# Centralized and Decentralized Data-Sampling Principles for Outer-Synchronization of Fractional-Order Neural Networks

Authors: Jin-E Zhang
Journal: Complexity
 (2017)

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## Abstract

This paper aims to investigate the outer-synchronization of fractional-order neural networks. Using centralized and decentralized data-sampling principles and the theory of fractional differential equations, sufficient criteria about outer-synchronization of the controlled fractional-order neural networks are derived for structure-dependent centralized data-sampling, state-dependent centralized data-sampling, and state-dependent decentralized data-sampling, respectively. A numerical example is also given to illustrate the superiority of theoretical results.


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## Body

## 1. Introduction

Fractional operator has become visible in application domains [1–15]. As the demanding performance expectations with uncertainty, fractional operator offers more degrees of freedom to designers to meet some predefined performance indexes. After gradually recognizing the importance of fractional operator, it is found that the description of fractional-order model is more accurate and totally different from that of the corresponding integer-order model. As a direct application, the characteristic of fractional-order model can be used to identify possible behavior of electrical signals from neurons. In physical implementation of neurodynamic systems, arbitrary order analog fractance circuit is most appropriate, which reveals profoundly the relationships among neural circuit elements [9–11]. In that way, real neurodynamic systems should be addressed by fractional-order models. Fractional-order neurodynamic systems can better describe how action potentials in neurons are launched and spread. In addition, fractional-order neurodynamic systems possess infinite memory, and yet, integer-order neurodynamic systems are not of such feature [3–8, 12–15]. Therefore, fractional-order neurodynamic systems have the potential to accomplish what integer-order ones can not do. More feasible analysis methods and easy-to-use techniques to be deal with fractional-order neurodynamic systems are worth looking into.As a coherent behavior within nonlinear systems, synchronization of nonlinear systems has attracted phenomenal worldwide attention. Many studies have shown that synchronization mechanism is a universal phenomenon and has a wide range of applications in engineering systems. Generally, two schemes for synchronization are frequently used: inner-synchronization and outer-synchronization. For inner-synchronization, all nodes within a network will achieve a coherent behavior. However, for outer-synchronization, all individuals in two networks will achieve identical behaviors. In many application fields, outer-synchronization may seem practical [16–23]. For example, in heuristic computational intelligence, it is known that outer-synchronization is rooted in brain-inspired computing from evolutionary strategies to cognitive tasks. Nevertheless, results focusing on outer-synchronization of complex control systems have seldom been reported [19]. Control strategy for outer-synchronization deserves more investigation.Sampled-data control through only using the local information has recently generated significant research interest [24–38]. Unlike continuous-time control, which requires the continuous communication data, sampled-data control is more appropriate under networked environment. For control systems, once we can give effective sampling policies and schedule, then the sampled-data control will reduce communication data and save energy dramatically. Thus, how to develop high-efficiency, heuristic information-based sampled-data control with the ultimate aim of maximizing the data collected is worth studying [38]. However, relevant studies of the data-sampling strategy for control systems are still in early stage.Motivated by the above discussions, in this paper, we introduce the centralized and decentralized data-sampling principles to achieve outer-synchronization between coupled fractional-order neural networks. The efficient allocation of the limited energy resources of centralized and decentralized data-sampling principles that maximizes the information value of the data collected is clearly a step forward. Meanwhile, to more efficiently design the sampling method, we merge the structure and state clusters through centralized and decentralized data-sampling principles and then select the best sampling time. On the basis of some analytical tools of fractional differential equations, a series of criteria on outer-synchronization are derived. It should be noted that such criteria capture the information on sampling pattern and may have much wider application range.The rest of the paper is organized as follows. In Section2, we present the preliminaries and problem formulation. In Section 3, we state main results in detail. In Section 4, simulation example is illustrated. Finally, Section 5 concludes the paper.


