Cluster lag synchronization of delayed heterogeneous complex dynamical networks via intermittent pinning control

Abstract

This paper investigates the problem of cluster lag synchronization in the heterogeneous dynamical networks by using an intermittent pinning control strategy. Previous related works mainly focused on the time-varying delays in the self-dynamics, which was not consistent with the real world. The transmission delay in the communication channels is considered in this paper. We present several criteria to guarantee cluster lag synchronization without assuming the coupling matrix being symmetric and irreducible. A decentralized adaptive intermittent pinning control scheme is employed to reduce the control cost. An effective pinned-cluster selection scheme is adopted to guide what kind of clusters should be pinned preferentially. Two simulations are proposed to verify the correctness of the theoretical results.

Appendices

Appendix 1

1.1 Proof of Theorem 1

Let $E\left( t \right) = \left( {E_{1}^{\rm T} \left( t \right),E_{2}^{\rm T} \left( t \right), \ldots ,E_{m}^{\rm T} \left( t \right)} \right)^{\rm T}$, where $E_{1} \left( t \right) = \left( {e_{1,1}^{\rm T} \left( t \right), \ldots ,e_{{r_{1} ,1}}^{\rm T} \left( t \right)} \right)^{\rm T}$, $E_{2} \left( t \right) = \left( {e_{{r_{1} + 1,2}}^{\rm T} \left( t \right), \ldots ,e_{{r_{1} + r_{2} ,2}}^{\rm T} \left( t \right)} \right)^{\rm T} , \ldots .E_{m} \left( t \right) = \left( {e_{{r_{1} + r_{2} + \cdots + r_{m - 1} + 1,m}}^{\rm T} \left( t \right), \ldots ,e_{N,m}^{\rm T} \left( t \right)} \right)^{\rm T} .$

Construct a Lyapunov candidate as follows, where $\otimes$ is defined as Kronecker product.

$$\begin{aligned} W\left( t \right) & = \frac{1}{2}E^{\rm T} \left( t \right)\left( {I_{N} \otimes I_{n} } \right)E\left( t \right) = \frac{1}{2}\sum\nolimits_{k = 1}^{m} {E_{k}^{\rm T} \left( t \right)\left( {I_{{r_{k} }} \otimes I_{n} } \right)E_{k} \left( t \right)} \\ & = \frac{1}{2}\sum\nolimits_{k = 1}^{m} {\sum\nolimits_{{i \in C_{k} }} {e_{ik}^{\rm T} \left( t \right)e_{ik} \left( t \right)} } \\ \end{aligned}$$

When $t \in \left[ {\omega T,\left( {\omega + \theta } \right)T} \right)$, $\omega = 0,1,2, \ldots$, the time derivative of $W\left( t \right)$ along the trajectory of (9) at time $t$ can be given as

$$\begin{aligned} \dot{W}\left( t \right) & = \sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {e_{ik}^{\text{T}} \left( t \right)\tilde{f}_{k}^{{\tau_{k} ,\sigma }} \left( {t,x_{i} ,s_{k} } \right)} } + c\sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {\sum\limits_{p = 1}^{m} {\sum\limits_{{j \in C_{p} }} {b_{ij} e_{ik}^{\text{T}} \left( t \right)\varGamma e_{jp} \left( t \right)} } } } \\ & \quad - c\sum\limits_{k = 1}^{l} {E_{k}^{\text{T}} \left( t \right)\left( {d_{k} I_{{r_{k} }} \otimes \varGamma } \right)E_{k} \left( t \right)} . \\ \end{aligned}$$

(28)

