# Synchronization in pth Moment for Stochastic Chaotic Neural Networks with Finite-Time Control

Authors: Yuhua Xu; Jinmeng Wang; Wuneng Zhou; Xin Wang
Journal: Complexity
 (2019)

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## Abstract

Finite-time synchronization in pth moment is considered for time varying stochastic chaotic neural networks. Compared with some existing results about finite-time mean square stability of stochastic neural network, we obtain some useful criteria of finite-time synchronization in pth moment for chaotic neural networks based on finite-time nonlinear feedback control and finite-time adaptive feedback control, which are efficient and easy to implement in practical applications. Finally, a numerical example is given to illustrate the validity of the derived synchronization conditions.


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## Body

## 1. Introduction

In the past few decades, most of the studies on control of dynamical network were the infinite time control [1–9]. However, in practical engineering, the control of dynamical network is often required in a finite time. From the point of view of the time optimization of the controlled system, finite time convergent of dynamical system is the time optimal control method. Besides the advantages of the optimal convergence performance, the finite-time control of dynamical system has better robust performance and antidisturbance performance compared with the nonfinite-time control of complex system, because of the fractional power in the finite time controller. It makes the finite-time control theory more and more important, and many researchers have studied the finite-time control of dynamical network in detail [10–13].Moreover, there is inevitable delay in neural networks. The generation of time delays may affect the stability of neural network. Therefore, it is of great practical value and theoretical significance to study synchronization control of dynamic neural networks with time delays. In addition, neural networks with delays are easily affected by stochastic disturbances [14]. At present, some authors studied the finite-time stability of stochastic neural networks [15, 16], and some authors discussed the pth moment stability of stochastic neural networks [11, 17–19]. In these studies, LMI and a matrix equality constraint conditions often were applied to establish sufficient conditions about finite-time mean square asymptotic stability of the stochastic neural network [20–23]. As the LMI software cannot handle large-sized problems and it is not numerically stable, adaptive finite-time control method can be more useful for applying to finite-time mean square asymptotic stability of the stochastic neural network, and the finite-time stability of stochastic chaotic neural networks in pth moment is less studied by the existing works. As the quality of the output signal of the neural network can be measured by the speed of the output signal converging to 0 in pth moment. Therefore, it is of great value to study the finite-time stability in pth moment for stochastic neural networks.Motivated by the existing works, we considered finite-time synchronization in pth moment for time varying delayed stochastic neural networks in this paper, and we will establish synchronization criterion in pth moment for stochastic delayed neural networks in finite time.