## 2. Preliminaries and Problem Formulation

First, some preliminaries of fractional operator are given.Fractional integralIq(·) for H(t) with order q>0 is described as(1)IqHt=1Γq∫t0tt-sq-1Hsds,t≥t0,where Γ(·) is Gamma function and t0 is the initial time.Caputo fractional derivativeDCt0q(·) for H(t)∈Cm+1([t0,+∞),R) with order q>0 is described as(2)DCt0qHt=1Γm-q∫t0tHmst-sq-m+1ds,t≥t0,where Γ(·) is Gamma function, m-1<q<m,m is a positive integer, and t0 is the initial time.One-parameter Mittag-Leffler functionEq(·) is described as(3)Eqs=∑k=0+∞skΓkq+1,where Γ(·) is Gamma function, q>0, and s is a complex number.Consider a class of fractional-order neural networks(4)DCt0qxit=-aitxit+∑j=1nbijtfjxjt+uit,i=1,2,…,n,where 0<q<1, ai(t)>0, bij(t) and ui(t) are piecewise continuous and bounded, and feedback function fj(·) satisfies(5)0≤fjU1-fjU2U1-U2≤Fj,∀U1,U2∈R,U1≠U2,in which Fj>0,j=1,2,…,n.For the centralized data-sampling principle, (4) is rewritten as(6)DCt0qxit=-aitxitk+∑j=1nbijtfjxjtk+uit,i=1,2,…,n,where tk is simple notion of tk(t) with k(t)=max⁡K:tK≤t and 0=t0<t1<⋯<tk<⋯ is uniform for all the system states. Every neuron intersperses its state to its out-neighbors and receives the state information from its in-neighbors at the same time point tk.For the decentralized data-sampling principle, (4) is rewritten as(7)DCt0qxit=-aitxitki+∑j=1nbijtfjxjtkj+uit,i=1,2,…,n,where tki is simple notion of tk(t)i with k(t)=max⁡K:tKi≤t and 0=t0i<t1i<⋯<tki<⋯ is distributed for i∈1,2,…,n. Each neuron i pushes its state information to its out-neighbors at time tki when it updates its state. It receives the information of in-neighbor state at time tkj when the neighbor neuron j updates its state.Now, we state definition and problem formulation.Definition 1 (see [19]).
For any two trajectoriesw(t)=(w1(t),w2(t),…,wn(t))T and w¯(t)=(w¯1(t),w¯2(t),…,w¯n(t))T of (4) starting from different initial values w(0) and w¯(0), if there exists some control scheme such that(8)limt→+∞⁡wt-w¯t=0,then we call system (4) can achieve outer-synchronization, where · denotes norm.Letu(t)=(u1(t),u2(t),…,un(t))T and v(t)=(v1(t),v2(t),…,vn(t))T be two trajectories of (6) starting from different initial values u(0) and v(0). Defining zi(t)=ui(t)-vi(t),i=1,2,…,n, it follows that(9)DCt0qzit=-aitzitk+∑j=1nbijthjtk,i=1,2,…,n,where hj(t)=fj(uj(t))-fj(vj(t)),j=1,2,…,n, for all t∈[tk,tk+1),k=0,1,2,….When we adopt the centralized data-sampling principle via structure to achieve outer-synchronization of (6), according to Definition 1, we need to design control strategy based on system structure of (9) such that(10)limt→+∞⁡zt=0,where · denotes norm, z(t)=(z1(t),z2(t),…,zn(t))T.When we adopt the centralized data-sampling principle via state to achieve outer-synchronization of (6), in this case, consider state measurement error(11)eit=zitk-zit,i=1,2,…,n,where t∈[tk,tk+1),k=0,1,2,…. According to Definition 1, we need to design control strategy based on state measurement error (11) such that(12)limt→+∞⁡zt=0,where · denotes norm, z(t)=(z1(t),z2(t),…,zn(t))T.Letu(t)=(u1(t),u2(t),…,un(t))T and v(t)=(v1(t),v2(t),…,vn(t))T be two trajectories of (7) starting from different initial values u(0) and v(0). Defining zi(t)=ui(t)-vi(t),i=1,2,…,n, it follows that(13)DCt0qzit=-aitzitki+∑j=1nbijthjtkj,i=1,2,…,n,where hj(t)=fj(uj(t))-fj(vj(t)),j=1,2,…,n, for all t∈[tkj,tk+1j),k=0,1,2,….When we adopt the decentralized data-sampling principle via state to achieve outer-synchronization of (7), in this case, consider state measurement error(14)eit=zitki-zit,i=1,2,…,n,where t∈[tki,tk+1i),k=0,1,2,…. According to Definition 1, we need to design control strategy based on state measurement error (14) such that(15)limt→+∞⁡zt=0,where · denotes norm, z(t)=(z1(t),z2(t),…,zn(t))T.Next, we present relevant lemmas.Lemma 2 (see [1]).
Let0<q<1. If H(t)∈C1[t0,+∞], then(16)DCt0qHt≤sgn⁡HtDCt0qHt,t≥t0,where(17)DCt0qHt=1Γ1-q∫t0td/dsHst-sqds.Lemma 3 (see [39]).
Givenq>0, let A(t) be nonnegative and locally integrable on [a,b); let G(t) be continuous, bounded, nonnegative, and nondecreasing on [a,b). Assuming Y(t) to be nonnegative and locally integrable on [a,b) with(18)Yt≤At+Gt∫0tt-sq-1Ysds,t∈a,b,then(19)Yt≤At+∫0t∑k=1+∞GtΓqkΓkqt-skq-1Asds,t∈a,b.
Moreover, ifA(t) is nondecreasing on [a,b), then(20)Yt≤AtEqGtΓqtq,t∈a,b,where Γ(·) is Gamma function and Eq(·) is one-parameter Mittag-Leffler function.In the following, we end this section with some notations that are needed later.Letβi>0(i=1,2,…,n) be positive constants, throughout this paper; denote(21)θit=ait-Fibii+t,λjβ,t=θjt-Fj∑j=1,j≠inβiβjbijt,ρt=max1≤i≤n⁡ait-Fibii-t,(22)M=max1≤j≤n⁡supt≥t0⁡ajt+∑i=1nβiβjbijtFj,where bii+(t)=max0,bii(t),bii-(t)=min0,bii(t). For vector V=(V1,V2,…,Vn)T, vector norm V=∑i=1nβiVi. In addition, by the boundedness of ai(t) and bij(t), there exist positive constants B and C such that(23)supt≥t0⁡ρt≤B,max1≤j≤n⁡supt≥t0⁡λjβ,t≤C.


