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Aperiodically intermittent synchronization for switching complex networks dependent on topology structure
Advances in Difference Equations volumeÂ 2017, ArticleÂ number:Â 244 (2017)
Abstract
This paper investigates the aperiodically intermittent synchronization for a class of directed complex networks with switching network topologies. The assumption that all of the switching topologies contain a directed spanning tree is removed which is necessary in the previous related literature. That is, only some switching topologies contain a directed spanning tree when zeroindegree nodes are pinned. By using Mmatrix theory and constructing multiple Lyapunov functions, some sufficient conditions are derived to achieve the aperiodically intermittent synchronization of switching complex networks. Finally, some numerical simulations are given to demonstrate the theoretical results.
1 Introduction
In the past few years, the study of complex networks has attracted an increasing interest from various research areas. The main reason is that many systems in nature and human society can be described as complex networks such as World Wide Web [1], epidemic spreading networks [2], collaborative networks [3], biological neural networks [4], and so on. In complex networks, one of the interesting phenomena is synchronization, which is very important in many research and application fields [5â€“12].
However, complex networks are not always able to synchronize by themselves. Hence, various effective control protocols have been proposed to achieve synchronization, such as feedback control [13â€“15], adaptive control [16, 17], impulsive control [18, 19], pinning control [20, 21] and intermittent control [22, 23] and so on. Compared with the continuous control, intermittent control is more economic and it has attracted great interests [24, 25]. In [26], the synchronization problem for a class of complex delayed dynamical networks is investigated by periodically intermittent control. The topology structure of the considered complex networks is timeinvariant and the intermittent controller is periodical. In [27], synchronization problem for nonlinear coupled networks is investigated via aperiodically intermittent pinning control. Both an aperiodically constant intermittent control strategy and an aperiodically adaptive intermittent control strategy are designed. However, the network topology considered is undirected and timeinvariant. In most of the aforementioned works, it is commonly assumed that the network topology contains a directed spanning tree. There is very little research on switching complex networks containing no directed spanning tree.
Unfortunately, the common assumption that each possible network topology contains a directed spanning tree is not always satisfied. In [28], the common assumption is removed. It is theoretically shown that the global pinning synchronization in such switching complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected. Nevertheless, the switching signal is periodical and the zeroindegree nodes are pinned all the time. It should be pointed out that, to the best of our knowledge, there have been few results about the switching complex networks containing no directed spanning tree via aperiodically intermittent control, which is the motivation of this paper.
This paper aims to solve the challenging issue of pinning synchronization for a class of switching complex networks via aperiodically intermittent control. By using tools from Mmatrix theory and constructing multiple Lyapunov functions, the global pinning synchronization can be realized if the coupling strength and the switching time are satisfied some inequations.
The main contributions of this paper can be highlighted as follows. Firstly, the assumption that all of the switching topologies contain a directed spanning tree is removed which is necessary in the previous related literature. That is, only some switching topologies contain a directed spanning tree when zeroindegree nodes are pinned. Secondly, there is no node pinned when the switching topologies contain no directed spanning tree. The pinned nodes are timevarying, which are dependent on the switching topologies. Moreover, all the possible topologies switch aperiodically. That is, the zeroindegree nodes are pinned aperiodically intermittently.
The remainder of the paper is organized as follows. In Section 2, some preliminaries on graph theory and the problem formulation are provided. In Section 3, by using Mmatrix theory and constructing multiple Lyapunov functions, some sufficient conditions are derived to achieve the aperiodically intermittent synchronization of switching complex networks. Numerical simulations are given to demonstrate the effectiveness of the main results in Section 4. Finally, Section 5 concludes the whole work.
Notation
Let N, R, \(\mathbf{R}^{N}\), \(\mathbf{R}^{N\times N}\) be the sets of nonnegative integers, real numbers, Ndimensional real column vectors and \(N\times N\) real matrices, respectively. \(1_{N}\) represents the Ndimensional column vector with each element being 1. The superscript T means the transpose for matrices. Notation \(\operatorname{diag}\{x_{1},\ldots,x_{n}\}\) represents a diagonal matrix with \(x_{i}\) (\(i=1,\ldots,n\)), being its ith diagonal element. âŠ— and \(\Vert \cdot \Vert \) denote the Kronecker product and the Euclidean norm, respectively. For a real symmetric matrix Q, \(\lambda _{\min }(Q)\) represents the smallest eigenvalue of Q.
2 Preliminaries and problem formulation
In this section, we provide some useful preliminaries on algebraic graph theory and matrix theory.
