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)D Ct0qxt=-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 constant α1>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=diagQ1+Q1-,Q2+Q2-,…,Qn+Qn-,M2=diagQ1++Q1-2,Q2++Q2-2,…,Qn++Qn-2,then, for the impulsive time sequence thh=1+∞ admitting supth-th-1≤α1, system (12) can achieve static multisynchronization.
Proof.
From (A1)–(A3), according to Lemma 1, every subnetwork of system (1) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n.
Let Yi(t)=xi(t)-S, i=1,2,…,N, where S∈Sħ, ħ=1,2,…,(r+1)n, from (12), and then(16)D Ct0qYt=-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.
Let Zi(t)=Yi(t)-YN(t), i=1,2,…,N-1, and system (16) can be reformulated as(17)D Ct0qZt=-IN-1⊗DZt+IN-1⊗BFZt, t≠th,D Ct0qYNt=-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, for i=1,2,…,N-1,(18)Zit=Yit-YNt=xit-S-xNt-S=xit-xNt.
According to Lemma 3, (13) is equivalent to(19)IN-1-aSTIN-1-aS≤α2IN-1.
Define a Lyapunov function(20)Vt=ZTtIN-1⊗RZt.
When t=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).
When t≠th, we can get(25)D Ct0qVt≤2ZTtIN-1⊗RZt+2ZTtIN-1⊗R-IN-1⊗DZt+IN-1⊗BFZt.
Together with (23)–(25),(26)D Ct0qVt≤HTtIN-1⊗χHt-α2+α3α1Vt<-α2+α3α1Vt,where(27)HTt=ZTt,FTZtT,χ=-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W.
According to Lemma 4,(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 the Nth subnetwork of (12):(29)D Ct0qxNt=-DxNt+BFxNt, t≠th,ΔxNt=∑j=1N-1cNjxjt--xNt-, t=th, t≥t0≥0, h=1,2,….
Based on the above analysis, we have ΔxN(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 signal ρ(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)D Ct0qxt=-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 constant α1>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=diagQ1+Q1-,Q2+Q2-,…,Qn+Qn-,M2=diagQ1++Q1-2,Q2++Q2-2,…,Qn++Qn-2,then, for the impulsive time sequence thh=1+∞ admitting supth-th-1≤α1, system (31) can achieve static multisynchronization.
Proof.
From (A1)–(A3), according to Lemma 1, every subnetwork of system (1) has (r+1)n locally Mittag-Leffler stable equilibria S1,S2,…,S(r+1)n.
Let Yi(t)=xi(t)-S, i=1,2,…,N, where S∈Sħ, ħ=1,2,…,(r+1)n; from (31), then(35)D Ct0qYt=-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.
Let Zi(t)=Yi(t)-YN(t), i=1,2,…,N-1, and system (35) can be reformulated as(36)D Ct0qZt=-IN-1⊗DZt+IN-1⊗BFZt, t≠th,D Ct0qYNt=-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, for i=1,2,…,N-1,(37)Zit=Yit-YNt=xit-S-xNt-S=xit-xNt.
According to Lemma 3, (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.
When t=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).
When t≠th, we can get(44)D Ct0qVt≤2ZTtIN-1⊗RZt+2ZTtIN-1⊗R-IN-1⊗DZt+IN-1⊗BFZt.
Together with (42)–(44),(45)D Ct0qVt≤HTtIN-1⊗χHt-α2+α3α1Vt<-α2+α3α1Vt,where(46)HTt=ZTt,FTZtT,χ=-RD-DTR+α2+α3α1R-WM1RB+WM2RB+WM2T-W.
According to Lemma 4,(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 the Nth subnetwork of (31):(48)D Ct0qxNt=-DxNt+BFxNt, t≠th,ΔxNt=∑j=1N-1cNjρtxjt--xNt-, t=th, t≥t0≥0, h=1,2,….
Based on the above analysis, we have ΔxN(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.