1 Introduction

Over the past few decades, neural networks have received considerable attention for their important applications in many fields of image processing, pattern recognition and associative memories, etc [1–3]. Since the switching speed of information processing and the inherent neuron communication is finite, time delays are always unavoidably existent in neural networks, which may lead to instability or significantly deteriorated performances. Then it is necessary to take into account the time delay in neural networks [4–6].

Also, much concern has been paid to the synchronization of coupled neural networks for its potential applications, such as secure communication, and so on [7–13]. He and Cao [9] discussed the global synchronization of coupled neural networks with one single time-varying delay coupling, and established several sufficient criteria. Song et al. [10] proposed a pinning synchronization control scheme for a class of linearly coupled neural networks. Tang and Wong [11] considered the distributed synchronization problem of coupled neural networks by randomly occurring control method. Employing discontinuous impulsive control scheme, [12] put forward some criteria for the synchronization of coupled networks by impulsive control. Liu and Chen [13] investigated the cluster synchronization problem of the directed networks by intermittent pinning controller. In fact, not all the complex networks can synchronize by themselves. Those, who can not achieve synchronization by themselves, need introducing proper controllers to force them to synchronize. Among some existing control methods, the pinning control is a very important one.

Pinning control strategy applies the local feedback injections to a small fraction of nodes to achieve the global performances of the entire networks, which is effective and practical, especially for the large-size networks. Nowadays, the pinning synchronization of complex networks has been widely investigated, and meanwhile several selection rules of pinned nodes have been proposed in the existing literatures. Wang and Chen [14] concluded that the most-highly connected nodes can be pinned to get the better performance for the undirected networks. Chen et al. [15] showed that even using one single pinned node could control the whole networks as long as the coupling strength was large enough. Song and Cao [16] proposed the pinned nodes selection rule according to the out-degree and in-degree of the nodes, and the synchronization problem was studied for undirected and directed networks. In addition, the pinning control problems of complex networks were investigated, and some sufficient pinning conditions were established in [17–21].

As far as the synchronization state is concerned, most previous studies on synchronization problem of complex networks choose a special solution of the homogeneous systems as the controlled synchronization state [10, 16, 17, 19, 20, 22, 23], usually described by \(\dot{s}(t)=f(t,s(t))\). As pointed out in [24, 25], the synchronization of the networks not only depends on the topology, but also is heavily related to the dynamic of each node. Then, the contributions and influences of all nodes in synchronization seeking processes of the dynamical networks should be taken into account. In [26], authors considered the average of all the node states \(s(t)=\sum _{i=1}^{N}\frac{1}{N}x_i(t)\) to be the synchronization state. For the purpose of practical control strategy, authors in [24, 27, 28] further investigated the synchronization of complex networks to a desired synchronization state, which was in term of the weighted average of all node states in the networks, that was \(s(t)=\sum _{i=1}^{N}\xi _ix_i(t)\) with \(\sum _{i=1}^{N}\xi _i=1~(\xi _i>0)\). In some degree, there research results are important and valuable to the realistic dynamical behaviors of complex works. To the best of authors’ knowledge, the synchronization of delayed coupled neural networks in the above sense has not been fully studied by pinning control yet, and it is just the motivation of this paper.

Based on the aforementioned discussions, this paper aims to analyze the synchronization problem for coupled neural networks with the time-varying delay. The synchronization in a different sense is focused on via pinning control and adaptive pinning control respectively. By employing the Lyapunov–Krasovskii (L–K) function theory and an appropriate pinning staggery, the pinning controlled synchronization conditions are derived. Compared with the existing results, the main novelty of this paper lies in two aspects. First, the synchronization state, in terms of the weighted average of all node states, is introduced in coupled neural networks with different time-varying delays. Second, the more simple linear matrix inequalities conditions are established, and then the effectiveness of the proposed method can be checked conveniently by numerical results.

The remainder of this paper is organized as follows. In Sect. 2, the considered coupled neural networks model is formulated and the pinning synchronization problem is introduced. The pinning synchronization conditions are obtained for concerned networks in Sect. 3. Section 4 considers the adaptive pinning synchronization. A numerical example is provided to illustrate the validity of the proposed method in Sect. 5. Some conclusions are made in Sect. 6, together with some potential future study.

Throughout this paper, the superscript \(^T\) stands for the transpose of a matrix. \(\mathbb {R}^{n}\) and \(\mathbb {R}^{n\times n}\) denote the n-dimension Euclidean space and set of all \(n\times n\) real matrices, respectively. A real symmetric matrix \(P>0~\) (\(\ge \)0) denotes P being a positive definite (positive semi-definite) matrix. For a matrix \(P\in \mathbb {R}^{n\times n}\), \(P^s=\frac{P+P^T}{2}\) is its symmetric part, \(\lambda _{max}(P)\) and \(\lambda _{min}(P)\) represent its maximum and minimum eigenvalues, respectively. \(\mathrm{diag}\{\ldots \}\) denotes a block-diagonal matrix, and \(\Vert \cdot \Vert \) represents the Euclidean norm of a vector and its induced norm of a matrix. \(I_n\) stands for an n-dimensional identity matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by \(*\).

