# Multisynchronization for Coupled Multistable Fractional-Order Neural Networks via Impulsive Control

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

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## Abstract

We show that every subnetwork of a class of coupled fractional-order neural networks consisting ofN identical subnetworks can have r+1n locally Mittag-Leffler stable equilibria. In addition, we give some algebraic criteria for ascertaining the static multisynchronization of coupled fractional-order neural networks with fixed and switching topologies, respectively. The obtained theoretical results characterize multisynchronization feature for multistable control systems. Two numerical examples are given to verify the superiority of the proposed results.


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## Body

## 1. Introduction

Fractional calculus has been drawing interest over the last decade due to its many applications [1–12]. In mathematics and theoretical physics, fractional calculus is a generalization of integer-order calculus that roots in classical analysis theory. For system modeling, logical mechanism of fractional calculus is more sufficient and more essential [7, 12]. Then the models involving fractional calculus have greater ability to describe some real phenomena. To go in with mathematical tool behind fractional calculus, we have more room to develop new-type fractional dynamical systems. Actually, long and short memory with power law in dynamic evolution process are usually governed by the so-called fractional dynamical systems [5–7]. Many studies have shown that fractional dynamical systems have an unlimited memory. On the contrary, the memory in integer-order dynamical systems is limited. Moreover, in the general case, the evolution characteristics of fractional dynamical systems cannot be deduced by conventional integer-order variation. Besides, some existing formulations of fractional variation also meets bottleneck: (1) unitarity of evolution operator and (2) temporal fractality. Developing theoretical framework for a breakthrough in analysis and synthesis of fractional dynamical systems is even more important.Synchronization phenomenon within complex networks is one of the most intriguing and valuable issues [13–16]. Recently, as a special case of an online learning situation, synchronization of neurodynamic systems have been found to be useful in characterizing and validating the consistency of cooperation [1, 8, 9, 11, 12]. In addition, neural synchronization under setting control rules offers an appealing alternative to neuron architectures and information exchange mechanism during neural mutual learning [16]. However, dynamics of neural synchronization process are intricate. The rhythms of biological brain emerge via synchronization between individual firings by activity-dependent coupling. To get a broader understanding of neural synchronization, it is reasonable to take a closer look at the dynamics of synchronous activity. When neurodynamic systems possess multiple locally stable equilibria, what structure should we use to describe the synchronization manifolds? Moreover, for multistable neurodynamic systems, how to implement this new synchronization control? As far as we know, there is little study. In addition, Lyapunov method [1, 8], Razumikhin-type stability theory [9, 16], infinitesimal generator on analytic semigroup principle [12], adaptive control [17], and mathematical induction method [18] are not very good at estimating the multisynchronization effect. Therefore, for the multisynchronization of control systems, there would still be a lot of room left.It is found that impulsive control is very effective in a wide variety of applications for performance improvement of control process [17–31]. Ayati and Khaloozadeh [19] study the adaptive impulsive control to design an observer for nonlinear continuous systems. Chen et al. [20] address the delayed impulsive control for exponential stability of Takagi-Sugeno fuzzy systems. Chen et al. [21] show that complex networks can synchronize under the impulsive control policy. It is also demonstrated that second-order consensus can be achieved via impulsive control algorithms [23]. In [24], impulsive control scheme is utilized to improve the performance of differential evolution. In [26], by using the impulsive control method, the ultimate boundedness problem of nonautonomous complex networks is investigated. Li and Song [27] take full advantage of delay-dependent impulsive control to analyze the stabilization problem of time-delay systems. Liu and Zhang [29] establish the impulsive control principle for stabilization of discrete-time nonlinear systems. Together with comparison criterion, fuzzy impulsive control is used for stabilization of chaotic systems [30]. Impulsive control algorithms are designed for droop-based secondary distributed control in islanded microgrids [31]. In [17], by the impulsive control schemes, exponential synchronization of complex dynamical networks in the presence of stochastic perturbations is formulated. In [18], under delayed impulsive control, stochastic synchronization problem for complex dynamical networks is addressed. Nevertheless, it should be pointed out that the impulsive control for multistable complex systems is still in early stage.In view of the above discussions, the main objective of this paper is to investigate the multisynchronization for coupled multistable fractional-order neural networks via impulsive control. Several sufficient conditions are obtained to ensure that every subnetwork of coupled fractional-order neural networks has2r+1n equilibria and r+1n equilibria are locally Mittag-Leffler stable. The initial-value-related static multisynchronization is introduced to characterize the synchronous behaviors of the controlled coupled multistable fractional-order neural networks. The main emphasis will be then on impulsive control strategy to guarantee multisynchronization for coupled multistable fractional-order neural networks with fixed/switching topology. It is also shown that multisynchronization manifolds can sustain and maintain high levels for long running process. The proposed impulsive control strategy has a number of benefits: (1) saving communication bandwidth; (2) reducing communication cost; (3) good execution performance.The rest of this paper is organized as follows. In Section2, preliminaries and problem formulation are given. In Section 3, main results are derived to ascertain multisynchronization for coupled multistable fractional-order neural networks with fixed/switching topology. In Section 4, two numerical examples are presented to show the effectiveness of the obtained results. In Section 5, concluding remarks are stated.


## 2. Preliminaries and Problem Formulation

### 2.1. Notations

Throughout this paper,DCt0q(·) denotes Caputo fractional derivative operator. 1N is the N-dimensional column vector with its elements equal to 1. Ik is the k-dimensional identity matrix. A1⊗A2 denotes the Kronecker product of matrices A1 and A2. Matrix A<0(>0; ≤0; ≥0) denotes that A is negative definite (positive definite, negative semidefinite, and positive semidefinite). A-1 represents the inverse matrix of A. For the vector norm or the matrix norm, · stands for the Euclidean norm.