In virtue of Assumption 2, we can get

$$\begin{aligned} & \sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {e_{ik}^{\text{T}} \left( t \right)\tilde{f}_{k}^{{\tau_{k} ,\sigma }} \left( {t,x_{i} ,s_{k} } \right)} } \\ & \quad \le \sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {L_{k}^{0} e_{ik}^{\text{T}} \left( t \right)\varGamma e_{ik} \left( t \right)} } + \sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {L_{k}^{\tau } e_{ik}^{\text{T}} \left( {t - \tau_{k} \left( t \right)} \right)\varGamma e_{ik} \left( {t - \tau_{k} \left( t \right)} \right)} } \\ & \quad = \sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {L_{k}^{0} I_{{r_{k} }} \otimes \varGamma } \right)E_{k} \left( t \right)} + \sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( {t - \tau_{k} \left( t \right)} \right)\left( {L_{k}^{\tau } I_{{r_{k} }} \otimes \varGamma } \right)E_{k} \left( {t - \tau_{k} \left( t \right)} \right)} . \\ \end{aligned}$$

(29)

Noticing that $\sum\nolimits_{p = 1}^{m} {\sum\nolimits_{{i = C_{p} }} {b_{ij} } } = \sum\nolimits_{j = 1}^{N} {b_{ij} } = 0$, $i = 1,2, \ldots ,N$ with $\tilde{\varGamma } = \text{diag} \left( {\sqrt {\gamma_{1} } ,\sqrt {\gamma_{2} } , \ldots ,\sqrt {\gamma_{n} } } \right)$, one has

$$\begin{aligned} & \sum\limits_{k = 1}^{m} {\sum\limits_{{i = C_{k} }} {\sum\limits_{p = 1}^{m} {\sum\limits_{{j \in C_{p} }} {b_{ij} e_{ik}^{\text{T}} \left( t \right)\varGamma e_{jp} \left( t \right)} } } } = \sum\limits_{k = 1}^{m} {\sum\limits_{p = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {B_{kp} \otimes \varGamma } \right)E_{p} \left( t \right)} } \\ & \quad = \sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {B_{kk}^{s} \otimes \varGamma } \right)E_{k} \left( t \right)} + \sum\limits_{k = 1}^{m} {\sum\limits_{p = 1,p \ne k}^{m} {\sum\limits_{{i = C_{k} }} {\sum\limits_{{j \in C_{p} }} {b_{ij} \left( {\tilde{\varGamma }e_{ik} \left( t \right)} \right)^{\text{T}} \left( {\tilde{\varGamma }e_{jp} \left( t \right)} \right)} } } } \\ & \quad \le \sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {B_{kk}^{s} \otimes \varGamma } \right)E_{k} \left( t \right)} + \frac{1}{2}\sum\limits_{k = 1}^{m} {\sum\limits_{p = 1,p \ne k}^{m} {\sum\limits_{{i = C_{k} }} {\sum\limits_{{j \in C_{p} }} {b_{ij} \left( {e_{ik}^{\text{T}} \left( t \right)\varGamma e_{ik} \left( t \right) + e_{jp}^{\text{T}} \left( t \right)\varGamma e_{jp} \left( t \right)} \right)} } } } \\ & \quad = \sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {B_{kk}^{s} \otimes \varGamma } \right)E_{k} \left( t \right)} - \frac{1}{2}\sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {A_{k} \otimes \varGamma } \right)E_{k} \left( t \right)} + \frac{1}{2}\sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {\varPsi_{k} \otimes \varGamma } \right)E_{k} \left( t \right)} \\ & \quad = \sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( t \right)\left( {\left( {B_{kk}^{s} + \frac{1}{2}\varPsi_{k} - \frac{1}{2}A_{k} } \right) \otimes \varGamma } \right)E_{k} \left( t \right)} \\ \end{aligned}$$

(30)

Substituting (29) and (30) into (28), we get

$$\dot{W}\left( t \right) \le cE^{\rm T} \left( t \right)\left( {\left( {Z - D} \right) \otimes \varGamma } \right)E\left( t \right) + \frac{q}{2}\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} \sum\limits_{k = 1}^{m} {E_{k}^{\text{T}} \left( s \right)E_{k} \left( s \right)} } \right).$$

(31)

According to condition $\left( \text{i} \right)$ and the properties of the Kronecker product of matrix [42], from (31), one has

$$\begin{aligned} \dot{W}\left( t \right) & \le E^{\rm T} \left( t \right)\left( {\left( {cZ - cD + a_{1} I_{N} } \right) \otimes \varGamma } \right)E\left( t \right) - a_{1} E^{\rm T} \left( t \right)\left( {I_{N} \otimes \varGamma } \right)E\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right) \\ & \le - 2a_{1} \lambda_{\hbox{min} } \left( \varGamma \right)W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right) = - p_{1} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right). \\ \end{aligned}$$

