Preassigned-Time Projective Lag Synchronization of Octonion-Valued BAM Neural Networks via Exponential Quantized Event-Triggered Control

Abstract

This study addresses the preassigned-time (PDT) projective lag synchronization of octonion-valued BAM neural networks (OV-BAMNNs) through exponential quantized event-triggered control (ETC). First, an OV-BAMNN model incorporating discontinuous activation functions and time-varying delays is established. Subsequently, by introducing the octonion-valued sign function, several exponential quantized ETC schemes are designed, which employ solely a single exponential term while eliminating traditional linear and power-law components. Compared with conventional ETC designs, the proposed control schemes are simpler in form. Furthermore, within the framework of the non-separation method, several criteria for PDT projective lag synchronization are derived based on the Lyapunov method and Taylor expansion, proving that Zeno behavior is excluded. Finally, two simulation examples are given to verify the correctness of the theoretical results and to apply these results to image encryption.

1. Introduction

Over the past several years, high-dimensional neural networks (NNs), particularly complex-valued and quaternion-valued architectures [1,2], have demonstrated significant potential in domains such as image processing and computer vision [3]. However, their capacity for high-dimensional feature representation remains fundamentally constrained. As an advanced extension, octonion-valued neural networks (OVNNs) exhibit superior performance in information storage and high-dimensional data processing by virtue of their distinctive eight-dimensional algebraic structure [4]. Given the non-associative and non-commutative nature of octonion algebra, existing studies predominantly employ decomposition methods for analyzing OVNNs [5]. Nevertheless, such approaches typically lead to an eightfold expansion of the original system’s dimensionality while potentially compromising critical information integrity. Consequently, research on the dynamic characteristics of OVNNs based on non-separation approaches carries substantial significance and has attracted growing scholarly attention [6,7,8]. In neural network architectures, bidirectional associative memory NNs (BAMNNs) serve as a classical two-layer structure that demonstrates exceptional performance in associative memory tasks [9]. Although research on real-valued BAMNNs has reached considerable maturity, investigations into octonion-valued BAMNNs (OV-BAMNNs) remain relatively scarce [10,11], which has sparked our research interest.

Synchronization, as a characteristic dynamical behavior in NNs, has significant applications in bioengineering and image encryption [12,13,14]. The scaling factor $\rho$ enables projective synchronization between drive-response NNs, where $\rho =1$ and $\rho =-1$ correspond to complete synchronization and anti-synchronization, respectively. Given unavoidable transmission delays in practical systems, investigating projective lag synchronization is crucial [15]. Moreover, discontinuous activation functions are particularly effective for nonlinear problems with ultra-high slopes and show unique advantages in electronic switching circuits. Therefore, research on projective lag synchronization in OV-BAMNNs with discontinuous activation functions has both theoretical importance and practical value.

To further minimize convergence time and enhance the precision of synchronization settling time (ST) estimation, synchronization research has evolved through four distinct generations: commencing with asymptotic synchronization [16], advancing to finite-time synchronization [17], progressing to fixed-time (FDT) synchronization [18,19], and ultimately achieving preassigned-time (PDT) synchronization [20,21]. Compared with other synchronization types, PDT synchronization allows the convergence time to be predetermined according to requirements and is independent of any system parameters. In [20], Hu et al. established several innovative FDT stability inequalities by employing specific functions, then based on these inequalities and designed concise control strategies, sufficient conditions were derived for realizing both FDT and PDT synchronization in complex networks. In [21], by employing aperiodic intermittent pinning control, the FDT synchronization and PDT synchronization of quaternion-valued fuzzy BAMNNs were discussed. In [22], some general PDT stability lemmas are established to analyze the PDT synchronization of memristive complex-valued BAMNNs. By using event-triggered control (ETC), the practical PDT synchronization of complex networks with Markov switching topology under random DoS attacks was investigated in [23].

In the synchronization study of NNs, controllers play a pivotal role. To date, various control strategies have been successively developed, including quantized control [24,25], intermittent control [26,27], and ETC [28,29,30]. Notably, ETC executes updates only when predefined triggering conditions are met, thereby significantly reducing communication and computational overhead [31]. So far, numerous outstanding works have been reported on PDT synchronization of systems based on ETC [32,33,34,35,36,37]. For instance, in [32], by designing an ETC scheme, Zhang et al. achieved PDT synchronization for second-order NNs with distributed delays using the reduced-order approach. Based on PDT stability theory, the PDT anti-synchronization of unified chaotic systems was investigated by developing an ETC strategy in [33]. Additionally, the latest research progress on ETC is detailed in

Table 1. Research progress on PDT Synchronization under ETC.

Ref.	Number Field	ETC Strategy
[34]	$\mathbf{R}$	$-{\gamma }_1{\xi }_{1p}(t_{\sigma })-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}({\alpha }_{1p}{\xi }_{1p}^{{\displaystyle \frac{m_1}{m_2}}}(t_{\sigma })+{\beta }_{1p}{\xi }_{1p}^{{\displaystyle \frac{m_3}{m_4}}}(t_{\sigma }))$
[35]	$\mathbf{R}$	$-{\gamma }_{1}Q({\xi }_{1p}(t_{\sigma }))-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\left({\alpha }_{1p}sign(Q({\xi }_{1p}(t_{\sigma })))|Q({\xi }_{1p}(t_{\sigma })){|}^w+{\beta }_{1p}sign(Q({\xi }_{1p}(t_{\sigma }))\right)$
[36,37]	$\mathbf{Q}$	$-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}({\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))+{\beta }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma })){|{\xi }_{1p}(t_{\sigma })|}_1^{\widehat{w}})$
This paper	$\mathbf{O}$	$-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1^w}$

. It should be emphasized that existing findings primarily focus on real and quaternion domains. However, due to the non-commutative and non-associative algebraic structure of octonions, which entails more complex operational rules, conventional ETC schemes are not applicable to OVNNs. Moreover, signal quantization before transmission can effectively conserve bandwidth resources, while the integration of quantized control with ETC further enhances communication efficiency. Regrettably, no studies have yet discussed the PDT projective lag synchronization of OV-BAMNNs under quantized ETC, which has motivated our research.

Building upon the aforementioned discussions, this study investigates the PDT projective lag synchronization of OV-BAMNNs with discontinuous activation functions and time-varying delays via quantized ETC strategies. The main contributions are summarized as follows:

(1)

An OV-BAMNN model with discontinuous activation functions and time-varying delays is established. By adopting a non-separation approach that treats the original OVNNs as a unified entity instead of decomposing it into eight real-valued NNs [5,10,11], the PDT projective lag synchronization of the considered networks is discussed.

(2)

Compared with ETC schemes in [32,36] that contain at least two terms and multiple control parameters, the proposed control strategy consists solely of a single exponential term requiring only one power-law parameter. This demonstrates that the developed controller not only has a more compact structure but also offers greater convenience in engineering implementation.

(3)

By building two classes of ETC schemes based on the 2-norm and 1-norm, the synchronization problem of OV-BAMNNs is explored. Compared with studies considering only the 1-norm [7,18], the conclusions of this paper are more comprehensive and flexible. Furthermore, results such as FDT/PDT complete synchronization, FDT/PDT lag synchronization, or FDT/PDT anti-synchronization all be viewed as special cases of this study.

To further underscore the innovations of this work,

Table 2. Comparisons with the state-of-art works.

Ref.	Number Field	Convergence Rate	Synchronization Type	ETC	Quantized Control	Non-Separation Approach
[36]	$\mathbf{Q}$	FDT/PDT	Complete synchronization	×	×	√
[37]	$\mathbf{Q}$	FDT/PDT	Complete synchronization	√	×	√
[5]	$\mathbf{O}$	FDT	Complete synchronization	×	×	×
[6,18]	$\mathbf{O}$	FDT	Complete synchronization	×	×	√
[7,8]	$\mathbf{O}$	FDT/PDT	Complete synchronization	×	×	√
This paper	$\mathbf{O}$	FDT/PDT	Projective lag synchronization	√	√	√

provides a comparative analysis with existing works. For brevity, the quaternion and octonion fields are denoted by $\mathbf{Q}$ and $\mathbf{O}$, respectively; a checkmark √ indicates that a feature is included, whereas a cross × marks its omission.