### 2.2. Model

Consider a class of coupled fractional-order neural networks consisting ofN identical subnetworks(1)DCt0qxit=-Dxit+BFxit+uit,t≥t0≥0,i=1,2,…,N,where fractional-order 0<q<1, xi(t)∈Rn is the state, D=diag⁡(d1,d2,…,dn) is self-feedback term with dk>0,k=1,2,…,n, B=(bkl)n×n represents synaptic strength, F(xi(t))=(f1(xi1(t)),f2(xi2(t)),…,fn(xin(t)))T is the feedback function, and ui(t)∈Rn denotes the input. The initial value of (1) is given by xi(t0)=(xi1(t0),xi2(t0),…,xin(t0))T.Now, we start to make the following assumptions for (1).(A1) The feedback functionsF(xi(t))=(f1(xi1(t)),f2(xi2(t)),…,fn(xin(t)))T,i=1,2,…,N, satisfy(2)Pk≤fks≤P~k,for∀s∈R,(3)Qk-≤fks1-fks2s1-s2≤Qk+,for∀s1,s2∈R,s1≠s2,where Pk, P~k, Qk-, and Qk+,k=1,2,…,n, are constants.(A2)D-B~ is a nonsingular M-matrix, where(4)B~=b~kln×nwith(5)b~kk=max⁡bkkQk-,bkkQk+,b~kl=max⁡bklQl-,bklQl+,k≠l.(A3) To divideR into 2r+1 intervals (i.e., R=-∞,qj1∪[qj1,pj1]∪(pj1,qj2)∪[qj2,pj2]∪⋯∪[qjr,pjr]∪ (pjr,+∞)), then(6)-djqjl+bjjfjqjl+∑k=1,k≠jnmin⁡bjkPk,bjkP~k+ujt>0,-djpjl+bjjfjpjl+∑k=1,k≠jnmax⁡bjkPk,bjkP~k+ujt<0,for j=1,2,…,n, l=1,2,…,r, where -∞≤qj1<pj1<qj2<pj2<⋯<qjr<pjr≤+∞.Under (A1)–(A3), every subnetwork of system (1) is multistable in Mittag-Leffler sense.Lemma 1.
Let (A1)–(A3) hold, and every subnetwork of system (1) has 2r+1n equilibria and r+1n equilibria are locally Mittag-Leffler stable.
Using standard arguments as Theorem1 in [2], Lemma 1 can be proved.


### 2.3. Properties

In this subsection, we give necessary definition and lemmas.DenoteS1,S2,…,Sr+1n as r+1n locally Mittag-Leffler stable equilibria of every subnetwork of system (1).Definition 2.
System (1) is said to achieve static multisynchronization if the following holds:
(1) For any initial valuex(t0)=(x1T(t0),x2T(t0),…,xNT(t0))T of (1), where xi(t0)=(xi1(t0),xi2(t0),…,xin(t0))T, i=1,2,…,N, there exists Sl∈Rn such that limt→+∞⁡xi(t)=Sl,i∈1,2,…,N,l∈1,2,…,r+1n.
(2) The synchronization manifolds1N⊗Sl and 1N⊗Sl~ starting from different initial values x(t0)=(x1T(t0),x2T(t0),…,xNT(t0))T and x^(t0)=(x^1T(t0),x^2T(t0),…,x^NT(t0))T, respectively, satisfy the following: there exists ε>0 such that(7)∀x~t∈xt:0<xt-1N⊗Sl<ε,xt∈RNn,t≥t0,where x~(t) is not a point on 1N⊗Sl.Lemma 3.
Linear matrix inequality(8)A1A2A2TA3<0is equivalent to(1)
A1<0 and A3-A2TA1-1A2<0,or(2)
A3<0 and A1-A2A3-1A2T<0,where A1=A1T and A3=A3T.Lemma 4.
LetA(t) be a continuous function on [t0,+∞); if there exists constant κ such that(9)DCt0qAt≤κAt,t≥t0≥0,then(10)At≤At0Eqκt-t0q,t≥t0≥0,where 0<q<1, Eq(·) is one-parameter Mittag-Leffler function.


## 2.1. Notations

Throughout this paper,DCt0q(·) denotes Caputo fractional derivative operator. 1N is the N-dimensional column vector with its elements equal to 1. Ik is the k-dimensional identity matrix. A1⊗A2 denotes the Kronecker product of matrices A1 and A2. Matrix A<0(>0; ≤0; ≥0) denotes that A is negative definite (positive definite, negative semidefinite, and positive semidefinite). A-1 represents the inverse matrix of A. For the vector norm or the matrix norm, · stands for the Euclidean norm.


## 2.2. Model

Consider a class of coupled fractional-order neural networks consisting ofN identical subnetworks(1)DCt0qxit=-Dxit+BFxit+uit,t≥t0≥0,i=1,2,…,N,where fractional-order 0<q<1, xi(t)∈Rn is the state, D=diag⁡(d1,d2,…,dn) is self-feedback term with dk>0,k=1,2,…,n, B=(bkl)n×n represents synaptic strength, F(xi(t))=(f1(xi1(t)),f2(xi2(t)),…,fn(xin(t)))T is the feedback function, and ui(t)∈Rn denotes the input. The initial value of (1) is given by xi(t0)=(xi1(t0),xi2(t0),…,xin(t0))T.Now, we start to make the following assumptions for (1).(A1) The feedback functionsF(xi(t))=(f1(xi1(t)),f2(xi2(t)),…,fn(xin(t)))T,i=1,2,…,N, satisfy(2)Pk≤fks≤P~k,for∀s∈R,(3)Qk-≤fks1-fks2s1-s2≤Qk+,for∀s1,s2∈R,s1≠s2,where Pk, P~k, Qk-, and Qk+,k=1,2,…,n, are constants.(A2)D-B~ is a nonsingular M-matrix, where(4)B~=b~kln×nwith(5)b~kk=max⁡bkkQk-,bkkQk+,b~kl=max⁡bklQl-,bklQl+,k≠l.(A3) To divideR into 2r+1 intervals (i.e., R=-∞,qj1∪[qj1,pj1]∪(pj1,qj2)∪[qj2,pj2]∪⋯∪[qjr,pjr]∪ (pjr,+∞)), then(6)-djqjl+bjjfjqjl+∑k=1,k≠jnmin⁡bjkPk,bjkP~k+ujt>0,-djpjl+bjjfjpjl+∑k=1,k≠jnmax⁡bjkPk,bjkP~k+ujt<0,for j=1,2,…,n, l=1,2,…,r, where -∞≤qj1<pj1<qj2<pj2<⋯<qjr<pjr≤+∞.Under (A1)–(A3), every subnetwork of system (1) is multistable in Mittag-Leffler sense.Lemma 1.
Let (A1)–(A3) hold, and every subnetwork of system (1) has 2r+1n equilibria and r+1n equilibria are locally Mittag-Leffler stable.
Using standard arguments as Theorem1 in [2], Lemma 1 can be proved.