Similarly, when $t \in \left[ {\left( {\omega + \theta } \right)T,\left( {\omega + 1} \right)T} \right)$, $\omega = 0,1,2, \ldots$, using condition $\left( {\text{ii} } \right)$, we can deduce

$$\begin{aligned} \dot{W}\left( t \right) & \le E^{\rm T} \left( t \right)\left( {\left( {cZ - a_{2} I_{N} } \right) \otimes \varGamma } \right)E\left( t \right) + a_{2} E^{\rm T} \left( t \right)\left( {I_{N} \otimes \varGamma } \right)E\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right) \\ & \le 2a_{2} \lambda_{\hbox{max} } \left( \varGamma \right)W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right) = p_{2} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right). \\ \end{aligned}$$

As a result, we have

$$\left\{ {\begin{array}{ll} {\dot{W}\left( t \right) \le - p_{1} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right),} & {\omega T \le t < \left( {\omega + \theta } \right)T,} \\ {\dot{W}\left( t \right) \le p_{2} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right),} & {\left( {\omega + \theta } \right)T \le t < \left( {\omega + 1} \right)T.} \\ \end{array} } \right.$$

(32)

Next, we will prove that conditions (iii)–(iv) imply that, for $t \ge 0$, we have $\dot{W}\left( t \right) \le \left( {\mathop {\sup }\nolimits_{ - \tau \le s \le 0} W\left( s \right)} \right)e^{ - \varpi t}$.

Define $\zeta \left( \lambda \right) = \lambda - p_{1} + qe^{\lambda \tau }$, where $p_{1} > q > 0$ such that $\zeta \left( 0 \right) < 0$, $\zeta \left( { + \infty } \right) > 0$, $\zeta^{\prime}\left( \lambda \right) > 0$. Utilizing the monotonicity and continuity of $\zeta \left( \lambda \right)$, the equation $\lambda - p_{1} + qe^{\lambda \tau } = 0$ has a unique positive solution $\lambda > 0$. Let $M_{0} = \mathop {\sup }\nolimits_{ - \tau \le s \le 0} W\left( s \right)$, $V\left( t \right) = e^{\lambda t} W\left( t \right)$, $t \ge 0$. Let $S\left( t \right) = V\left( t \right) - hM_{0}$, where $h > 1$ is a constant. Apparently, for any $t \in \left[ { - \tau ,0} \right]$, one has

$$S\left( t \right) < 0.$$

(33)

Then, we indicate for all $t \in \left[ {0,\theta T} \right]$, that

$$S\left( t \right) < 0.$$

(34)

From (33), there exist a $t_{0} \in \left[ {0,\theta T} \right]$ such that

$$S\left( {t_{0} } \right) = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot{S}\left( {t_{0} } \right) \ge 0,$$

(35)

$$S\left( t \right) < 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \tau \le t < t_{0} .$$

(36)

By (32), (35) and (36), we have

$$\dot{S}\left( {t_{0} } \right) = \lambda V\left( {t_{0} } \right) + e^{{\lambda t_{0} }} \dot{W}\left( {t_{0} } \right) \le \left( {\lambda - p_{1} } \right)V\left( {t_{0} } \right) + qe^{{\lambda t_{0} }} \left( {\mathop {\sup }\limits_{{t_{0} - \tau \le s \le t_{0} }} W\left( s \right)} \right).$$

(37)

On the other hand, from (35) and (36), we have $V\left( t \right) < hM_{0}$, $- \tau \le t < t_{0}$ and $V\left( {t_{0} } \right) = hM_{0}$. As a result, we get $W\left( t \right) < hM_{0} e^{ - \lambda t}$, $- \tau \le t < t_{0}$, it leads to $\left( {\sup_{{t_{0} - \tau \le s \le t_{0} }} W\left( s \right)} \right) < e^{\lambda \tau } hM_{0} e^{{ - \lambda t_{0} }}$. This means $e^{{\lambda t_{0} }} \left( {\sup_{{t_{0} - \tau \le s \le t_{0} }} W\left( s \right)} \right) < e^{\lambda \tau } hM_{0} = e^{\lambda \tau } V\left( {t_{0} } \right)$.