Let \(\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A})\) be a weighted directed graph of order N, where \(\mathcal{V}=\{v_{1},v_{2},\ldots ,v_{N}\}\) is the set of nodes, \(\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V} \) is the set of edges, and \(\mathcal{A}=[a_{ij}]_{N\times N}\) with \(a_{ij}\geq 0\) (\(i,j=1,2,\ldots,N\)) is a weighted adjacency matrix. An edge of \(\mathcal{G}\) is denoted by \(e_{ij}=(v_{i},v_{j}) \), where \(v_{i}\) and \(v_{j}\) are called the tail and head of the edge, and \(e_{ij}\in \mathcal{E}\) if and only if \(a_{ij}>0 \). Moreover, only simple graph is considered in this paper, that is, selfloops and multiple links are not allowed in \(\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A})\). Correspondingly, the Laplacian matrix of \(\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A})\) is defined as \({L}=[l_{ij}]_{N\times N}\), where \(l _{ij}=a_{ij}\), \(i\neq j\) and \(l_{ii}=\sum_{k=1,k\neq i}^{N}a_{ik}\) for \(i=1,2,\ldots,N\). A directed path is an order sequence of vertices such that any two consecutive vertices are an edge of digraph. If there is a directed path from every node to every other node, the graph is said to be strongly connected for directed graph. A digraph has a directed spanning tree if it has N vertices and \(N1\) edges and there exists a root vertex with directed paths to all other vertices and the Laplacian matrix L with a directed spanning tree has the following properties.
Lemma 1
[29]
Suppose that the directed graph \(\mathcal{G}\) contains a directed spanning tree. Then 0 is a simple eigenvalue of its Laplacian matrix L, and all the other eigenvalues of L have positive real parts.
Definition 1
[30]
Let \(\mathbf{Z}_{N}=\{L=[l_{ij}]_{N\times N}\in \mathbf{R}^{N\times N}:l_{ij}\leq 0 \mbox{ if } i\neq j, i,j=1,2,\ldots,N\}\) denote the set of real matrices whose offdiagonal elements are all nonpositive.
Definition 2
[30]
A matrix \(L\in \mathbf{R}^{{N\times N}}\) is called a nonsingular Mmatrix if \(L\in \mathbf{Z}_{N}\) and all the leading principal minors of L are positive.
Lemma 2
[30]
Suppose that matrix \(L=[l_{ij}]_{N\times N}\in \mathbf{R}^{N\times N}\) has \(l_{ij}\leq 0\), for all \(i\neq j\), \(i,j=1,2,\ldots,N\). Then the following statements are equivalent.

(1)
L is a nonsingular Mmatrix;

(2)
There exists a positive definite diagonal matrix \(\Phi =\operatorname{diag}\{\phi _{1},\phi _{2},\ldots,\phi _{n}\}\in \mathbf{R}^{N\times N}\) such that \(L^{T}\Phi +\Phi L>0\);

(3)
All the eigenvalues of L have positive real parts.
Definition 3
[31]
The function \(f(\cdot )\) is said to satisfy \(f(\cdot )\in \operatorname{QUAD}(P,\Delta )\), if there exist two positive definite diagonal matrices \(P=\operatorname{diag}(p_{1},\ldots,p_{n})\) and \(\Delta =\operatorname{diag}(\delta _{1},\ldots,\delta _{n})\), such that, for any \(x,y\in \mathbf{R}^{n}\), the following condition holds: \((xy)^{T}P(f(x)f(y)\Delta x+\Delta y)\leq 0\).
The QUAD assumption can be satisfied for several wellknown chaotic oscillators, such as cellular neural networks, the Lorenz system, and so on. Furthermore, it is easy to verify that the QUAD assumption holds if the nonlinear function f satisfies the global Lipschitz condition.
In [27], the authors have investigated the aperiodically intermittent control for the timeinvariant network and the pinning control was only imposed on the first node with a constant control gain. The model in [27] was described as
where \(x_{i}(t)=(x_{i1},x_{i2},\ldots,x_{in})^{T}\in \mathbf{R}^{n}\), \(f(x_{i}(t))=(f_{1}(x_{i1}(t)),f_{2}(x_{i2}(t)),\ldots,f_{n}(x_{in}(t)))^{T}\), \(c>0\) is the coupling strength, \(\Gamma =\operatorname{diag}(\gamma _{1},\ldots,\gamma _{n})\in \mathbf{R}^{n\times n}\) is a positive semidefinite matrix denoting the inner coupling matrix, \(A=[a_{ij}]_{N\times N}\) is the outer coupling matrix reflecting the network topology, the network topology is timeinvariant and \(\pi (t)\) is the target trajectory satisfying \(\dot{\pi }(t)=f(\pi (t))\).
