# Hybrid Adaptive Pinning Control for Function Projective Synchronization of Delayed Neural Networks with Mixed Uncertain Couplings

Authors: T. Botmart; N. Yotha; P. Niamsup; W. Weera
Journal: Complexity
 (2017)

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## Abstract

This paper presents the function projective synchronization problem of neural networks with mixed time-varying delays and uncertainties asymmetric coupling. The function projective synchronization of this model via hybrid adaptive pinning controls and hybrid adaptive controls, composed of nonlinear and adaptive linear feedback control, is further investigated in this study. Based on Lyapunov stability theory combined with the method of the adaptive control and pinning control, some novel and simple sufficient conditions are derived for the function projective synchronization problem of neural networks with mixed time-varying delays and uncertainties asymmetric coupling, and the derived results are less conservative. Particularly, the control method focuses on how to determine a set of pinned nodes with fixed coupling matrices and strength values and randomly select pinning nodes. Based on adaptive control technique, the parameter update law, and the technique of dealing with some integral terms, the control may be used to manipulate the scaling functions such that the drive system and response systems could be synchronized up to the desired scaling function. Finally, numerical examples are given to illustrate the effectiveness of the proposed theoretical results.


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## Body

## 1. Introduction

Presently, neural networks are under extensive consideration because of their significant application in various fields such as image processing, pattern recognition, and associative memories because the switching speed of information processing and the inherent neuron communication is limited [1, 2]. Global asymptotic stability of a general class of recurrent neural networks with time-varying delays is discussed in [3]. Zhang and Han [4] investigated the global asymptotical stability analysis for delayed neural networks using a matrix-based quadratic convex approach. It is well know that the time delay is continually necessarily existent in neural networks because it might lead to inconstancy or considerably inferior performances. So, the neural networks with time delays have attracted considerable attention of many researchers [5–7].In addition, much attention has been paid to the potential applications of the synchronization of coupled neural networks, for example, secure communication [8–10]. The synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay were discussed in [11]. Chen and Cao [12] suggested projective synchronization of neural networks with mixed time-varying delays and parameter mismatch. Moreover, there is synchronization problem that is called function projective synchronization that has received increasing attention in [13–15]. FPS is the driver and response system that can be synchronized up to a scaling function. Many researches mentioned that function projective synchronization (FPS) is the greater general definition of chaotic synchronization [16–18]. It is obvious that the definition of FPS includes comprehensive synchronization and projective synchronization. In order for scaling function to achieve unity or be constant, only one complete synchronization or projective synchronization could be obtained since the unpredictability of the scaling function in FPS can also raise the security of communication. Thus, FPS has drawn the attention of many researchers in various fields. In Gao et al. [19] the generalized function projective synchronization of weighted cellular neural networks with multiple time-varying coupling delays was studied. Adaptive projective synchronization in complex networks with time-varying coupling delay discussed in [20]. Tang and Wong [21] studied on the distributed synchronization problem of coupled neural networks by randomly occurring control method. Hu et al. [22] suggested a pinning synchronization control scheme for a class of linearly coupled neural networks. Huang et al. [23] investigated the stabilization of delayed chaotic neural networks by periodically intermittent control. Further, Cai et al. [24] considered the outer synchronization between two hybrid-coupled delayed dynamical networks via aperiodically adaptive intermittent pinning control. However, not all of the neural networks could synchronize by themselves. So, they need to bring the suitable controllers in order to make them synchronize. One of the most important existing control methods is the pinning control.Pinning control is the strategy that employs the local feedback injection to a small fraction of nodes to carry out the global performances of the total networks. It is a competent and useful strategy especially for the large size networks. The pinning synchronization of neural networks has been generally examined at the present [25–33]. Meanwhile, various selection rules of pinned nodes have been introduced in the existing literatures. The pinned nodes selection rules according to the out-degree and in-degree of the nodes and the synchronization problem were studied for undirected and directed networks which was presented in [25, 26, 34]. The pinning control problem of neural networks was considered and then some unexceptional pinning conditions were found out [27–29], while Chen et al. [35] expressed that, even applying one single pinned node, the whole networks could be controlled as long as the coupling strength was large enough. Furthermore, Wang and Chen [36] summarized that the most highly connected nodes are pinned in order to get the better performance for the undirected networks.As discussions mentioned above, hybrid adaptive pinning control for FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling is an interesting topic for investigating. Therefore, this paper will be focused on this topic in order to facilitate clear comprehension and the purposes of this paper are given as follows:(i)
The mixed time-varying delays with discrete and distributed time-varying delays are considered in the dynamical nodes and in uncertainties asymmetric coupling, simultaneously, which are different from time-delay case in [23, 27–29]. So, our systems are general ones.(ii)
For the control method, FPS is studied by using the nonlinear and adaptive pinning controls and using the nonlinear and adaptive controls which contain error linear term, time-varying delay error linear term, and distributed time-varying delay error linear term.(iii)
The FPS of this paper focuses on how to determine a set of pinned nodes for a linearly coupled delayed neural network with fixed coupling matrices and strength values. Moreover, this paper used random selection of pinning nodes which is different from the pinning control method in [13, 37]. Based on constructing a novel Lyapunov-Krasovskii functional, adaptive control technique, the parameter update law, and the technique of dealing with Jensen’s and Cauchy inequalities, some novel sufficient conditions for guaranteeing the existence of the FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling are derived. Finally, numerical examples are included to show the effectiveness of using the nonlinear and adaptive pinning controls and the nonlinear and adaptive controls.The rest of the paper is organized as follows. Section2 provides some mathematical preliminaries and network model. Section 3 presents FPS of neural network with mixed time-varying delays and hybrid uncertainties asymmetric coupling by hybrid adaptive control and hybrid adaptive pinning control, respectively. In Section 4, some numerical examples illustrate given theoretical results. The paper ends with conclusions in Section 5 and cited references.