## 2.3. Properties

In this subsection, we give necessary definition and lemmas.DenoteS1,S2,…,Sr+1n as r+1n locally Mittag-Leffler stable equilibria of every subnetwork of system (1).Definition 2.
System (1) is said to achieve static multisynchronization if the following holds:
(1) For any initial valuex(t0)=(x1T(t0),x2T(t0),…,xNT(t0))T of (1), where xi(t0)=(xi1(t0),xi2(t0),…,xin(t0))T, i=1,2,…,N, there exists Sl∈Rn such that limt→+∞⁡xi(t)=Sl,i∈1,2,…,N,l∈1,2,…,r+1n.
(2) The synchronization manifolds1N⊗Sl and 1N⊗Sl~ starting from different initial values x(t0)=(x1T(t0),x2T(t0),…,xNT(t0))T and x^(t0)=(x^1T(t0),x^2T(t0),…,x^NT(t0))T, respectively, satisfy the following: there exists ε>0 such that(7)∀x~t∈xt:0<xt-1N⊗Sl<ε,xt∈RNn,t≥t0,where x~(t) is not a point on 1N⊗Sl.Lemma 3.
Linear matrix inequality(8)A1A2A2TA3<0is equivalent to(1)
A1<0 and A3-A2TA1-1A2<0,or(2)
A3<0 and A1-A2A3-1A2T<0,where A1=A1T and A3=A3T.Lemma 4.
LetA(t) be a continuous function on [t0,+∞); if there exists constant κ such that(9)DCt0qAt≤κAt,t≥t0≥0,then(10)At≤At0Eqκt-t0q,t≥t0≥0,where 0<q<1, Eq(·) is one-parameter Mittag-Leffler function.


## 3. Main Results

When the coupled fractional-order neural networks (1) generate multistable behavior, finding out the related multisynchronization control strategy is a challenging issue. In the following, we introduce impulsive control for multisynchronization in (1).


### 3.1. Fixed Topology Case

For the coupled fractional-order neural networks (1) with fixed topology, the impulsive control design is given below:(11)uit=a∑h=1+∞∑j=1,j≠iNcijxjt-xitδt-th,for i=1,2,…,N, where a>0 is the coupled gain, cij denotes the element of the weighted adjacency matrix of digraph D, digraph D possesses a directed spanning tree, δ(·) is the Dirac Delta function, and the impulsive time sequence thh=1+∞ satisfies 0<t1<t2<⋯<th<⋯ and limh→∞⁡th=+∞.Combining with (1) and (11), we get(12)DCt0qxt=-IN⊗Dxt+IN⊗BFxt,t≠th,Δxt=-aL⊗Inxt-,t=th,t≥t0≥0,h=1,2,…,where x(t)=(x1T(t),x2T(t),…,xNT(t))T, F(x(t))=(FT(x1(t)),FT(x2(t)),…,FT(xN(t)))T, and L=(Lij)N×N is the Laplacian matrix associated with digraph D.Theorem 5.
Let (A1)–(A3) hold; for any given constantα1>0, if there exist constant 0<α2<1 and matrices R>0 and W=diag⁡(W1,W2,…,Wn)>0 such that(13)-α2IN-1IN-1-aSTIN-1-aS-IN-1<0,(14)-RD-DTR+α2α1R-WM1RB+WM2RB+WM2T-W<0,where(15)S=Lij-LNjN-1×N-1,M1=diag⁡Q1+Q1-,Q2+Q2-,…,Qn+Qn-,M2=diag⁡Q1++Q1-2,Q2++Q2-2,…,Qn++Qn-2,then, for the impulsive time sequence thh=1+∞ admitting sup⁡th-th-1≤α1, system (12) can achieve static multisynchronization.Proof.
From (A1)–(A3), according to Lemma1, every subnetwork of system (1) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n.
LetYi(t)=xi(t)-S,i=1,2,…,N, where S∈Sħ,ħ=1,2,…,(r+1)n, from (12), and then(16)DCt0qYt=-IN⊗DYt+IN⊗BFYt,t≠th,ΔYt=-aL⊗InYt-,t=th,t≥t0≥0,h=1,2,…,where Y(t)=(Y1T(t),Y2T(t),…,YNT(t))T and F(Y(t))=(FT(Y1(t)),FT(Y2(t)),…,FT(YN(t)))T.
LetZi(t)=Yi(t)-YN(t),i=1,2,…,N-1, and system (16) can be reformulated as(17)DCt0qZt=-IN-1⊗DZt+IN-1⊗BFZt,t≠th,DCt0qYNt=-DYNt+BFYNt,t≠th,ΔZt=-aS⊗InZt-,t=th,ΔYNt=-aS^⊗InYNt-,t=th,t≥t0≥0,h=1,2,…,where Z(t)=(Z1T(t),Z2T(t),…,ZN-1T(t))T, F(Z(t))=(FT(Z1(t)),FT(Z2(t)),…,FT(ZN-1(t)))T, and S^=(LN1,LN2,…,LN(N-1)).
Obviously, fori=1,2,…,N-1,(18)Zit=Yit-YNt=xit-S-xNt-S=xit-xNt.
According to Lemma3, (13) is equivalent to(19)IN-1-aSTIN-1-aS≤α2IN-1.
Define a Lyapunov function(20)Vt=ZTtIN-1⊗RZt.
Whent=th, we can obtain(21)Vth=ZTthIN-1⊗RZth=ZTth-IN-1-aS⊗InTIN-1⊗R×IN-1-aS⊗InZth-≤α2Vth-.
On the other hand, by (3) in (A1), for any given W=diag⁡(W1,W2,…,Wn)>0, we have(22)0≤∑j=1nWjQj+Zjt-FZjtFZjt-Qj-Zjt=-ZiTtWM1Zit+2ZiTtWM2FZit-FTZitWFZit,and thus(23)0≤-ZTtIN-1⊗WM1Zt+2ZTtIN-1⊗WM2FZt-FTZtIN-1⊗WFZt.By (14), it follows that(24)-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W<0,where constant α3∈(0,1-α2).
Whent≠th, we can get(25)DCt0qVt≤2ZTtIN-1⊗RZt+2ZTtIN-1⊗R-IN-1⊗DZt+IN-1⊗BFZt.
Together with (23)–(25),(26)DCt0qVt≤HTtIN-1⊗χHt-α2+α3α1Vt<-α2+α3α1Vt,where(27)HTt=ZTt,FTZtT,χ=-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W.
According to Lemma4,(28)Vt≤Vt0Eq-α2+α3α1t-t0q,t≥t0≥0,so V(t)→0 as t→+∞, which implies that, for any given initial value of (1), x1(t)=x2(t)=⋯=xN(t) as t→+∞, and, hence, system (12) can reach complete synchronization.Moreover, consider theNth subnetwork of (12):(29)DCt0qxNt=-DxNt+BFxNt,t≠th,ΔxNt=∑j=1N-1cNjxjt--xNt-,t=th,t≥t0≥0,h=1,2,….Based on the above analysis, we haveΔxN(t)=0 as t→+∞. Through Lemma 1, the Nth subnetwork of (12) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n. Therefore, it follows that x1(t)=x2(t)=⋯=xN(t)=S as t→+∞, S∈Sħ,ħ=1,2,…,(r+1)n. To sum up, system (12) can achieve static multisynchronization.