The remainder of this paper is organized as follows. Section 2 introduces the OV-BAMNN model and provides the necessary prerequisite knowledge. Section 3 discusses the PDT projective lag synchronization of OV-BAMNNs under two exponential quantized ETC schemes. Numerical simulations are presented in Section 4 to validate the theoretical analysis. Section 5 applies the theoretical results to image encryption, and the work is concluded in Section 6.

Notations. Denote $\mathbf{Z}=\{0,1,2,\dots \}$, $\tilde{\mathbf{r}}=\{1,2,\dots ,n_1\}$ and $\tilde{\mathbf{p}}=\{1,2,\dots ,n_2\}$. Let $\mathbf{R}$, $\mathbf{O}$ and ${\mathbf{O}}^n$ represent the set of real constants, the set of octonions and the n-dimensional octonion-valued vectors, respectively.

2. Preliminaries and Model Description

2.1. Octonion Algebra

The set of octonions is defined as $\mathbf{O}=\{\vartheta ={\sum }_{j=0}^7{\vartheta }^{(j)}{\mathbf{u}}_{(j)}\mid {\vartheta }^0,{\vartheta }^1,{\vartheta }^2,{\vartheta }^3,{\vartheta }^4,{\vartheta }^5,{\vartheta }^7\in \mathbf{R}\}$, where $j=0,1,\dots ,7$, ${\mathbf{u}}_{(j)}$ is the unit element of the octonions and meets the following properties

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathbf{u}}_0=1,\hfill \\ {\mathbf{u}}_j{\mathbf{u}}_k=-{\mathbf{u}}_k{\mathbf{u}}_j,j\ne k,j,k=1,2,\dots ,7,\hfill \\ {\mathbf{u}}_j^2=-1,j=1,2,\dots ,7.\hfill \end{array}\right.\end{array}$$

For any $\vartheta ={\sum }_{j=0}^7{\vartheta }^{(j)}{\mathbf{u}}_j\in \mathbf{O}$, $\varsigma ={\sum }_{j=0}^7{\varsigma }^{(j)}{\mathbf{u}}_j\in \mathbf{O}$ and $F\in \mathbf{R}$, $F\vartheta ={\sum }_{j=0}^{7}F{\vartheta }^{(j)}{\mathbf{u}}_j$, $\vartheta +\varsigma ={\sum }_{j=0}^7({\vartheta }^{(j)}+{\varsigma }^{(j)}){\mathbf{u}}_j$, the conjugate of $\vartheta$ is denoted by $\overline{\vartheta }={\vartheta }^{(0)}{\mathbf{u}}_0-{\sum }_{j=1}^7{\vartheta }^{(j)}{\mathbf{u}}_j$. Moreover, the 1-norm and 2-norm of $\vartheta$ are denoted by ${|\vartheta |}_1={\sum }_{j=0}^7|{\vartheta }^{(j)}|$ and ${|\vartheta |}_2=\sqrt{\overline{\vartheta }\vartheta }=\sqrt{{\sum }_{j=0}^7{({\vartheta }^{(j)})}^2}$. For any $\vartheta (t)={\sum }_{j=0}^7{\vartheta }^{(j)}(t){\mathbf{u}}_j:\mathbf{R}\to \mathbf{O}$, where ${\vartheta }^{(j)}(t):\mathbf{R}\to \mathbf{R}$, one has ${\displaystyle \frac{\mathrm{d}(\vartheta (t))}{\mathrm{d}t}}={\sum }_{j=0}^7{\displaystyle \frac{\mathrm{d}({\vartheta }^{(j)}(t))}{\mathrm{d}t}}{\mathbf{u}}_j$.

Definition 1

([8]). For any $\vartheta ={\sum }_{j=0}^7{\vartheta }^{(j)}{\mathbf{u}}_j\in \mathbf{O}$, the sign function of ϑ is given by

$$\mathbf{Sign}(\vartheta )\triangleq \mathrm{sign}({\vartheta }^{(0)}){\mathbf{u}}_0+\mathrm{sign}({\vartheta }^{(1)}){\mathbf{u}}_1+\dots +\mathrm{sign}({\vartheta }^{(7)}){\mathbf{u}}_7.$$

Moreover, the convex hull of $\mathbf{Sign}(\vartheta )$ is characterized by

$$\overline{\mathrm{co}}(\mathbf{Sign}(\vartheta ))\triangleq \overline{\mathrm{co}}(\mathrm{sign}({\vartheta }^{(0)})){\mathbf{u}}_0+\overline{\mathrm{co}}(\mathrm{sign}({\vartheta }^{(1)})){\mathbf{u}}_1\dots +\overline{\mathrm{co}}(\mathrm{sign}({\vartheta }^{(7)})){\mathbf{u}}_7,$$

and

$$\overline{\mathrm{co}}(\mathrm{sign}({\vartheta }^{(j)}))=\left\{\begin{array}{cc}1,\hfill & \mathrm{sign}({\vartheta }^{(j)})>0,j=0,1,\dots ,7,\hfill \\ [-1,1],\hfill & \mathrm{sign}({\vartheta }^{(j)})=0,j=0,1,\dots ,7,\hfill \\ -1,\hfill & \mathrm{sign}({\vartheta }^{(j)})<0,j=0,1,\dots ,7.\hfill \end{array}\right.$$

Lemma 1

([8]). For any ϑ, $\varsigma \in \mathbf{O}$, then

$$(1)\overline{\vartheta }\varsigma +\overline{\varsigma }\vartheta \le {2|\vartheta |}_2{|\varsigma |}_2;$$

$$(2)\vartheta +\overline{\vartheta }=2{\vartheta }^{(0)}\le {2|\vartheta |}_2\le 2{|\vartheta |}_1;$$

$${(3)|\vartheta \varsigma |}_1\le {|\vartheta |}_1{|\varsigma |}_1{;|\vartheta \varsigma |}_2={|\vartheta |}_2{|\varsigma |}_2.$$

Lemma 2

([8]). For any ${\mathcal{A}}_1(t)$, ${\mathcal{A}}_2(t):\mathbf{R}\to \mathbf{O}$ and any measurable selection $\varpi (t)\in \overline{\mathrm{co}}(\mathbf{Sign}({\mathcal{A}}_1(t)))$, then

$$(1)\overline{\mathbf{Sign}({\mathcal{A}}_1(t))}{\mathcal{A}}_2(t)+\overline{{\mathcal{A}}_2(t)}\mathbf{Sign}({\mathcal{A}}_1(t))\le 2{|{\mathcal{A}}_2(t)|}_1;$$

$$(2)\overline{\mathbf{Sign}({\mathcal{A}}_1(t))}\varpi (t)+\overline{\varpi (t)}\mathbf{Sign}({\mathcal{A}}_1(t))=2{|\mathbf{Sign}({\mathcal{A}}_1(t))|}_1;$$

$$(3)\overline{\mathbf{Sign}({\mathcal{A}}_1(t))}{\mathcal{A}}_1(t)+\overline{{\mathcal{A}}_1(t)}\mathbf{Sign}({\mathcal{A}}_1(t))=2|{\mathcal{A}}_1{(t)|}_1\ge 2{|{\mathcal{A}}_1(t)|}_2;$$

$$(4)D^{+}{|{\mathcal{A}}_1(t)|}_1={\displaystyle \frac{1}{2}}\left(\overline{\mathbf{Sign}({\mathcal{A}}_1(t))}{D}^{+}{\mathcal{A}}_1(t)+D^{+}\overline{{\mathcal{A}}_1(t)}\mathbf{Sign}({\mathcal{A}}_1(t))\right).$$

where $D^{+}$ denotes the Dini derivative.

Lemma 3.