## 2. Model Description and Preliminaries

Notations. The following notation will be used in this paper: Rn denotes the n-dimensional space and · denotes the Euclidean vector norm; AT denotes the transpose of matrix A; A is symmetric if A=AT; IN denotes an N-dimensional identity matrix; for the matrix A∈RN×RN, the ith row and the ith column of A are called the ith row-column pair of A. Al∈R(N-l)×(N-l) is the minor matrix of A∈RN×N by removing arbitrary l(1≤l≤N) row-column pairs of A. The symbol ⊗ denotes the Kronecker product.Consider an array of delayed neural networks consisting ofN identical nodes with uncertainties asymmetric coupling(1)xi˙t=-Axit+W1f1xit+W2f2xit-ht+W3∫t-kttf3xiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1xjt+c2∑j=1NG2ijΓ2+ΔΓ2xjt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttxjθdθ+Uit,i=1,2,…,N,where N is the number of coupled nodes, xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn is the neuron state vector of the ith node, n denotes the number of neurons in a neural network, A=diag(a1,a2,…,an)>0 denote the rate with which the cell i resets its potential to the resting state when being isolated from other cells and inputs, W1,W2, and W3 are connection weight matrices, h(t) and k(t) are the time-varying delays, fr(xi(·))=[fr1(xi1(·)),…,frn(xin(·))]T,r=1,2,3, denote the neuron activation function vector, the positive constants c1,c2, and c3 are the strength values for the constant coupling and delayed couplings, respectively, Ui(t)∈Rm are the control input of the node i, Γ1,Γ2, and Γ3∈Rn×n are constant inner-coupling matrices and it is assumed that Γ1,Γ2, and Γ3 are positive definite matrix, ΔΓ1,ΔΓ2, and ΔΓ3 are the uncertainties of inner-coupling matrices, and Gr=(Grij)N×N(r=1,2,3) are the outer-coupling matrices and satisfy the following conditions:(2)Grij≥0,i≠j,r=1,2,3,Grii=-∑j=1,j≠iNGrij,i,j=1,2,…,N.Assumption 1.
The time-varying delay functionh(t) is differential function and k(t) satisfies the conditions 0≤h(t)≤h,0≤k(t)≤k, and 0≤h˙(t)≤μ<1.Assumption 2.
The activation functionsfri(·),r=1,2,3,i=1,2,…,n, satisfy the Lipschitz constants fri>0: (3)f1ixt-f1iαtyt≤F1xt-αtyt,f2ixt-ht-f2iαtyt-ht≤F2xt-ht-αtyt-ht,f3ixθ-f3iαtyθ≤F3xθ-αtyθ,where Fr(r=1,2,3) are positive constant matrices and we denote (4)F1=diagf1i,i=1,2,…,n,F2=diagf2i,i=1,2,…,n,F3=diagf3i,i=1,2,…,n.Assumption 3.
The parameter uncertainties are assumed to satisfy the following conditions:(5)ΔΓ1tΔΓ2tΔΓ3t=M∇tE1E2E3,where M,E1,E2, and E3 are known real constant matrices and ∇(t) is and unknown time-varying matrix function satisfying ∇T(t)∇(t)≤I.The isolated dynamic network is(6)s˙t=-Ast+W1f1st+W2f2st-ht+W3∫t-kttf3sθdθ,where s(t)=(s1(t),s2(t),…,sn(t))T∈Rn with x0∈Rn and the parameters A,W1,W2, and W3 and the nonlinear functions f(·) have the same definitions as in (1).Definition 4 (FPS).
Network (1) with time delay is said to achieve function projective synchronization if there exists a continuously differentiable scaling function α(t) such that (7)limt→∞eit=limt→∞xit-αtst,i=1,2,…,N,where · stands for the Euclidean vector norm and s(t)∈Rn can be an equilibrium point, or a (quasi-)periodic orbit, or an orbit of a chaotic attractor.To investigate the stability of the synchronized states (1), we set the synchronization error ei(t) in the form ei(t)=xi(t)-α(t)s(t),i=1,…,N. Then, substituting it into (1), it is easy to get the following:(8)e˙it=x˙it-α˙tst-αts˙t=-Aeit+W1f1xit-αtf1st+W2f2xit-ht-αtf2st-ht+W3∫t-kttf3xiθ-αtf3sθdθ+c1∑j=1NG1ijΓ1+ΔΓ1ejt+c2∑j=1NG2ijΓ2+ΔΓ2ejt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttejθdθ-α˙tst+Uit.The initial condition of (8) is defined by(9)eir=ϕir,-τ≤r≤0,where τ=max{k,h} and ϕi(r)∈C([-τ,0],Rn),i=1,2,…,N.Remark 5.
If neural networks (1) are without parameter uncertainties and c3=0, the networks model turns into the neural networks proposed in [27–29]: (10)xi˙t=-Axit+W1f1xit+W2f2xit-ht+W3∫t-kttf3xiθdθ+c1∑j=1NG1ijΓ1xjt+c2∑j=1NG2ijΓ2xjt-ht,i=1,2,…,N.Hence, our network model (1) includes previous network model, which can be regarded as a special case of neural network (1).Lemma 6 (see [5] (Cauchy inequality)).
For any symmetric positive definite matrixesN∈Mn×n and x,y∈Rn we have (11)±2xTy≤xTNx+yTN-1y.Lemma 7 (see [5]).
For any constant symmetric matrixesM∈Rm×m,M=MT>0, and γ>0, vector function ω:[0,γ]→Rm such that the integrations concerned are well defined, and then (12)∫0γωTθdθTM∫0γωθdθ≤γ∫0γωTsMωθdθ.Lemma 8 (see [38]).
Letc∈R and A, B, C, and D be matrices with appropriate dimensions. Then (i)
cA⊗B=cA⊗B=A⊗cB,(ii)
(A⊗B)T=AT⊗BT,(iii)
(A⊗B)(C⊗D)=(AC)⊗(BD),(iv)
A⊗(B+C)=(A⊗B)+(A⊗C),(v)
A⊗B⊗C=(A⊗B)⊗C=A⊗(B⊗C).Lemma 9 (see [25]).
Assume thatA and B are the N×N Hermitian matrices. Suppose that α1≥α2≥⋯≥αN, β1≥β2≥⋯≥βN, and γ1≥γ2≥⋯≥γN are eigenvalues of matrices A,B, and A+B, respectively. Then one has αi+βN≤γi≤αi+β1,i=1,2,…,N.Lemma 10 (see [35]).
IfA=(aij)N×N is irreducible and satisfies aij=aji≥0,i≠j, aii=-∑j=1,i≠jNaij,i,j=1,2,…,N. Then, for any constant ξ>0, all eigenvalues of the matrix A-Ξ are negative definite, where Ξ=diag(ξ,0,…,0).Lemma 11 (see [25]).
For a symmetric matrixM∈RN×N and a diagonal matrix D=diag(d1,…,dl,0,…,0︸N-l) with di>0,i=1,2,…,l(1≤l<N), let (13)M-D=A-D~BBTMl,where Ml is the minor matrix of M by removing its first l row-column pairs, A and B are matrices with appropriate dimensions, and D~=diag(d1,…,dl). If di>λmax(A-BMl-1BT),i=1,…,l,M-D<0 is equivalent to Ml<0.


## 3. Main Results

In this section, we present hybrid control scheme to synchronize neural networks (1) to the homogenous trajectory (6). Then, we will give some sufficient conditions in FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling.