2 Problem Formulation

Consider an array of linearly coupled neural networks consisting of N identical nodes in which the dynamics of the ith \((1\leqslant i \leqslant N)\) node is described by

$$\begin{aligned} \dot{x}_i(t)= & {} -Cx_i(t)+Af(x_i(t))+B f(x_i(t-\tau _1(t)))+J(t)+c_1\sum _{j=1}^{N}G_{1,ij}\Gamma _1 x_j(t)\nonumber \\&+\,c_2\sum _{j=1}^{N}G_{2,ij}\Gamma _2 x_j(t-\tau _2(t)), \end{aligned}$$
(1)

where \(x_i(\cdot )=[x_{i1}(\cdot ),\ldots , x_{in}(\cdot )]^T\in \mathbb {R}^n\) is the state vector, \(C=\mathrm{diag}\{c_1,\ldots ,c_n\}\) is a positive diagonal matrix. \(A=(a_{ij})_{n\times n}\) and \(B=(b_{ij})_{n\times n}\) represent the connection weight matrix and the delayed connection weight matrix, respectively. \(f(x_i(\cdot ))=[f_{1}(x_{i1}(\cdot )),\ldots ,f_{n}(x_{in}(\cdot ))]^T\in \mathbb {R}^n\) denotes the neuron activation function. \(J(t)=[J_1(t),\ldots ,J_n(t)]^T\in \mathbb {R}^n\) is the external input vector. \(\tau _1(t)\) and \(\tau _2(t)\) are the time-varying delays, which can be constant or bounded and differentiable functions. The positive constants \(c_1\) and \(c_2\) are the strengths for the constant coupling and delayed coupling, respectively. \( G_1=(G_{1,ij})_{N\times N}\) is the constant coupling configuration matrix defined to be diffusive, i.e., \(G_{1,ij}\geqslant 0~(i\ne j)\) and \(G_{1,ii} =-\sum _{j=1,j\ne i}^{N}G_{1,ij}\). The delayed coupling configuration matrix \( G_2=(G_{2,ij})_{N\times N}\) is also diffusive with \(G_{2,ij}\geqslant 0~(i\ne j)\) and \(G_{2,ii} =-\sum _{j=1,j\ne i}^{N}G_{2,ij}\). The coupling matrices \(G_1\) and \(G_2\) are nor required to be symmetric or irreducible.

Assumption 2.1

\(\tau _1(t)\) and \(\tau _2(t)\) are bounded and differentiable with \(\bar{\tau }_1=\sup _{t\geqslant 0}\tau _1(t)\) and \(\bar{\tau }_2=\sup _{t\geqslant 0}\tau _2(t)\). Meanwhile, \(\dot{\tau }_1(t)\leqslant \mu _1<1\) and \(\dot{\tau }_2(t)\leqslant \mu _2<1\) for all \(t\geqslant 0\) with \(\mu _1>0\) and \(\mu _2>0\).

Assumption 2.2

There exists a constant matrix L such that

$$\begin{aligned} \Vert f(x)-f(y)\Vert \leqslant \Vert L(x-y)\Vert , ~\forall x, y\in \mathbb {R}. \end{aligned}$$
(2)

Remark 2.1

Assumptions 2.1 and 2.2 impose restrictions on the time-varying delays and the activation function, and are widely used in the literature [16, 20].

The initial conditions for coupled networks (1) are given by

$$\begin{aligned} x_{ij}(t)=\varphi _{ij}(t)\in C([-\bar{\tau }, 0],R), ~\forall j=1,\ldots ,n,~ i=1,\ldots ,N, \end{aligned}$$
(3)

where \(\bar{\tau }=\max \{\bar{\tau }_1,\bar{\tau }_2\}\), \(C([-\bar{\tau }, 0],\mathbb {R})\) denotes the set of continuous functions from \([-\bar{\tau }, 0]\) to \(\mathbb {R}\). Since the synchronization problem is considered, we now give the definition.

Definition 2.1

The coupled neural networks (1) is said to be globally asymptotically synchronizable to the goal trajectory s(t) if the following relations

$$\begin{aligned} \lim _{t \rightarrow \infty }\Vert x_i(t)-s(t)\Vert =0, ~i=1,\ldots ,N \end{aligned}$$

hold for the initial conditions (3).