### 3.2. Switching Topology Case

Without loss of generality, we introduce a switching signalρ(t):[t0,+∞)→1,2,…,K and K digraphs indexed by digraph D1, digraph D2,…, digraph DK.For the coupled fractional-order neural networks (1) with switching topology, the impulsive control design is given below:(30)uit=aρt∑h=1+∞∑j=1,j≠iNcijρtxjt-xitδt-th,for i=1,2,…,N, where the switching signal ρ(t):[t0,+∞)→1,2,…,K, aρ(t)>0 is the coupled gain, cijρ(t) denotes the element of the weighted adjacency matrix of digraph Dρ(t), digraph Dρ(t) possesses a directed spanning tree, δ(·) is the Dirac Delta function, and the impulsive time sequence thh=1+∞ satisfies 0<t1<t2<⋯<th<⋯ and limh→∞⁡th=+∞.Combining with (1) and (30), we get(31)DCt0qxt=-IN⊗Dxt+IN⊗BFxt,t≠th,Δxt=-aρtLρt⊗Inxt-,t=th,t≥t0≥0,h=1,2,…,where x(t)=(x1T(t),x2T(t),…,xNT(t))T, F(x(t))=(FT(x1(t)),FT(x2(t)),…,FT(xN(t)))T, and Lρ(t)=(Lijρ(t))N×N is the Laplacian matrix associated with digraph Dρ(t).Theorem 6.
Let (A1)–(A3) hold; for any given constantα1>0, if there exist constant 0<α2<1 and matrices R>0 and W=diag⁡(W1,W2,…,Wn)>0 such that(32)-α2IN-1IN-1-aCSCTIN-1-aCSC-IN-1<0,C=1,2,…,K,(33)-RD-DTR+α2α1R-WM1RB+WM2RB+WM2T-W<0,where(34)SC=LijC-LNjCN-1×N-1,C=1,2,…,K,M1=diag⁡Q1+Q1-,Q2+Q2-,…,Qn+Qn-,M2=diag⁡Q1++Q1-2,Q2++Q2-2,…,Qn++Qn-2,then, for the impulsive time sequence thh=1+∞ admitting sup⁡th-th-1≤α1, system (31) can achieve static multisynchronization.Proof.
From (A1)–(A3), according to Lemma1, every subnetwork of system (1) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n.
LetYi(t)=xi(t)-S,i=1,2,…,N, where S∈Sħ,ħ=1,2,…,(r+1)n; from (31), then(35)DCt0qYt=-IN⊗DYt+IN⊗BFYt,t≠th,ΔYt=-aρtLρt⊗InYt-,t=th,t≥t0≥0,h=1,2,…,where Y(t)=(Y1T(t),Y2T(t),…,YNT(t))T and F(Y(t))=(FT(Y1(t)),FT(Y2(t)),…,FT(YN(t)))T.
LetZi(t)=Yi(t)-YN(t),i=1,2,…,N-1, and system (35) can be reformulated as(36)DCt0qZt=-IN-1⊗DZt+IN-1⊗BFZt,t≠th,DCt0qYNt=-DYNt+BFYNt,t≠th,ΔZt=-aρtSρt⊗InZt-,t=th,ΔYNt=-aρtS^ρt⊗InYNt-,t=th,t≥t0≥0,h=1,2,…,where Z(t)=(Z1T(t),Z2T(t),…,ZN-1T(t))T, F(Z(t))=(FT(Z1(t)),FT(Z2(t)),…,FT(ZN-1(t)))T, and S^ρ(t)=(LN1ρ(t),LN2ρ(t),…,LN(N-1)ρ(t)).
Obviously, fori=1,2,…,N-1,(37)Zit=Yit-YNt=xit-S-xNt-S=xit-xNt.
According to Lemma3, (32) is equivalent to(38)IN-1-aCSCTIN-1-aCSC≤α2IN-1,C=1,2,…,K.
Define a Lyapunov function(39)Vt=ZTtIN-1⊗RZt.
Whent=th, we can obtain(40)Vth=ZTthIN-1⊗RZth=ZTth-IN-1-aCSC⊗InTIN-1⊗R×IN-1-aCSC⊗InZth-≤α2Vth-.
On the other hand, by (3) in (A1), for any given W=diag⁡(W1,W2,…,Wn)>0, we have(41)0≤∑j=1nWjQj+Zjt-FZjtFZjt-Qj-Zjt=-ZiTtWM1Zit+2ZiTtWM2FZit-FTZitWFZit,and thus(42)0≤-ZTtIN-1⊗WM1Zt+2ZTtIN-1⊗WM2FZt-FTZtIN-1⊗WFZt.
By (33), it follows that(43)-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W<0,where constant α3∈(0,1-α2).
Whent≠th, we can get(44)DCt0qVt≤2ZTtIN-1⊗RZt+2ZTtIN-1⊗R-IN-1⊗DZt+IN-1⊗BFZt.
Together with (42)–(44),(45)DCt0qVt≤HTtIN-1⊗χHt-α2+α3α1Vt<-α2+α3α1Vt,where(46)HTt=ZTt,FTZtT,χ=-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W.
According to Lemma4,(47)Vt≤Vt0Eq-α2+α3α1t-t0q,t≥t0≥0,so V(t)→0 as t→+∞, which implies that, for any given initial value of (1), x1(t)=x2(t)=⋯=xN(t) as t→+∞, and, hence, system (31) can reach complete synchronization.Moreover, consider theNth subnetwork of (31):(48)DCt0qxNt=-DxNt+BFxNt,t≠th,ΔxNt=∑j=1N-1cNjρtxjt--xNt-,t=th,t≥t0≥0,h=1,2,….Based on the above analysis, we haveΔxN(t)=0 as t→+∞. Through Lemma 1, the Nth subnetwork of (31) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n. Therefore, it follows that x1(t)=x2(t)=⋯=xN(t)=S as t→+∞, S∈Sħ,ħ=1,2,…,(r+1)n. To sum up, system (31) can achieve static multisynchronization.Remark 7.
Linear matrix inequality has emerged as a very powerful tool and design technique for a lot of the control problems. From the viewpoint of mathematics, the linear matrix inequality is a convex constraint. So the computational procedure scheme for linear matrix inequality can be processed efficiently. A much more effective computing method for solving these kinds of issues is the interior point method [32]. By using Newton’s method, the interior point method transforms the constrained optimization problem into an unconstrained optimization problem to be solved. Accordingly, reducing a control design problem to the linear matrix inequality may be a practical approach to this problem [33]. Given that, we have proposed the linear matrix inequality based design method for multisynchronization control problem. From the previous discussion, linear matrix inequalities (13) and (14) in Theorem 5 and linear matrix inequalities (32) and (33) in Theorem 6 can be effectively solved.Remark 8.
When nonlinear systems generate multiple locally stable equilibria, finding appropriate control strategy and effective method to deal with such nonlinear systems is difficult. For example, as shown in [34], how to achieve the new synchronization scheme for multistable nonlinear systems is a very intractable problem. However, this issue may be solved if the effective impulsive control strategy is adopted.Remark 9.
Note that the impulsive control strategy (11) or (30) samples the state information only at impulsive times th; namely, each subnetwork takes only the sampling information of its neighbors. Hence, compared with the continuous control law, the impulsive control strategy then has strong pertinence, low energy consumption, and high response speed. As revealed in (12) and (31), the impulsive control system integrates the advantages of impulsive control and continuous control.Remark 10.
Under the framework of Filippov solution, Gu et al. [1] investigate the global synchronization of fractional-order memristive neural networks based on comparison principle and Lyapunov method. Together with fractional-order differential inequality and Lyapunov theory, Xiao et al. [8] analyze the finite-time synchronization of fractional-order memristive bidirectional associative memory neural networks. By employing Holder inequality, Cp inequality, and Gronwall-Bellman inequality, Yang et al. [9] formulate the quasi-uniform synchronization for fractional-order memristive neural networks. Based on Barbalat lemma and Razumikhin-type stability theorem, Zhang et al. [11] establish projective synchronization for fractional-order memristive neural networks. By using the infinitesimal generator on analytic semigroup principle and inequality techniques, Zhou et al. [12] study exponential synchronization of stochastic neural networks driven by fractional Brownian motion. By introducing the concept of joint connectivity and sequential connectivity, Chen et al. [21] show that complex networks can synchronize even if the topology is not connected at any time instant. By combining adaptive control and impulsive control, Yang et al. [17] discuss the global exponential synchronization of complex dynamical networks with nonidentical nodes and stochastic perturbation. By using the mathematical induction method, Zhang et al. [18] achieve the stochastic exponential synchronization for a class of delayed dynamical networks under delayed impulsive control, whereas the above works are all concerned about the global synchronization (or monosynchronization). These analytical methods for global synchronization (or monosynchronization) can not be migrated well to the multisynchronization problem. Using the impulsive control strategy and the Razumikhin-type technique, Wang et al. [16] study the multisynchronization problem of coupled neural networks with directed topology. Nevertheless, the controlled system in [16] is integer-order model. As have often been noted in most existing publications, analytical approach for integral-order systems could not be directly extended and applied to deal with fractional-order systems. In this light, this paper extends and renews the relative results.