From a practical viewpoint, it is impossible that the network topology is timeinvariant forever and it is inevitable some links may be lost or added as the networked systems evolve with time. Then the directed complex networks with switching topologies considered in this paper are more significant than [27]. Now, we firstly introduce an aperiodically switching signal \(r(t):[0,+\infty )\rightarrow \{1,2,\ldots,p\}\) to describe the evolution of the network topologies. Suppose there exists an infinite sequence \(\{\bar{t}_{\rho },\rho =0,1,2,\ldots\}\) with \(\bar{t}_{0}=0\), \(\omega _{1}>\bar{t}_{\rho +1}\bar{t}_{\rho }>\omega _{0}>0\), and \([\bar{t}_{\rho },\bar{t}_{\rho +1})\), \(\rho \in \mathbf{N}\) are uniformly bounded nonoverlapping time intervals where \(\bar{t}_{\rho }\) denotes the switching time. For each \(\rho \in \mathbf{N}\), the network topology is timeinvariant for all \(t\in [\bar{t}_{\rho },\bar{t}_{\rho +1})\). Take \(\pi (t)\) as a virtual root into consider and let \(\mathcal{G}(\bar{\mathcal{A}}^{r(t)})\) denote the augmented interaction graph consisting of \(N+1\) nodes. \(\overline{\mathcal{G}}=\{\mathcal{G}(\bar{\mathcal{A}}^{1}),\mathcal{G}(\bar{\mathcal{A}}^{2}),\ldots,\mathcal{G}(\bar{\mathcal{A}}^{p})\}\) is the set of all possible augmented interaction graph. It is not necessary that each possible network topologies contains a directed spanning tree with the virtual root. Suppose \(\widehat{\mathcal{G}}=\{\mathcal{G}(\bar{\mathcal{A}}^{\varsigma _{1}}),\mathcal{G}(\bar{\mathcal{A}}^{\varsigma _{2}}),\ldots,\mathcal{G}(\bar{\mathcal{A}}^{\varsigma _{q}})\}\) is the set of augmented interaction graphs containing a directed spanning tree with \(\{\varsigma _{1},\ldots,\varsigma _{q}\}=\mathcal{Q}\subseteq \mathcal{P}=\{1,2,\ldots,p\}\). In general, it is assumed that the directed network topology contains a directed spanning tree at the beginning and some links will be lost or added as the network topology evolves with time.
In order to observe the switching time, we introduce a function \(h:\mathcal{P}\rightarrow \{0,1\}\) as follows:
Obviously, if the image of h equals the set \(\{1\}\), each possible network topology contains a directed spanning tree, that is, \(\mathcal{Q}=\mathcal{P}\), if h maps into \(\{0\}\), each possible network topology does not have any directed spanning tree, that is, \(\mathcal{Q}=\Phi \). In fact, the function h can be regarded as an observer to measure the switching network topologies containing a directed spanning tree. Without loss of generality, we assume \(h(\cdot )\) does not always equal 0 or 1, that is, \(\mathcal{Q}\subset \mathcal{P}\). Then, using the function h, we can get a time sequence:
and for \(k\geq 0\),
It is easy to see that \(\{\tilde{t}_{k},k\geq 0\}\) is the set of switching time containing directed spanning tree and \(\{\tilde{s}_{k},k\geq 0\}\) is the set of switching time when the directed network does not contain any directed spanning tree. That is, for all \(t\in [\tilde{t}_{k},\tilde{s}_{k})\), the network topologies contain a directed spanning tree, and for all \(t\in [\tilde{s}_{k},\tilde{t}_{k+1})\), the network topologies contain no directed spanning tree. Moreover, \([\tilde{t}_{k},\tilde{s}_{k})\) and \([\tilde{s}_{k},\tilde{t}_{k+1})\) have no multiple switching. Suppose \(\tilde{t}_{k}=\tilde{t}_{k}^{1}<\tilde{t}_{k}^{2}<\cdots<\tilde{t}_{k}^{\theta _{k}+1} =\tilde{s}_{k}\) and \(\tilde{s}_{k}=\tilde{s}_{k}^{1}<\tilde{s}_{k}^{2}<\cdots<\tilde{s}_{k}^{\vartheta _{k}+1} =\tilde{t}_{k+1}\) where \(\tilde{t}_{k}^{j}\) (\(j=1,\ldots,\theta _{k}, \theta _{k}\leq q\)) and \(\tilde{s}_{k}^{j}\) (\(j=1,\ldots,\vartheta _{k}, \vartheta _{k}\leq pq\)) are all switching time in \([\tilde{t}_{k},\tilde{s}_{k})\) and \([\tilde{s}_{k},\tilde{t}_{k+1})\), respectively (see Figure 1).
Now, we introduce the switching network model with aperiodically intermittent pinning control. The dynamics of the ith node are described as
where \(x_{i}(t)\), \(f(x_{i}(t))\), Î“ and c are the same as defined in (1), \(r(t)\in \mathcal{Q}\) represents the mode at time t, \(A^{r(t)}=[a_{ij}^{r(t)}]_{N\times N}\) is the outer coupling matrix reflecting the switched network topology at mode \(r(t)\), \(D^{r(t)}=\operatorname{diag}\{d_{1}^{r(t)},d_{2}^{r(t)},\ldots,d_{N}^{r(t)}\}\), \(d_{i}^{r(t)}\in \{0,1\}\) and \(d_{i}^{r(t)}=1\) if and only if the ith node is pinned at mode \(r(t)\). \(\pi (t)\) is the target trajectory satisfying \(\dot{\pi }(t)=f(\pi (t))\). \(\pi (t)\) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Our aim in this paper is to pin some nodes with zero indegree in some switching topologies via aperiodically intermittent control such that all the nodes can approach \(\pi (t)\) as time t approaches +âˆž, that is, \(\lim_{t\rightarrow +\infty }\Vert x_{i}(t)\pi (t) \Vert =0\), \(i=1,2,\ldots,N\).
Remark 1
Although aperiodically intermittent control has been investigated in [27], the network topology was timeinvariant and the pinning control was only imposed on the first node all the time. In this paper, we investigate switching complex networks and choose the pinned nodes with zero indegree in \(\mathcal{G}(\bar{\mathcal{A}}^{\varsigma _{i}})\) (\(i=1,\ldots,q\)) by using Tarjanâ€™s algorithm [32]. Moreover, the nodes which are pinned may be different at different modes. The pinned nodes are controlled aperiodically intermittently only in some topologies which contain a directed spanning tree, while in other topologies no node is pinned. Our model (3) is more general than the model in [27].