For $\vartheta (t)\in \mathbf{O}$, then $D^{+}{|\vartheta (t)|}_1\le {|\dot{\vartheta }(t)|}_1$.

Proof. 

According to Lemma 2,

$$\begin{array}{cc}\hfill D^{+}{|\vartheta (t)|}_1=& {\displaystyle \frac{1}{2}}\left(\overline{\mathbf{Sign}(\vartheta (t))}{D}^{+}{\mathcal{A}}_1(t)+D^{+}\overline{\vartheta (t)}\mathbf{Sign}(\vartheta (t))\right)\hfill \\ \hfill =& \sum _{j=0}^7\mathrm{sign}({\vartheta }^{(j)}(t)){\dot{\vartheta }}^{(j)}(t)\le \sum _{j=0}^7|{\dot{\vartheta }}^{(j)}(t)|.\hfill \end{array}$$

In addition, by the definition of 1-norm, $|\dot{\vartheta }{(t)|}_1={\sum }_{j=0}^7|{\dot{\vartheta }}^{(j)}(t)|$. Hence, $D^{+}{|\vartheta (t)|}_1\le {|\dot{\vartheta }(t)|}_1$. □

2.2. Model Description

Consider the following discontinuous OV-BAMNN model with time-varying delays:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}X-layer:{\dot{z}}_{1p}(t)=\hfill & -d_{1p}{z}_{1p}(t)+{\displaystyle \sum _{r=1}^{n_1}}{\omega }_{pr}{h}_r(z_{2r}(t))+W_{1p}\hfill \\ \hfill & +{\displaystyle \sum _{r=1}^{n_1}}{a}_{pr}{h}_r(z_{2r}(t-{\Im }_{2r}(t))),p\in \tilde{\mathbf{p}},\hfill \\ Y-layer:{\dot{z}}_{2r}(t)=\hfill & -d_{2r}{z}_{2r}(t)+{\displaystyle \sum _{p=1}^{n_2}}{e}_{rp}{m}_p(z_{1p}(t))+W_{2r}\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^{n_2}}{l}_{rp}{m}_p(z_{1p}(t-{\Im }_{1p}(t))),r\in \tilde{\mathbf{r}},\hfill \end{array}\right.\end{array}$$

(1)

where $z_{1p}(t)$, $z_{2r}(t)\in \mathbf{O}$ are the states of the pth neuron in $X-layer$ and rth neurons in $Y-layer$, respectively, $d_{1p},d_{2r}>0$, ${\omega }_{pr},a_{pr},e_{rp},l_{rp}\in \mathbf{O}$ are the connections weights, $h_r(·)$ and $m_p(·):\mathbf{O}\to \mathbf{O}$ are the discontinuous activation functions, ${\Im }_{2r}(t)$ and ${\Im }_{1p}(t)$ denote the time-varying delays and $0<{\Im }_{2r}(t)\le {\Im }_{2r}$, $0<{\Im }_{1p}(t)\le {\Im }_{1p}$. $W_{1p}$ and $W_{2r}\in \mathbf{O}$ are the external inputs. The initial conditions of system (1) are $z_{1p}(𝚤)={\wp }_{1p}(𝚤)$ and $z_{2r}(𝚤)={\wp }_{2r}(𝚤)$ for $𝚤\in [-\Im ,0]$, where $\Im =max\{{max}_{p\in \tilde{\mathbf{p}}}\left\{{\Im }_{1p}\right\},{max}_{r\in \tilde{\mathbf{r}}}\left\{{\Im }_{2r}\right\}\}$.

System (1) is referred to as the driving system, and its corresponding response system is depicted as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}X-layer:{\dot{\tilde{z}}}_{1p}(t)=\hfill & -d_{1p}{\tilde{z}}_{1p}(t)+{\displaystyle \sum _{r=1}^{n_1}}{\omega }_{pr}{h}_r({\tilde{z}}_{2r}(t))+W_{1p}\hfill \\ \hfill & +{\displaystyle \sum _{r=1}^{n_1}}{a}_{pr}{h}_r({\tilde{z}}_{2r}(t-{\Im }_{2r}(t)))+U_{1p}(t),p\in \tilde{\mathbf{p}},\hfill \\ Y-layer:{\dot{\tilde{z}}}_{2r}(t)=\hfill & -d_{2r}{\tilde{z}}_{2r}(t)+{\displaystyle \sum _{p=1}^{n_2}}{e}_{rp}{m}_p({\tilde{z}}_{1p}(t))+W_{2r}\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^{n_2}}{l}_{rp}{m}_p({\tilde{z}}_{1p}(t-{\Im }_{1p}(t)))+U_{2r}(t),r\in \tilde{\mathbf{r}},\hfill \end{array}\right.\end{array}$$

(2)

where ${\tilde{z}}_{1p}(t)$, ${\tilde{z}}_{2r}(t)\in \mathbf{O}$ are the neuron states in the response system, $U_{1p}(t)$ and $U_{2r}(t)$ are the controllers, and the remaining parameters are consistent with system (1). The initial conditions of system (2) are ${\tilde{z}}_{1p}(𝚤)={\tilde{\wp }}_{1p}(𝚤)$ and ${\tilde{z}}_{2r}(𝚤)={\tilde{\wp }}_{2r}(𝚤)$ for $𝚤\in [-\Im ,0]$.

Assumption 1.

For any $p\in \tilde{\mathbf{p}}$ and $r\in \tilde{\mathbf{r}}$, $h_r(·)$ and $m_p(·)\in \mathbf{O}$ are continuous except at the countable isolated point sets $\left\{{\beth }_q^r\right\}$ and $\left\{{\tilde{\beth }}_q^p\right\}$, where the left and right limits of $h_r^{(j)}({\beth }_q^r)$ and $m_p^{(j)}({\tilde{\beth }}_q^p)$ exist. Furthermore, on every bounded compact subset of $\mathbf{O}$, $h_r(·)$ and $m_p(·)$ have at most finitely many jump discontinuities.

Assumption  2.

For any $p\in \tilde{\mathbf{p}}$, $r\in \tilde{\mathbf{r}}$ and $𝚥=1,2$, there exist positive constants ${\daleth }_{r𝚥}$, $F_{p𝚥}$ and nonnegative constants ${\tilde{\daleth }}_{r𝚥}$ and ${\tilde{F}}_{p𝚥}$ such that

$$\begin{array}{c}\hfill |{\phi }_r(t)-{\psi }_r{(t)|}_{𝚥}\le {\daleth }_{r𝚥}{|{\tilde{z}}_{2r}(t)-z_{2r}(t)|}_{𝚥}+{\tilde{\daleth }}_{r𝚥},\\ \hfill |{\tilde{\phi }}_p(t)-{\tilde{\psi }}_p{(t)|}_j\le F_{p𝚥}{|{\tilde{z}}_{1p}(t)-z_{1p}(t)|}_{𝚥}+{\tilde{F}}_{p𝚥},\end{array}$$

hold for ${\tilde{z}}_{2r}(t),z_{2r}(t),{\tilde{z}}_{1p}(t),z_{1p}(t)\in \mathbf{O}$, where ${\phi }_r(t)\in \overline{\mathrm{co}}\left[h_r({\tilde{z}}_{2r}(t))\right]$, ${\psi }_r(t)\in \overline{\mathrm{co}}\left[h_r(z_{2r}(t))\right]$, ${\tilde{\phi }}_p(t)\in \overline{\mathrm{co}}\left[m_p({\tilde{z}}_{1p}(t))\right]$, ${\tilde{\psi }}_p(t)\in \overline{\mathrm{co}}\left[m_p(z_{1p}(t))\right]$.

Remark 1.

The constants in Assumption 2 possess definite physical interpretations and are verifiable in practice. Specifically,

(1)

The positive constants ${\daleth }_{r𝚥}$ and $F_{p𝚥}$ bound the rate of change of the activation functions, analogous to standard Lipschitz constants. They can be determined from parameters of physical components such as transistor transconductance, linear-region amplifier gain, or operational amplifier slew rates.