### 3.1. FPS under Hybrid Adaptive Pinning Control

We design nonlinear and adaptive pinning controls to realize FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling. In order to stabilize the origin of neural networks (1) by means of nonlinear and adaptive pinning controls Ui(t) such as(14)Uit=ui1t+ui2t,i=1,2,…,N,where (15)ui1t=α˙tst-W1αtf1st+f1αtst-W2αtf2st-ht+f2αtst-ht-W3∫t-kttαtf3sθ-f3αtsθdθ,i=1,2,…,N,ui2t=-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,l,0i=l+1,l+2,…,N,the updating laws are(16)d˙1it=βi1eiTtΓ1eit,d˙2it=βi2eiTtΓ2eit-ht,d˙3it=βi3eiTtΓ3∫t-ktteiθdθ,where βi1,βi2, and βi3 are positive constants and s(t) is a solution of an isolated node. The controller Ui(t) is different type of controller, and ui1(t) is the nonlinear control and ui2(t) is the adaptive linear pinning control. Then, substituting (14) into (8), it can be derived that(17)e˙it=-Aeit+W1f~1eit+W2f~2eit-ht+W3∫t-kttf~3eiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1ejt+c2∑j=1NG2ijΓ2+ΔΓ2ejt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttejθdθ-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,l,e˙it=-Aeit+W1f~1eit+W2f~2eit-ht+W3∫t-kttf~3eiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1ejt+c2∑j=1NG2ijΓ2+ΔΓ2ejt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttejθdθ,i=l+1,l+2,…,N,d˙1it=βi1eiTtΓ1eit,i=1,2,…,l,d˙2it=βi2eiTtΓ2eit-ht,i=1,2,…,l,d˙3it=βi3eiTtΓ3∫t-ktteiθdθ,i=1,2,…,l,where (18)f~1eit=f1xit-f1αtst,f~2eit-ht=f2xit-ht-f2αtst-ht,f~3eiθ=f3xiθ-f3αtsθ,and, for convenience, we denote (19)Π1=1λminIN⊗Γ1-λmaxIN⊗A+12ε1λmaxIN⊗W1W1T+ε12λmaxIN⊗F1TF1+c3k22λmaxIN⊗Γ3+12ε2λmaxIN⊗W2W2T+12ε3λmaxIN⊗W3W3T+c1ε62λmaxIN⊗E1TE1+c12ε6λmaxG1G1TλmaxMMT+c22ε7λmaxG2G2TλmaxΓ2Γ2T+c22ε8λmaxG2G2TλmaxMMT+c32ε9λmaxG3G3TλmaxΓ3Γ3T+c32ε10λmaxG3G3TλmaxMMT+c221-μλmaxIN⊗Γ2+c2d¯2∗2ε4λmaxIN⊗Γ2Γ2T+c3d¯3∗2ε5λmaxIN⊗Γ3Γ3T,Π2=12λminIN⊗Γ2ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2+c2d¯2∗ε4,Π3=12λminIN⊗Γ3ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3+c3d¯3∗ε5,ξt=eTt,eTt-ht,∫t-ktteθdθTT.Theorem 12.
For some given synchronization scaling functionα(t), neural networks (1) satisfying Assumptions 1, 2, and 3, and target system can realize function projective synchronization by the nonlinear and adaptive pinning control law as shown in (14) if there exist positive constants εi,i=1,2,…,10, and by taking appropriate d1i∗(i=1,2,…,l),d¯2∗, and d¯3∗ such that(20)λmaxG1+G1T2l<-Π1c1,(21)d¯2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,(22)d¯3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.Then controlled neural network (1) is function projective synchronization.Proof.
Construct the following Lyapunov-Krasovskii functional candidate:(23)Vt=V1t+V2t+V3t+V4t,where (24)V1t=12∑i=1NeiTteit,V2t=c221-μ∑i=1N∫t-htteiTθΓ2eiθdθ,V3t=c3k2∑i=1N∫-k0∫t+steiTθΓ3eiθdθds,V4t=c12∑i=1l1βi1d1it-d1i∗2+c22∑i=1l1βi2d2it-d2i∗2+c32∑i=1l1βi3d3it-d3i∗2.By taking the derivative of V(t) along the trajectories of system (17), we have the following: (25)V˙1t=-∑i=1NeiTtAeit+∑i=1NeiTtW1f~1eit+∑i=1NeiTtW2f~2eit-ht+∑i=1NeiTtW3∫t-kttf~3eiθdθ+c1∑i=1N∑j=1NeiTtG1ijΓ1+ΔΓ1ejt+c2∑i=1N∑j=1NeiTtG2ijΓ2+ΔΓ2ejt-τt+c3∑i=1N∑j=1NeiTtG3ijΓ3+ΔΓ3∫t-kttejθdθ-∑i=1NeiTtc1d1itΓ1eit-∑i=1NeiTtc2d2itΓ2eit-ht-∑i=1NeiTtc3d3itΓ3∫t-ktteiθdθ.V˙2t=c221-μ∑i=1NeiTtΓ2eit-c21-h˙t21-μ∑i=1NeiTt-htΓ2eit-ht.V˙3t=c3k22∑i=1NeiTtΓ3eit-c3k2∑i=1N∫t-ktteiTθΓ3eiθdθ.V˙4t=∑i=1lc1βi1d1it-d1i∗βi1eiTtΓ1eit+∑i=1lc2βi2d2it-d2i∗βi2eiTtΓ2eit-ht+∑i=1lc3βi3d3it-d3i∗βi3eiTtΓ3∫t-ktteiθdθ.After V(t) were calculated, we will get that(26)V˙t≤-∑i=1NeiTtAeit+∑i=1NeiTtW1f~1eit+∑i=1NeiTtW2f~2eit-ht+∑i=1NeiTtW3∫t-kttf~3eiθdθ+c1∑i=1N∑j=1NeiTtG1ijΓ1+ΔΓ1ejt+c2∑i=1N∑j=1NeiTtG2ijΓ2+ΔΓ2ejt-τt+c3∑i=1N∑j=1NeiTtG3ijΓ3+ΔΓ3×∫t-kttejθdθ+c221-μ∑i=1NeiTtΓ2eit-c221-μ∑i=1NeiTt-htΓ2eit-ht+c3k22∑i=1NeiTtΓ3eit-c3k2∑i=1N∫t-ktteiTθΓ3eiθdθ-c1∑i=1ld1i∗eiTtΓ1eit-c2∑i=1ld2i∗eiTtΓ2eit-htds-c3∑i=1ld3i∗eiTtΓ3∫t-ktteiθdθ≤-eTtIN⊗Aet+eTtIN⊗W1f~1et+eTtIN⊗W2f~2et-ht+eTtIN⊗W3∫t-kttf~3eθdθ+c1eTtG1⊗Γ1+ΔΓ1et+c2eTtG2⊗Γ2+ΔΓ2et-ht+c3eTtG3⊗Γ3+ΔΓ3∫t-ktteθdθ+c221-μeTtIN⊗Γ2et-c221-μeTt-htIN⊗Γ2et-ht+c3k22eTtIN⊗Γ3et-c3k2∫t-ktteTθIN⊗Γ3eθdθ-c1eTtD1∗⊗Γ1et-c2d¯2∗eTtIN⊗Γ2et-ht-c3d¯3∗eTtIN⊗Γ3∫t-ktteθdθ,where e(t)=(e1(t),…,eN(t))∈Rn×N, e(t-h(t))=(e1(t-h(t)),…,eN(t-h(t)))∈Rn×N, ∫t-k(t)te(θ)dθ=∫t-k(t)t(e1(θ),e2(θ),…,eN(θ))dθ∈Rn×N, D1∗=diag(d11∗,d12∗,…,d1l∗,0,…,0)∈RN×N, d¯2∗=max1≤i≤l{d2i∗}, and d¯3∗=max1≤i≤l{d3i∗}.
From Assumption2, we obtain the following three inequalities: (27)f~1ieit=f~1ixt-f~1iαtst≤F1ieit,f~2ieit-ht=f~2ixt-ht-f~2iαtst-ht≤F2ieit-ht,f~3ieiθ=f~3ixθ-f~3iαtsθ≤F3ieiθ.Applying Lemmas 6 and 7, we have (28)eTtIN⊗W1f~1et≤12ε1eTtIN⊗W1W1Tet+ε12f~1TetIN⊗Inf~1et,≤12ε1eTtIN⊗W1W1Tet+ε12eTtIN⊗F1TF1et,eTtIN⊗W2f~2et-ht≤12ε2eTtIN⊗W2W2Tet+ε22f~2Tt-htIN⊗Inf~2et-ht,≤12ε2eTtIN⊗W2W2Tet+ε22eTt-htIN⊗F2TF2et-ht,eTtIN⊗W3∫t-kttf~3eθdθ≤12ε3eTtIN⊗W3W3Tet+ε32∫t-kttf~3TeθdθTIN⊗In∫t-kttf~3Teθdθ,≤12ε3eTtIN⊗W3W3Tet+ε32×∫t-ktteTθdθTIN⊗F3TF3∫t-ktteTθdθ,-c2d¯2∗eTtIN⊗Γ2et-ht≤c2d¯2∗2ε4eTtIN⊗Γ2Γ2Tet+c2d¯2∗ε42eTt-htIN⊗Inet-ht,-c3d¯3∗eTtIN⊗Γ3∫t-ktteθdθ≤c3d¯3∗2ε5eTtIN⊗Γ3Γ3Tet+c3d¯3∗ε52×∫t-ktteTθdθTIN⊗In∫t-ktteTθdθ,c1eTtG1⊗Γ1+ΔΓ1et=c1eTtG1⊗Γ1et+c1eTtG1⊗M∇tE1et,≤c1eTtG1⊗Γ1et+c12ε6eTtG1G1T⊗MMTet+c1ε62eTtIN⊗E1TE1et,c2eTtG2⊗Γ2+ΔΓ2et-ht=c2eTtG2⊗Γ2et-ht+c2eTtG2⊗M∇tE2et-ht,≤c22ε7eTtG2G2T⊗Γ2Γ2Tet+c2ε72eTt-htIN⊗Inet-ht+c22ε8eTtG2G2T⊗MMTet+c2ε82eTt-htIN⊗E2TE2t-ht,c3eTtG3⊗Γ3+ΔΓ3∫t-ktteθdθ=c3eTtG3⊗Γ3∫t-ktteθdθ+c3eTtG3⊗M∇tE3∫t-ktteθdθ,≤c32ε9eTtG3G3T⊗Γ3Γ3Tet+c3ε92∫t-ktteTθdθTIN⊗In∫t-ktteTθdθ+c32ε10eTtG3G3T⊗MMTet+c3ε102×∫t-ktteTθdθTIN⊗E3TE3∫t-ktteTθdθ.Therefore, we have(29)eTt-c1D1∗⊗Γ1+c1G1⊗Γ1+c3k22IN⊗Γ3+12ε2IN⊗W2W2T+12ε3IN⊗W3W3T+c12ε6G1G1T⊗MMT+c1ε62IN⊗E1TE1+c22ε7G2G2T⊗Γ2Γ2T+c22ε8G2G2T⊗MMT+c32ε9G3G3T⊗Γ3Γ3T+c32ε10G3G3T⊗MMT+c2d¯2∗2ε4IN⊗Γ2Γ2T+c3d¯3∗2ε5IN⊗Γ3Γ3Tet-eTt-htc22IN⊗Γ2-Π2IN⊗Γ2et-ht-∫t-ktteTθc3k2IN⊗Γ3-Π3IN⊗Γ3eθdθ,≤eTtΠ1IN+c1G1-c1D1∗⊗Γ1et-eTt-htc22-Π2IN⊗Γ2et-ht-∫t-ktteTθc3k2-Π3IN⊗Γ3eθdθ,=eTtM-c1D1∗⊗Γ1et-eTt-htc22-Π2IN⊗Γ2et-ht-∫t-ktteTθc3k2-Π3IN⊗Γ3eθdθ,where M=Π1IN+c1G1. Note that the matrix M is symmetric. Let (30)M-D1∗=M~1-D~1l∗M~3M~3TM~2,where M~2 is the minor matrix of M by removing its first l(1≤l<N) row-column pairs, M~1 and M~3 are matrices with appropriate dimensions, and D~1l∗=diag(d11∗,d12∗,…,d1l∗). If λmax((G1+G1T)/2)l<-Π1/c1 and with Lemma 9, we have M~2<0. Therefore, one can choose suitable positive constants d1i∗>0,i=1,2,…,l, such that d1i∗>λmax(M~1-M~3M~2-1M~3T). It follows from Lemma 11 and M~2<0 that M-c1D1∗<0. Then, by Γ1>0 and (29), we can conclude that(31)V˙t≤-eTt-htc22-Π2IN⊗Γ2et-ht-∫t-ktteθdθTc3k2-Π3IN⊗Γ3∫t-ktteθdθ.We only need to choose the suitable positive constants d¯2∗ and d¯3∗ such that(32)d¯2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,(33)d¯3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.We can choose d¯2∗ and d¯3∗ satisfying (32) and (33), respectively. Since Γ1,Γ2, and Γ3 are positive definite matrix, we know that V˙(t)≤0. Therefore, pinning controlled delayed neural networks (1) globally asymptotically synchronize to trajectory (6) if conditions (20), (21), and (22) hold. Then the controlled neural networks (1) are function projective synchronization.Remark 13.
If there are no uncertain parameters in coupled delayed neural networks (1), ΔΓ1=ΔΓ2=ΔΓ3=0. One can obtain similar synchronization results.Remark 14.
The nodes pinned for directed networks are chosen as follows.
Step
1. Choose some appropriate parameters εi,i=1,2,…,10, by taking appropriate d¯1i∗,i=1,2,…,l, d¯2∗, and d¯3∗ such that the conditions in Theorem 12 are feasible.
Step
2. The l pinned nodes are sorted according to the pinned-node selection scheme studied [26], for the pinning controlled error neural network (17); so, the nodes to be pinned are chosen in the particular order. Let l=1, if the first inequalities of Theorem 12 are satisfied, and then the least number is 1; otherwise, go to the next step.
Step
3. If condition (20) is not satisfied, increase l(l=l+1) gradually with more network nodes to the pinned node based on the order in Step 2 particularly until condition (20) holds.
For undirected networks, for example, the small-world network [31], the scale-free network [32], and the Watts-Strogatz network [39], we can randomly choose a set of pinned nodes to satisfy condition (20) by increasing the number of pinned nodes l.