Let \(e_{i}(t)=x_{i}(t)\pi (t)\) (\(i=1,2,\ldots,N\)) denote the synchronization error and \(E(t)=(e_{1}^{T}(t),e_{2}^{T}(t),\ldots,e_{N}^{T}(t))^{T}\). Then we can get the synchronization error systems as follows:
where \({F}(E(t))=(({f}(x_{1}(t))f(\pi (t)))^{T},({f}(x_{2}(t))f(\pi (t)))^{T},\ldots,({f}(x_{N}(t))f(\pi (t)))^{T})^{T}\), \(\widehat{L}^{r(t)}=L^{r(t)}+cD^{r(t)}\). According to the above statement and Lemma 2, it is found that \(\widehat{L}^{r(t)}\) (\(t\in [\tilde{t}_{k}^{j},\tilde{t}_{k}^{j+1}), j=1,\ldots,\theta _{k}\)) is a nonsingular Mmatrix. Then, using Theorem 2.3 of [30], we can get the following result.
Lemma 3
For \(t\in [\tilde{t}_{k}^{j},\tilde{t}_{k}^{j+1})\), \(j\in \{1,2,\ldots,\theta _{k}\}\), \(k\in \mathbf{N}\), there exist positive vectors \(\xi ^{r(\tilde{t}_{k}^{j})}=(\xi _{1}^{r(\tilde{t}_{k}^{j})},\xi _{2}^{r(\tilde{t}_{k}^{j})},\ldots, \xi _{N}^{r(\tilde{t}_{k}^{j})})^{T}\in \mathbf{R}^{N}\), such that \((\widehat{L}^{r(\tilde{t}_{k}^{j})})^{T}\xi ^{r(\tilde{t}_{k}^{j})}=\mathbf{1}_{N}\) and \(\Xi ^{r(\tilde{t}_{k}^{j})}\widehat{L}^{r(\tilde{t}_{k}^{j})}+(\widehat{L}^{r(\tilde{t}_{k}^{j})})^{T}\Xi ^{r(\tilde{t}_{k}^{j})}>0\), where \(\Xi ^{r(\tilde{t}_{k}^{j})}=\operatorname{diag}\{1/\xi _{1}^{r(\tilde{t}_{k}^{j})},1/\xi _{2}^{r(\tilde{t}_{k}^{j})}, \ldots,1/\xi _{N}^{r(\tilde{t}_{k}^{j})}\}\).
For notational convenience, denote \(\mu _{k}=\tilde{s}_{k}\tilde{t}_{k}\) and \(\eta _{k}=\tilde{t}_{k+1}\tilde{s}_{k}\) as the ith control width and the ith rest width, respectively. Let \(\lambda _{0}^{r(\tilde{t}_{k}^{j})}=\lambda _{\min }^{r(\tilde{t}_{k}^{j})}\xi _{\min }^{r(\tilde{t}_{k}^{j})}\), where \(\lambda _{\min }^{r(\tilde{t}_{k}^{j})}\) is the smallest eigenvalue of \(\Xi ^{r(\tilde{t}_{k}^{j})}\widehat{L}^{r(\tilde{t}_{k}^{j})}+(\widehat{L}^{r(\tilde{t}_{k}^{j})})^{T} \Xi ^{r(\tilde{t}_{k}^{j})}\), \(\xi _{\min }^{r(\tilde{t}_{k}^{j})}=\min_{i=1,2,\ldots ,N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), \(\xi ^{r(\tilde{t}_{k}^{j})}=(\xi _{1}^{r(\tilde{t}_{k}^{j})},\xi _{2}^{\tilde{t}_{k}^{j}},\ldots, \xi _{N}^{\tilde{t}_{k}^{j}})^{T}\) is defined in Lemma 3, and \(\chi _{k}=\min_{1\leq j\leq \vartheta _{k}}\tilde{\lambda }^{r(\tilde{s}_{k}^{j})}\) where \(\tilde{\lambda }_{\min }^{r(\tilde{s}_{k}^{j})}\) is the smallest eigenvalue of \((\Xi ^{r(\tilde{t}_{k}^{\theta _{k}})})^{1}(\Xi ^{r(\tilde{t}_{k}^{\theta _{k}})}{L}^{r(\tilde{s}_{k}^{j})} +({L}^{r(\tilde{s}_{k}^{j})})^{T} \Xi ^{r(\tilde{t}_{k}^{\theta _{k}})})\).
Assumption 1
For any \(k\in \mathbf{N}\), there is no repetitively switching topology in \([\tilde{t}_{k},\tilde{s}_{k})\) or \([\tilde{s}_{k},\tilde{t}_{k+1})\). The switching times in \([\tilde{t}_{k},\tilde{s}_{k})\) and \([\tilde{s}_{k},\tilde{t}_{k+1})\) are not more than q and \(pq\), respectively.