(2)

The nonnegative constants ${\tilde{\daleth }}_{r𝚥}$ and ${\tilde{F}}_{p𝚥}$ quantify the effects of discontinuities, representing the maximum output deviations at discontinuity points, which are independent of input variations. In electronic implementations, these correspond to measurable characteristics including voltage gaps in switching circuits, hysteresis widths in magnetic components, and quantization errors in digital-analog interfaces. The discontinuous activation functions degrade into the continuous scenario when ${\tilde{\daleth }}_{r𝚥}={\tilde{F}}_{p𝚥}=0$.

Assumption 3.

For any $p\in \tilde{\mathbf{p}}$, $r\in \tilde{\mathbf{r}}$ and $𝚥=1,2$, there are positive constants ${\widehat{\daleth }}_{r𝚥}$, ${\widehat{F}}_{p𝚥}$ such that $|h_r{(·)|}_{𝚥}\le {\widehat{\daleth }}_{r𝚥}$ and $|m_p{(·)|}_{𝚥}\le {\widehat{F}}_{p𝚥}$.

For convenience, the following symbols are adopted.

$$\Theta (t)={(z_{11}(t),\dots ,z_{1n_2}(t),z_{21}(t),\dots ,z_{2n_1}(t))}^T,$$

$$\tilde{\Theta }(t)={({\tilde{z}}_{11}(t),\dots ,{\tilde{z}}_{1n_2}(t),{\tilde{z}}_{21}(t),\dots ,{\tilde{z}}_{2n_1}(t))}^T,$$

$$\widehat{\Theta }(𝚤)={(z_{11}(𝚤),\dots ,z_{1n_2}(𝚤),z_{21}(𝚤),\dots ,z_{2n_1}(𝚤))}^T,$$

$$\check{\Theta }(𝚤)={({\tilde{z}}_{11}(𝚤),\dots ,{\tilde{z}}_{1n_2}(𝚤),{\tilde{z}}_{21}(𝚤),\dots ,{\tilde{z}}_{2n_1}(𝚤))}^T.$$

Definition 2.

A continuous function vector $\Theta (t):[-\Im ,T_0)\to {\mathbf{O}}^n$ is called a solution of system (1) on $[-\Im ,T_0)$ if

(1)

$\Theta (t)$ is absolutely continuous on $[0,T_0)$;

(2)

There exist measurable choices ${\psi }_r(t)\in \overline{\mathrm{co}}\left[h_r(z_{2r}(t))\right]$ and ${\tilde{\psi }}_p(t)\in \overline{\mathrm{co}}\left[m_p(z_{1p}(t))\right]$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{z}}_{1p}(t)=\hfill & -d_{1p}{z}_{1p}(t)+{\displaystyle \sum _{r=1}^{n_1}}{\omega }_{pr}{\psi }_r(t)+W_{1p}\hfill \\ \hfill & +{\displaystyle \sum _{r=1}^{n_1}}{a}_{pr}{\psi }_r(t-{\Im }_{2r}(t)),p\in \tilde{\mathbf{p}},\hfill \\ {\dot{z}}_{2r}(t)=\hfill & -d_{2r}{z}_{2r}(t)+{\displaystyle {\displaystyle \sum _{p=1}^{n_2}}}{e}_{rp}{\tilde{\psi }}_p(t)+W_{2r}\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^{n_2}}{l}_{rp}{\tilde{\psi }}_p(t-{\Im }_{1p}(t)),r\in \tilde{\mathbf{r}},\hfill \end{array}\right.\end{array}$$

(3)

for almost everywhere $t\in [0,T_0)$.

Analogously, the response system (2) is governed by measurable functions ${\phi }_r(t)\in \overline{\mathrm{co}}\left[h_r({\tilde{z}}_{2r}(t))\right]$ and ${\tilde{\phi }}_p(t)\in \overline{\mathrm{co}}\left[m_p({\tilde{z}}_{1p}(t))\right]$, such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{\tilde{z}}}_{1p}(t)=\hfill & -d_{1p}{\tilde{z}}_{1p}(t)+{\displaystyle \sum _{r=1}^{n_1}}{\omega }_{pr}{\phi }_r(t)+W_{1p}\hfill \\ \hfill & +{\displaystyle \sum _{r=1}^{n_1}}{a}_{pr}{\phi }_r(t-{\Im }_{2r}(t))+U_{1p}(t),p\in \tilde{\mathbf{p}},\hfill \\ {\dot{\tilde{z}}}_{2r}(t)=\hfill & -d_{2r}{\tilde{z}}_{2r}(t)+{\displaystyle \sum _{p=1}^{n_2}}{e}_{rp}{\tilde{\phi }}_p(t)+W_{2r}\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^{n_2}}{l}_{rp}{\tilde{\phi }}_p(t-{\Im }_{1p}(t))+U_{2r}(t),r\in \tilde{\mathbf{r}},\hfill \end{array}\right.\end{array}$$

(4)

for almost everywhere $t\in [0,T_0)$.

Let ${\xi }_{1p}(t)={\tilde{z}}_{1p}(t)-\rho z_{1p}(t-\epsilon )$ and ${\xi }_{2r}(t)={\tilde{z}}_{2r}(t)-\rho z_{2r}(t-\epsilon )$ be the projective lag synchronization error, by subtracting (3) from (4), the error system is derived as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{\xi }}_{1p}(t)=\hfill & -d_{1p}{\xi }_{1p}(t)+{\displaystyle \sum _{r=1}^{n_1}}{\omega }_{pr}{G}_r({\xi }_{2r}(t))+{\displaystyle \sum _{r=1}^{n_1}}{a}_{pr}{F}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))+U_{1p}(t)\hfill \\ \hfill & +{\displaystyle \sum _{r=1}^{n_1}}{\omega }_{pr}{L}_r({\xi }_{2r}(t))+{\displaystyle \sum _{r=1}^{n_1}}{a}_{pr}{S}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))+(1-\rho )W_{1p},p\in \tilde{\mathbf{p}},\hfill \\ {\dot{\xi }}_{2r}(t)=\hfill & -d_{2r}{\xi }_{2r}(t)+{\displaystyle \sum _{p=1}^{n_2}}{e}_{rp}{\tilde{G}}_p({\xi }_{1p}(t))+{\displaystyle \sum _{p=1}^{n_2}}{l}_{rp}{\tilde{F}}_p({\xi }_{1p}(t-{\Im }_{1p}(t)))+U_{2r}(t)\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^{n_2}}{e}_{rp}{\tilde{L}}_p({\xi }_{1p}(t))+{\displaystyle \sum _{p=1}^{n_2}}{l}_{rp}{\tilde{S}}_p({\xi }_{1p}(t-{\Im }_{1p}(t)))+(1-\rho )W_{2r},r\in \tilde{\mathbf{r}}.\hfill \end{array}\right.\end{array}$$

(5)

where $\rho \in \mathbf{R}$, $\epsilon >0$, ${\eta }_r(t-\epsilon )\in \overline{\mathrm{co}}\left[h_r(\rho z_{2r}(t-\epsilon ))\right]$, ${\tilde{\eta }}_p(t-\epsilon )\in \overline{\mathrm{co}}\left[m_p(\rho z_{1p}(t-\epsilon ))\right]$, $G_r({\xi }_{2r}(t))={\phi }_r(t)-{\eta }_r(t-\epsilon )$, $F_r({\xi }_{2r}(t-{\Im }_{2r}(t)))={\phi }_r(t-{\Im }_{2r}(t))-{\eta }_r(t-{\Im }_{2r}(t)-\epsilon )$, $L_r({\xi }_{2r}(t))={\eta }_r(t-\epsilon )-\rho {\psi }_r(t-\epsilon )$, $S_r({\xi }_{2r}(t-{\Im }_{2r}(t)))={\eta }_r(t-{\Im }_{2r}(t)-\epsilon )-\rho {\psi }_r(t-{\Im }_{2r}(t)-\epsilon )$, ${\tilde{G}}_p({\xi }_{1p}(t))={\tilde{\phi }}_p(t)-{\tilde{\eta }}_p(t-\epsilon )$, ${\tilde{F}}_p({\xi }_{1p}(t-{\Im }_{2r}(t)))={\tilde{\phi }}_p(t-{\Im }_{1p}(t))-{\tilde{\eta }}_p(t-{\Im }_{1p}(t)-\epsilon )$, ${\tilde{L}}_p({\xi }_{1p}(t))={\tilde{\eta }}_p(t-\epsilon )-\rho {\tilde{\psi }}_p(t-\epsilon )$, ${\tilde{S}}_p({\xi }_{1p}(t-{\Im }_{1p}(t)))={\tilde{\eta }}_p(t-{\Im }_{1p}(t)-\epsilon )-\rho {\tilde{\psi }}_p(t-{\Im }_{1p}(t)-\epsilon )$.