### 3.2. FPS under Hybrid Adaptive Control

The nonlinear and adaptive controls are designed to realize FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling. Then we have the following controlled form:(34)Uit=ui1t+ui2t,i=1,2,…,N,where (35)ui1t=α˙tst-W1αtf1st+f1αtst-W2αtf2st-ht+f2αtst-ht-W3∫t-kttαtf3sθ-f3αtsθdθ,i=1,2,…,N,ui2t=-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,N.Then, substituting (34) into (8), we have(36)e˙it=-Aeit+W1f~1eit+W2f~2eit-ht+W3∫t-kttf~3eiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1ejt+c2∑j=1NG2ijΓ2+ΔΓ2ejt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttejθdθ-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,N,d˙1it=βi1eiTtΓ1eit,i=1,2,…,N,d˙2it=βi2eiTtΓ2eit-ht,i=1,2,…,N,d˙3it=βi3eiTtΓ3∫t-ktteiθdθ,i=1,2,…,N,where (37)Ω1=1λminIN⊗Γ1-λmaxIN⊗A+12ε1λmaxIN⊗W1W1T+ε12λmaxIN⊗F1TF1+c1λmaxG1λmaxΓ1+c3k22λmaxIN⊗Γ3+12ε2λmaxIN⊗W2W2T+12ε3λmaxIN⊗W3W3T+c1ε62λmaxIN⊗E1TE1+c12ε6λmaxG1G1TλmaxMMT+c22ε7λmaxG2G2TλmaxΓ2Γ2T+c22ε8λmaxG2G2TλmaxMMT+c32ε9λmaxG3G3TλmaxΓ3Γ3T+c32ε10λmaxG3G3TλmaxMMT+c221-μλmaxIN⊗Γ2+c2d2∗2ε4λmaxIN⊗Γ2Γ2T+c3d3∗2ε5λmaxIN⊗Γ3Γ3T,Ω2=12λminIN⊗Γ2ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2+c2d2∗ε4,Ω3=12λminIN⊗Γ3ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3+c3d3∗ε5,ξt=eTt,eTt-ht,∫t-ktteθdθTT.By using adaptive controlling method, we get the following theorem.Theorem 15.
For some given synchronization scaling functionα(t), neural networks (1) satisfying Assumptions 1, 2, and 3, and target system can realize function projective synchronization by the adaptive control law as shown in (34) if there exist positive constants εi,i=1,2,…,10, and by taking appropriate d1∗, d2∗, and d3∗ such that(38)d1∗-Ω1c1>0,d2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,d3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.Then controlled neural networks (1) are function projective synchronization.Proof.
Construct the following Lyapunov-Krasovskii functional candidate:(39)Vt=V1t+V2t+V3t+V4t,where (40)V1t=12∑i=1NeiTteit,V2t=c221-μ∑i=1N∫t-htteiTθΓ2eiθdθ,V3t=c3k2∑i=1N∫-k0∫t+steiTθΓ3eiθdθds,V4t=c12∑i=1N1βi1d1it-d1∗2+c22∑i=1N1βi2d2t-d2i∗2+c32∑i=1N1βi3d3t-d3i∗2.By taking the derivative of V(t) along the trajectories of system (36) that is similar to the proof of Theorem 12, we obtain(41)V˙t≤eTt-IN⊗A+12ε1IN⊗W1W1T+ε12IN⊗F1TF1+c221-μIN⊗Γ2-c1d1∗IN⊗Γ1+c1G1⊗Γ1+c3k22IN⊗Γ3+12ε2IN⊗W2W2T+12ε3IN⊗W3W3T+c12ε6G1G1T⊗MMT+c1ε62IN⊗E1TE1+c22ε7G2G2T⊗Γ2Γ2T+c22ε8G2G2T⊗MMT+c32ε9G3G3T⊗Γ3Γ3T+c32ε10G3G3T⊗MMT+c2d2∗2ε4IN⊗Γ2Γ2T+c3d3∗2ε5IN⊗Γ3Γ3Tet-eTt-htc22IN⊗Γ2-Ω2IN⊗Γ2et-ht-∫t-ktteTθc3k2IN⊗Γ3-Ω3IN⊗Γ3eθdθ,≤eTtΩ1-c1d1∗IN⊗Γ1et-eTt-htc22-Ω2IN⊗Γ2et-ht-∫t-ktteθdθTc3k2-Ω3IN⊗Γ3∫t-ktteθdθ.It is obvious that there exist sufficiently large positive constants d1∗, d2∗, and d3∗ such that (42)d1∗-Ω1c1>0,(43)d2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,(44)d3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.We can choose d1∗, d2∗, and d3∗ satisfying (42), (43), and (44), respectively. The remaining proof is similar to Theorem 12 and omitted.