Lemma 4
[33]
Suppose that \(P\in \mathbf{R}^{N\times N}\) is a positive definite matrix and \(M\in \mathbf{R}^{N\times N}\) is symmetric. Then, for any vector \(x\in \mathbf{R}^{N}\), the following inequality holds:
where \(\lambda _{\min }(P^{1}M)\) and \(\lambda _{\max }(P^{1}M)\) are the minimum and maximum eigenvalues of \(P^{1}M\), respectively.
3 Main results
In the following section, we aim to find some sufficient synchronization criteria for synchronizing all the nodes in the switching networks (3) with the target trajectory \(\pi (t)\).
Theorem 1
Under the QUAD assumption, the pinning synchronization in switching networks (3) with target trajectory \(\pi (t)\) could be achieved if there exists a positive scalar \(\varepsilon _{0}\) such that, for each \(k\in \mathbf{N}\), \(j=1,2,\ldots,\theta _{k}\), the following conditions are satisfied.

(1)
\({\lambda }_{0}^{r(\tilde{t}_{k}^{j})}>2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{j})}\),

(2)
\(\sum_{j=1}^{\theta _{k}}\alpha ^{r(\tilde{t}_{k}^{j})} (\tilde{t}_{k}^{j+1}\tilde{t}_{k}^{j})>{\sum_{j=1}^{\vartheta _{k}} \beta ^{r(\tilde{s}_{k}^{j})}(\tilde{s}_{k}^{j+1}\tilde{s}_{k}^{j})}+p\ln\zeta _{k}+\varepsilon _{0}\),
where \(\alpha ^{r(\tilde{t}_{k}^{j})}={\lambda }_{0}^{r(\tilde{t}_{k}^{j})}2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{j})}\), \(\beta ^{r(\tilde{s}_{k}^{j})}=2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{\theta _{k}})}\tilde{\lambda }_{\min }^{r(\tilde{s}_{k}^{j})}\), \(\lambda _{0}^{r(\tilde{t}_{k}^{j})}=\lambda _{\min }^{r(\tilde{t}_{k}^{j})} \xi _{\min }^{r(\tilde{t}_{k}^{j})}\), \(\delta _{\max }=\max_{1\leq i\leq n}\delta _{i}\), \(\zeta ^{r(\tilde{t}_{k}^{j})}=\xi _{\max }^{r(\tilde{t}_{k}^{j})}/\xi _{\min }^{r(\tilde{t}_{k}^{j})}\), \(\xi _{\max }^{r(\tilde{t}_{k}^{j})}=\max_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), and \(\xi _{\min }^{r(\tilde{t}_{k}^{j})}=\min_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), \(\zeta _{k}=\max \{\varsigma _{1,k},\varsigma _{2,k}\}\), \(\varsigma _{1,k}=\max_{1\leq j\leq \theta _{k}}\{\zeta ^{r(\tilde{t}_{k}^{j})}\}\), \(\varsigma _{2,k}=\max_{1\leq j\leq \vartheta _{k}}\{\zeta ^{r(\tilde{s}_{k}^{j})}\}\).
Proof
Note that global pinning synchronization of the switching networks (3) with the target trajectory \(\pi (t)\) is achieved if and only if the zero equilibrium point of the error systems (4) is globally attractive.
For each \(\tilde{t}_{k}^{j}\), \(j=1,\ldots,\theta _{k}\), \(k\in \mathbf{N}\), the interaction digraph \(\mathcal{G}(\bar{\mathcal{A}}^{r(\tilde{t}_{k}^{j})})\) contain a directed spanning tree rooted at the virtual node. Thus, there exists a positive definite matrix \(\Xi ^{r(\tilde{t}_{k}^{j})}\) as defined in Lemma 3 such that \(\Xi ^{r(\tilde{t}_{k}^{j})}\widehat{L}^{r(\tilde{t}_{k}^{j})}+ (\widehat{L}^{r(\tilde{t}_{k}^{j})})^{T}\Xi ^{r(\tilde{t}_{k}^{j})}>0\). Then we construct the following multiple Lyapunov functions for the switching systems (4):
For \(t\in [\tilde{t}_{k}^{j},\tilde{t}_{k}^{j+1})\), \(j=1,\ldots,\theta _{k}\), \(k\in \mathbf{N}\), the derivative of \(V(t)\) with respect to time t along with solutions of (4) can be calculated as follows:
Using Lemma 4, it follows from (6) that