We will address the pinning synchronization of (1) by feedback controllers, and the controlled networks model of the ith \((1\leqslant i \leqslant N)\) node follows

$$\begin{aligned} \dot{x}_i(t)= & {} -Cx_i(t)+Af(x_i(t))+B f(x_i(t-\tau _1(t)))+J(t)+c_1\sum _{j=1}^{N}G_{1,ij}\Gamma _1 x_j(t)\nonumber \\&+\,c_2\sum _{j=1}^{N}G_{2,ij}\Gamma _2 x_j(t-\tau _2(t))+u_i(t). \end{aligned}$$
(4)

This paper aims to design and implement an appropriate pinning controller for (4) such that all the states of the controlled networks can globally asymptotically synchronize to a desired synchronization s(t), which is presented in terms of a weighted average of all the states in the networks. More specifically, that is

$$\begin{aligned} s(t)=\sum _{k=1}^N\xi _kx_k(t), \end{aligned}$$
(5)

where \(x_i(t)\) is the state of the delayed dynamical networks (4), and \(\xi _k\) are chosen to satisfy \(\sum _{k=1}^N\xi _k=1\) with \(\xi _k>0\), \(i=1,\ldots , N\). Such synchronization strategy takes into account the contributions and confluences of all the nodes of the dynamical networks, compared with a special solution of the homogeneous systems being the synchronization state in most existing literatures. Moreover, the synchronization state s(t) may also be an equilibrium point, a periodic orbit, or a chaotic attractor, among others [24, 27].

The following pinning feedback controller is adopted,

$$\begin{aligned} u_i(t)=-d_i(x_i(t)-s(t)),~i=1,\ldots ,N \end{aligned}$$
(6)

where the feedback control gain \(d_i>0\) if the ith node is pinned, otherwise \(d_i=0\). The next section will provide several sufficient conditions for global asymptotical synchronization, where some special cases are also considered in corollaries.

3 Sufficient Conditions for Global Asymptotical Synchronization

For convenience, we denote that \(\Lambda =\mathrm{diag}\{\xi _1,\ldots ,\xi _N\}\), \(\Omega _1=(\Lambda ^{\frac{1}{2}}G_1\Lambda ^{-\frac{1}{2}})\otimes \Gamma _1\) and \(\Omega _2=(\Lambda ^{\frac{1}{2}}G_2\Lambda ^{-\frac{1}{2}})\otimes \Gamma _2\). By defining the synchronization error \(e_i(t)=x_i(t)-s(t)\), the error dynamical system is governed as follows

$$\begin{aligned} \dot{e}_i(t)= & {} -Ce_i(t)+A[f(x_i(t))-f(s(t))]+B[f(x_i(t-\tau _1(t)))-f(s(t-\tau _1(t)))]+\Pi (t)\nonumber \\&+\,c_1\sum _{j=1}^N G_{1,ij}\Gamma _1e_j(t) +c_2\sum _{j=1}^N G_{2,ij}\Gamma _2e_j(t-\tau _2(t))-d_ie_i(t),~i=1,\ldots ,N \end{aligned}$$
(7)

where

$$\begin{aligned} \Pi (t)= & {} A\left[ f(s(t))-\sum _{k=1}^N\xi _kf(x_k(t))\right] +B\left[ f(s(t-\tau _1(t)))-\sum _{k=1}^N\xi _kf(x_k(t-\tau _1(t)))\right] \\&-\,c_1\sum _{k=1}^N\xi _k\sum _{j=1}^N G_{1,kj}\Gamma _1e_j(t)-c_2\sum _{k=1}^N\xi _k\sum _{j=1}^N G_{2,kj}\Gamma _2e_j(t-\tau _2(t))+\sum _{j=1}^{N}\xi _jd_je_j(t). \end{aligned}$$

Before giving the main results, two lemmas are provided first.

Lemma 3.1

([29, Schur Complement]). The linear matrix inequality \( \left[ \begin{array}{cc}S_{11} &{} S_{12} \\ *&{} S_{22}\end{array}\right] <0\) is equivalent to \(S_{22}<0\) and \(S_{11}-S_{12}S_{22}^{-1}S_{12}^T<0,\) where \(S_{11}=S_{11}^T\), \(S_{22}=S_{22}^T\).

Lemma 3.2

([10]). For a symmetric matrix \(M\in \mathbb {R}^{N\times N}\) and a diagonal matrix \(D=\mathrm{diag}\{\bar{d}_1,\ldots ,\bar{d}_l,\underbrace{0,\ldots ,0}_{N-l}\}\) with \(\bar{d}_i>0, i=1,\ldots ,l~(1\leqslant l\leqslant N)\). Set \(M-D=\left[ \begin{array}{cc}E-\widetilde{D} &{}F\\ F^T &{} M_l\end{array}\right] \), where \(M_l\) is the minor matrix of M by removing its first l row-column pairs, E and F are matrices with appropriate dimensions, \(\widetilde{D}=\mathrm{diag}\{\bar{d}_1,\ldots ,\bar{d}_l\}.\) If \(\bar{d}_i>\lambda _{\max }(E-FM_l^{-1}B^T),i=1,\ldots ,l\), then \(M-D<0\) is equivalent to \(M_l<0\).