## 3.1. FPS under Hybrid Adaptive Pinning Control

We design nonlinear and adaptive pinning controls to realize FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling. In order to stabilize the origin of neural networks (1) by means of nonlinear and adaptive pinning controls Ui(t) such as(14)Uit=ui1t+ui2t,i=1,2,…,N,where (15)ui1t=α˙tst-W1αtf1st+f1αtst-W2αtf2st-ht+f2αtst-ht-W3∫t-kttαtf3sθ-f3αtsθdθ,i=1,2,…,N,ui2t=-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,l,0i=l+1,l+2,…,N,the updating laws are(16)d˙1it=βi1eiTtΓ1eit,d˙2it=βi2eiTtΓ2eit-ht,d˙3it=βi3eiTtΓ3∫t-ktteiθdθ,where βi1,βi2, and βi3 are positive constants and s(t) is a solution of an isolated node. The controller Ui(t) is different type of controller, and ui1(t) is the nonlinear control and ui2(t) is the adaptive linear pinning control. Then, substituting (14) into (8), it can be derived that(17)e˙it=-Aeit+W1f~1eit+W2f~2eit-ht+W3∫t-kttf~3eiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1ejt+c2∑j=1NG2ijΓ2+ΔΓ2ejt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttejθdθ-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,l,e˙it=-Aeit+W1f~1eit+W2f~2eit-ht+W3∫t-kttf~3eiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1ejt+c2∑j=1NG2ijΓ2+ΔΓ2ejt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttejθdθ,i=l+1,l+2,…,N,d˙1it=βi1eiTtΓ1eit,i=1,2,…,l,d˙2it=βi2eiTtΓ2eit-ht,i=1,2,…,l,d˙3it=βi3eiTtΓ3∫t-ktteiθdθ,i=1,2,…,l,where (18)f~1eit=f1xit-f1αtst,f~2eit-ht=f2xit-ht-f2αtst-ht,f~3eiθ=f3xiθ-f3αtsθ,and, for convenience, we denote (19)Π1=1λminIN⊗Γ1-λmaxIN⊗A+12ε1λmaxIN⊗W1W1T+ε12λmaxIN⊗F1TF1+c3k22λmaxIN⊗Γ3+12ε2λmaxIN⊗W2W2T+12ε3λmaxIN⊗W3W3T+c1ε62λmaxIN⊗E1TE1+c12ε6λmaxG1G1TλmaxMMT+c22ε7λmaxG2G2TλmaxΓ2Γ2T+c22ε8λmaxG2G2TλmaxMMT+c32ε9λmaxG3G3TλmaxΓ3Γ3T+c32ε10λmaxG3G3TλmaxMMT+c221-μλmaxIN⊗Γ2+c2d¯2∗2ε4λmaxIN⊗Γ2Γ2T+c3d¯3∗2ε5λmaxIN⊗Γ3Γ3T,Π2=12λminIN⊗Γ2ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2+c2d¯2∗ε4,Π3=12λminIN⊗Γ3ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3+c3d¯3∗ε5,ξt=eTt,eTt-ht,∫t-ktteθdθTT.Theorem 12.
For some given synchronization scaling functionα(t), neural networks (1) satisfying Assumptions 1, 2, and 3, and target system can realize function projective synchronization by the nonlinear and adaptive pinning control law as shown in (14) if there exist positive constants εi,i=1,2,…,10, and by taking appropriate d1i∗(i=1,2,…,l),d¯2∗, and d¯3∗ such that(20)λmaxG1+G1T2l<-Π1c1,(21)d¯2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,(22)d¯3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.Then controlled neural network (1) is function projective synchronization.Proof.
Construct the following Lyapunov-Krasovskii functional candidate:(23)Vt=V1t+V2t+V3t+V4t,where (24)V1t=12∑i=1NeiTteit,V2t=c221-μ∑i=1N∫t-htteiTθΓ2eiθdθ,V3t=c3k2∑i=1N∫-k0∫t+steiTθΓ3eiθdθds,V4t=c12∑i=1l1βi1d1it-d1i∗2+c22∑i=1l1βi2d2it-d2i∗2+c32∑i=1l1βi3d3it-d3i∗2.By taking the derivative of V(t) along the trajectories of system (17), we have the following: (25)V˙1t=-∑i=1NeiTtAeit+∑i=1NeiTtW1f~1eit+∑i=1NeiTtW2f~2eit-ht+∑i=1NeiTtW3∫t-kttf~3eiθdθ+c1∑i=1N∑j=1NeiTtG1ijΓ1+ΔΓ1ejt+c2∑i=1N∑j=1NeiTtG2ijΓ2+ΔΓ2ejt-τt+c3∑i=1N∑j=1NeiTtG3ijΓ3+ΔΓ3∫t-kttejθdθ-∑i=1NeiTtc1d1itΓ1eit-∑i=1NeiTtc2d2itΓ2eit-ht-∑i=1NeiTtc3d3itΓ3∫t-ktteiθdθ.V˙2t=c221-μ∑i=1NeiTtΓ2eit-c21-h˙t21-μ∑i=1NeiTt-htΓ2eit-ht.V˙3t=c3k22∑i=1NeiTtΓ3eit-c3k2∑i=1N∫t-ktteiTθΓ3eiθdθ.V˙4t=∑i=1lc1βi1d1it-d1i∗βi1eiTtΓ1eit+∑i=1lc2βi2d2it-d2i∗βi2eiTtΓ2eit-ht+∑i=1lc3βi3d3it-d3i∗βi3eiTtΓ3∫t-ktteiθdθ.After V(t) were calculated, we will get that(26)V˙t≤-∑i=1NeiTtAeit+∑i=1NeiTtW1f~1eit+∑i=1NeiTtW2f~2eit-ht+∑i=1NeiTtW3∫t-kttf~3eiθdθ+c1∑i=1N∑j=1NeiTtG1ijΓ1+ΔΓ1ejt+c2∑i=1N∑j=1NeiTtG2ijΓ2+ΔΓ2ejt-τt+c3∑i=1N∑j=1NeiTtG3ijΓ3+ΔΓ3×∫t-kttejθdθ+c221-μ∑i=1NeiTtΓ2eit-c221-μ∑i=1NeiTt-htΓ2eit-ht+c3k22∑i=1NeiTtΓ3eit-c3k2∑i=1N∫t-ktteiTθΓ3eiθdθ-c1∑i=1ld1i∗eiTtΓ1eit-c2∑i=1ld2i∗eiTtΓ2eit-htds-c3∑i=1ld3i∗eiTtΓ3∫t-ktteiθdθ≤-eTtIN⊗Aet+eTtIN⊗W1f~1et+eTtIN⊗W2f~2et-ht+eTtIN⊗W3∫t-kttf~3eθdθ+c1eTtG1⊗Γ1+ΔΓ1et+c2eTtG2⊗Γ2+ΔΓ2et-ht+c3eTtG3⊗Γ3+ΔΓ3∫t-ktteθdθ+c221-μeTtIN⊗Γ2et-c221-μeTt-htIN⊗Γ2et-ht+c3k22eTtIN⊗Γ3et-c3k2∫t-ktteTθIN⊗Γ3eθdθ-c1eTtD1∗⊗Γ1et-c2d¯2∗eTtIN⊗Γ2et-ht-c3d¯3∗eTtIN⊗Γ3∫t-ktteθdθ,where e(t)=(e1(t),…,eN(t))∈Rn×N, e(t-h(t))=(e1(t-h(t)),…,eN(t-h(t)))∈Rn×N, ∫t-k(t)te(θ)dθ=∫t-k(t)t(e1(θ),e2(θ),…,eN(θ))dθ∈Rn×N, D1∗=diag(d11∗,d12∗,…,d1l∗,0,…,0)∈RN×N, d¯2∗=max1≤i≤l{d2i∗}, and d¯3∗=max1≤i≤l{d3i∗}.