where \(\delta _{\max }=\max_{1\leq i\leq n}\delta _{i}\), \(\zeta ^{r(\tilde{t}_{k}^{j})}=\xi _{\max }^{r(\tilde{t}_{k}^{j})}/\xi _{\min }^{r(\tilde{t}_{k}^{j})}\), \(\xi _{\max }^{r(\tilde{t}_{k}^{j})}=\max_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), and \(\xi _{\min }^{r(\tilde{t}_{k}^{j})} = \min_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), \(\tilde{\lambda }_{0}^{r(\tilde{t}_{k}^{j})}\) is the smallest eigenvalue of \((\Xi ^{r(\tilde{t}_{k}^{j})})^{1}(\Xi ^{r(\tilde{t}_{k}^{j})}\widehat{L}^{r(\tilde{t}_{k}^{j})}+ (\widehat{L}^{r(\tilde{t}_{k}^{j})})^{T}\Xi ^{r(\tilde{t}_{k}^{j})})\). Note that, for each \(\tilde{t}_{k}^{j}\), \(j=1,\ldots,\theta _{k}\), the following inequality is always satisfied: \(\tilde{\lambda }_{0}^{r(\tilde{t}_{k}^{j})}\geq \lambda _{\min }^{r(\tilde{t}_{k}^{j})} \xi _{\min }^{r(\tilde{t}_{k}^{j})}\), where \(\lambda _{\min }^{r(\tilde{t}_{k}^{j})}\) is the smallest eigenvalue of \(\Xi ^{r(\tilde{t}_{k}^{j})}\widehat{L}^{r(\tilde{t}_{k}^{j})}+(\widehat{L}^{r(\tilde{t}_{k}^{j})})^{T} \Xi ^{r(\tilde{t}_{k}^{j})}\), \(\xi _{\min }^{r(\tilde{t}_{k}^{j})}=\min_{i=1,2,\ldots, N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), \(\xi ^{r(\tilde{t}_{k}^{j})}=(\xi _{1}^{r(\tilde{t}_{k}^{j})},\xi _{2}^{\tilde{t}_{k}^{j}},\ldots, \xi _{N}^{\tilde{t}_{k}^{j}})^{T}\) is defined in Lemma 3. Then we find that
where \(\alpha ^{r(\tilde{t}_{k}^{j})}={\lambda }_{0}^{r(\tilde{t}_{k}^{j})}2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{j})}\), \(\lambda _{0}^{r(\tilde{t}_{k}^{j})}=\lambda _{\min }^{r(\tilde{t}_{k}^{j})} \xi _{\min }^{r(\tilde{t}_{k}^{j})}\). It can be seen from condition (1) that \(\alpha ^{r(\tilde{t}_{k}^{j})}>0\), \(j=1,\ldots,\theta _{k}\), \(k\in \mathbf{N}\).
Therefore,
On the other hand, it follows from (5) that
where \(t\in [\tilde{t}_{k}^{j},\tilde{t}_{k}^{j+1})\), \(j=1,\ldots,\theta _{k}, k\in \mathbf{N}\). Then we have
where \(\varsigma _{1,k}=\max_{1\leq j\leq \theta _{k}}\{\zeta ^{r(\tilde{t}_{k}^{j})}\}\).
For \(t\in [\tilde{s}_{k}^{j},\tilde{s}_{k}^{j+1}), j=1,\ldots,\vartheta _{k}, k\in \mathbf{N}\), we get
where \(\beta ^{r(\tilde{s}_{k}^{j})}=2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{\theta _{k}})}\tilde{\lambda }_{\min }^{r(\tilde{s}_{k}^{j})}\), \(\tilde{\lambda }_{\min }^{r(\tilde{s}_{k}^{j})}\) is the smallest eigenvalue of
Similarly, we get
where \(\varsigma _{2,k}=\max_{1\leq j\leq \vartheta _{k}}\{\zeta ^{r(\tilde{s}_{k}^{j})}\}\).
Therefore,
where \(\zeta _{k}=\max \{\varsigma _{1,k},\varsigma _{2,k}\}\), \(\varsigma _{1,k}=\max_{1\leq j\leq \theta _{k}}\{\zeta ^{r(\tilde{t}_{k}^{j})}\}\), \(\varsigma _{2,k}=\max_{1\leq j\leq \vartheta _{k}}\{\zeta ^{r(\tilde{s}_{k}^{j})}\}\).
Then it can be directly found from (14) and the condition (2) that
for any given \(k\in \mathbf{N}\). Furthermore, we can obtain by recursion that
Used to the dwell time \(\omega _{0}>0\), we know that there is no Zeno behavior as the switching networks evolve with time [34]. Thus, for any given time \(t>0\), there exists a positive integer m such that \(\tilde{t}_{m}< t\leq \tilde{t}_{m+1}\).
When \(t\in [0,\tilde{t}_{1}^{2})\), we get
where \(\alpha ^{r(\tilde{t}_{1}^{1})}=\lambda _{0}^{r(\tilde{t}_{1}^{1})}2\delta _{\max }\zeta ^{r(\tilde{t}_{1}^{1})}>0\).
When \(t\in [\tilde{t}_{1}^{\theta _{1}},\tilde{t}_{1}^{\theta _{1}+1})\), it can be found that
When \(t\in [\tilde{s}_{1}^{1},\tilde{s}_{1}^{2})\), it can be similarly calculated that
When \(t\in [\tilde{s}_{1}^{\vartheta _{1}},\tilde{s}_{1}^{\vartheta _{1}+1})\), we get
When \(t\in [\tilde{t}_{m}^{j},\tilde{t}_{m}^{j+1})\), \(j=1,2,\ldots,\theta _{m}, m\geq 2\), we have
where \(\hat{\iota }_{0}=\max_{1\leq k\leq m}\{\mu _{k}+\eta _{k}\}\).