## 2. Problem Formulation

Consider the following chaotic neural networks:(1)u˙it=Cuit+Afuit+Bfuit-τt+∑j=1NdijΓujt-τt,where ui=(ui1,ui2,…,uin)T∈Rn, f(ui(t))=(f1(ui1(t)),f2(ui2(t)),…,fn(uin(t)))T, C=diagc1,c2,…,cn, A=(apq)n×n, B=(bpq)n×n, and Γ=(λij)n×n. D=(dij)N×N, dij≥0, for i≠j and dii=-∑j=1,j≠iNdij,i=1,2,…N. τ(t) satisfies 0<τ(t)≤τ-, τ˙(t)≤τ⌢<1, where τ-,τ⌢ are constants.Letu(t)=((u1(t))T,(u2(t))T,…,(uN(t))T)T∈RnN and F(u(t))=(f(u1(t))T,f(u2(t))T,…,f(uN(t))T)T; then(2)dut=IN⊗Cut+IN⊗AFut+IN⊗BFut-τt+D⊗Γut-τtdt.The corresponding response chaotic neural network is(3)dvt=IN⊗Cvt+IN⊗AFvt+IN⊗BFvt-τt+D⊗Γvt-τt+rtdt+σt,vt-ut,vt-τt-ut-τtdωt,where v(t) is the state vector, ω(t)=(ω1(t),ω2(t),…,ωn(t))T is an n-dimensional Brown moment, σ:R+×Rn×Rn→Rn×n is the noise intensity matrix, and r(t)=(r1(t),r2(t),…,rn(t))T is the controller.Let the errorswi(t)=vi(t)-ui(t), w(t)=((w1(t))T,(w2(t))T,…,(wN(t))T)T∈RnN, and w(t-τ(t))=wτ(t); then (4)dwt=IN⊗Cwt+IN⊗AFvt-Fut+IN⊗BFvt-τt-Fut-τt+D⊗Γwτt+rtdt+σt,wt,wτtdωt,Definition 1.
The trivial solutionw(t,ξ(s)) of the error system (4) is said to be stability in pth moment if (5)limsupt→∞⁡1tlog⁡Εwt,ξsp<0,where p≥2,p∈Z.Obviously, Definition1 is stability in mean square for p=2.Definition 2 : (finite-time stable).
For the error system (4), if there exists a constant t∗>0 such that (6)limt→t∗wt=0,and w(t)≡0, if t≥t∗, then error system (4) is called stable in finite time.Assumption 3.
Supposef satisfies the following conditions:(7)fvit-fuit≤ηvit-uit,η∈R.Assumption 4.
Letχ,γ>0; then(8)traceσTt,w,wτσt,w,wτ≤χw2+γwτ2.Lemma 5 (see [24]).
Lettingx∈Rn and y∈Rn, then(9)xTy+yTx≤ςxTx+ς-1yTy,ς>0.Lemma 6 (see [25]).
IfV(t,u(t)) and LV(t,u(t)) are bounded on t∈[τ1,τ2] with probability 1, then(10)ΕVτ2,uτ2-ΕVτ1,uτ1=Ε∫τ1τ2LVs,us,uτsds,where τ2≥τ1≥0.Lemma 7 (see [26]).
For some constantsψ,β,(11)Φt≤ψ+β∫0tΦsds,∀0≤t≤T,then(12)Φt≤ψexp⁡βt,∀0≤t≤T.Lemma 8 (see [27]).
Fordu(t)=f(u)dt+g(u)dω(t), T0(x0)=inf⁡T≥0:x(t,x0)=0,∀t≥T. Assume that du(t)=f(u)dt+g(u)dω(t) has the unique global solution, if there is a positive definite, twice continuously differentiable, radially unbounded Lyapunov function V:Rn→R+, and real numbers k>0,0<ρ<1, such that(13)LVu≤-kVuρ,then the origin of system du(t)=f(u)dt+g(u)dω(t) is globally stochastically finite-time stable.