## 3. Main Results

For problem formulation in preceding section, in this section, we propose the corresponding control schemes for centralized data-sampling principle and decentralized data-sampling principle, respectively.To facilitate the narrative, we first address the control designs, then review, and analyze the theoretical results.


### 3.1. Centralized Data-Sampling Principle

Theorem 4.
Let0<ε<1 and ι>0 be positive constants with Bε≤ι and Cε≤ι(2-ε). Assume that there exist positive constants βi>0(i=1,2,…,n) such that λj(β,t)≥ι for all j=1,2,…,n and t∈[t0,+∞). Set tk+1 as a time point such that(24)tk+1=supτ≥tk⁡τ:min1≤j≤n⁡1Γq∫tktt-sq-1λjβ,sds≤ε,∀t∈tk,τfor k=0,1,2,…. Then system (6) reaches outer-synchronization.Proof.
Fromλj(β,t)≥ι for all j=1,2,…,n and t∈[t0,+∞), together with (23), it follows that(25)ιΓq∫tktt-sq-1ds≤1Γq∫tktt-sq-1λjβ,sds≤CΓq∫tktt-sq-1ds;then(26)ιt-tkqΓq+1≤1Γq∫tktt-sq-1λjβ,sds≤Ct-tkqΓq+1,for all j=1,2,…,n and any t∈[tk,tk+1). According to (9), z(t)=(z1(t),z2(t),…,zn(t))T will not update until(27)min1≤j≤n⁡1Γq∫tktt-sq-1λjβ,sds=εat time point t=tk+1. Thus, we get ι(tk+1-tk)q/Γ(q+1)≤ε≤C(tk+1-tk)q/Γ(q+1), which implies(28)εC≤tk+1-tkqΓq+1≤ει,for k=0,1,2,…. Then(29)tk≥t0+kΓq+1εC1/q;therefore, the Zeno behavior can be excluded. Combining with (24) and (28),(30)1Γq∫tktk+1tk+1-sq-1λjβ,sds≤Cει≤2-ε,1Γq∫tktk+1tk+1-sq-1λjβ,sds≥min1≤j≤n⁡1Γq∫tktk+1tk+1-sq-1λjβ,sds=ε,so(31)ε≤1Γq∫tktk+1tk+1-sq-1λjβ,sds≤2-ε,for k=0,1,2,…. By the definition of vector norm in this paper, from (9), now let us consider zi(t)(i=1,2,…,n) at time t=tk+1,(32)ztk+1=∑i=1nβizitk+1=∑i=1nβizitk+1Γq∫tktk+1tk+1-sq-1-aiszitk+∑j=1nbijshjtkds=∑i=1nβizitk+1Γq∫tktk+1tk+1-sq-1-aiszitk+biismitkzitk+∑j=1,j≠inbijsmjtkzjtkds=∑i=1nβizitk1-1Γq∫tktk+1tk+1-sq-1ais-biismitkds+1Γq∑j=1,j≠in∫tktk+1tk+1-sq-1bijsmjtkzjtkds,where(33)mit=hitzit,zit≠0,0,zit=0.
According to (5), obviously, 0≤mi(t)≤Fi for all i=1,2,…,n,t>t0, and(34)bii-sFi≤biismitk≤bii+sFi.
Notice thatBε≤ι; then for any t∈[tk,tk+1],(35)1Γq∫tktk+1tk+1-sq-1ais-biismitkds≤1Γq∫tktk+1tk+1-sq-1ais-bii-sFids≤BΓq∫tktk+1tk+1-sq-1ds=Btk+1-tkqΓq+1≤Bει≤1;thus(36)1-1Γq∫tktk+1tk+1-sq-1ais-biismitkds≥0.