From Assumption2, we obtain the following three inequalities: (27)f~1ieit=f~1ixt-f~1iαtst≤F1ieit,f~2ieit-ht=f~2ixt-ht-f~2iαtst-ht≤F2ieit-ht,f~3ieiθ=f~3ixθ-f~3iαtsθ≤F3ieiθ.Applying Lemmas 6 and 7, we have (28)eTtIN⊗W1f~1et≤12ε1eTtIN⊗W1W1Tet+ε12f~1TetIN⊗Inf~1et,≤12ε1eTtIN⊗W1W1Tet+ε12eTtIN⊗F1TF1et,eTtIN⊗W2f~2et-ht≤12ε2eTtIN⊗W2W2Tet+ε22f~2Tt-htIN⊗Inf~2et-ht,≤12ε2eTtIN⊗W2W2Tet+ε22eTt-htIN⊗F2TF2et-ht,eTtIN⊗W3∫t-kttf~3eθdθ≤12ε3eTtIN⊗W3W3Tet+ε32∫t-kttf~3TeθdθTIN⊗In∫t-kttf~3Teθdθ,≤12ε3eTtIN⊗W3W3Tet+ε32×∫t-ktteTθdθTIN⊗F3TF3∫t-ktteTθdθ,-c2d¯2∗eTtIN⊗Γ2et-ht≤c2d¯2∗2ε4eTtIN⊗Γ2Γ2Tet+c2d¯2∗ε42eTt-htIN⊗Inet-ht,-c3d¯3∗eTtIN⊗Γ3∫t-ktteθdθ≤c3d¯3∗2ε5eTtIN⊗Γ3Γ3Tet+c3d¯3∗ε52×∫t-ktteTθdθTIN⊗In∫t-ktteTθdθ,c1eTtG1⊗Γ1+ΔΓ1et=c1eTtG1⊗Γ1et+c1eTtG1⊗M∇tE1et,≤c1eTtG1⊗Γ1et+c12ε6eTtG1G1T⊗MMTet+c1ε62eTtIN⊗E1TE1et,c2eTtG2⊗Γ2+ΔΓ2et-ht=c2eTtG2⊗Γ2et-ht+c2eTtG2⊗M∇tE2et-ht,≤c22ε7eTtG2G2T⊗Γ2Γ2Tet+c2ε72eTt-htIN⊗Inet-ht+c22ε8eTtG2G2T⊗MMTet+c2ε82eTt-htIN⊗E2TE2t-ht,c3eTtG3⊗Γ3+ΔΓ3∫t-ktteθdθ=c3eTtG3⊗Γ3∫t-ktteθdθ+c3eTtG3⊗M∇tE3∫t-ktteθdθ,≤c32ε9eTtG3G3T⊗Γ3Γ3Tet+c3ε92∫t-ktteTθdθTIN⊗In∫t-ktteTθdθ+c32ε10eTtG3G3T⊗MMTet+c3ε102×∫t-ktteTθdθTIN⊗E3TE3∫t-ktteTθdθ.Therefore, we have(29)eTt-c1D1∗⊗Γ1+c1G1⊗Γ1+c3k22IN⊗Γ3+12ε2IN⊗W2W2T+12ε3IN⊗W3W3T+c12ε6G1G1T⊗MMT+c1ε62IN⊗E1TE1+c22ε7G2G2T⊗Γ2Γ2T+c22ε8G2G2T⊗MMT+c32ε9G3G3T⊗Γ3Γ3T+c32ε10G3G3T⊗MMT+c2d¯2∗2ε4IN⊗Γ2Γ2T+c3d¯3∗2ε5IN⊗Γ3Γ3Tet-eTt-htc22IN⊗Γ2-Π2IN⊗Γ2et-ht-∫t-ktteTθc3k2IN⊗Γ3-Π3IN⊗Γ3eθdθ,≤eTtΠ1IN+c1G1-c1D1∗⊗Γ1et-eTt-htc22-Π2IN⊗Γ2et-ht-∫t-ktteTθc3k2-Π3IN⊗Γ3eθdθ,=eTtM-c1D1∗⊗Γ1et-eTt-htc22-Π2IN⊗Γ2et-ht-∫t-ktteTθc3k2-Π3IN⊗Γ3eθdθ,where M=Π1IN+c1G1. Note that the matrix M is symmetric. Let (30)M-D1∗=M~1-D~1l∗M~3M~3TM~2,where M~2 is the minor matrix of M by removing its first l(1≤l<N) row-column pairs, M~1 and M~3 are matrices with appropriate dimensions, and D~1l∗=diag(d11∗,d12∗,…,d1l∗). If λmax((G1+G1T)/2)l<-Π1/c1 and with Lemma 9, we have M~2<0. Therefore, one can choose suitable positive constants d1i∗>0,i=1,2,…,l, such that d1i∗>λmax(M~1-M~3M~2-1M~3T). It follows from Lemma 11 and M~2<0 that M-c1D1∗<0. Then, by Γ1>0 and (29), we can conclude that(31)V˙t≤-eTt-htc22-Π2IN⊗Γ2et-ht-∫t-ktteθdθTc3k2-Π3IN⊗Γ3∫t-ktteθdθ.We only need to choose the suitable positive constants d¯2∗ and d¯3∗ such that(32)d¯2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,(33)d¯3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.We can choose d¯2∗ and d¯3∗ satisfying (32) and (33), respectively. Since Γ1,Γ2, and Γ3 are positive definite matrix, we know that V˙(t)≤0. Therefore, pinning controlled delayed neural networks (1) globally asymptotically synchronize to trajectory (6) if conditions (20), (21), and (22) hold. Then the controlled neural networks (1) are function projective synchronization.Remark 13.
If there are no uncertain parameters in coupled delayed neural networks (1), ΔΓ1=ΔΓ2=ΔΓ3=0. One can obtain similar synchronization results.Remark 14.
The nodes pinned for directed networks are chosen as follows.
Step
1. Choose some appropriate parameters εi,i=1,2,…,10, by taking appropriate d¯1i∗,i=1,2,…,l, d¯2∗, and d¯3∗ such that the conditions in Theorem 12 are feasible.
Step
2. The l pinned nodes are sorted according to the pinned-node selection scheme studied [26], for the pinning controlled error neural network (17); so, the nodes to be pinned are chosen in the particular order. Let l=1, if the first inequalities of Theorem 12 are satisfied, and then the least number is 1; otherwise, go to the next step.
Step
3. If condition (20) is not satisfied, increase l(l=l+1) gradually with more network nodes to the pinned node based on the order in Step 2 particularly until condition (20) holds.
For undirected networks, for example, the small-world network [31], the scale-free network [32], and the Watts-Strogatz network [39], we can randomly choose a set of pinned nodes to satisfy condition (20) by increasing the number of pinned nodes l.