Definition 3.

Systems (1) and (2) can attain FDT projective lag synchronization, if for any solutions of (1) and (2) denoted by $\Theta (t)$ and $\tilde{\Theta }(t)$ with different initial values $\widehat{\Theta }(𝚤)$ and $\check{\Theta }(𝚤)$, there are a constant $\breve{\mathcal{T}}>0$ which is not related to the initial conditions and a constant $\tilde{\mathcal{T}}(\widehat{\Theta }(𝚤),\check{\Theta }(𝚤))>0$ called the ST, such that

$$\begin{array}{c}\hfill \underset{t\to \tilde{\mathcal{T}}(\widehat{\Theta }(𝚤),\check{\Theta }(𝚤))}{lim}\parallel \tilde{\Theta }{(t)-\rho \Theta (t-\epsilon )\parallel }_{𝚥}=0,{\parallel \tilde{\Theta }(t)-\rho \Theta (t-\epsilon )\parallel }_{𝚥}\equiv 0fort\ge \tilde{\mathcal{T}}(\widehat{\Theta }(𝚤),\check{\Theta }(𝚤)),\end{array}$$

and $\tilde{\mathcal{T}}(\widehat{\Theta }(𝚤),\check{\Theta }(𝚤))\le \breve{\mathcal{T}}$, for any $(\widehat{\Theta }(𝚤),\check{\Theta }(𝚤))\in {\mathbf{O}}^{2(n_1+n_2)}$ and $𝚥=1,2$. Furthermore, systems (1) and (2) can attain PDT projective lag synchronization, if there is a pre-determined positive value ${\mathcal{T}}_{pre}>0$ such that $\parallel \tilde{\Theta }{(t)-\rho \Theta (t-\epsilon )\parallel }_{𝚥}=0$ for all $t\ge {\mathcal{T}}_{pre}$, where ${\mathcal{T}}_{pre}$ is independent to any parameters.

Lemma 4

([21]). Let ${\tilde{\zeta }}_1,{\tilde{\zeta }}_2,\dots ,{\tilde{\zeta }}_m\ge 0$, $0<{\tau }_1\le 1$ and ${\tau }_2>1$, one has

$$\sum _{p=1}^m{\tilde{\zeta }}_p^{{\tau }_1}\ge {(\sum _{p=1}^m{\tilde{\zeta }}_p)}^{{\tau }_1},\sum _{p=1}^m{\tilde{\zeta }}_p^{{\tau }_2}\ge m^{1-{\tau }_2}{(\sum _{p=1}^m{\tilde{\zeta }}_p)}^{{\tau }_2}.$$

Lemma 5

([20]). Suppose that $V(\varpi ):{\mathbf{R}}^n\to {\mathbf{R}}^{+}$ is a regular, positive definite and radially unbounded function Lyapunov function, there exist ${\lambda }_1\in \mathbf{R}$, ${\lambda }_2,{\lambda }_3>0$, $0\le {\zeta }_1<1$, ${\zeta }_2>1$ and ${\mathcal{T}}_{pre}>0$ satisfying

$$D^{+}V(\varpi (t))\le \left\{\begin{array}{c}-{\displaystyle \frac{\breve{\mathcal{T}}}{{\mathcal{T}}_{pre}}}\left({\lambda }_2{V}^{{\zeta }_1}(\varpi (t))+{\lambda }_3{V}^{{\zeta }_2}(\varpi (t))\right),{\lambda }_1\le 0,\hfill \\ {\displaystyle \frac{\breve{\mathcal{T}}}{{\mathcal{T}}_{pre}}}\left({\lambda }_{1}V(\varpi (t))-{\lambda }_2{V}^{{\zeta }_1}(\varpi (t))-{\lambda }_3{V}^{{\zeta }_2}(\varpi (t))\right),0<{\lambda }_1<min\{{\lambda }_2,{\lambda }_3\},\hfill \end{array}\right.$$

the zero solution of the QV-BAMNNs (5) is PDT stable in preassigned time ${\mathcal{T}}_{pre}$, where

$$\breve{\mathcal{T}}=\left\{\begin{array}{cc}{\breve{\mathcal{T}}}^{*}=\hfill & {\displaystyle \frac{\pi }{{\lambda }_3({\zeta }_2-{\zeta }_1)}}{({\displaystyle \frac{{\lambda }_3}{{\lambda }_2}})}^{\varrho }csc(\varrho \pi ),{\lambda }_1\le 0,\hfill \\ {\breve{\mathcal{T}}}^{**}=\hfill & {\displaystyle \frac{\pi csc(\varrho \pi )}{{\lambda }_3({\zeta }_2-{\zeta }_1)}}{({\displaystyle \frac{{\lambda }_3}{{\lambda }_2-{\lambda }_1}})}^{1-\varrho }\mathbf{I}({\displaystyle \frac{{\lambda }_3}{{\lambda }_2+{\lambda }_3-{\lambda }_1}},\varrho ,1-\varrho )\hfill \\ \hfill & +{\displaystyle \frac{\pi csc(\varrho \pi )}{{\lambda }_2({\zeta }_2-{\zeta }_1)}}{({\displaystyle \frac{{\lambda }_2}{{\lambda }_3-{\lambda }_1}})}^{\varrho }\mathbf{I}({\displaystyle \frac{{\lambda }_2}{{\lambda }_2+{\lambda }_3-{\lambda }_1}},1-\varrho ,\varrho ),0<{\lambda }_1<min\{{\lambda }_2,{\lambda }_3\},\hfill \end{array}\right.$$

and $\varrho ={\displaystyle \frac{1-{\zeta }_1}{{\zeta }_2-{\zeta }_1}}$, $\mathbf{I}(·,·,·)$ denotes the incomplete beta function ratio given in [20].

Lemma 6.

For any $♭\in [0,+\infty )$, one has

$$\begin{array}{c}\hfill e^{♭}\ge 1+♭+{\displaystyle \frac{{♭}^2}{2}}.\end{array}$$

3. Preassigned-Time Projective Lag Synchronization

In order to facilitate the description, the relevant parameters are given as follows.