Theorem 3.1

Under Assumptions 2.1–2.2, the pinning controlled networks (4) with controller (6) can globally asymptotically synchronize to the desired synchronization state s(t) given by (5), if there exist positive scalars \(\epsilon \) and \(\gamma _2\) such that the following inequality holds

$$\begin{aligned} \left[ \begin{array}{cc} I_N\otimes \Phi +c_1\Omega _1^s-D\otimes I_{n} &{} \frac{1}{2}c_2\Omega _2 \\ *&{} -\gamma _2(1-\mu _2)I_{nN}\end{array}\right] <0, \end{aligned}$$
(8)

where \(\Phi =-C+\frac{AA^T}{2}+\frac{BB^T}{2}+\frac{L^TL}{2}+(\gamma _1+\gamma _2)I_n\) with \(\gamma _1=\frac{\lambda _1+\epsilon }{1-\mu _1}\) and \(\lambda _1=\frac{1}{2}\lambda _{\max }(L^TL)\).

Proof

Firstly, a Lyapunov–Krasovskii function is constructed for (7) as follows

$$\begin{aligned} V(t)= & {} \frac{1}{2}\sum _{i=1}^{N}\xi _ie_i^T(t)e_i(t)+\gamma _1\sum _{i=1}^{N}\xi _i \int _{t-\tau _1(t)}^te_i^T(s)e_i(s)ds+\gamma _2\sum _{i=1}^{N}\xi _i \int _{t-\tau _2(t)}^te_i^T(s)e_i(s)ds, \end{aligned}$$

where \(\gamma _1\) and \(\gamma _2\) are positive scalars to be specialized below. According to Assumptions 2.2 and using Kronecker product technique, the derivative of V(t) along the trajectory of system (4) can be calculated as

$$\begin{aligned} \dot{V}(t)= & {} \sum _{i=1}^{N}\xi _ie_i^T(t)\{-Ce_i(t)\nonumber \\&+\,A[f(x_i(t))-f(s(t))]+B[f(x_i(t-\tau _1(t)))-f(s(t-\tau _1(t)))]\nonumber \\&+\,\gamma _1\sum _{i=1}^{N}\xi _ie_i^T(t)e_i(t)-\gamma _1(1-\dot{\tau }_1(t))\sum _{i=1}^{N}\xi _ie_i^T(t-\tau _1(t))e_i(t-\tau _1(t))\nonumber \\&+\,\gamma _2\sum _{i=1}^{N}\xi _ie_i^T(t)e_i(t)-\gamma _2(1-\dot{\tau }_2(t))\sum _{i=1}^{N}\xi _ie_i^T(t-\tau _2(t))e_i(t-\tau _2(t))\nonumber \\&+\,\sum _{i=1}^{N}\xi _ie_i^T(t)[c_1\sum _{j=1}^N G_{1,ij}\Gamma _1e_j(t)+c_2\sum _{j=1}^N G_{2,ij}\Gamma _2e_j(t-\tau _2(t))] \nonumber \\&-\sum _{i=1}^{N}d_i\xi _ie_i^T(t)e_i(t)+\sum _{i=1}^{N}\xi _ie_i^T(t)\Pi (t). \end{aligned}$$
(9)

Noticing that \(\sum _{i=1}^{N}\xi _ie_i(t)=0\), we have \(\sum _{i=1}^{N}\xi _ie_i^T(t)\Pi (t)=0\). Based on Assumption 2.2 and the fact \(2x^Ty\leqslant x^Tx+y^Ty\), \(\forall x,y\in \mathbb {R}^n\), one can make the following estimation

$$\begin{aligned}&\sum _{i=1}^{N}\xi _ie_i^T(t)A[f(x_i(t))-f(s(t))]\nonumber \\&\quad =\sum _{i=1}^{N}\xi _ke_i^T(t)A[f(e_i(t)+s(t))-f(s(t))]\nonumber \\&\quad \leqslant \sum _{i=1}^{N}\xi _i\left[ e_i^T(t)\frac{AA^T}{2}e_i(t)+\frac{1}{2}\Vert f(e_i(t)+s(t))-f(s(t))\Vert ^2\right] \nonumber \\&\quad \leqslant \sum _{i=1}^{N}\xi _ie_i^T(t)\frac{AA^T}{2}e_i(t)+\sum _{i=1}^{N}\xi _ie_i^T(t)\frac{L^TL}{2}e_i(t)\nonumber \\&\quad =\sum _{i=1}^{N}\xi _ie_i^T(t)\left[ \frac{AA^T}{2}+\frac{L^TL}{2}\right] e_i(t). \end{aligned}$$
(10)

Similarly, it follows that

$$\begin{aligned}&\sum _{i=1}^{N}\xi _ie_i^T(t)B[f(x_i(t-\tau _1(t)))-f(s(t-\tau _1(t)))]\nonumber \\&\quad \leqslant \sum _{i=1}^{N}\xi _ie_i^T(t)\frac{BB^T}{2}e_i(t)+\sum _{i=1}^{N}\xi _ie_i^T(t-\tau _1(t))\frac{L^TL}{2}e_i(t-\tau _1(t)). \end{aligned}$$
(11)