## 3.1. Fixed Topology Case

For the coupled fractional-order neural networks (1) with fixed topology, the impulsive control design is given below:(11)uit=a∑h=1+∞∑j=1,j≠iNcijxjt-xitδt-th,for i=1,2,…,N, where a>0 is the coupled gain, cij denotes the element of the weighted adjacency matrix of digraph D, digraph D possesses a directed spanning tree, δ(·) is the Dirac Delta function, and the impulsive time sequence thh=1+∞ satisfies 0<t1<t2<⋯<th<⋯ and limh→∞⁡th=+∞.Combining with (1) and (11), we get(12)DCt0qxt=-IN⊗Dxt+IN⊗BFxt,t≠th,Δxt=-aL⊗Inxt-,t=th,t≥t0≥0,h=1,2,…,where x(t)=(x1T(t),x2T(t),…,xNT(t))T, F(x(t))=(FT(x1(t)),FT(x2(t)),…,FT(xN(t)))T, and L=(Lij)N×N is the Laplacian matrix associated with digraph D.Theorem 5.
Let (A1)–(A3) hold; for any given constantα1>0, if there exist constant 0<α2<1 and matrices R>0 and W=diag⁡(W1,W2,…,Wn)>0 such that(13)-α2IN-1IN-1-aSTIN-1-aS-IN-1<0,(14)-RD-DTR+α2α1R-WM1RB+WM2RB+WM2T-W<0,where(15)S=Lij-LNjN-1×N-1,M1=diag⁡Q1+Q1-,Q2+Q2-,…,Qn+Qn-,M2=diag⁡Q1++Q1-2,Q2++Q2-2,…,Qn++Qn-2,then, for the impulsive time sequence thh=1+∞ admitting sup⁡th-th-1≤α1, system (12) can achieve static multisynchronization.Proof.
From (A1)–(A3), according to Lemma1, every subnetwork of system (1) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n.
LetYi(t)=xi(t)-S,i=1,2,…,N, where S∈Sħ,ħ=1,2,…,(r+1)n, from (12), and then(16)DCt0qYt=-IN⊗DYt+IN⊗BFYt,t≠th,ΔYt=-aL⊗InYt-,t=th,t≥t0≥0,h=1,2,…,where Y(t)=(Y1T(t),Y2T(t),…,YNT(t))T and F(Y(t))=(FT(Y1(t)),FT(Y2(t)),…,FT(YN(t)))T.
LetZi(t)=Yi(t)-YN(t),i=1,2,…,N-1, and system (16) can be reformulated as(17)DCt0qZt=-IN-1⊗DZt+IN-1⊗BFZt,t≠th,DCt0qYNt=-DYNt+BFYNt,t≠th,ΔZt=-aS⊗InZt-,t=th,ΔYNt=-aS^⊗InYNt-,t=th,t≥t0≥0,h=1,2,…,where Z(t)=(Z1T(t),Z2T(t),…,ZN-1T(t))T, F(Z(t))=(FT(Z1(t)),FT(Z2(t)),…,FT(ZN-1(t)))T, and S^=(LN1,LN2,…,LN(N-1)).
Obviously, fori=1,2,…,N-1,(18)Zit=Yit-YNt=xit-S-xNt-S=xit-xNt.
According to Lemma3, (13) is equivalent to(19)IN-1-aSTIN-1-aS≤α2IN-1.
Define a Lyapunov function(20)Vt=ZTtIN-1⊗RZt.
Whent=th, we can obtain(21)Vth=ZTthIN-1⊗RZth=ZTth-IN-1-aS⊗InTIN-1⊗R×IN-1-aS⊗InZth-≤α2Vth-.
On the other hand, by (3) in (A1), for any given W=diag⁡(W1,W2,…,Wn)>0, we have(22)0≤∑j=1nWjQj+Zjt-FZjtFZjt-Qj-Zjt=-ZiTtWM1Zit+2ZiTtWM2FZit-FTZitWFZit,and thus(23)0≤-ZTtIN-1⊗WM1Zt+2ZTtIN-1⊗WM2FZt-FTZtIN-1⊗WFZt.By (14), it follows that(24)-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W<0,where constant α3∈(0,1-α2).
Whent≠th, we can get(25)DCt0qVt≤2ZTtIN-1⊗RZt+2ZTtIN-1⊗R-IN-1⊗DZt+IN-1⊗BFZt.
Together with (23)–(25),(26)DCt0qVt≤HTtIN-1⊗χHt-α2+α3α1Vt<-α2+α3α1Vt,where(27)HTt=ZTt,FTZtT,χ=-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W.
According to Lemma4,(28)Vt≤Vt0Eq-α2+α3α1t-t0q,t≥t0≥0,so V(t)→0 as t→+∞, which implies that, for any given initial value of (1), x1(t)=x2(t)=⋯=xN(t) as t→+∞, and, hence, system (12) can reach complete synchronization.Moreover, consider theNth subnetwork of (12):(29)DCt0qxNt=-DxNt+BFxNt,t≠th,ΔxNt=∑j=1N-1cNjxjt--xNt-,t=th,t≥t0≥0,h=1,2,….Based on the above analysis, we haveΔxN(t)=0 as t→+∞. Through Lemma 1, the Nth subnetwork of (12) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n. Therefore, it follows that x1(t)=x2(t)=⋯=xN(t)=S as t→+∞, S∈Sħ,ħ=1,2,…,(r+1)n. To sum up, system (12) can achieve static multisynchronization.


