. ��Remark 3.1 Since the equilibrium of (2.1) is globallyMittag-Leffler stable, then it is globally asymptoticallystable.

Remark 3.2 The impulsive fractional-order DCNN(2.1) is the corresponding closed-loop system to thecontrol system

Ct0 D

q

txi (t) = −ci xi (t)+

n∑

j=1

ai j f j(x j (t)

)

+n∑

j=1

bi j f j (x j (t − τ j (t)))

+Ii + ui (t), t > t0,

where

ui (t)=∞∑

k=1

Pik(xi (t))δ(t−tk), i = 1, 2, . . . , n (3.3)

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Synchronization of impulsive fractional-order neural networks

is the control input, δ(t) is the Dirac impulsive function.The controller u(t) = (u1(t), . . . , un(t))T has an effecton suddenly change of the states of (2.1) at the timeinstants tk due to which the states xi (t) of units changefrom the position xi (tk) into the position xi (t

+k ), Pik

are the functions, which characterize the magnitudesof the impulse effects on the units xi at the moments tk ,i.e., u(t) is an impulsive control of the fractional-orderDCNN

Ct0 D

q

txi (t) = −ci xi (t)+

n∑

j=1

ai j f j(x j (t)

)

+n∑

j=1

bi j f j (x j (t − τ j (t)))+ Ii , t > t0.

(3.4)

Therefore, Theorem 3.1 presents a general designmethod of impulsive control law (3.3) for the impulsefree fractional-order DCNN model (3.4). The constantsσik in condition 3 of Theorem 3.1 characterize the con-trol gains of synchronizing impulses. Hence, our resultscan be used to design impulsive control law underwhich the controlled neural networks (2.1) are glob-ally asymptotically synchronized onto system (3.4).

4 Synchronization of impulsive fractional chaoticnetworks

In this section, we shall investigate drive-responseimpulsive fractional-order chaotic neural networksachieve synchronization under linear control. We willconsider neural network system (2.1) as the master sys-tem, and the slave system is given by⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Ct0 D

qt

yi (t) = −ci yi (t)+n∑

j=1

ai j f j(y j (t)

)

+n∑

j=1

bi j f j (y j (t − τ j (t)))

+Ii + ui (t), t �= tk, t > t0,�yi (tk)= yi (t

+k )−yi (tk)= Pik(yi (tk)), k =1, 2, . . . ,

(4.1)

where ui (t) is a suitable non-impulsive controller.We shall use a linear feedback scheme to realize

synchronization between impulsive DCNNs (2.1) and(4.1), i.e., the controller ui (t) is defined by

ui (t) = −di (yi (t)− xi (t)),

where di > 0, i = 1, 2, . . . , n represents the controlgain.

Let ei (t) = yi (t) − xi (t), i = 1, 2, . . . , n. Thenfrom (2.1) and (4.1) the error system is given by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Ct0 D

qtei (t) = −ci ei (t)+

n∑

j=1

ai j g j (e j (t))

+n∑

j=1

bi j g j (e j (t − τ j (t)))− di ei (t), t �= tk,

�ei (tk) = Pik(ei (tk)), k = 1, 2, . . . ,

where g j (e j (t)) = f j (y j (t))− f j (x j (t)) and g j (e j (t−τ j (t))) = f j (y j (t − τ j (t)))− f j (x j (t − τ j (t))).

Introduce the following definition.

Definition 4.1 Master system (2.1) and slave system(4.1) are said to be globally Mittag-Leffler synchro-nized, if for ϕ0, φ0 ∈ PC there exist constants c > 0and d > 0 such that

||x(t; t0, ϕ0)− y(t; t0, φ0)||≤ {m(ϕ0 − φ0)Eq(−c(t − t0)

q)}d , t ≥ t0,

where m(0) = 0, m(ϕ) ≥ 0, and m(ϕ) is Lipschitzianwith respect to ϕ ∈ PC .

The proof of the next theorem is similar to the proof ofTheorem 3.1.

Theorem 4.1 Assume that:

1. Conditions H2.1-H2.4 are fulfilled.2. The system parameters ai j , bi j , ci , di (i, j = 1,

2, . . . , n) satisfy

k1 = min1≤i≤n

⎛

⎝ci + di − Li

n∑

j=1

|a ji |⎞

⎠

> max1≤i≤n

⎛

⎝Li

n∑

j=1

|b ji |⎞

⎠ = k2 > 0.

3. The functions Pik are such that

Pik(ei (tk)) = −σikei (tk), 0 < σik < 2,

i = 1, 2, . . . , n, k = 1, 2, . . . .

Then the master system (2.1) and slave system (4.1)are globally Mittag-Leffler (globally asymptotically)synchronized.

5 Examples

In the following, we shall give examples to illustratethe main results.