When \(t\in [\tilde{s}_{m}^{l},\tilde{s}_{m}^{l+1})\), \(l=1,2,\ldots,\vartheta _{m}\), \(m\geq 2\), we have
Combining with (21) and (22), we have the following. As time \(t\rightarrow +\infty \), \(V(t)\rightarrow 0\). Then \(e_{i}(t)\rightarrow 0\), \(j=1,\ldots,N\). That is pinning synchronization of the switched networks (3) with the target trajectory \(\pi (t)\) could be realized. The proof is completed.â€ƒâ–¡
Remark 2
From the above analysis, we can see that the network topologies contain a directed spanning tree in the time periods \([\tilde{t}_{k}^{j},\tilde{t}_{k}^{j+1})\subseteq [\tilde{t}_{k},\tilde{s}_{k})\), \(j=1,2,\ldots,\theta _{k}\), \(k\in \mathbf{N}\) when zeroindegree nodes are pinned. While \(t\in [\tilde{s}_{k}^{l},\tilde{s}_{k}^{l+1})\subseteq [\tilde{s}_{k},\tilde{t}_{k+1})\), \(l=1,2,\ldots\vartheta _{k}, k\in \mathbf{N}\), no node is pinned and the network topologies contain no directed spanning tree. Moreover, \([\tilde{t}_{k}^{j},\tilde{t}_{k}^{j+1})\) are called control spans benefitial to pinning synchronization, thus they are as long as possible; while the rest spans \([\tilde{s}_{k}^{l},\tilde{s}_{k}^{l+1})\), harmful to synchronization, should be as short as possible.
Remark 3
It is easy to see that c plays a very important role which affect the realization of the conditions in Theorem 1. Given the network topology, the bigger c the bigger the value of \({\lambda }_{0}^{r(\tilde{t}_{k}^{j})}\), then the second condition is easier to be satisfied. Moreover, the control spans become shorter and the rest spans become longer.
Remark 4
In [28], the topologies switch periodically and some nodes are pinned all the time though the topologies contain no directed spanning tree. In this paper, all the topologies switch aperiodically and there exist no two topologies simultaneously which is different from jointly connected graphs [35]. Moreover, no node is pinned when the topologies contain no directed spanning tree and the pinned nodes are dependent on the topologies. Our method is more reasonable than [28].
When all the possible switching topologies contain a directed spanning tree, some nodes with zero indegree are pinned all the time though the pinned nodes may be different in different modes.
Corollary 1
Under the QUAD assumption, if all the network topologies contain a directed spanning tree, that is, \(\mathcal{P}=\mathcal{Q}\), the global pinning synchronization of switching networks (3) with the target trajectory \(\pi (t)\) can be achieved if \({\lambda }_{0}^{r(\tilde{t}_{k}^{j})}>2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{j})}\), where \(\delta _{\max }=\max_{1\leq i\leq n}\delta _{i}\), \(\zeta ^{r(\tilde{t}_{k}^{j})}=\xi _{\max }^{r(\tilde{t}_{k}^{j})}/\xi _{\min }^{r(\tilde{t}_{k}^{j})}\), \(\xi _{\max }^{r(\tilde{t}_{k}^{j})}=\max_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), and \(\xi _{\min }^{r(\tilde{t}_{k}^{j})}=\min_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), \(\tilde{\lambda }_{0}^{r(\tilde{t}_{k}^{j})}\) is the smallest eigenvalue of \((\Xi ^{r(\tilde{t}_{k}^{j})})^{1}(\Xi ^{r(\tilde{t}_{k}^{j})}\widehat{L}^{r(\tilde{t}_{k}^{j})}+ (\widehat{L}^{r(\tilde{t}_{k}^{j})})^{T}\Xi ^{r(\tilde{t}_{k}^{j})})\).
Remark 5
Since the controller is dependent on the topologies according to (3), the controller becomes continuous instead of the aperiodically intermittent controller when \(\mathcal{P}=\mathcal{Q}\). The conditions (2) in Theorem 1 is apparently satisfied. Then, provided the conditions (1) in Theorem 1 could be satisfied under some \(c>0\), the switching network (3) can synchronize with the target trajectory \(\pi (t)\). Although the controller becomes continuous when \(\mathcal{P}=\mathcal{Q}\), different nodes are pinned in different topologies, not as in [36].
Corollary 2
Suppose function \(f\in \operatorname{QUAD}(P,\Delta )\) and \(\mathcal{P}\subset \mathcal{Q}\). The global pinning synchronization of switched networks (3) with the target trajectory \(\pi (t)\) can be achieved if there exists a positive scalar \(\varepsilon _{k}\) for each \(k\in \mathbf{N}\), such that the following conditions hold.