By (32) and (36),(37)ztk+1=∑i=1nβizitk+1≤∑i=1nβizitk1-1Γq∫tktk+1tk+1-sq-1ais-biismitkds+1Γq∑i=1nβi∑j=1,j≠in∫tktk+1tk+1-sq-1bijsmjtkzjtkds≤∑j=1nβjzjtk1-1Γq∫tktk+1tk+1-sq-1ajs-bjj+sFjds+1Γq∑i=1n∑j=1,j≠inβjzjtk∫tktk+1tk+1-sq-1βiβjbijsFjds≤∑j=1nβjzjtk1-1Γq∫tktk+1tk+1-sq-1λjβ,sds≤1-ε∑j=1nβjzjtk=1-εztk,which leads to(38)ztk=otk-q-1,tk⟶+∞,hence(39)limtk→+∞⁡ztk=0.
Recalling system (9), we have(40)limt→+∞⁡zt=limt→+∞⁡∑i=1nβizit-zitk+zitk=limt→+∞⁡ztk+limt→+∞⁡∑i=1nβi1Γq∫tktk+1t-sq-1-aiszitk+∑j=1nbijshjtkds≤limk→+∞⁡ztk+limt→+∞⁡M∑j=1nβjzjtk1Γq∫tktt-sq-1ds≤limk→+∞⁡ztk+Mlimt→+∞⁡tk+1-tkqΓq+1ztk=0;where M is defined in (22). It can be concluded that outer-synchronization of system (6) is proved.Remark 5.
From inequality (28), we can see(41)Γq+1εC1/q≤tk+1-tk≤Γq+1ει1/qfor all k=0,1,2,…, which excludes the Zeno behavior for rule (24).Theorem 6.
Letφ(t) be a positive and continuous function on [t0,+∞). Set tk+1 as a time point such that(42)tk+1=supτ≥tk⁡τ:et≤φt,∀t∈tk,τfor all k=0,1,2,…, where e(t)=(e1(t),e2(t),…,en(t))T is defined in (11). If there exist positive constants βi>0(i=1,2,…,n) such that min1≤j≤n⁡λj(β,t)≥N for some N>0 and all t≥t0, supt≥t0⁡1/Γ(q)∫t0t(t-s)q-1φ(s)ds<+∞, then system (6) reaches outer-synchronization.Proof.
According to Lemma2, from (9) and (42),(43)DCtkqzt=∑i=1nβiDCtkqzit≤∑i=1nsgn⁡zitβiDCtkqzit=∑i=1nsgn⁡zitβi-aitzitk+∑j=1nbijthjtk=∑i=1nsgn⁡zitβi-aitzitk-zit+zit+∑j=1nbijtmjtkzjtk-mjtzjt+mjtzjt=∑i=1nsgn⁡zitβi-aiteit+∑j=1nbijtmjtkejt-aitzit+biitmitkzit+∑j=1,j≠inbijtmjtkzjt≤∑j=1najt+∑i=1nβiβjbijtFjβjejt-∑j=1najt-bjj+tFj-∑i=1,i≠jβiβjbijtFjβjzjt≤-Nzt+Met≤-Nzt+Mφt,where M is defined in (22), and(44)mit=hitzit,zit≠0,0,zit=0.
On the other hand, by (43),(45)zt≤zt0+1Γq∫t0tt-sq-1-Nzs+Mφsds=zt0-1Γq∫t0tt-sq-1Nzsds+1Γq∫t0tt-sq-1Mφsds≤zt0-1Γq∫t0tt-sq-1Nzsds+Mδ,for s∈[tk,t),t∈[tk,tk+1), where 1/Γ(q)∫t0t(t-s)q-1φ(s)ds≤δ<+∞.
Using Lemma3, from (45), it follows(46)limt→+∞⁡zt≤limt→+∞⁡ΘEq-Nt-t0q=0,t≥t0,where Θ=z(t0)+Mδ, which implies that z(t) converges to 0 by the sampling time sequence {tk}k=0+∞. Therefore, system (6) reaches out-synchronization.