By denoting \(\nu (t)=[\sqrt{\xi _1}e_1^T(t),\ldots ,\sqrt{\xi _N}e_N^T(t)]^T\), the transformations can be made as

$$\begin{aligned}&\sum _{i=1}^{N}\xi _ie_i^T(t)c_1\sum _{j=1}^N G_{1,ij}\Gamma _1e_j(t)\nonumber \\&=c_1\sum _{i=1}^{N}\xi _ie_i^T(t)\sum _{j=1}^{N}G_{1,ij}\Gamma _1 e_j(t)=c_1e^T(t)[(\Lambda G_1)\otimes \Gamma _1]e(t)\nonumber \\&=c_1\nu ^T(t)[(\Lambda ^{\frac{1}{2}}G_1\Lambda ^{-\frac{1}{2}})\otimes \Gamma _1]\nu (t)=c_1\nu ^T(t)\Omega _1^s\nu (t), \end{aligned}$$
(12)

and

$$\begin{aligned}&\sum _{i=1}^{N}\xi _ie_i^T(t)c_2\sum _{j=1}^N G_{2,ij}\Gamma _2e_j(t-\tau _2(t))\nonumber \\&\quad =c_2\sum _{i=1}^{N}\xi _ie_i^T(t)\sum _{j=1}^{N}G_{2,ij}\Gamma _2 e_j(t-\tau _2(t))=c_2e^T(t)[(\Lambda G_2)\otimes \Gamma _2]e(t-\tau _2(t))\nonumber \\&\quad =c_2\nu ^T(t)[(\Lambda ^{\frac{1}{2}}G_2\Lambda ^{-\frac{1}{2}})\otimes \Gamma _2]\nu (t-\tau _2(t))=c_2\nu ^T(t)\Omega _2\nu (t-\tau _2(t)). \end{aligned}$$
(13)

Submitting the above inequalities (10)–(13) into (9) yields that

$$\begin{aligned} \dot{V}(t)\leqslant & {} \sum _{i=1}^{N}\xi _ie_i^T(t)\left[ -C+\frac{AA^T}{2}\right. \nonumber \\&\left. +\frac{BB^T}{2}+\frac{L^TL}{2}\right] e_i(t)\nonumber \\&+\,\nu ^T(t)[c_1\Omega _1^s+(\gamma _1+\gamma _2)I_N\otimes I_n]\nu (t) \nonumber \\&+\,\sum _{i=1}^{N}\xi _ie_i^T(t-\tau _1(t))\frac{L^TL}{2} e_i(t-\tau _1(t))-\gamma _1(1-\mu _1)\nu ^T(t-\tau _1(t))\nu (t-\tau _1(t))\nonumber \\&+\,c_2\nu ^T(t)\Omega _2\nu (t-\tau _2(t))-\gamma _2(1-\mu _2)\nu ^T(t-\tau _2(t))\nu (t-\tau _2(t))\nonumber \\&-\nu ^T(t)(D\otimes I_N)\nu (t)\nonumber \\\leqslant & {} \nu ^T(t)[I_N\otimes \Phi +c_1\Omega _1^s-D\otimes I_{n}]\nu (t)-\gamma _2(1-\mu _2)\nu ^T(t-\tau _2(t))\nu (t-\tau _2(t))\nonumber \\&+\,c_2\nu ^T(t)\Omega _2\nu (t-\tau _2(t))+ [\lambda _1-\gamma _1(1-\mu _1)]\nu ^T(t-\tau _1(t))\nu (t-\tau _1(t)). \end{aligned}$$
(14)

Choose \(\gamma _1=\frac{\lambda _1+\epsilon }{1-\mu _1}\) such that \(\lambda _1-\gamma _1(1-\mu _1)=-\epsilon <0.\) Then, the above inequality (14) degenerates into

$$\begin{aligned} \dot{V}(t)\leqslant & {} \nu ^T(t)[I_N\otimes \Phi +c_1\Omega _1^s-D\otimes I_{N}]\nu (t) +c_2\nu ^T(t)\Omega _2\nu (t-\tau _2(t))\nonumber \\&\gamma _2(1-\mu _2)\nu ^T(t-\tau _2(t))\nu (t-\tau _2(t))\nonumber \\= & {} \eta ^T(t)\left[ \begin{array}{cc} I_N\otimes \Phi +c_1\Omega _1^s-D\otimes I_{n} &{} \frac{1}{2}c_2\Omega _2 \\ *&{} -\gamma _2(1-\mu _2)I_{nN}\end{array}\right] \eta (t), \end{aligned}$$
(15)

where \(\eta ^T(t)=\left[ \begin{array}{cc}\nu ^T(t)&\nu ^T(t-\tau _2(t))\end{array}\right] ^T\). Furthermore, from the condition (8) in Theorem 3.1 and (15), it follows that \(\dot{V}(t)<0\), which indicates that the error system (7) is globally asymptotically stable, i.e., \(\lim _{t\rightarrow \infty }e_i(t)=0\). Therefore, the pinning controlled networks (4) globally asymptotically synchronize to the desired synchronization state s(t) in (5). Now, the proof follows. \(\square \)

Remark 3.1

The condition (8) in Theorem 3.1 is given in form of linear matrix inequality (LMI) with respect to the invariants \(\epsilon \) and \(\gamma _2\), which can be checked by Matlab Toolbox. In addition, the minimization of the estimations of control gains \(d_i\) can be conducted. For example, it is assumed that the same control gain d is chosen, and then \(D=d\Sigma \) in Theorem 3.1, where \(\Sigma =\mathrm{diag}\{\sigma _1,\ldots ,\sigma _N\}\) represents the pinning matrix, and \(\sigma _i=1\) when the ith node is pinned, otherwise \(\sigma _i=0\). The minimization of the estimation of control gain d can be formulated as

$$\begin{aligned} \min _{\epsilon ,\gamma _2} ~~ d,~~~s.t.~~ LMI ~(8)~ holds. \end{aligned}$$

In fact, the single control gain also can be optimized by similar method.