## 3.2. Switching Topology Case

Without loss of generality, we introduce a switching signalρ(t):[t0,+∞)→1,2,…,K and K digraphs indexed by digraph D1, digraph D2,…, digraph DK.For the coupled fractional-order neural networks (1) with switching topology, the impulsive control design is given below:(30)uit=aρt∑h=1+∞∑j=1,j≠iNcijρtxjt-xitδt-th,for i=1,2,…,N, where the switching signal ρ(t):[t0,+∞)→1,2,…,K, aρ(t)>0 is the coupled gain, cijρ(t) denotes the element of the weighted adjacency matrix of digraph Dρ(t), digraph Dρ(t) possesses a directed spanning tree, δ(·) is the Dirac Delta function, and the impulsive time sequence thh=1+∞ satisfies 0<t1<t2<⋯<th<⋯ and limh→∞⁡th=+∞.Combining with (1) and (30), we get(31)DCt0qxt=-IN⊗Dxt+IN⊗BFxt,t≠th,Δxt=-aρtLρt⊗Inxt-,t=th,t≥t0≥0,h=1,2,…,where x(t)=(x1T(t),x2T(t),…,xNT(t))T, F(x(t))=(FT(x1(t)),FT(x2(t)),…,FT(xN(t)))T, and Lρ(t)=(Lijρ(t))N×N is the Laplacian matrix associated with digraph Dρ(t).Theorem 6.
Let (A1)–(A3) hold; for any given constantα1>0, if there exist constant 0<α2<1 and matrices R>0 and W=diag⁡(W1,W2,…,Wn)>0 such that(32)-α2IN-1IN-1-aCSCTIN-1-aCSC-IN-1<0,C=1,2,…,K,(33)-RD-DTR+α2α1R-WM1RB+WM2RB+WM2T-W<0,where(34)SC=LijC-LNjCN-1×N-1,C=1,2,…,K,M1=diag⁡Q1+Q1-,Q2+Q2-,…,Qn+Qn-,M2=diag⁡Q1++Q1-2,Q2++Q2-2,…,Qn++Qn-2,then, for the impulsive time sequence thh=1+∞ admitting sup⁡th-th-1≤α1, system (31) can achieve static multisynchronization.Proof.
From (A1)–(A3), according to Lemma1, every subnetwork of system (1) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n.
LetYi(t)=xi(t)-S,i=1,2,…,N, where S∈Sħ,ħ=1,2,…,(r+1)n; from (31), then(35)DCt0qYt=-IN⊗DYt+IN⊗BFYt,t≠th,ΔYt=-aρtLρt⊗InYt-,t=th,t≥t0≥0,h=1,2,…,where Y(t)=(Y1T(t),Y2T(t),…,YNT(t))T and F(Y(t))=(FT(Y1(t)),FT(Y2(t)),…,FT(YN(t)))T.
LetZi(t)=Yi(t)-YN(t),i=1,2,…,N-1, and system (35) can be reformulated as(36)DCt0qZt=-IN-1⊗DZt+IN-1⊗BFZt,t≠th,DCt0qYNt=-DYNt+BFYNt,t≠th,ΔZt=-aρtSρt⊗InZt-,t=th,ΔYNt=-aρtS^ρt⊗InYNt-,t=th,t≥t0≥0,h=1,2,…,where Z(t)=(Z1T(t),Z2T(t),…,ZN-1T(t))T, F(Z(t))=(FT(Z1(t)),FT(Z2(t)),…,FT(ZN-1(t)))T, and S^ρ(t)=(LN1ρ(t),LN2ρ(t),…,LN(N-1)ρ(t)).
Obviously, fori=1,2,…,N-1,(37)Zit=Yit-YNt=xit-S-xNt-S=xit-xNt.
According to Lemma3, (32) is equivalent to(38)IN-1-aCSCTIN-1-aCSC≤α2IN-1,C=1,2,…,K.
Define a Lyapunov function(39)Vt=ZTtIN-1⊗RZt.
Whent=th, we can obtain(40)Vth=ZTthIN-1⊗RZth=ZTth-IN-1-aCSC⊗InTIN-1⊗R×IN-1-aCSC⊗InZth-≤α2Vth-.
On the other hand, by (3) in (A1), for any given W=diag⁡(W1,W2,…,Wn)>0, we have(41)0≤∑j=1nWjQj+Zjt-FZjtFZjt-Qj-Zjt=-ZiTtWM1Zit+2ZiTtWM2FZit-FTZitWFZit,and thus(42)0≤-ZTtIN-1⊗WM1Zt+2ZTtIN-1⊗WM2FZt-FTZtIN-1⊗WFZt.
By (33), it follows that(43)-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W<0,where constant α3∈(0,1-α2).
Whent≠th, we can get(44)DCt0qVt≤2ZTtIN-1⊗RZt+2ZTtIN-1⊗R-IN-1⊗DZt+IN-1⊗BFZt.
Together with (42)–(44),(45)DCt0qVt≤HTtIN-1⊗χHt-α2+α3α1Vt<-α2+α3α1Vt,where(46)HTt=ZTt,FTZtT,χ=-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W.
According to Lemma4,(47)Vt≤Vt0Eq-α2+α3α1t-t0q,t≥t0≥0,so V(t)→0 as t→+∞, which implies that, for any given initial value of (1), x1(t)=x2(t)=⋯=xN(t) as t→+∞, and, hence, system (31) can reach complete synchronization.Moreover, consider theNth subnetwork of (31):(48)DCt0qxNt=-DxNt+BFxNt,t≠th,ΔxNt=∑j=1N-1cNjρtxjt--xNt-,t=th,t≥t0≥0,h=1,2,….Based on the above analysis, we haveΔxN(t)=0 as t→+∞. Through Lemma 1, the Nth subnetwork of (31) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n. Therefore, it follows that x1(t)=x2(t)=⋯=xN(t)=S as t→+∞, S∈Sħ,ħ=1,2,…,(r+1)n. To sum up, system (31) can achieve static multisynchronization.Remark 7.