From (37), it can be deduced that $\dot{S}\left( {t_{0} } \right) < \left( {\lambda - p_{1} + qe^{\lambda \tau } } \right)V\left( {t_{0} } \right) = 0$. This is a contradiction with inequality (35), thus, (34) is established. With (33), we get that

$$W\left( t \right) < hM_{0} e^{ - \lambda t} ,\quad \forall t \in \left[ { - \tau ,\theta T} \right]$$

(38)

Let $\vartheta = p_{1} + p_{2}$, for $t \in \left[ {\theta T,T} \right)$, we have $Q\left( t \right) = V\left( t \right) - hM_{0} e^{{\vartheta \left( {t - \theta T} \right)}} < 0$. Otherwise, there exist a $t_{1} \in \left[ {\theta T,T} \right)$, such that

$$Q\left( {t_{1} } \right) = 0,\quad \dot{Q}\left( {t_{1} } \right) \ge 0,$$

(39)

$$Q\left( t \right) < 0,\quad \theta T \le t < t_{1} .$$

(40)

For $\tau > 0$, according to (38)–(40), we always have

$$\mathop {\sup }\limits_{{t_{1} - \tau \le s \le t_{1} }} W\left( s \right) < e^{\lambda \tau } W\left( {t_{1} } \right).$$

Then, one has $\dot{Q}\left( {t_{1} } \right) < 0$, which is contradicts with (39). Thus, for $t \in \left[ {\theta T,T} \right)$,

$$V\left( t \right) < hM_{0} e^{{\vartheta \left( {1 - \theta } \right)T}} .$$

With (33) and (34), we have

$$V\left( t \right) < hM_{0} e^{{\vartheta \left( {1 - \theta } \right)T}} ,\quad {\text{for}}\quad t \in \left[ { - \tau ,T} \right).$$

Similarly, we can prove

$$V\left( t \right) < hM_{0} e^{{\vartheta \left( {1 - \theta } \right)T}} ,\;\;{\text{for}}\;\;t \in \left[ {T,\left( {1 + \theta } \right)T} \right)\;\;{\text{and}}\;\;V\left( t \right) < hM_{0} e^{{\vartheta \left( {t - 2\theta } \right)T}} ,\;\;{\text{for}}\;\;t \in \left[ {\left( {1 + \theta } \right)T,2T} \right).$$

Through a mathematical induction, for any integer $\omega = 0,1,2, \ldots$, we can derive the estimates of $V\left( t \right)$ as

$$\left\{ {\begin{array}{ll} {V\left( t \right) < hM_{0} e^{{\omega \vartheta \left( {1 - \theta } \right)T}} \le hM_{0} e^{{\vartheta \left( {1 - \theta } \right)t}} ,} & {\omega T \le t < \left( {\omega + \theta } \right)T,} \\ {V\left( t \right) < hM_{0} e^{{\vartheta \left[ {t - \left( {m + 1} \right)\theta T} \right]}} \le hM_{0} e^{{\vartheta \left( {1 - \theta } \right)t}} ,} & {\left( {\omega + \theta } \right)T \le t < \left( {\omega + 1} \right)T.} \\ \end{array} } \right.$$

Let $h \to 1$, from the definition of $V\left( t \right)$, we have

$$W\left( t \right) \le M_{0} e^{{ - \left[ {\lambda - \vartheta \left( {1 - \theta } \right)} \right]t}} = \left( {\mathop {\sup }\limits_{ - \tau \le s \le 0} W\left( s \right)} \right)e^{ - \varpi t} ,\quad t \ge 0.$$

Therefore, for the error dynamical system (9) the zero solution is globally exponentially stable. The proof of Theorem 1 is thus completed.