## 3.2. FPS under Hybrid Adaptive Control

The nonlinear and adaptive controls are designed to realize FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling. Then we have the following controlled form:(34)Uit=ui1t+ui2t,i=1,2,…,N,where (35)ui1t=α˙tst-W1αtf1st+f1αtst-W2αtf2st-ht+f2αtst-ht-W3∫t-kttαtf3sθ-f3αtsθdθ,i=1,2,…,N,ui2t=-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,N.Then, substituting (34) into (8), we have(36)e˙it=-Aeit+W1f~1eit+W2f~2eit-ht+W3∫t-kttf~3eiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1ejt+c2∑j=1NG2ijΓ2+ΔΓ2ejt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttejθdθ-c1d1itΓ1eit-c2d2itΓ2eit-ht-c3d3itΓ3∫t-ktteiθdθ,i=1,2,…,N,d˙1it=βi1eiTtΓ1eit,i=1,2,…,N,d˙2it=βi2eiTtΓ2eit-ht,i=1,2,…,N,d˙3it=βi3eiTtΓ3∫t-ktteiθdθ,i=1,2,…,N,where (37)Ω1=1λminIN⊗Γ1-λmaxIN⊗A+12ε1λmaxIN⊗W1W1T+ε12λmaxIN⊗F1TF1+c1λmaxG1λmaxΓ1+c3k22λmaxIN⊗Γ3+12ε2λmaxIN⊗W2W2T+12ε3λmaxIN⊗W3W3T+c1ε62λmaxIN⊗E1TE1+c12ε6λmaxG1G1TλmaxMMT+c22ε7λmaxG2G2TλmaxΓ2Γ2T+c22ε8λmaxG2G2TλmaxMMT+c32ε9λmaxG3G3TλmaxΓ3Γ3T+c32ε10λmaxG3G3TλmaxMMT+c221-μλmaxIN⊗Γ2+c2d2∗2ε4λmaxIN⊗Γ2Γ2T+c3d3∗2ε5λmaxIN⊗Γ3Γ3T,Ω2=12λminIN⊗Γ2ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2+c2d2∗ε4,Ω3=12λminIN⊗Γ3ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3+c3d3∗ε5,ξt=eTt,eTt-ht,∫t-ktteθdθTT.By using adaptive controlling method, we get the following theorem.Theorem 15.
For some given synchronization scaling functionα(t), neural networks (1) satisfying Assumptions 1, 2, and 3, and target system can realize function projective synchronization by the adaptive control law as shown in (34) if there exist positive constants εi,i=1,2,…,10, and by taking appropriate d1∗, d2∗, and d3∗ such that(38)d1∗-Ω1c1>0,d2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,d3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.Then controlled neural networks (1) are function projective synchronization.Proof.
Construct the following Lyapunov-Krasovskii functional candidate:(39)Vt=V1t+V2t+V3t+V4t,where (40)V1t=12∑i=1NeiTteit,V2t=c221-μ∑i=1N∫t-htteiTθΓ2eiθdθ,V3t=c3k2∑i=1N∫-k0∫t+steiTθΓ3eiθdθds,V4t=c12∑i=1N1βi1d1it-d1∗2+c22∑i=1N1βi2d2t-d2i∗2+c32∑i=1N1βi3d3t-d3i∗2.By taking the derivative of V(t) along the trajectories of system (36) that is similar to the proof of Theorem 12, we obtain(41)V˙t≤eTt-IN⊗A+12ε1IN⊗W1W1T+ε12IN⊗F1TF1+c221-μIN⊗Γ2-c1d1∗IN⊗Γ1+c1G1⊗Γ1+c3k22IN⊗Γ3+12ε2IN⊗W2W2T+12ε3IN⊗W3W3T+c12ε6G1G1T⊗MMT+c1ε62IN⊗E1TE1+c22ε7G2G2T⊗Γ2Γ2T+c22ε8G2G2T⊗MMT+c32ε9G3G3T⊗Γ3Γ3T+c32ε10G3G3T⊗MMT+c2d2∗2ε4IN⊗Γ2Γ2T+c3d3∗2ε5IN⊗Γ3Γ3Tet-eTt-htc22IN⊗Γ2-Ω2IN⊗Γ2et-ht-∫t-ktteTθc3k2IN⊗Γ3-Ω3IN⊗Γ3eθdθ,≤eTtΩ1-c1d1∗IN⊗Γ1et-eTt-htc22-Ω2IN⊗Γ2et-ht-∫t-ktteθdθTc3k2-Ω3IN⊗Γ3∫t-ktteθdθ.It is obvious that there exist sufficiently large positive constants d1∗, d2∗, and d3∗ such that (42)d1∗-Ω1c1>0,(43)d2∗-1c2ε4ε2λmaxIN⊗F2TF2+c2ε7+c2ε8λmaxIN⊗E2TE2-λminIN⊗Γ2>0,(44)d3∗-1c3ε5ε3λmaxIN⊗F3TF3+c3ε9+c3ε10λmaxIN⊗E3TE3-λminIN⊗Γ3>0.We can choose d1∗, d2∗, and d3∗ satisfying (42), (43), and (44), respectively. The remaining proof is similar to Theorem 12 and omitted.