Linear matrix inequality has emerged as a very powerful tool and design technique for a lot of the control problems. From the viewpoint of mathematics, the linear matrix inequality is a convex constraint. So the computational procedure scheme for linear matrix inequality can be processed efficiently. A much more effective computing method for solving these kinds of issues is the interior point method [32]. By using Newton’s method, the interior point method transforms the constrained optimization problem into an unconstrained optimization problem to be solved. Accordingly, reducing a control design problem to the linear matrix inequality may be a practical approach to this problem [33]. Given that, we have proposed the linear matrix inequality based design method for multisynchronization control problem. From the previous discussion, linear matrix inequalities (13) and (14) in Theorem 5 and linear matrix inequalities (32) and (33) in Theorem 6 can be effectively solved.Remark 8.
When nonlinear systems generate multiple locally stable equilibria, finding appropriate control strategy and effective method to deal with such nonlinear systems is difficult. For example, as shown in [34], how to achieve the new synchronization scheme for multistable nonlinear systems is a very intractable problem. However, this issue may be solved if the effective impulsive control strategy is adopted.Remark 9.
Note that the impulsive control strategy (11) or (30) samples the state information only at impulsive times th; namely, each subnetwork takes only the sampling information of its neighbors. Hence, compared with the continuous control law, the impulsive control strategy then has strong pertinence, low energy consumption, and high response speed. As revealed in (12) and (31), the impulsive control system integrates the advantages of impulsive control and continuous control.Remark 10.
Under the framework of Filippov solution, Gu et al. [1] investigate the global synchronization of fractional-order memristive neural networks based on comparison principle and Lyapunov method. Together with fractional-order differential inequality and Lyapunov theory, Xiao et al. [8] analyze the finite-time synchronization of fractional-order memristive bidirectional associative memory neural networks. By employing Holder inequality, Cp inequality, and Gronwall-Bellman inequality, Yang et al. [9] formulate the quasi-uniform synchronization for fractional-order memristive neural networks. Based on Barbalat lemma and Razumikhin-type stability theorem, Zhang et al. [11] establish projective synchronization for fractional-order memristive neural networks. By using the infinitesimal generator on analytic semigroup principle and inequality techniques, Zhou et al. [12] study exponential synchronization of stochastic neural networks driven by fractional Brownian motion. By introducing the concept of joint connectivity and sequential connectivity, Chen et al. [21] show that complex networks can synchronize even if the topology is not connected at any time instant. By combining adaptive control and impulsive control, Yang et al. [17] discuss the global exponential synchronization of complex dynamical networks with nonidentical nodes and stochastic perturbation. By using the mathematical induction method, Zhang et al. [18] achieve the stochastic exponential synchronization for a class of delayed dynamical networks under delayed impulsive control, whereas the above works are all concerned about the global synchronization (or monosynchronization). These analytical methods for global synchronization (or monosynchronization) can not be migrated well to the multisynchronization problem. Using the impulsive control strategy and the Razumikhin-type technique, Wang et al. [16] study the multisynchronization problem of coupled neural networks with directed topology. Nevertheless, the controlled system in [16] is integer-order model. As have often been noted in most existing publications, analytical approach for integral-order systems could not be directly extended and applied to deal with fractional-order systems. In this light, this paper extends and renews the relative results.