$$\begin{array}{cc}\hfill & d_1=\underset{p\in \tilde{\mathbf{p}}}{max}\{-d_{1p}+\sum _{r=1}^{n_1}|e_{rp}{|}_1{F}_{p1}\},d_2=\underset{r\in \tilde{\mathbf{r}}}{max}\{-d_{2r}+\sum _{p=1}^{n_2}|{\omega }_{pr}{|}_1{\daleth }_{r1}\},\hfill \\ \hfill & d=max\{d_1,d_2\},\delta =\underset{p\in \tilde{\mathbf{p}},r\in \tilde{\mathbf{r}}}{max}\{{\delta }_{1p},{\delta }_{2r}\},k_1=max\{d,\delta \},\hfill \\ \hfill & {\acute{k}}_2=\underset{p\in \tilde{\mathbf{p}}}{min}\{{\alpha }_{1p}{\beta }_{1p}{(1-ϵ)}^w,{\gamma }_{1p}\},{\stackrel{‘}{k}}_2=\underset{r\in \tilde{\mathbf{l}}}{min}\{{\alpha }_{2r}{\beta }_{2r}{(1-ϵ)}^w,{\gamma }_{2r}\},\hfill \\ \hfill & {\acute{k}}_3=\underset{p\in \tilde{\mathbf{p}}}{min}\{{\displaystyle \frac{1}{2}}{\alpha }_{1p}{\beta }_{1p}^2{(1-ϵ)}^{2w},{\mu }_{1p}\},{\stackrel{‘}{k}}_3=\underset{r\in \tilde{\mathbf{l}}}{min}\{{\displaystyle \frac{1}{2}}{\alpha }_{2r}{\beta }_{2r}^2{(1-ϵ)}^{2w},{\mu }_{2r}\},\hfill \\ \hfill & {\upsilon }_{1p}=\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_1({\tilde{\daleth }}_{r1}+{\widehat{\daleth }}_{r1}(1+|\rho |))+\sum _{r=1}^{n_1}|{\alpha }_{pr}{|}_1{\widehat{\daleth }}_{r1}(3+|\rho |)+(1+|\rho |)|W_{1p}{|}_1,\hfill \\ \hfill & {\upsilon }_{2r}=\sum _{p=1}^{n_2}|e_{rp}{|}_1({\tilde{F}}_{p1}+{\widehat{F}}_{p1}(1+|\rho |))+\sum _{p=1}^{n_2}|l_{rp}{|}_1{\widehat{F}}_{p1}(3+|\rho |)+(1+|\rho |)|W_{2r}{|}_1,\hfill \\ \hfill & k_2=min\{{\acute{k}}_2,{\stackrel{‘}{k}}_2\},k_3=min\{n_2^{1-2w}{\acute{k}}_3,n_1^{1-2w}{\stackrel{‘}{k}}_3\},{\Phi }_1=n_2\Phi +n_1\Phi .\hfill \end{array}$$

Firstly, the exponential quantized ETC scheme based on 1-norm for $t\in [t_{\sigma },t_{\sigma +1})$ can be designed as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{U}_{1p}(t)=\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1^w},\hfill \\ U_{2r}(t)=\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma }))e^{{\beta }_{2r}{|Q({\xi }_{2r}(t_{\sigma }))|}_1^w},\hfill \end{array}\right.\end{array}$$

(6)

where ${\alpha }_{1p},{\beta }_{1p},{\alpha }_{2r},{\beta }_{2r}>0$, ${\displaystyle \frac{1}{2}}<w<1$, $p\in \tilde{\mathbf{p}}$, $r\in \tilde{\mathbf{r}}$, ${\left\{t_{\sigma }\right\}}_{\sigma \in \mathbf{Z}}$ denotes an increasing sequence of release instants and $t_0=0$, ${\mathcal{T}}_{pre}>0$ and

$$\begin{array}{cc}\hfill {\mathcal{T}}_{max}={\mathcal{T}}_{max}^{**}=& {\displaystyle \frac{\pi csc(\hslash \pi )}{k_3{4}^{1-2w}w}}{({\displaystyle \frac{k_3{4}^{1-2w}}{k_2-k_1}})}^{1-\hslash }\mathbf{I}({\displaystyle \frac{k_3{4}^{1-2w}}{k_2+k_3{4}^{1-2w}-k_1}},\hslash ,1-\hslash )\hfill \\ \hfill & +{\displaystyle \frac{\pi csc(\hslash \pi )}{k_{2}w}}{({\displaystyle \frac{k_2}{k_3{4}^{1-2w}-k_1}})}^{\hslash }\mathbf{I}({\displaystyle \frac{k_2}{k_2+k_3{4}^{1-2w}-k_1}},1-\hslash ,\hslash ),\hfill \end{array}$$

and $\hslash ={\displaystyle \frac{1-w}{w}}$. Besides, $Q({\xi }_{1p}(t))={\sum }_{j=0}^7\widehat{Q}({\xi }_{1p}^{(j)}(t))$, $Q({\xi }_{2r}(t))={\sum }_{j=0}^7\widehat{Q}({\xi }_{2r}^{(j)}(t))$, and the quantization function $\widehat{Q}(\diamond ):\mathbf{R}\to \mho$ ($\diamond \in \mathbf{R}$) is defined as

$$\widehat{Q}(\diamond )=\left\{\begin{array}{cc}{\kappa }_{\Lambda },\hfill & {\displaystyle \frac{{\kappa }_{\Lambda }}{1+ϵ}}<\diamond \le {\displaystyle \frac{{\kappa }_{\Lambda }}{1-ϵ}},\hfill \\ 0,\hfill & \diamond =0,\hfill \\ -\widehat{Q}(-\diamond ),\hfill & \diamond <0,\hfill \end{array}\right.$$

where $\mho =\{\pm {\kappa }_{\Lambda }={\nu }^{\Lambda }{\kappa }_0:0<\nu <1,\Lambda =\pm 1,\pm 2,\dots \}\cup \left\{0\right\}$ with ${\Lambda }_0>0$, $\diamond \in \mathbf{R}$ and $ϵ={\displaystyle \frac{1-\nu }{1+\nu }}$. According to the analysis in [25], there is a Filippov solution $\Omega \in [-ϵ,ϵ)$ which satisfies $Q(\diamond )=(1+\Omega )\diamond$.

The triggering condition is exploited as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{t}_{\sigma +1}=inf\{t>t_{\sigma }|B_{1p}(t)\wedge B_{2r}(t)\ge 0\},\hfill \\ B_{1p}(t)={\upsilon }_{1p}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}+{|{\mathcal{E}}_{1p}(t)|}_1-{\delta }_{1p}{\Gamma }_{1p}(t),\hfill \\ B_{2r}(t)={\upsilon }_{2r}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}+{|{\mathcal{E}}_{2r}(t)|}_1-{\delta }_{2r}{\Gamma }_{2r}(t),\hfill \end{array}\right.\end{array}$$

(7)

where ${\delta }_{1p},{\delta }_{2r}>0$, ${\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}>{\upsilon }_{1p}$, ${\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}>{\upsilon }_{2r}$, ${\mathcal{E}}_{1p}(t)$ and ${\mathcal{E}}_{2r}(t)$ are measurable errors, which are described as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{E}}_{1p}(t)={\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1^w}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t))|}_1^w},\hfill \\ {\mathcal{E}}_{2r}(t)={\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma }))e^{{\beta }_{2r}{|Q({\xi }_{2r}(t_{\sigma }))|}_1^w}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}\mathbf{Sign}({\xi }_{2r}(t))e^{{\beta }_{2r}{|Q({\xi }_{2r}(t))|}_1^w},\hfill \end{array}\right.\end{array}$$

(8)

for $t\in [t_{\sigma },t_{\sigma +1}]$. Furthermore, dynamic variables ${\Gamma }_{1p}(t)$ and ${\Gamma }_{2r}(t)$ are designed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\Gamma }}_{1p}(t)=-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\gamma }_{1p}{\Gamma }_{1p}^w(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\mu }_{1p}{\Gamma }_{1p}^{2w}(t),\hfill \\ {\dot{\Gamma }}_{2r}(t)=-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\gamma }_{2r}{\Gamma }_{2r}^w(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\mu }_{2r}{\Gamma }_{2r}^{2w}(t),\hfill \end{array}\right.\end{array}$$

(9)

where ${\Gamma }_{1p}(0),{\Gamma }_{2r}(0)>0$, ${\gamma }_{1p},{\mu }_{1p},{\gamma }_{2r}$, ${\mu }_{2r}>0$ and the remaining undefined parameters are consistent with those in (6).

Remark 2.

The preassigned time ${\mathcal{T}}_{pre}$ is a user-specified parameter that sets the desired synchronization time. Typically, the selected ${\mathcal{T}}_{pre}$ should satisfy ${\mathcal{T}}_{pre}\le {\mathcal{T}}_{max}$. However, choosing an excessively short ${\mathcal{T}}_{pre}$ will raise the control gain through the term ${\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}$ in the controller (6), increasing control energy consumption and implementation costs. This may result in violations of practical constraints, such as actuator saturation or energy limitations. Hence, the selection of ${\mathcal{T}}_{pre}$ should strike a balance between synchronization speed and feasible control inputs, ensuring that ${\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}$ remains within acceptable system limits.