When there does not exist the uncoupled discrete delay or the coupling delay in networks model, the controlled system (4) will degenerate into the following forms respectively:

$$\begin{aligned} \dot{x}_i(t)= & {} -C(t)x_i(t)+Af(x_i(t))+J(t)+c_1\sum _{j=1}^{N}G_{1,ij}\Gamma _1 x_j(t)\nonumber \\&+\,c_2\sum _{j=1}^{N}G_{2,ij}\Gamma _2 x_j(t-\tau _2(t))+u_i(t), \end{aligned}$$
(16)
$$\begin{aligned} \dot{x}_i(t)= & {} -C(t)x_i(t)+Af(x_i(t))+B f(x_i(t-\tau _1(t)))+J(t)\nonumber \\&+\,c_1\sum _{j=1}^{N}G_{1,ij}\Gamma _1 x_j(t)+u_i(t). \end{aligned}$$
(17)

Based on Theorem 3.1, we can derive the corresponding results for the pinning synchronization of the above systems (16) and (17). Now, the following corollaries are proposed, and the detailed proofs are omitted.

Corollary 3.1

Under Assumptions 2.1–2.2, the pinning controlled networks (16) with controller (6) can globally asymptotically synchronize to the desired synchronization state s(t) in (5), if there exists a positive scalar \(\gamma _2\) such that the following inequality holds

$$\begin{aligned} \left[ \begin{array}{cc} I_N\otimes \bar{\Phi }+c_1\Omega _1^s-D\otimes I_{n} &{} \frac{1}{2}c_2\Omega _2 \\ *&{} -\gamma _2(1-\mu _2)I_{nN}\end{array}\right] <0 \end{aligned}$$

with \(\bar{\Phi }=-C+\frac{AA^T}{2}+\frac{L^TL}{2}+\gamma _2I_n\).

Corollary 3.2

Under Assumption 2.1–2.2, the pinning controlled networks (17) with controller (6) can globally asymptotically synchronize to the desired synchronization state s(t) in (5), if there exists a positive scalar \(\epsilon \) such that the following inequality holds

$$\begin{aligned} I_N\otimes \bar{\bar{\Phi }}+c_1\Omega _1^s-D\otimes I_{n} <0, \end{aligned}$$

where \(\bar{\bar{\Phi }}=-C+\frac{AA^T}{2}+\frac{BB^T}{2}+\frac{L^TL}{2}+\gamma _1I_n\) with \(\gamma _1=\frac{\lambda _1+\epsilon }{1-\mu _1}\) and \(\lambda _1\) is defined in Theorem 3.1.

Remark 3.2

By setting \(B=0,~\tau _1(t)=0,~\gamma _1=0\) in Theorem 3.1, we can get Corollary 3.1 on the pinning synchronization problem for system (16). Moreover, letting \(G_2=0,~\Gamma _2=0,~\gamma _2=0\), Corollary 3.2 can be concluded for system (17). Furthermore, choosing \(\mu _1=\mu _2=0\) in Theorem 3.1, one can also derive the corresponding results for the constant delays case. With \(\xi _i=\frac{1}{N}~(1\leqslant i\leqslant N)\), the achieved results are the ones for the case that the average of all node states is the desired synchronization state.

Corollary 3.3

Under Assumptions 2.1–2.2, the pinning controlled networks (4) with controller (6) can globally asymptotically synchronize to the desired synchronization state \(s(t)=\frac{1}{N}\sum \limits _{k=1}^{N}x_k(t)\), if there exist positive scalars \(\epsilon \) and \(\gamma _2\) such that the following inequality holds

$$\begin{aligned} \left[ \begin{array}{cc} I_N\otimes \Phi +c_1(G_1\otimes \Gamma _1)^s-D\otimes I_{n} &{} \frac{1}{2}c_2G_2\otimes \Gamma _2 \\ *&{} -\gamma _2(1-\mu _2)I_{nN}\end{array}\right] <0, \end{aligned}$$

where \(\Phi \) is the same as the one in Theorem 3.1.

Remark 3.3

Compared with the recent literatures [24] and [27], which focus on the weighted average synchronization problems, the results in this paper have a better application to some extent. First, only the coupling delay is considered in [24], and it is a constant delay. As for the networks in [27], the coupling delay is not involved. Second, from a computational point of view, the criteria in [24, 27] are related to some eigenvalues, while our proposed conditions are in form of linear matrix inequalities, which can be checked conveniently by Matlab LMI Toolbox.

