YANG Zhanying,LI Jingwen,CHEN Yongrong

(College of Mathematics and Statistics,South-Central University for Nationalities,Wuhan 430074,China)

Abstract The stabilization problem of a class of fractional-order neural networks with the orders lying in the interval (1,2) is studied. Based on the feedback control,a sufficient condition is obtained to realize the finite-time stabilization of networks. This condition is an algebraic inequality that can be easily calculated in applications. Different from those in earlier works,our proof mainly depends on the Cauchy-Schwartz inequality and the Gronwall inequality. Finally,an example is presented to verify the effectiveness of theoretical result.

Keywords fractional-order neural networks; finite-time;Gronwall inequality

Nowadays,fractional-order neural networks have attracted more and more interests from researchers due to their wide applications. Meanwhile,various fractional-order neural networks have been proposed. It is well-known that the powerful applications of neural networks heavily depend on the dynamical behavior of networks,such as stability and synchronization. In the existing literature,there have been a lot of works on the stability and synchronization of fractional-order neural networks,see Ref[1-4] and the references therein. Notice that the orders of fractional-order neural networks considered in these works lie in the interval (0,1). However,there have been many systems which can be well described by various fractional differential equations with the orders lying in the interval (1,2). Hence,it is very important to consider the study on fractional-order neural networks with the orders lying in the interval (1,2).

In the last decade,there have been a few works on the stability of fractional-order neural networks with the orders lying in the interval (1,2). For example,Wu et al. considered the finite-time stability of fractional-order neural networks with time delays[5]. In Ref[6],Cao and Bai studied the finite-time stability of fractional-order BAM neural networks with distributed delay. Their proof mainly depend on the generalized Gronwall-Bellman inequality and some properties of Mittag-Leffler functions.

Based on the feedback control,we give a sufficient condition to realize the finite-time stabilization of systems. Different from those in Ref[5,6],our proof mainly depends on the Cauchy-Schwartz inequality and the Gronwall inequality. Finally,we give an example to verify the effectiveness of theoretical result.

1 Preliminaries

Definition1[7]The fractional integral with non-integer orderq>0 of a functionf(t)∈Cn([0,+∞),R) is defined as

Definition2[7]The Caputo derivative of fractional orderqof a functionf(t)∈Cn([t0,+∞),R) is defined by

wheret≥t0,q>0 andnis a positive integer satisfyingn-1

Proposition1[8]Suppose thatnis a positive integer satisfyingn-10. Iff(t)∈Cn([0,+∞),R),then

Proposition2[9]Letn∈Nand letx1,x2,…,xnbe non-negative real numbers. Forp>1,we have

t∈[0,T).

Describe the network model as follows:

(1)

where 10 anddj>0 are the self-regulating parameters of the neurons,aij(t) andbji(t) are the connection functions.ui(t) andvj(t) represent the external controls.fjandgistand for the activation functions satisfyingfj(0)=0 andgi(0)=0. The initial conditions of system (1) are given asx(k)(0)=ψ(k)andy(k)(0)=φ(k)(k=0,1),whereψ(k)andφ(k)are two given constants.

Assume thataij(t) andbji(t) (i=1,2,…,n;j=1,2,…,m) are continuous and bounded on [0,+∞).fj(x) andgi(x) (i=1,2,…,n;j=1,2,…,m) satisfy the Lipschitz conditions,that is,there exist positive constantsζandθsuch that

|fj(x)-fj(y)|≤ζ|x-y|,

|gi(x)-gi(y)|≤θ|x-y|,

?x,y∈,

Definition3 Suppose thatδ,εandTare positive constants,andδ<ε. System (1) can achieve the finite-time stabilization with respect to {δ,ε,T} under the external control if ‖x(t)‖+‖y(t)‖<ε,?t∈[0,T],holds for max{‖ψ(0)‖+‖φ(0)‖,‖ψ(1)‖+‖φ(1)‖}<δ.

2 Main results

In this section,we will investigate the finite-time stabilization of fractional-order neural networks and obtain a sufficient condition guaranteeing the finite-time stabilization. The theoretical proof mainly depends on the Cauchy-Schwartz inequality and the Gronwall inequality.