### 3.2. Decentralized Data-Sampling Principle

Theorem 7.
Letϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t))T be positive and continuous on [t0,+∞). Set tk+1 as a time point such that(47)tk+1=supτ≥tk⁡τ:eit≤ϕit,∀t∈tki,τfor i=1,2,…,n and all k=0,1,2,…, where e(t)=(e1(t),e2(t),…,en(t))T is defined in (14). If there exist positive constants βi>0(i=1,2,…,n) such that min1≤j≤n⁡λj(β,t)≥K for some K>0 and all t≥t0, and supt≥t0⁡1/Γ(q)∫t0t(t-s)q-1ϕ(s)ds<+∞, then system (7) reaches outer-synchronization.Proof.
According to Lemma2, from (13) and (47),(48)DCtkqzt=∑i=1nβiDCtkqzit≤∑i=1nsgn⁡zitβiDCtkqzit=∑i=1nsgn⁡zitβi-aitzitki+∑j=1nbijthjtkj=∑i=1nsgn⁡zitβi-aitzitki-zit+zit+∑j=1nbijtmjtkjzjtkj-mjtkjzjt+mjtkjzjt=∑i=1nsgn⁡zitβi-aiteit-aitzit+biitmitkizit+∑j=1nbijtmjtkjejt+∑j=1,j≠inbijtmjtkjzjt≤∑j=1najt+∑i=1nβiβjbijtFjβjejt-∑j=1najt-bjj+tFj-∑i=1,i≠jβiβjbijtFjβjzjt≤-Kzt+Met≤-Kzt+Mϕt,where M is defined in (22), and(49)mjtkj=hjtkjzjtkj,zjtkj≠0,0,zjtkj=0.
On the other hand, by (48),(50)zt≤zt0+1Γq∫t0tt-sq-1-Kzs+Mϕsds=zt0-1Γq∫t0tt-sq-1Kzsds+1Γq∫t0tt-sq-1Mϕsds≤zt0-1Γq∫t0tt-sq-1Kzsds+Mρ,for s∈[tki,t), t∈[tki,tk+1i), where 1/Γ(q)∫t0t(t-s)q-1ϕ(s)ds≤ρ<+∞.
Using Lemma3, from (50), it follows(51)limt→+∞⁡zt≤limt→+∞⁡ΠEq-Kt-t0q=0,t≥t0,where Π=z(t0)+Mρ, which implies that z(t) converges to 0 by the sampling time sequence {tki}k=0+∞, i=1,2,…,n. Therefore, system (7) reaches out-synchronization.Remark 8.
As Theorem5 in [19], under the data-sampling rule in Theorem 6 or Theorem 7, the interevent interval of each system state is strictly positive and possesses a common positive lower bound. Furthermore, the Zeno behavior is excluded.Remark 9.
For the sampled-data control, how to choose the proper scheme with the ultimate aim of maximizing the data collected to control the system is challenging. For example, as revealed in [9, 10], it is extremely difficult to design the sampling time point inherited from the sampled-data control strategy. However, according to Theorems 4–7, this situation can be effectively solved if the centralized and decentralized data-sampling principles are cleverly utilized.Remark 10.
For three control schemes in Theorems4–7, these are just the type and level of points, not the merits of good points of difference. Theorem 4 is entirely focused around the centralized data-sampling principle via structure. Theorem 6 is concerned with the centralized data-sampling principle via state. Theorem 7 is to place emphasis on the decentralized data-sampling principle via state.Remark 11.
Note that the sampled-data control in Theorems4–7 exerts only at the sampling time point, that is, every system state employs only its neighbors’ information at tk or tki. Thus, compared with the continuous-time control strategy, the control schemes in Theorems 4–7 can effectively save the bandwidth and reduce the communication cost. Moreover, the results obtained here are the first ones on centralized and decentralized data-sampling principles for outer-synchronization of fractional-order neural networks.Remark 12.
The key features of outer-synchronization in Theorems4–7 are follows. (1) Each outer-synchronization scheme is closely related to the sampling time point. Once the sampling time point is given, the states of the controlled fractional-order neural networks will achieve outer-synchronization. (2) Centralized data-sampling principle via structure makes full use of the characteristic of system itself, while centralized or decentralized data-sampling principle via state skillfully combines the feature of state measurement error.Remark 13.
The analytical methods for outer-synchronization in Theorems4–7 are quite different from conventional complete synchronization, projective synchronization, phase synchronization, distributed synchronization, pinning synchronization, and cluster synchronization.