Lemma 7.

For $t\ge 0$, $p\in \tilde{\mathbf{p}}$, $r\in \tilde{\mathbf{r}}$, ${\Gamma }_{1p}(t)\ge 0$, ${\Gamma }_{2r}(t)\ge 0$.

Proof. 

Firstly, it is proved that ${\Gamma }_{1p}(t)\ge 0$ holds for $t\ge 0$. Otherwise, ${\Gamma }_{1p}(\widehat{t}a)<0$ must hold for some $\widehat{t}a$. Because ${\Gamma }_{1p}(t)$ is continuous, there are some ${\widehat{t}}_b\in [0,{\widehat{t}}_a)$ such that ${\Gamma }_{1p}({\widehat{t}}_b)=0$. Denote ${\widehat{t}}_c=inf\{{\widehat{t}}_b|{\Gamma }_{1p}({\widehat{t}}_b)=0,{\widehat{t}}_b\in [0,{\widehat{t}}_a)\}$. Because of ${\Gamma }_{1p}(0)>0$, and by (9),

$$\begin{array}{c}\hfill {\dot{\Gamma }}_{1p}(t)=-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\left({\gamma }_{1p}{\Gamma }_{1p}^{w-1}(t)+{\mu }_{1p}{\Gamma }_{1p}^{2w-1}(t)\right){\Gamma }_{1p}(t),t\in [0,{\widehat{t}}_c).\end{array}$$

Integrating the above formula from 0 to ${\widehat{t}}_c$, one gets

$$\begin{array}{c}\hfill 0={\Gamma }_{1p}(t_c)=e^{-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_0^{{\widehat{t}}_c}\left({\gamma }_{1p}{\Gamma }_{1p}^{w-1}(t)+{\mu }_{1p}{\Gamma }_{1p}^{2w-1}(t)\right)\mathrm{dv}}·{\Gamma }_{1p}(0)>0,\end{array}$$

This results in a contradiction. Hence, ${\Gamma }_{1p}(t)\ge 0$ for any $t\ge 0$. By analogous reasoning, we have ${\Gamma }_{2r}(t)\ge 0$ for any $t\ge 0$. □

Theorem 1.

Under Assumptions 1–3, the quantized control protocol (6) and the dynamic ETC mechanism (7)–(9), if $k_1<min\{k_2,k_3{4}^{1-2w}\}$, systems (1) and (2) can attain PDT projective lag synchronization within ${\mathcal{T}}_{pre}$, where ${\mathcal{T}}_{pre}\le {\mathcal{T}}_{max}$ and ${\mathcal{T}}_{max}$ is given in (6).

Proof. 

The Lyapunov function is constructed as

$$\begin{array}{cc}\hfill V(t)=& V_1(t)+V_2(t)+V_3(t)+V_4(t),\hfill \\ \hfill V_1(t)=& \sum _{p=1}^{n_2}{|{\xi }_{1p}(t)|}_1,V_2(t)=\sum _{p=1}^{n_2}{\Gamma }_{1p}(t),\hfill \\ \hfill V_3(t)=& \sum _{p=1}^{n_1}{|{\xi }_{2r}(t)|}_1,V_4(t)=\sum _{p=1}^{n_1}{\Gamma }_{2r}(t).\hfill \end{array}$$

Calculating the upper right Dini derivative $V_1(t)$ along the error trajectories of (5) for $t\in [t_{\sigma },t_{\sigma +1})$, one derives

$$\begin{array}{cc}\hfill D^{+}{V}_1(t)=& {\displaystyle \frac{1}{2}}\sum _{p=1}^{n_2}(\overline{\mathbf{Sign}({\xi }_{1p}(t))}(-d_{1p}{\xi }_{1p}(t)+\sum _{r=1}^{n_1}{\omega }_{pr}{G}_r({\xi }_{2r}(t))\hfill \\ \hfill & +\sum _{r=1}^{n_1}{a}_{pr}{F}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))+\sum _{r=1}^{n_1}{\omega }_{pr}{L}_r({\xi }_{2r}(t))\hfill \\ & +\sum _{r=1}^{n_1}{a}_{pr}{S}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))+(1-\rho )W_{1p}+U_{1p}(t))\hfill \\ \hfill & +(\sum _{r=1}^{n_1}\overline{G_r({\xi }_{2r}(t))}\overline{{\omega }_{pr}}+\sum _{r=1}^{n_1}\overline{F_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}\hfill \\ & +\sum _{r=1}^{n_1}\overline{L_r({\xi }_{2r}(t))}\overline{{\omega }_{pr}}+\sum _{r=1}^{n_1}\overline{S_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +(1-\rho )\overline{W_{1p}}+\overline{U_{1p}(t)}-d_{1p}\overline{{\xi }_{1p}(t)})\mathbf{Sign}({\xi }_{1p}(t))).\hfill \end{array}$$

(10)

From Assumptions 2 and 3, and by means of Lemmas 1 and 2, one gets that

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{\mathbf{Sign}({\xi }_{1p}(t))}{\omega }_{pr}{G}_r({\xi }_{2r}(t))\hfill \\ \hfill & +\overline{G_r({\xi }_{2r}(t))}\overline{{\omega }_{pr}}\mathbf{Sign}({\xi }_p(t)))\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 2\sum _{r=1}^{n_1}{|{\omega }_{pr}|}_1\left({\daleth }_{r1}{|{\xi }_{2r}(t)|}_1+{\tilde{\daleth }}_{r1}\right)\hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{\mathbf{Sign}({\xi }_{1p}(t))}{a}_{pr}{F}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))\hfill \\ \hfill & +\overline{F_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}\mathbf{Sign}({\xi }_{1p}(t)))\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 4\sum _{r=1}^{n_1}{|a_{pr}|}_1{\widehat{\daleth }}_{r1}\hfill \end{array}$$

(12)

and

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{\mathbf{Sign}({\xi }_{1p}(t))}{\omega }_{pr}{L}_{2r}({\xi }_{2r}(t))\hfill \\ \hfill & +\overline{L_{2r}({\xi }_{2r}(t))}\overline{{\omega }_{pr}}\mathbf{Sign}({\xi }_{1p}(t)))\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 2\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_1{\widehat{\daleth }}_{r1}(1+|\rho |)\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{\mathbf{Sign}({\xi }_{1p}(t))}{a}_{pr}{S}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))\hfill \\ \hfill & +\overline{S_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}\mathbf{Sign}({\xi }_{1p}(t)))\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 2\sum _{r=1}^{n_1}|a_{pr}{|}_1{\widehat{\daleth }}_{r1}(1+|\rho |)\hfill \end{array}$$

(14)

and

$$\begin{array}{c}\hfill \overline{\mathbf{Sign}({\xi }_{1p}(t))}(1-\rho )W_{1p}+(1-\rho )\overline{W_{1p}}\mathbf{Sign}({\xi }_{1p}(t))\le 2(1+|\rho |)|W_{1p}{|}_1.\end{array}$$

(15)

By the measure error ${\mathcal{E}}_{1p}(t)$, one has

$$\begin{array}{c}\hfill U_{1p}(t)=-{\mathcal{E}}_{1p}(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t))|}_1^w}.\end{array}$$