## 3. Main Results

Theorem 9.
Under Assumptions3 and 4, p≥2 and meet the following conditions:(14)κwτp≤πwp,(15)π>11-τ⌢κ,where π=pH-Σ0-η1Σ1-1/2η2Σ2-1/2Θ-1/2(p-1)[χ+p-2/pγ], Σ0=λmax(IN⊗C), Σ1=λmaxIN⊗A, Σ2=λmax(IN⊗B)(IN⊗B)T, Θ=λmax(D⊗Γ)(D⊗Γ)T, and κ=(p-1)γ+1/2η2+1/2.
Then error system (4) is finite-time stability in pth moment by nonlinear feedback controller(16)rt=-Hwt-αsign⁡wtwtθ,ifwt≠00,ifwt=0,where H is a positive constant which is to be determined, 0<θ<1, and α=1/pw(t)p-2.Proof.
We prove Theorem9 in two steps.
The First Step. Error system (4) is finite-time stability under nonlinear feedback controller (16).
Let(17)Vt,w=wp.
By computingLV(t,w(t)),(18)LVt,w,wτ=Vtt,w+Vwt,wIN⊗Cwt+IN⊗AFvt-Fut+IN⊗BFvt-τt-Fut-τt+D⊗Γwτt+rt+12traceσTt,w,wτVwwt,wσt,w,wτ=pwp-2wTIN⊗Cwt+IN⊗AFvt-Fut+IN⊗BFvt-τt-Fut-τt+D⊗Γwτt-Hwt-αsign⁡wtwtθ+12traceσTt,w,wτpp-1wp-2σt,w,wτ.By using Assumptions 3 and 4 and Lemma 5, (19)LV≤pwp-2wTIN⊗Cwt+η1IN⊗Awt+η2IN⊗Bwτ+D⊗Γwτt-Hwt-αsign⁡wtwtθ+12pp-1wp-2χw2+γwτ2.=pwp-2wTIN⊗Cwt+η1IN⊗Awt-Hwt-αsign⁡wtwtθ+pwp-2wTη2IN⊗Bwτ+D⊗Γwτt+12pp-1γwp-2wτ2+12pp-1χwp≤pwp-2wTIN⊗Cwt+η1IN⊗Awt-Hwt-αsign⁡wtwtθ+12pp-1γwp-2wτ2+12pp-1χwp+pwp-2η212wTIN⊗BIN⊗BTw+12wτTwτ+12wTD⊗ΓD⊗ΓTw+12wτTwτ≤pwp-2wTIN⊗Cwt+η1IN⊗Awt-Hwt-αsign⁡wtwtθ+12pp-1γp-2pwp+2pwτp+12pp-1χwp+pwp-2η212wTIN⊗BIN⊗BTw+12wτTwτ+12wTD⊗ΓD⊗ΓTw+12wτTwτ≤pΣ0+η1Σ1+12η2Σ2+12Θ-H+12p-1χ+p-2pγwp+p-1γ+12η2+12wτp-pwp-2wTαsign⁡wtwtθ.Meanwhile, (20)-αpwtp-2wTtsign⁡wtwtθ=-αpwtp-2sign⁡wtwtθTwt=-αpwtp-2wTtwtθ≤-αpwtp-2wTtwtθ+1/2.Therefore, (21)LV≤pΣ0+η1Σ1+12η2Σ2+12Θ-H+12p-1χ+p-2pγwp+p-1γ+12η2+12wτp-αpwtp-2wTtwtθ+1/2=-πwp+κwτp-αpwtp-2wTtwtθ+1/2.where π=pH-Σ0-η1Σ1-1/2η2Σ2-1/2Θ-1/2(p-1)[χ+p-2/pγ], Σ0=λmax(IN⊗C), Σ1=λmaxIN⊗A, Σ2=λmax(IN⊗B)(IN⊗B)T, Θ=λmax(D⊗Γ)(D⊗Γ)T, and κ=(p-1)γ+1/2η2+1/2.
Fromκwτp≤πwp, (22)LV≤-αpwtp-2wTtwtθ+1/2=-wtpθ+1/p=-Vt,wθ+1/p.By using Lemma 8, the error system (4) is stability in finite-time.
The Second Step. The stability in pth moment of error system (4) is achieved under nonlinear feedback controller (16).
From (21), we have(23)LV≤-πwp+κwτp-αpwtp-2wTtwtθ+1/2≤-πwp+κwτp,so(24)Εwp≤ΕVwt≤ΕV0+Ε∫0tLVds≤ΕV0+Ε∫0t-πwp+κwτpds.From the literature [12](25)∫0twτpds≤τ-1-τ⌢maxτ⌢≤s≤0⁡ξsp+11-τ⌢∫0twspds,so(26)Εwp≤ΕV0+κτ-1-τ⌢maxτ-≤s≤0⁡Εξsp+∫0tκ11-τ⌢-δΕwpds.From Lemma 8, (27)Εwp≤ΕV0+κτ-1-τ⌢maxτ-≤s≤0⁡Εξspexp⁡-δ-κ11-τ⌢t.Therefore(28)limt→∞sup⁡1tlog⁡Εwt,ξp≤-π-κ11-τ⌢<0.The stability in pth moment of error system (4) is achieved by using Definition 1.
Combining the first step and the second step, one can get that the fixed-time stability in pth moment of error system (4) is finally realized by nonlinear feedback controller (16).In fact,H may be very large to ensure the stability of error systems (4), and it can be meaningless in some practical applications. Therefore, adaptive feedback control is selected by the following Theorem 10.Theorem 10.
Under Assumptions3 and 4, p≥2 and meet the following conditions:(29)κwτp≤π∗wp,(30)π∗>11-τ⌢κ,where π∗=pk0-Σ0-η1Σ1-1/2η2Σ2-1/2Θ-1/2(p-1)[χ+p-2/pγ], Σ0=λmax(IN⊗C), Σ1=λmaxIN⊗A, Σ2=λmax(IN⊗B)(IN⊗B)T, Θ=λmax(D⊗Γ)(D⊗Γ)T, κ=(p-1)γ+1/2η2+1/2, and H is positive constant.
Then, the stability in pth moment of error system (4) is completed by the following adaptive feedback controller:(31)rit=-kiwit-αsign⁡witwitθ,ifwit≠00,ifwit≠0,(32)k˙i=pwip-2wi2-κsign⁡kit-k0kit-k0θ,where k0 is a positive constant which is to be determined, 0<θ<1 and α=1/pw(t)p-2.Proof.
Similar to the proofs of Theorem9.
The First Step. The finite-time stability of error system (4) is achieved under adaptive feedback controller (31).
Let(33)Vt,w=wp+∑i=1N12ki-k02.Computing LV(t,w(t)), (34)
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.From κwτp≤π∗wp, have(35)LV≤-αpwtp-2wTtwtθ+1/2-∑i=1Nκkit-k0θ+1=-wtpθ+1/p+∑i=1Nκkit-k0pθ+1/p=-wtpθ+1/p+∑i=1N12kit-k02θ+1/p≤-wtp+∑i=1N12kit-k02θ+1/p=-Vt,wθ+1/p,where κ=1/21/ki(t)-k0p-2θ+1/p.
By using Lemma8, error system (4) is stability in finite-time.
The Second Step. The stability in pth moment of error system (4) is achieved under adaptive feedback controller (31).
Form (34), we have(36)LV≤-π∗wp+κwτp-αpwtp-2wTtwtθ+1/2-∑i=1Nκkit-k0θ+1≤-π∗wp+κwτp.The remaining reasoning is similar to the proof process of Theorem 9; we have(37)limt→∞⁡sup⁡1tlogΕwt,ξp≤-π∗-κ11-τ⌢<0.By using Definition 1, error system (4) is stability in pth moment.
Combining the first step and the second step, error system (4) is finite-time stability in pth moment by adaptive feedback controller (31).Remark 11.
It can be seen from Theorems9 and 10 that the greater the noise intensity, the greater the control intensity, so the noisy environment may affect robustness.Remark 12.
In [20–23], research efforts have concentrated on studying finite-time mean stability of stochastic networks based on LMI. In this paper, finite-time nonlinear feedback control and finite-time adaptive feedback control are considered, which can be simpler than LMI.