Appendix 2

2.1 Proof of Theorem 3

Construct a piecewise Lyapunov candidate function

$$W\left( t \right) = \frac{1}{2}\sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {e_{ik}^{\text{T}} \left( t \right)e_{ik} \left( t \right)} } + \frac{1}{2}\varPi \left( t \right)\sum\limits_{k = 1}^{l} {ce^{{ - p_{1} t}} \frac{{\left( {d_{k} \left( t \right) - d_{k}^{*} } \right)^{2} }}{{h_{k} }}} ,$$

where $d_{k}^{ * } > 0$, $k = 1,2, \ldots ,l$ are constants, and $\varPi \left( \cdot \right)$ is a piecewise function defined as

$$\varPi \left( s \right) = \left\{ {\begin{array}{ll} {1,} & { - \tau \le s \le 0,} \\ {e^{{p_{1} \omega T}} ,} & {\omega T \le s < \left( {\omega + 1} \right)T,\omega = 0,1,2, \ldots } \\ \end{array} } \right.$$

Obviously, while $\omega = 0,1,2, \ldots$, we have

$$W\left( t \right) = \left\{ {\begin{array}{ll} {\frac{1}{2}\sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {e_{ik}^{\text{T}} \left( t \right)e_{ik} \left( t \right)} } + \frac{1}{2}e^{{ - p_{1} \left( {t - \omega T} \right)}} \sum\limits_{k = 1}^{l} {\frac{c}{{h_{k} }}\left( {d_{k} \left( t \right) - d_{k}^{*} } \right)^{2} } ,} & {\omega T \le t < \left( {\omega + \theta } \right)T,} \\ {\frac{1}{2}\sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {e_{ik}^{\text{T}} \left( t \right)e_{ik} \left( t \right)} } + \frac{1}{2}e^{{ - p_{1} \left( {t - \omega T} \right)}} \sum\limits_{k = 1}^{l} {\frac{c}{{h_{k} }}d_{k}^{{*^{2} }} } ,} & {\left( {\omega + \theta } \right)T \le t < \left( {\omega + 1} \right)T.} \\ \end{array} } \right.$$

With Assumptions 1 and 2, the time derivative of $W\left( t \right)$ along the trajectory of (9) can be calculated as follows. When $t \in \left[ {\omega T,\left( {\omega + \theta } \right)T} \right)$, $\omega = 0,1,2, \ldots$, we have

$$\begin{aligned} \dot{W}\left( t \right) & \le E^{\rm T} \left( t \right)\left( {\left( {cZ - cD^{ * } + a_{1}^{0} I_{N} } \right) \otimes \varGamma } \right)E\left( t \right) + \frac{q}{2}\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} \sum\nolimits_{k = 1}^{m} {E_{k}^{\rm T} \left( s \right)E_{k} \left( s \right)} } \right) \\ & \quad - a_{1}^{0} E^{\rm T} \left( t \right)\left( {I_{N} \otimes \varGamma } \right)E\left( t \right) - \frac{{p_{1} }}{2}e^{{ - p_{1} \left( {t - \omega T} \right)}} \sum\nolimits_{k = 1}^{l} {\frac{c}{{h_{k} }}\left( {d_{k} \left( t \right) - d_{k}^{ * } } \right)^{2} } , \\ \end{aligned}$$

(41)

where $D^{ * }$ is a modified block diagonal matrix of $D$ through replacing the principal-diagonal sub-matrices $d_{k} I_{{r_{k} }}$ by $\left( {e^{{ - p_{1} \theta T}} d_{k}^{ * } } \right)I_{{r_{k} }} ,k = 1,2, \ldots ,l$.