Design the external controlsui(t) andvj(t) as follows:

ui(t)=-kixi(t),vj(t)=-ljyj(t),

(2)

wherekiandlj(i=1,2,…,n;j=1,2,…,m) are positive constants.

Theorem1 Letδ,εandTbe positive constants. System (1) can achieve the finite-time stabilization with respect to {δ,ε,T} under the control (2),if max{‖ψ(0)‖+‖φ(0)‖,‖ψ(1)‖+‖φ(1)‖}<δand

ProofMaking use of Proposition 1,we get

and

Hence,

In view of the Cauchy-Schwartz inequality,we obtain

Together with the following inequality

we have

‖x(t)‖+‖y(t)‖≤‖φ‖+‖φ‖t+

where ‖φ‖=max{‖ψ(0)‖+‖φ(0)‖,‖ψ(1)‖+

‖φ(1)‖}.

In view of Proposition 2,we obtain

(‖x(t)‖+‖y(t)‖)2e-2t≤

(3e-2t+3t2e-2t)‖φ‖2+

According to Proposition 3,we get

Thus,

(‖x(t)‖+‖y(t)‖)2≤

In view of the assumptions of Theorem 1,we get ‖x(t)‖+‖y(t)‖<ε. Hence,system (1) can achieve the finite-time stabilization with respect to {δ,ε,T}. The proof is completed.

If the external controlsui(t) andvj(t) are designed as:

ui(t)=-kixi(t),vj(t)=0,

(3)

whereki(i=1,2,…,n) is any positive constant,then we have the following result.

Corollary1 Letδ,εandTbe positive constants. System (1) can achieve the finite-time stabilization with respect to {δ,ε,T} under the control (3),if max{‖ψ(0)‖+‖φ(0)‖,‖ψ(1)‖+‖φ(1)‖}<δand

If the external controlsui(t) andvj(t) are designed as:

ui(t)=0,vj(t)=-ljyj(t),

(4)

wherelj(j=1,2,…,m) is any positive constant,then we have the following result.

Corollary2 Letδ,εandTbe positive constants. System (1) can achieve the finite-time stabilization with respect to {δ,ε,T} under the control (4),if max{‖ψ(0)‖+‖φ(0)‖,‖ψ(1)‖+‖φ(1)‖}<δand

3 Numerical simulations

In this section,we will give an example to illustrate the effectiveness of our results. The network model is described as follows:

(5)

whereq=1.3,C=diag(0.02,0.03),D=diag(0.01,0.02,0.01),f(y(t))=(tanhy1(t),tanhy2(t),tanhy3(t))T,g(x(t))=(tanhx1(t),tanhx2(t))T,ui(t)=-kixi(t),vj(t)=-ljyj(t) and

By calculating,we can obtaina*=2.4,b*=1. The activation functionsfj(x) andgi(x) satisfy the Lipschitz conditions withζ=θ=1. The initial condition of system (5) are given as follows:x(0)=(0.015,0.03)T,x′(0)=(-0.015,0)Tandy(0)=(0,0.02,-0.01)T,y′(0)=(0.02,0.03,0.01)T. Fig. 1 shows the evolution of system (5) without control.

Fig.1 The evolution of system(5) without control圖1 網(wǎng)絡(luò)(5)不加控制的演化

Fig.2 The evolution of system (5) with the control (2)圖2 網(wǎng)絡(luò)(5)加控制(2)后的演化

Takingki=3 andlj=2 (i=1,2;j=1,2,3),we can calculateβ=32.3667. From the initial conditions,we takeδ=0.09>max{‖ψ(0)‖+‖φ(0)‖,‖ψ(1)‖+‖φ(1)‖}. Letε=1,according to Theorem 1,we obtain the estimated settling timeT=0.1095. Fig. 2 shows the evolution of system (5) with the control (2).

4 Conclusions

In this paper,we studied the finite-time stabilization of a class of fractional-order neural networks with the orders lying in the interval (1,2). Based on the feedback control,we obtained a sufficient condition to realize finite-time stabilization. The sufficient condition is a simple inequality which can be easily calculated in applications. Different from those in earlier works,our theoretical result mainly depends on the Gronwall inequality and some elementary inequalities. In addition,we directly give two conditions to stabilize the systems based on partial feedback control. Finally,we give an example to present the effectiveness of our result.