(1)
\({\lambda }_{0}^{r(\tilde{t}_{k}^{j})}>2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{j})}\),

(2)
\(\sum_{j=1}^{\theta _{k}}\alpha ^{r(\tilde{t}_{k}^{j})}(\tilde{t}_{k}^{j+1}\tilde{t}_{k}^{j})>{\sum_{j=1}^{\vartheta _{k}} \beta ^{r(\tilde{s}_{k}^{j})}(\tilde{s}_{k}^{j+1}\tilde{s}_{k}^{j})}+p\ln\zeta _{k}+\varepsilon _{k}\),
where \(\alpha ^{r(\tilde{t}_{k}^{j})}={\lambda }_{0}^{r(\tilde{t}_{k}^{j})}2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{j})}\), \(\beta ^{r(\tilde{s}_{k}^{j})}=2\delta _{\max }\zeta ^{r(\tilde{t}_{k}^{\theta _{k}})}\tilde{\lambda }_{\min }^{r(\tilde{s}_{k}^{j})}\), \(\lambda _{0}^{r(\tilde{t}_{k}^{j})}=\lambda _{\min }^{r(\tilde{t}_{k}^{j})} \xi _{\min }^{r(\tilde{t}_{k}^{j})}\), \(\delta _{\max }=\max_{1\leq i\leq n}\delta _{i}\), \(\zeta ^{r(\tilde{t}_{k}^{j})}=\xi _{\max }^{r(\tilde{t}_{k}^{j})}/\xi _{\min }^{r(\tilde{t}_{k}^{j})}\), \(\xi _{\max }^{r(\tilde{t}_{k}^{j})}=\max_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), and \(\xi _{\min }^{r(\tilde{t}_{k}^{j})}=\min_{1\leq i\leq N}\xi _{i}^{r(\tilde{t}_{k}^{j})}\), \(\zeta _{k}=\max \{\varsigma _{1,k},\varsigma _{2,k}\}\), \(\varsigma _{1,k}=\max_{1\leq j\leq \theta _{k}}\{\zeta ^{r(\tilde{t}_{k}^{j})}\}\), \(\varsigma _{2,k}=\max_{1\leq j\leq \vartheta _{k}}\{\zeta ^{r(\tilde{s}_{k}^{j})}\}\).
Remark 6
In this paper, if the switching topologies become periodic, that is, \(\tilde{s}_{k}\tilde{t}_{k}=\varpi \) and \(\tilde{t}_{k+1}\tilde{t}_{k}=\omega\) (\(k=1,2,\ldots\)), where Ï– and Ï‰ are positive scalars, the obtained results still hold. Then it becomes periodically intermittent pinning synchronization.
4 An illustrative example
In this section, some numerical simulations are provided to illustrate the effectiveness of the obtained theoretical results.
Assume the target trajectory \(\pi (t)\) is given by a 3D cellular neural network as follows:
where \(\pi (t)=(\pi _{1}(t),\pi _{2}(t),\pi _{3}(t))^{T}\), \(L(\pi (t))=(l(\pi _{1}(t)),l(\pi _{2}(t)),l(\pi _{3}(t)))^{T}\), \(l(v)=(\vert v+1 \vert \vert v1 \vert )/2\). This neural network has a doublescrolling chaotic attractor shown in Figure 2, which is the same as in [37]. It is easy to see that the function \(f(\cdot )\) satisfies Assumption 1 [38]. Moreover, let \(\Gamma =I\).
Suppose the switching time as in Figure 1. According to Figure 1, we can see that \(\overline{\mathcal{G}}=\{\mathcal{G}(\bar{\mathcal{A}^{1}}),\mathcal{G}(\bar{\mathcal{A}^{2}}),\mathcal{G}(\bar{\mathcal{A}^{3}}),\mathcal{G}(\bar{\mathcal{A}^{4}})\}\) and \(\widehat{\mathcal{G}}=\{\mathcal{G}(\bar{\mathcal{A}^{1}}),\mathcal{G}(\bar{\mathcal{A}^{2}})\}\). The four switching topologies are given in Figures 3 and 4 where the weight on each edge is assumed to be one. In Figures 3 and 4, the neighboring relationships between the nodes in network (3) and the single target node (labeled node 9) are depicted by dashed lines with arrows.
Choose \(c=19.25\), we can obtain from Theorem 1 that the aperiodically intermittent synchronization of switching complex networks dependent on topology structure can be ensured with the target trajectory \(\pi (t)\). The synchronization errors of the switching networks (4) are, respectively, shown in Figures 5, 6 and 7. Let \(\Vert e(t) \Vert =1/8(\sum_{i=1}^{8}\Vert x_{i}(t)\pi (t) \Vert ^{2})^{1/2}\) denote the synchronization error of the considered network. It can be seen from Figure 8 that the global pinning synchronization problem is indeed solved.
5 Conclusions
In this paper, eventtriggered schemes and selftriggered schemes are investigated to realize the exponential synchronization of the networked dynamical systems. The coupled information under these schemes is updated only when the triggering conditions are violated. The next observation time these nodes is predicted only based on the latest observations of their neighborhood and the virtual leader. Thus, continuous communication can be avoided and the quantity of information transmission is largely reduced. Moreover, a positive lower bound for interevent intervals is achieved and the Zeno behavior can be excluded. Finally, two numerical simulation examples are provided to illustrate the effectiveness of the proposed results. In the future, we will focus on the related applications of the eventtriggered scheme in the coupled neural networks with timedelays and quantization.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (11501391, 11601449) and supported by Key Program of Sichuan Provincial Department of Education (16ZA0066).
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Gao, J., Cao, J. Aperiodically intermittent synchronization for switching complex networks dependent on topology structure. Adv Differ Equ 2017, 244 (2017). https://doi.org/10.1186/s1366201712618
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DOI: https://doi.org/10.1186/s1366201712618