Set $Q^{ * } = a_{1}^{0} I_{N} + cZ$, $a_{1}^{0} I_{N} + cZ - cD^{ * } = Q^{ * } - cD^{ * } = \left( {\begin{array}{*{20}c} {G^{ * } - c\tilde{D}^{ * } } & 0 \\ {0^{\rm T} } & {Q_{r}^{ * } } \\ \end{array} } \right)$, where $G^{ * } = \left( {\begin{array}{llll} {cZ_{11} + a_{1}^{0} I_{{r_{1} }} } & 0 & \cdots & 0 \\ 0 & {cZ_{22} + a_{1}^{0} I_{{r_{2} }} } & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & {cZ_{ll} + a_{1}^{0} I_{{r_{l} }} } \\ \end{array} } \right)$, $\tilde{D}^{ * } = e^{{ - p_{1} \theta T}} \left( {\begin{array}{llll} {d_{1}^{ * } I_{{r_{1} }} } & 0 & \cdots & 0 \\ 0 & {d_{2}^{ * } I_{{r_{2} }} } & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & {d_{l}^{ * } I_{{r_{l} }} } \\ \end{array} } \right)$, and $Q_{r}^{ * }$ is the minor matrix of $Q^{ * }$ by removing its first $\left( {r_{1} + r_{2} + \cdots + r_{l} } \right)$ row–column pairs. It is obviously that $Q^{ * }$ is a real symmetric matrix. From the pinning condition $\left( {{\text{i}}^{\prime \prime } } \right)$, we conclude that

$$\begin{aligned} \lambda_{\hbox{max} } \left( {cZ_{kk} + a_{1}^{0} I_{{r_{k} }} } \right) & = \lambda_{\hbox{max} } \left( {c\varTheta_{kk} + L_{k}^{0} I_{{r_{k} }} + a_{1}^{0} I_{{r_{k} }} } \right) \\ & < c\lambda_{\hbox{max} } \left( {\varTheta_{kk} } \right) + L_{k}^{0} + a_{1}^{0} < 0,\quad l + 1 \le k \le m. \\ \end{aligned}$$

This means $Q_{r}^{ * } < 0$. As a result, for $k = 1,2, \cdots ,l$, when $d_{k}^{ * } > 0$ are large enough such that $d_{k}^{ * } > \frac{{e^{{p_{1} \theta T}} \lambda_{\hbox{max} } \left( {G^{ * } } \right)}}{c}$. It is easy to get $Q^{ * } - cD^{ * } < 0$. This follows Lemma 2 directly.

With (41), we can deduce that

$$\begin{aligned} \dot{W}\left( t \right) & \le - \frac{{p_{1} }}{2}\sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {e_{ik}^{\text{T}} \left( t \right)e_{ik} \left( t \right)} } - \frac{{p_{1} }}{2}e^{{ - p_{1} \left( {t - \omega T} \right)}} \sum\limits_{k = 1}^{l} {\frac{c}{{h_{k} }}\left( {d_{k} \left( t \right) - d_{k}^{*} } \right)^{2} } \\ & \quad + \frac{q}{2}\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} \sum\limits_{k = 1}^{m} {\sum\limits_{{i \in C_{k} }} {e_{ik}^{\text{T}} \left( s \right)e_{ik} \left( s \right)} } } \right) \\ & \le - p_{1} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right). \\ \end{aligned}$$

In the same way, when $t \in \left[ {\left( {\omega + \theta } \right)T,\left( {\omega + 1} \right)T} \right)$, $\omega = 0,1,2, \ldots$, we obtain that

$$\dot{W}\left( t \right) \le p_{2} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right).$$

Namely,

$$\left\{ {\begin{array}{ll} {\dot{W}\left( t \right) \le - p_{1} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right),} & {\omega T \le t < \left( {\omega + \theta } \right)T,} \\ {\dot{W}\left( t \right) \le p_{2} W\left( t \right) + q\left( {\mathop {\sup }\limits_{t - \tau \le s \le t} W\left( s \right)} \right),} & {\left( {\omega + \theta } \right)T \le t < \left( {\omega + 1} \right)T.} \\ \end{array} } \right.$$

Then, by a similar proof of Theorem 1, we can prove that condition (15) means

$$\dot{W}\left( t \right) \le \left( {\mathop {\sup }\limits_{ - \tau \le s \le 0} W\left( s \right)} \right)e^{ - \varpi t} ,\quad t \ge 0.$$

Therefore, the global cluster lag synchronization of the delayed dynamical network (6) under the adaptive intermittent pinned controllers (16)–(17) is realizable. The proof of Theorem 3 is thus completed.