4 Adaptive Pinning Synchronization

Based on the above pinning synchronization analysis, we will further study the adaptive pinning synchronization of the networks (4). Without loss of generality, let the nodes \(i_1,\ldots ,i_l\) be selected as the pinned nodes and rearrange the order of the nodes such that the first l nodes are pinned or controlled. Therefore, it indicates that \(d_i=0\) for \(i=l+1,\ldots , N\) in the pinning controller (6). If \(d_i\) for \(i=1,\ldots ,l\) are designed to be time-variable, the details are as follows

$$\begin{aligned} u_i(t)=-d_i(t)e_i(t),~~\dot{d}_i(t)=q_ie_i^T(t)e_i(t),~~~i=1,\ldots ,l, \end{aligned}$$
(18)

where \(q_i\) are positive constants. The controller \(u_i(t)\) in (18) is said to be an adaptive pinning controller. In the following content, we will derive some conditions under which the coupled neural networks (4) can achieve the global synchronization to the desired state s(t) in (5) with the adaptive pinning controller (18). To avoid unnecessary duplication with Theorem 3.1, we will omit some details and give a sketch of the proof.

Theorem 4.1

Under Assumptions 2.1–2.2, the coupled neural networks (4) can globally asymptotically synchronizes to the desired orbit s(t) in (5) with the adaptive pinning controller (18), if there exist scalars \(\epsilon >0\) and \(\gamma _2>0\) such that the following inequality holds

$$\begin{aligned} \left[ \begin{array}{cc} I_N\otimes \Phi +c_1\Omega _1^s &{} \frac{1}{2}c_2\Omega _2 \\ *&{} -\gamma _2(1-\mu _2)I_{nN}\end{array}\right] _{nl}<0, \end{aligned}$$
(19)

where \(\Phi \) is the one in Theorem 3.1.

Proof

Construct a Lyapunov function for coupled networks (4) as follows

$$\begin{aligned} \bar{V}(t)= & {} V(t) +\sum _{i=1}^l\xi _i\frac{1}{2q_i}(d_i(t)-d_i^{*})^2, \end{aligned}$$

where \(q_i>0\) are defined in (18), and \(d_i^{*}>0\) are constants to be determined below. Using Kronecker product technique and Assumption 2.1, the time derivative of \(\bar{V}(t)\) along the trajectory of system (4) can be computed and estimated as

$$\begin{aligned} \dot{\bar{V}}(t)= & {} \dot{V}(t)+\sum _{i=1}^{l}\xi _i(d_i(t)-d_i^{*})e_i^T(t)e_i(t)\nonumber \\\leqslant & {} \nu ^T(t)[I_N\otimes \Phi +c_1\Omega _1^s-D^{*}\otimes I_{n}]\nu (t)-\gamma _2(1-\mu _2)\nu ^T(t-\tau _2(t))\nu (t-\tau _2(t))\nonumber \\&+\,c_2\nu ^T(t)\Omega _2\nu (t-\tau _2(t))+ [\lambda _1-\gamma _1(1-\mu _1)]\nu ^T(t-\tau _1(t))\nu (t-\tau _1(t)), \end{aligned}$$
(20)

where \(D^{*}=\mathrm{diag}\{d_1^{*},\ldots ,d_l^{*},\underbrace{0,\ldots ,0}_{N-l}\}\), \(\nu (t)\) and \(\lambda _1\) are the ones in Theorem 3.1. Letting \(\gamma _1=\frac{\lambda _1+\epsilon }{1-\mu }\) and from (20), it is known that

$$\begin{aligned} \dot{\bar{V}}(t)\leqslant & {} \eta ^T(t)\left[ \begin{array}{cc} I_N\otimes \Phi +c_1\Omega _1^s-D^{*}\otimes I_{n} &{} \frac{1}{2}c_2\Omega _2 \\ *&{} -\gamma _2(1-\mu _2)I_{nN}\end{array}\right] \eta (t), \end{aligned}$$
(21)

where \(\eta (t)\) has been defined in Theorem 3.1, and \(\bar{D}=\mathrm{diag}\{d_1^{*},\ldots ,d_l^{*},\underbrace{0,\ldots ,0}_{2N-l}\}\).