## 3.1. Centralized Data-Sampling Principle

Theorem 4.
Let0<ε<1 and ι>0 be positive constants with Bε≤ι and Cε≤ι(2-ε). Assume that there exist positive constants βi>0(i=1,2,…,n) such that λj(β,t)≥ι for all j=1,2,…,n and t∈[t0,+∞). Set tk+1 as a time point such that(24)tk+1=supτ≥tk⁡τ:min1≤j≤n⁡1Γq∫tktt-sq-1λjβ,sds≤ε,∀t∈tk,τfor k=0,1,2,…. Then system (6) reaches outer-synchronization.Proof.
Fromλj(β,t)≥ι for all j=1,2,…,n and t∈[t0,+∞), together with (23), it follows that(25)ιΓq∫tktt-sq-1ds≤1Γq∫tktt-sq-1λjβ,sds≤CΓq∫tktt-sq-1ds;then(26)ιt-tkqΓq+1≤1Γq∫tktt-sq-1λjβ,sds≤Ct-tkqΓq+1,for all j=1,2,…,n and any t∈[tk,tk+1). According to (9), z(t)=(z1(t),z2(t),…,zn(t))T will not update until(27)min1≤j≤n⁡1Γq∫tktt-sq-1λjβ,sds=εat time point t=tk+1. Thus, we get ι(tk+1-tk)q/Γ(q+1)≤ε≤C(tk+1-tk)q/Γ(q+1), which implies(28)εC≤tk+1-tkqΓq+1≤ει,for k=0,1,2,…. Then(29)tk≥t0+kΓq+1εC1/q;therefore, the Zeno behavior can be excluded. Combining with (24) and (28),(30)1Γq∫tktk+1tk+1-sq-1λjβ,sds≤Cει≤2-ε,1Γq∫tktk+1tk+1-sq-1λjβ,sds≥min1≤j≤n⁡1Γq∫tktk+1tk+1-sq-1λjβ,sds=ε,so(31)ε≤1Γq∫tktk+1tk+1-sq-1λjβ,sds≤2-ε,for k=0,1,2,…. By the definition of vector norm in this paper, from (9), now let us consider zi(t)(i=1,2,…,n) at time t=tk+1,(32)ztk+1=∑i=1nβizitk+1=∑i=1nβizitk+1Γq∫tktk+1tk+1-sq-1-aiszitk+∑j=1nbijshjtkds=∑i=1nβizitk+1Γq∫tktk+1tk+1-sq-1-aiszitk+biismitkzitk+∑j=1,j≠inbijsmjtkzjtkds=∑i=1nβizitk1-1Γq∫tktk+1tk+1-sq-1ais-biismitkds+1Γq∑j=1,j≠in∫tktk+1tk+1-sq-1bijsmjtkzjtkds,where(33)mit=hitzit,zit≠0,0,zit=0.
According to (5), obviously, 0≤mi(t)≤Fi for all i=1,2,…,n,t>t0, and(34)bii-sFi≤biismitk≤bii+sFi.
Notice thatBε≤ι; then for any t∈[tk,tk+1],(35)1Γq∫tktk+1tk+1-sq-1ais-biismitkds≤1Γq∫tktk+1tk+1-sq-1ais-bii-sFids≤BΓq∫tktk+1tk+1-sq-1ds=Btk+1-tkqΓq+1≤Bει≤1;thus(36)1-1Γq∫tktk+1tk+1-sq-1ais-biismitkds≥0.
By (32) and (36),(37)ztk+1=∑i=1nβizitk+1≤∑i=1nβizitk1-1Γq∫tktk+1tk+1-sq-1ais-biismitkds+1Γq∑i=1nβi∑j=1,j≠in∫tktk+1tk+1-sq-1bijsmjtkzjtkds≤∑j=1nβjzjtk1-1Γq∫tktk+1tk+1-sq-1ajs-bjj+sFjds+1Γq∑i=1n∑j=1,j≠inβjzjtk∫tktk+1tk+1-sq-1βiβjbijsFjds≤∑j=1nβjzjtk1-1Γq∫tktk+1tk+1-sq-1λjβ,sds≤1-ε∑j=1nβjzjtk=1-εztk,which leads to(38)ztk=otk-q-1,tk⟶+∞,hence(39)limtk→+∞⁡ztk=0.
Recalling system (9), we have(40)limt→+∞⁡zt=limt→+∞⁡∑i=1nβizit-zitk+zitk=limt→+∞⁡ztk+limt→+∞⁡∑i=1nβi1Γq∫tktk+1t-sq-1-aiszitk+∑j=1nbijshjtkds≤limk→+∞⁡ztk+limt→+∞⁡M∑j=1nβjzjtk1Γq∫tktt-sq-1ds≤limk→+∞⁡ztk+Mlimt→+∞⁡tk+1-tkqΓq+1ztk=0;where M is defined in (22). It can be concluded that outer-synchronization of system (6) is proved.Remark 5.
From inequality (28), we can see(41)Γq+1εC1/q≤tk+1-tk≤Γq+1ει1/qfor all k=0,1,2,…, which excludes the Zeno behavior for rule (24).Theorem 6.
Letφ(t) be a positive and continuous function on [t0,+∞). Set tk+1 as a time point such that(42)tk+1=supτ≥tk⁡τ:et≤φt,∀t∈tk,τfor all k=0,1,2,…, where e(t)=(e1(t),e2(t),…,en(t))T is defined in (11). If there exist positive constants βi>0(i=1,2,…,n) such that min1≤j≤n⁡λj(β,t)≥N for some N>0 and all t≥t0, supt≥t0⁡1/Γ(q)∫t0t(t-s)q-1φ(s)ds<+∞, then system (6) reaches outer-synchronization.Proof.
According to Lemma2, from (9) and (42),(43)DCtkqzt=∑i=1nβiDCtkqzit≤∑i=1nsgn⁡zitβiDCtkqzit=∑i=1nsgn⁡zitβi-aitzitk+∑j=1nbijthjtk=∑i=1nsgn⁡zitβi-aitzitk-zit+zit+∑j=1nbijtmjtkzjtk-mjtzjt+mjtzjt=∑i=1nsgn⁡zitβi-aiteit+∑j=1nbijtmjtkejt-aitzit+biitmitkzit+∑j=1,j≠inbijtmjtkzjt≤∑j=1najt+∑i=1nβiβjbijtFjβjejt-∑j=1najt-bjj+tFj-∑i=1,i≠jβiβjbijtFjβjzjt≤-Nzt+Met≤-Nzt+Mφt,where M is defined in (22), and(44)mit=hitzit,zit≠0,0,zit=0.
On the other hand, by (43),(45)zt≤zt0+1Γq∫t0tt-sq-1-Nzs+Mφsds=zt0-1Γq∫t0tt-sq-1Nzsds+1Γq∫t0tt-sq-1Mφsds≤zt0-1Γq∫t0tt-sq-1Nzsds+Mδ,for s∈[tk,t),t∈[tk,tk+1), where 1/Γ(q)∫t0t(t-s)q-1φ(s)ds≤δ<+∞.
Using Lemma3, from (45), it follows(46)limt→+∞⁡zt≤limt→+∞⁡ΘEq-Nt-t0q=0,t≥t0,where Θ=z(t0)+Mδ, which implies that z(t) converges to 0 by the sampling time sequence {tk}k=0+∞. Therefore, system (6) reaches out-synchronization.