## 4. Two Numerical Examples

In the following numerical examples, we consider the coupled fractional-order neural networks consisting of three identical subnetworks; meanwhile, every subnetwork has two neuron states and the parameters of every subnetwork are characterized by(49)D=diag⁡1,1,B=30.20.12.5,f1s=f2s=tanh⁡s.When there is zero input in every subnetwork above, conditions (2)–(6) are satisfied. That is, (A1)–(A3) hold. According to Lemma 1, every subnetwork has 4 locally Mittag-Leffler stable equilibria. Figure 1 depicts the state evolutive characteristics.Figure 1
State evolutive characteristics.Example 11.
We investigate fixed topology case. The Laplacian matrixL=(Lij)3×3 associated with digraph D is described as(50)L=1-1001-1-101.To choose constantα1=0.3, in addition, we select α2=0.9, a=0.1, and matrices R=diag⁡(1,1) and W=diag⁡(0.5,0.5), and then conditions (13) and (14) are satisfied. That is, Theorem 5 holds. According to Theorem 5, the controlled system can achieve static multisynchronization. Figure 2 depicts the multisynchronization characteristics.Figure 2
Multisynchronization characteristics in fixed topology case.
(a)
(b)Example 12.
We investigate switching topology case. The Laplacian matricesLρ(t)=(Lijρ(t))3×3 associated with digraphs Dρ(t) are described as(51)L1=1-1001-1-101,L2=10-1-1100-11,L3=2-1-1-12-1-1-12.To choose constantα1=0.3, in addition, we select α2=0.9, a1=0.1, a2=0.1, a3=0.3, and matrices R=diag⁡(1,1) and W=diag⁡(0.5,0.5), and then conditions (32) and (33) are satisfied. That is, Theorem 6 holds. According to Theorem 6, the controlled system can achieve static multisynchronization. Figure 3 depicts the multisynchronization characteristics.Figure 3
Multisynchronization characteristics in switching topology case.
(a)
(b)


## 5. Concluding Remarks

Multisynchronization represents one of the most striking manifestations of multiconsistency in multistable systems. In this paper, we present the analytical results on the multisynchronization for coupled multistable fractional-order neural networks. Using the impulsive control principle, fractional-order Lyapunov method, and linear matrix inequality technique, several sufficient conditions are deduced to guarantee multisynchronization which characterize the cardinality of the set of synchronization manifolds. How to develop new multisynchronization schemes for multistable fractional-order systems would be the topic of future research.



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*Source: 9323172-2017-08-07.xml*