On the other hand, denote the matrix \(M=\left[ \begin{array}{cc} I_N\otimes \Phi +c_1\Omega _1^s &{} \frac{1}{2}c_2\Omega _2 \\ *&{} -\gamma _2(1-\mu _2)I_{nN}\end{array}\right] \), which is symmetric, and let

$$\begin{aligned} M-\bar{D}\otimes I_n=\left[ \begin{array}{cc}\widehat{A}-\widehat{D} &{} \widehat{B} \\ \widehat{B}^T &{} (M)_{nl}\end{array}\right] , \end{aligned}$$

where \(\widehat{D}=\widehat{\bar{D}}\otimes I_n\) with \(\widehat{\bar{D}}=\mathrm{diag}\{d_1^{*},\ldots ,d_l^{*}\}\), and \(\widehat{A}, \widehat{B}\) are matrices with appropriate dimensions. According to Lemma 2.1, if the positive constants \(d_i^{*}\) satisfy \(d_i^{*}>\lambda _{\max }(\widehat{A}-\widehat{B}(M)^{-1}_{nl}\widehat{B}^T) , (i=1,\ldots ,l)\), then \((M)_{nl}<0\) can give that \(M-\bar{D}\otimes I_n<0\). Together with (19) and (21), we have \(\dot{V}(t)\leqslant 0\) and \(\dot{V}(t)=0\) if and only if \(e(t)=e(t-\tau (t))=0_{nN}\). Therefore, the pinning controlled coupled neural networks (4) can globally asymptotically synchronize to the trajectory s(t) in (5) by the adaptive pinning controller (18) if the condition (19) holds. This completes the proof. \(\square \)

Remark 4.1

In the adaptive pinning synchronization design, we can rearrange network nodes in ascending order such that the first l nodes are pinned candidates. Based on the out-degree and in-degree of all the nodes, the pinned nodes can be selected in this paper. The details can be seen in [10] and [16].

Fig. 1
figure 1

Constant time delays case. a State trajectories, b state trajectories, c synchronization error, d adaptive pinning feedback gains

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Fig. 2
figure 2

Time-varying delays case. a State trajectories, b state trajectories, c synchronization error, d adaptive pinning feedback gains

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5 Numerical Example

We consider the pinning synchronization of coupled neural networks (4) with ten nodes (i.e., \(N=10\)), and the relevant parameters are given by [10, 30]:

$$\begin{aligned}&C=\left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] ,~A=\left[ \begin{array}{cc} 2.0 &{} -0.1 \\ -5.0 &{} 3.0 \end{array}\right] ,~B=\left[ \begin{array}{ccc} -1.5 &{}-0.1 \\ -0.2&{}-2.5 \end{array}\right] . \end{aligned}$$

The activation function is descried as \(f(x_i(t))=[\tanh (x_{i1}(t)),\tanh (x_{i2}(t))]^T\). It can be seen from Assumption 2.1 that \(L=\mathrm{diag}\{1.0,1.0\}\). The coupling matrix \(G_1\) is given by (22), which is determined by a directed topology (See Fig. 2 in [10]), and the delayed coupling matrix is set to be \(G_2=0.5G_1\). The coupling strengths are chosen as \(c_1=120,~c_2=5.\) The weighted average values \(\xi _i\) are chosen randomly in (0, 1). Letting \(e_i(t)=x_i(t)-s(t)\) with \(s(t)=\sum _{i=1}^{10}\xi _ix_i(t)\), and the synchronization error is measured by \(E(t)=\sqrt{\frac{1}{10}\sum _{i=1}^{10}e_i^T(t)e_i(t)}\).

$$\begin{aligned} G_1=\left[ \begin{array}{cccccccccc}-1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 \end{array}\right] \end{aligned}$$
(22)

First, we set the time delays are constant scalars \(\tau _1(t)=\tau _2(t)=1.0\) with \(\mu _1=\mu _2=0\), and the inner connecting matrices are defined as \(\Gamma _1=\Gamma _2=I_{n}\). For the case, the pinning synchronization to the isolate node is studied in [10]. Now, we deal with the pinning synchronization to the weighted average of all node states. Employing Theorem 4.1 and pinning the nodes 3, 4, 9, we can check the synchronization of the concerned system to s(t) by LMI Matlab toolbox with controller (18), and obtain that the feasible solutions are \(\epsilon =1.5894\) and \(\gamma _2=24.4658\). The state trajectories, the synchronization error E(t) and the adaptive feedback gains are illustrated in Fig. 1. It shows that the coupled neural networks with parameters in (22) can synchronize to the state s(t) under the adaptive pinning controller (18).

For the case that the time delays are different time-varying functions \(\tau _1(t)=0.7-0.3\cos (t)\) and \(\tau _2(t)=0.5-0.2\sin (t)\) and resorting to Matlab LMI toolbox, solving the inequality (19) for setting \(\mu _1=0.3\) and \(\mu _2=0.2\) leads to some feasible solutions \(\epsilon =1.9892\) and \(\gamma _2=21.8983\). The corresponding simulation results are given in Fig. 2.

Remark 5.1

Since the weighted average synchronization problem is concerned, the results in [10, 30] may not be employed here. In addition, for the case that \(\tau _1(t)\) and \(\tau _2(t)\) are different time-varying functions, the methods in [24, 27] may not work.

6 Concluding Remarks

This paper has investigated the problem of pinning synchronization for a class of coupled neural networks with time-varying delays. The Lyapunov–Krasovskii function method is developed to derive some sufficient criteria, which can guarantee that the pinning controlled networks globally asymptotically synchronize to the desired state that is in terms of the weight average of all node states in networks. Further study can focus on some advanced analysis technique to obtain the less conservative results, the synchronization of switched networks, and other control methods for complex networks.