## 3.2. Decentralized Data-Sampling Principle

Theorem 7.
Letϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t))T be positive and continuous on [t0,+∞). Set tk+1 as a time point such that(47)tk+1=supτ≥tk⁡τ:eit≤ϕit,∀t∈tki,τfor i=1,2,…,n and all k=0,1,2,…, where e(t)=(e1(t),e2(t),…,en(t))T is defined in (14). If there exist positive constants βi>0(i=1,2,…,n) such that min1≤j≤n⁡λj(β,t)≥K for some K>0 and all t≥t0, and supt≥t0⁡1/Γ(q)∫t0t(t-s)q-1ϕ(s)ds<+∞, then system (7) reaches outer-synchronization.Proof.
According to Lemma2, from (13) and (47),(48)DCtkqzt=∑i=1nβiDCtkqzit≤∑i=1nsgn⁡zitβiDCtkqzit=∑i=1nsgn⁡zitβi-aitzitki+∑j=1nbijthjtkj=∑i=1nsgn⁡zitβi-aitzitki-zit+zit+∑j=1nbijtmjtkjzjtkj-mjtkjzjt+mjtkjzjt=∑i=1nsgn⁡zitβi-aiteit-aitzit+biitmitkizit+∑j=1nbijtmjtkjejt+∑j=1,j≠inbijtmjtkjzjt≤∑j=1najt+∑i=1nβiβjbijtFjβjejt-∑j=1najt-bjj+tFj-∑i=1,i≠jβiβjbijtFjβjzjt≤-Kzt+Met≤-Kzt+Mϕt,where M is defined in (22), and(49)mjtkj=hjtkjzjtkj,zjtkj≠0,0,zjtkj=0.
On the other hand, by (48),(50)zt≤zt0+1Γq∫t0tt-sq-1-Kzs+Mϕsds=zt0-1Γq∫t0tt-sq-1Kzsds+1Γq∫t0tt-sq-1Mϕsds≤zt0-1Γq∫t0tt-sq-1Kzsds+Mρ,for s∈[tki,t), t∈[tki,tk+1i), where 1/Γ(q)∫t0t(t-s)q-1ϕ(s)ds≤ρ<+∞.
Using Lemma3, from (50), it follows(51)limt→+∞⁡zt≤limt→+∞⁡ΠEq-Kt-t0q=0,t≥t0,where Π=z(t0)+Mρ, which implies that z(t) converges to 0 by the sampling time sequence {tki}k=0+∞, i=1,2,…,n. Therefore, system (7) reaches out-synchronization.Remark 8.
As Theorem5 in [19], under the data-sampling rule in Theorem 6 or Theorem 7, the interevent interval of each system state is strictly positive and possesses a common positive lower bound. Furthermore, the Zeno behavior is excluded.Remark 9.
For the sampled-data control, how to choose the proper scheme with the ultimate aim of maximizing the data collected to control the system is challenging. For example, as revealed in [9, 10], it is extremely difficult to design the sampling time point inherited from the sampled-data control strategy. However, according to Theorems 4–7, this situation can be effectively solved if the centralized and decentralized data-sampling principles are cleverly utilized.Remark 10.
For three control schemes in Theorems4–7, these are just the type and level of points, not the merits of good points of difference. Theorem 4 is entirely focused around the centralized data-sampling principle via structure. Theorem 6 is concerned with the centralized data-sampling principle via state. Theorem 7 is to place emphasis on the decentralized data-sampling principle via state.Remark 11.
Note that the sampled-data control in Theorems4–7 exerts only at the sampling time point, that is, every system state employs only its neighbors’ information at tk or tki. Thus, compared with the continuous-time control strategy, the control schemes in Theorems 4–7 can effectively save the bandwidth and reduce the communication cost. Moreover, the results obtained here are the first ones on centralized and decentralized data-sampling principles for outer-synchronization of fractional-order neural networks.Remark 12.
The key features of outer-synchronization in Theorems4–7 are follows. (1) Each outer-synchronization scheme is closely related to the sampling time point. Once the sampling time point is given, the states of the controlled fractional-order neural networks will achieve outer-synchronization. (2) Centralized data-sampling principle via structure makes full use of the characteristic of system itself, while centralized or decentralized data-sampling principle via state skillfully combines the feature of state measurement error.Remark 13.
The analytical methods for outer-synchronization in Theorems4–7 are quite different from conventional complete synchronization, projective synchronization, phase synchronization, distributed synchronization, pinning synchronization, and cluster synchronization.