123

I. Stamova

Example 5.1 Consider the impulsive fractional-orderDCNN with time-varying delays

C0 D

qt xi (t) = −ci xi (t)+

n∑

j=1

ai j f j(x j (t)

)

+n∑

j=1

bi j f j (x j (t − τ j (t)))

+Ii , t �= tk, t > 0, (5.1)

where 0 < q < 1, n = 2, I1 = 0.66, I2 = 0.69, c1 =c2 = 1, fi (xi ) = 1

2(|xi + 1| − |xi − 1|), i = 1, 2,

0 ≤ τi (t) ≤ τ (τ = 1),

(ai j )2×2 =(

a11 a12

a21 a22

)

=(

0.2 0.2−0.2 0.2

)

,

(bi j )2×2 =(

b11 b12

b21 b22

)

=(

0.3 −0.10.1 0.2

)

,

with⎧⎪⎨

⎪⎩

x1(t+k ) = 1.5 + x1(tk)

4, k = 1, 2, . . . ,

x2(t+k ) = 1.8 + x2(tk)

3, k = 1, 2, . . . ,

(5.2)

where the impulsive moments are such that 0 < t1 <t2 < . . ., and lim

k→∞ tk = ∞.

It is easy to verify that the condition 2 of Theorem3.1 is satisfied for L1 = L2 = 1, k1 = 0.6, k2 = 0.4.Also we have that

0 < σ1k = 3

4< 2, 0 < σ2k = 2

3< 2.

According to Theorem 3.1 the unique equilibrium

x∗ = (x∗1 , x∗

2 )T = (0.5, 0.9)T (5.3)

of (5.1), (5.2) is globally Mittag-Leffler (globallyasymptotically) stable.

If we consider again the system (5.1) but with impul-sive perturbations of the form⎧⎪⎨

⎪⎩

x1(t+k ) = 1.5 − 2x1(tk), k = 1, 2, . . . ,

x2(t+k ) = 1.8 + x2(tk)

3, k = 1, 2, . . . ,

(5.4)

the point (5.3) will be again an equilibrium of (5.1),(5.4), but there is nothing we can say about its globalasymptotic stability, because σ1k = 3 > 2.

Remark 5.1 The example shows that by means ofappropriate impulsive perturbations we can control theneural network system’s dynamics.

Example 5.2 Consider the master impulsive fractional-order DCNN with time-varying delays (5.1) whereq = 0.98, n = 2, I1 = I2 = 1, c1 = c2 = 2,

fi (xi ) = 1

2(|xi +1|−|xi −1|), i = 1, 2, 0 ≤ τi (t) ≤ τ

(τ = 1),

(ai j )2×2 =(

a11 a12

a21 a22

)

=(

0.3 0.5−0.7 0.8

)

,

(bi j )2×2 =(

b11 b12

b21 b22

)

=(

0.2 −0.60.5 0.3

)

,

with⎧⎪⎪⎪⎨

⎪⎪⎪⎩

x1(t+k ) = 2x1(tk)

3, k = 1, 2, . . . ,

x2(t+k ) = 2x2(tk)

5, k = 1, 2, . . . ,

(5.5)

where the impulsive moments are such that 0 < t1 <t2 < . . ., and lim

k→∞ tk = ∞, and the response system

C0 D

qt yi (t) = −ci yi (t)+

n∑

j=1

ai j f j(y j (t)

)

+n∑

j=1

bi j f j (y j (t − τ j (t)))

+Ii − di (yi (t)− x(t)), t �= tk, t > 0

(5.6)

with the feedback gains d1 = d2 = 1.2 and impulsiveperturbations of the form

y1(t+k ) = 2y1(tk)

3, y2(t

+k ) = 2y2(tk)

5,

k = 1, 2, . . . , (5.7)

It is easy to verify that the condition 2 of Theorem 4.1is satisfied for L1 = L2 = 1, k1 = 1.9, k2 = 0.9. Alsowe have that

0 < σ1k = 1

3< 2, 0 < σ2k = 3

5< 2.

According to Theorem 4.1 the master system (5.1),(5.5) and response system (5.6), (5.7) are globallyMittag-Leffler (globally asymptotically) synchronized.

Remark 5.2 For d1 = d2 = 0 the condition 2 of Theo-rem 4.1 is not satisfied. Example 5.2 demonstrates thatthe linear controller −1.2(yi (t)− xi (t)) can realize thesynchronization goal, which verifies the Theorem 4.1.

123

Synchronization of impulsive fractional-order neural networks

6 Conclusions

This research serves as a first step in the presentationof sufficient conditions for global asymptotical stabil-ity of the equilibrium point for a fractional-order neuralnetwork model with impulsive effects and time-varyingdelays. We extend the Lyapunov function method to theimpulsive fractional-order case. The results are applica-ble to more general neuron activation functions thanboth the usual sigmoid activation functions in Hopfieldnetworks and the piecewise linear function in standardcellular networks. We show that by means of appro-priate impulsive perturbations we can control stabil-ity properties of the neural networks. In addition, theglobal asymptotic synchronization of fractional chaoticnetworks was proposed by constructing a linear con-troller.

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