## 4. Numerical Simulations

In this section, we provide several numerical examples to demonstrate the feasibility of the proposed method.Example 16.
Consider a two-dimensional neural network with time-varying delay presented in the following system:(45)s˙t=-Ast+W1f1st+W2f2st-ht+W3∫t-kttf3sθdθ,where s(t)=s1ts2tT∈R2 is the state vector of the network, the activation function fj(xi(t))=tanh(xi(t)),(i=1,2,j=1,2,3), the delays h(t)=1-0.9sin⁡(t) and k(t)=0.2sin(t), and the other matrices are as follows: (46)A=1001,W1=2-0.1-54.5,W2=-1.5-0.1-0.2-4,W3=-0.30.10.1-0.2. The dynamical chaotic trajectory of neural network (45) with initial conditions s1(r)=0.6 and s2(r)=0.5,∀r∈[-2,0], is shown in Figure 1.Figure 1
(a) Chaotic trajectory of neural network (45). (b) Phase portrait of the strange attractor.
(a)
(b)Afterwards, the FPS problems for the nonlinear and adaptive pinning controlled network consisting ofN two-dimensional neural networks are described as follows:(47)xi˙t=-Axit+W1f1xit+W2f2xit-ht+W3∫t-kttf3xiθdθ+c1∑j=1NG1ijΓ1+ΔΓ1xjt+c2∑j=1NG2ijΓ2+ΔΓ2xjt-ht+c3∑j=1NG3ijΓ3+ΔΓ3∫t-kttxjθdθ+Uit,i=1,2,…,N,where xi(t)=[xi1(t),xi2(t)]T∈R2 is the state variable of the ith node. The inner-coupling matrices with uncertainties are (48)Γ1=5005,Γ2=2002,Γ3=1001,M=0.2000.2,∇t=sint00cost,Er=0.1000.1r=1,2,3,and the other parameters are the same as those in (45).Directed Neural Network. We consider the directed neural networks as shown in Figure 2.Figure 2
Simple directed neural network with 6 nodes.Choose the following parameters: the time-varying scaling functionα(t)=1.3sin⁡(2π/15), the coupling strength values c1=16, c2=1, and c3=1, and the positive constants ε1=12.5, ε2=4.5, ε3=0.8, ε4=0.7, ε5=0.8, ε6=7.2, ε7=3.7, ε8=5, ε9=5, and ε10=12. According to Figure 2, the controller networks with 6 nodes are described with the outer-coupling matrices by (49)G1=-2000111-2001001-1000001-1000011-2000001-1,G2=-1001001-2001001-1000000-1101010-3100100-1,G3=-2001101-1000001-1000010-2100100-2100100-1.Solution. As presented in Figure 2, there are the out-degrees and the in-degree of nodes. According to the pinned-node selection scheme, we rearrange the nodes in G1 and the new order is 3,5,4,6,1,2. When node 3 is chosen as pinned node with l=1, we obtain d1l=5, d¯2∗=2.0125, and d¯3∗=3.4560. Thus, it is confirmed that pinning conditions (20)–(22) in Theorem 12 hold: (50)λmaxG1+G1T2l=-0.2989<-Π1c1=-4.565915=-0.2854.Figure3 shows the chaotic behavior of isolate node α(t)s(t) (45) and node xi(t) (47), with the time-varying scaling function α(t). Figure 4 shows the FPS errors between the states of isolate node α(t)s(t) (45) and node xi(t) (47), where eij(t)=xij(t)-αj(t)sj(t) for i=1,…,6,j=1,2, without nonlinear and adaptive pinning control (14). Figure 5 shows the FPS errors between the states of isolate node α(t)s(t) (45) and node xi(t) (47) where eij(t)=xij(t)-αj(t)sj(t) for i=1,…,6,j=1,2 with nonlinear and adaptive pinning control (14). Figure 6 gives the evolution of adaptive pinning feedback gains d11(t), d21(t), and d31(t).Figure 3
Chaotic behavior of isolate nodeα(t)s(t) (45) and node xi(t) (47) with the time-varying scaling function α(t).Figure 4
The FPS errors between the states of isolate nodeα(t)s(t) (45) and node xi(t) (47) without nonlinear and adaptive pinning control (14).Figure 5
The FPS errors between the states of isolate nodeα(t)s(t) (45) and node xi(t) (47) with nonlinear and adaptive pinning control (14).Figure 6
The evolution of adaptive pinning feedback gainsd11(t), d21(t), and d31(t).Undirected Neural Network. We consider the FPS for a large-scale undirected Watts-Strogatz network with 30 identical nodes of the isolated dynamic network with mixed time-varying delays which is given in (47). Choose the time-varying scaling function α(t)=1.3sin⁡(2π/15) and the coupling strength values c1=12, c2=1, and c3=1. For a Watts-Strogatz network here, we set the parameters [N=30, K=2, and ρ=0.65], [N=30, K=1, and ρ=1], and [N=30, K=1, and ρ=0.8], respectively. Then, the coupling matrices G1, G2, and G3 can be randomly generated by the Watts-Strogatz models and are shown in Figures 7–9, respectively.Figure 7
The topology structure of Watts-Strogatz neural network withN=30, K=2, and ρ=0.65.Figure 8
The topology structure of Watts-Strogatz neural network withN=30, K=1, and ρ=1.Figure 9
The topology structure of Watts-Strogatz neural network withN=30, K=1, and ρ=0.8.Solution. From conditions (20)–(22) in Theorem 12 and the positive constants ε1=12.5, ε2=4.5, ε3=0.8, ε4=0.7, ε5=0.8, ε6=7.2, ε7=3.7, ε8=5, ε9=5, and ε10=12, one can check that the last three conditions in Theorem 12 are satisfied. We study how to select pinned nodes of network. As G1 is an undirected Watts-Strogatz network, the pinned nodes can be randomly chosen for the convenience of practical applications by randomly choosing ten network nodes; that is, l=10, and the feedback control gains are chosen as d1i=30(i=1,2,…,10),d¯2∗=2.0125, and d¯3∗=3.4560. From simple numerical calculation, it can be seen that pinning condition (20) holds:(51)λmaxG1+G1T2l=-0.6407<-Π1c1=-7.111012=-0.5926.Figure10 shows the FPS errors between the states of isolate node α(t)s(t) (45) and node xi(t) (47), where eij(t)=xij(t)-αj(t)sj(t) for i=1,…,6,j=1,2, without nonlinear and adaptive pinning control (14). Figure 11 shows the FPS errors between the states of isolate node α(t)s(t) (45) and node xi(t) (47), where eij(t)=xij(t)-αj(t)sj(t) for i=1,…,6,j=1,2, with nonlinear and adaptive pinning control (14). Figure 12 gives the evolution of adaptive pinning feedback gains d1i(t), d2i(t), and d3i(t)(i=1,2,…,10).Figure 10
The FPS errors between the states of isolate nodeα(t)s(t) (45) and node xi(t) (47) without nonlinear and adaptive pinning control (14).Figure 11
The FPS errors between the states of isolate nodeα(t)s(t) (45) and node xi(t) (47) with nonlinear and adaptive pinning control (14).Figure 12
The evolution of adaptive pinning feedback gainsdi1(t), di2(t), and di3(t)(i=1,2,…,10).Example 17.
We consider the FPS problems for the nonlinear and adaptive controlled network consisting ofN two-dimensional neural networks (36). The directed neural networks are shown in Figure 13, where the constant coupling matrix is determined to be(52)G1=-100011-200101-100001-100011-2,G2=-200111-100000-101011-200101-2,G3=-100101-100011-200001-210100-1.
The inner-coupling matrices with uncertainties are the same as in (47), and the other parameters are the same as those in (45). Choose the time-varying scaling function α(t)=1.3sin⁡(2π/15) and the coupling strength values c1=2.5, c2=0.2, and c3=0.5.Figure 13
Simple directed neural network with 5 nodes.Solution. From conditions (38) in Theorem 15 and the positive constants ε1=0.1, ε2=10.5, ε3=3.5, ε4=0.5, ε5=0.7, ε6=10, ε7=3.7, ε8=6, ε9=4.2, and ε10=5, one can check that the last three conditions in Theorem 15 are satisfied. Then, we obtain d1∗>3.4425, d2∗>2.8000, and d3∗>5.5199.Figure14 shows the FPS errors between the states of isolate node α(t)s(t) of (45) and node xi(t) of (47), where eij(t)=xij(t)-αj(t)sj(t) for i=1,2,j=1,2,3, without nonlinear and adaptive control (34). Figure 15 shows the FPS errors between the states of isolate node α(t)s(t) of (45) and node xi(t) of (47), where eij(t)=xij(t)-αj(t)sj(t) for i=1,2,j=1,2,3, with nonlinear and adaptive control (34). Figure 16 gives the evolution of adaptive feedback gains d11(t), d21(t), and d31(t).Figure 14
The FPS errors between the states of isolate nodeα(t)s(t) (45) and node xi(t) (47) without nonlinear and adaptive control (14).Figure 15
The FPS errors between the states of isolate nodeα(t)s(t) (45) and node xi(t) (47) with nonlinear and adaptive control (14).Figure 16
The evolution of adaptive feedback gainsd11(t), d21(t), and d31(t).


## 5. Conclusions

In this paper, the hybrid adaptive pinning control for FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling were investigated. We have applied the use of nonlinear and adaptive pinning controls and the nonlinear and adaptive controls. Some sufficient conditions are derived to guarantee the FPS by use of the Lyapunov-Krasovskii function method. Moreover, the drive and response systems could be synchronized up to the desired scaling functions based on the adaptive control technique. Furthermore, numerical examples are given to illustrate the effectiveness of the proposed theoretical results in this paper as well.



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