## 4. A Numerical Example

In this section, a numerical example is utilized to show the effectiveness of the results obtained.Consider a class of fractional-order neural networks as follows:(52)DCt0qx1t=-a1tx1t+b11tf1x1t+b12tf2x2t+u1t,DCt0qx2t=-a2tx2t+b21tf1x1t+b22tf2x2t+u2t,where q=1/2, t0=0, A=a1t00a2t=1.5001.5, B=b11tb12tb21tb22t=0.3-0.7-1.2-0.1, u=u1tu2t=0.10.1, f1(X)=f2(X)=1/1+e-X.By direct calculation, we can obtain(53)λ1β,t=a1t-F1b11+t-F1β2β1b21t,λ2β,t=a2t-F2b22+t-F2β1β2b12t.To chooseε=0.2, ι=0.8, β1=β2=1, then it follows that B=1.6, C=1.05. Hence the following inequalities hold:(54)Bε-ι≤0,Cε-ι2-ε≤0.According to Theorem 4, system (52) reaches outer-synchronization. Figures 1 and 2 depict the dynamics of u1(t) and v1(t), u2(t) and v2(t) in the triggering time points as Theorem 4, respectively. Figure 3 describes the release time points and release intervals.Figure 1
Dynamics ofu1(t) and v1(t) in the triggering mechanism as Theorem 4.Figure 2
Dynamics ofu2(t) and v2(t) in the triggering mechanism as Theorem 4.Figure 3
The release time points and release intervals in the triggering mechanism as Theorem4.To selectφ(t)=1/(t+1)1/2, together with(55)supt≥0⁡1Γ1/2∫0tt-sq-11s+11/2ds<+∞,according to Theorem 6, system (52) reaches outer-synchronization. Figures 4 and 5 depict the dynamics of u1(t) and v1(t), u2(t) and v2(t) in the triggering time points as Theorem 6, respectively. Figure 6 describes the release time points and release intervals.Figure 4
Dynamics ofu1(t) and v1(t) in the triggering mechanism as Theorem 6.Figure 5
Dynamics ofu2(t) and v2(t) in the triggering mechanism as Theorem 6.Figure 6
The release time points and release intervals in the triggering mechanism as Theorem6.To selectϕ1(t)=1/(t+1)1/2,ϕ2(t)=1/(t+2)1/2, together with(56)supt≥0⁡1Γ1/2∫0tt-sq-11s+11/2ds<+∞,supt≥0⁡1Γ1/2∫0tt-sq-11s+21/2ds<+∞,according to Theorem 7, system (52) reaches outer-synchronization. Figures 7 and 8 depict the dynamics of u1(t) and v1(t), u2(t) and v2(t) in the triggering time points as Theorem 7, respectively. Figure 9 describes the release time points and release intervals.Figure 7
Dynamics ofu1(t) and v1(t) in the triggering mechanism as Theorem 7.Figure 8
Dynamics ofu2(t) and v2(t) in the triggering mechanism as Theorem 7.Figure 9
The release time points and release intervals in the triggering mechanism as Theorem7.Remark 14.
In existing publications, there has been no theoretic criterion to achieve outer-synchronization of (52). In addition, using centralized or decentralized data-sampling principle to analyze and control fractional-order systems is also rare.Remark 15.
According to simulation analysis in Figures1–9, there is no essential difference regarding outer-synchronization performance in three control schemes as Theorems 4–7. By comparative analysis of Figures 3, 6, and 9, the release intervals via control scheme as Theorem 4 are relatively minor, and the triggering time points via control scheme as Theorem 7 are spread more thinly.


## 5. Concluding Remarks

In this paper, we show that outer-synchronization of fractional-order neural networks can be achieved by applying appropriate centralized and decentralized data-sampling principles. Such theoretical results improve and supplement some existing related results. The results obtained here are sufficient conditions for outer-synchronization of fractional-order neural networks and may remain room for improvement. Further extensions would be welcome: (1) outer-synchronization of fractional-order neural networks considering both conservativeness and complexity; (2) analyzing the outer-synchronization of fractional-order neural networks subject to time-delay; (3) analyzing the outer-synchronization of fractional-order neural networks subject to stochastic disturbance.



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*Source: 6290646-2017-03-08.xml*