## 4. Illustrative Example

In the following, we present an example to illustrate the usefulness of Theorem10.The drive systems are(38)u˙it=-Cuit+Afuit+Bfuit-τt+∑j=14dijΓujt-τt,where (39)C=-100-1,A=380.32.5,B=-50.30.3-5,fx=tanh⁡x,Γ=I,τt=0.1,D=-12-102-1.50.5-1-10.50.500-101.The response systems are(40)v˙it=-Cvit+Afvit+Bfvit-τt+∑j=14dijΓvjt-τt+0.1wit+0.1wit-τt+rit.The adaptive controllers are(41)rit=-kiwit-αsign⁡witwitθ,k˙i=pwip-2wi2-κsign⁡kit-k0kit-k0θ,where p=2, α=0.5, k0=60, θ=0.15, and κ=0.50.575.We letui(0)=(-0.5i,i), vi(0)=(2i,-i), and ki(0)=(3i,-2i). Figure 1 shows synchronization errors. Figure 2 shows dynamic curve of the feedback gain. Let E(t)=∑i=14vi(t)-ui(t)2/4. Obviously, when E(t)→0, finite-time synchronization of neural networks is realized (See Figure 3). All numerical simulations illustrate the effectiveness of Theorem 10.Figure 1
Synchronization errors.Figure 2
Dynamic curve of the feedback gain.Figure 3
Dynamic curve ofE(t).


## 5. Conclusion

In this paper, we have discussed finite-time synchronization in pth moment for time-varying stochastic chaotic neural networks via finite-time nonlinear feedback control and finite-time adaptive feedback control, which are efficient and easy to implement in practical applications. Numerical simulations demonstrate the effectiveness of the main results obtained in this paper. Our future work is to study adaptive finite-time pining control in pth moment for time varying stochastic chaotic neural networks, which may help to save control cost and has more practical application in practical engineering.



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