According to Lemmas 2 and 6,

$$\begin{array}{cc}\hfill & \overline{\mathbf{Sign}({\xi }_{1p}(t))}{U}_{1p}(t)+\overline{U_{1p}(t)}\mathbf{Sign}({\xi }_{1p}(t))\hfill \\ \hfill =& -\left(\overline{\mathbf{Sign}({\xi }_{1p}(t))}{\mathcal{E}}_{1p}(t)+\overline{{\mathcal{E}}_{1p}(t)}\mathbf{Sign}({\xi }_{1p}(t))\right)\hfill \\ \hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}{e}^{{\beta }_{1p}{|Q({\xi }_{1p}(t))|}_1^w}\left(\overline{\mathbf{Sign}({\xi }_{1p}(t))}{{\rm Y}}_{1p}(t)+\overline{{{\rm Y}}_{1p}(t)}\mathbf{Sign}({\xi }_{1p}(t))\right)\hfill \\ \hfill \le & 2|{\mathcal{E}}_{1p}{(t)|}_1-2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}{e}^{{\beta }_{1p}{|Q({\xi }_{1p}(t))|}_1^w}{|\mathbf{Sign}({\xi }_{1p}(t))|}_1\hfill \\ \hfill \le & 2|{\mathcal{E}}_{1p}{(t)|}_1-2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}(1+{\beta }_{1p}{|Q({\xi }_{1p}(t))|}_1^w\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\beta }_{1p}^2|Q({\xi }_{1p}(t)){|}_1^{2w}){|\mathbf{Sign}({\xi }_{1p}(t))|}_1\hfill \\ \hfill \le & 2|{\mathcal{E}}_{1p}{(t)|}_1-2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}-2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}{\beta }_{1p}{(1-ϵ)}^w{|{\xi }_{1p}(t)|}_1^w\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}{\beta }_{1p}^2{(1-ϵ)}^{2w}{|{\xi }_{1p}(t)|}_1^{2w},\hfill \end{array}$$

(16)

where ${{\rm Y}}_{1p}(t)\in \overline{\mathrm{co}}(\mathbf{Sign}({\xi }_{1p}(t)))$. Substituting (11)–(16) into (10), and using the dynamic ETC mechanism under 1-norm (7)–(9), one can conclude that

$$\begin{array}{cc}\hfill D^{+}{V}_1(t)\le & -\sum _{p=1}^{n_2}{d}_{1p}|{\xi }_{1p}{(t)|}_1+\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_1{\daleth }_{r1}{|{\xi }_{2r}(t)|}_1+\sum _{p=1}^{n_2}({\upsilon }_{1p}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}+{|{\mathcal{E}}_{1p}(t)|}_1)\hfill \\ \hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\sum _{p=1}^{n_2}{\alpha }_{1p}{\beta }_{1p}{(1-ϵ)}^w|{\xi }_{1p}{(t)|}_1^w-{\displaystyle \frac{{\mathcal{T}}_{max}}{2{\mathcal{T}}_{pre}}}\sum _{p=1}^{n_2}{\alpha }_{1p}{\beta }_{1p}^2{(1-ϵ)}^{2w}{|{\xi }_{1p}(t)|}_1^{2w}\hfill \\ \hfill \le & -\sum _{p=1}^{n_2}{d}_{1p}|{\xi }_{1p}{(t)|}_1+\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_1{\daleth }_{r1}{|{\xi }_{2r}(t)|}_1+\sum _{p=1}^{n_2}{\delta }_{1p}{\Gamma }_{1p}(t)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\sum _{p=1}^{n_2}{\alpha }_{1p}{\beta }_{1p}{(1-ϵ)}^w|{\xi }_{1p}{(t)|}_1^w-{\displaystyle \frac{{\mathcal{T}}_{max}}{2{\mathcal{T}}_{pre}}}\sum _{p=1}^{n_2}{\alpha }_{1p}{\beta }_{1p}^2{(1-ϵ)}^{2w}{|{\xi }_{1p}(t)|}_1^{2w}.\hfill \end{array}$$

(17)

Analogously,

$$\begin{array}{cc}\hfill D^{+}{V}_3(t)\le & -\sum _{r=1}^{n_1}{d}_{2r}|{\xi }_{2r}{(t)|}_1+\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}|e_{rp}{|}_1{F}_{p1}{|{\xi }_{1p}(t)|}_1+\sum _{r=1}^{n_1}({\upsilon }_{2r}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}+{|{\mathcal{E}}_{2r}(t)|}_1)\hfill \\ \hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\sum _{r=1}^{n_1}{\alpha }_{2r}{\beta }_{2r}{(1-ϵ)}^w|{\xi }_{2r}{(t)|}_1^w-{\displaystyle \frac{{\mathcal{T}}_{max}}{2{\mathcal{T}}_{pre}}}\sum _{r=1}^{n_1}{\alpha }_{2r}{\beta }_{2r}^2{(1-ϵ)}^{2w}{|{\xi }_{2r}(t)|}_1^{2w}\hfill \\ \hfill \le & -\sum _{r=1}^{n_2}{d}_{2r}|{\xi }_{2r}{(t)|}_1+\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}|e_{rp}{|}_1{F}_{p1}{|{\xi }_{1p}(t)|}_1+\sum _{r=1}^{n_2}{\delta }_{2r}{\Gamma }_{2r}(t)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\sum _{r=1}^{n_1}{\alpha }_{2r}{\beta }_{2r}{(1-ϵ)}^w|{\xi }_{2r}{(t)|}_1^w-{\displaystyle \frac{{\mathcal{T}}_{max}}{2{\mathcal{T}}_{pre}}}\sum _{r=1}^{n_1}{\alpha }_{2r}{\beta }_{2r}^2{(1-ϵ)}^{2w}{|{\xi }_{2r}(t)|}_1^{2w}.\hfill \end{array}$$

(18)

Combining (17), (18), adaptive law (9), and using Lemma 4,

$$\begin{array}{cc}\hfill D^{+}V(t)\le & d_1\sum _{p=1}^{n_2}|{\xi }_{1p}{(t)|}_1+\delta \sum _{p=1}^{n_2}{\Gamma }_{1p}(t)+d_2\sum _{r=1}^{n_1}{|{\xi }_{2r}(t)|}_1+\delta \sum _{r=1}^{n_1}{\Gamma }_{2r}(t)\hfill \\ \hfill & -{\acute{k}}_2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{V}_1^w(t)-{\acute{k}}_2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{V}_2^w(t)-{\stackrel{‘}{k}}_2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{V}_3^w(t)-{\stackrel{‘}{k}}_2{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{V}_4^w(t)\hfill \\ \hfill & -{\acute{k}}_3{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{n}_2^{1-2w}{V}_1^{2w}(t)-{\acute{k}}_3{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{n}_2^{1-2w}{V}_2^{2w}(t)\hfill \\ \hfill & -{\stackrel{‘}{k}}_3{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{n}_2^{1-2w}{V}_3^{2w}(t)-{\stackrel{‘}{k}}_3{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{n}_2^{1-2w}{V}_4^{2w}(t)\hfill \\ \hfill \le & k_{1}V(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2{V}^w(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_3{4}^{1-2w}{V}^{2w}(t).\hfill \end{array}$$

Since $k_1=max\{d,\delta \}>0$ and ${\mathcal{T}}_{pre}<{\mathcal{T}}_{max}$,

$$\begin{array}{c}\hfill D^{+}V(t)\le {\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}(k_{1}V(t)-k_2{V}^w(t)-k_3{4}^{1-2w}{V}^{2w}(t)).\end{array}$$

Therefore, by virtue of Lemma 5, if $k_1<min\{k_2,k_3{4}^{1-2w}\}$, the networks (1) and (2) can attain PDT projective lag synchronization within ${\mathcal{T}}_{pre}$ via the controller (6) with the dynamic ETC mechanism (7)–(9). □

Remark 3.

In [36,37], the authors solved the PDT synchronization of quaternion-valued BAMNNs by designing the following ETC scheme

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{U}_{1p}(t)=\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\left({\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))+{\beta }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma })){|{\xi }_{1p}(t_{\sigma })|}_1^{\widehat{w}}\right),\hfill \\ U_{2r}(t)=\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\left({\alpha }_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma }))+{\beta }_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma })){|{\xi }_{2r}(t_{\sigma })|}_1^{\widehat{w}}\right),\hfill \end{array}\right.\end{array}$$

where $\widehat{w}>1$. In contrast, the controller proposed (6) in this paper introduces two key innovations. First, it establishes a novel hybrid control framework that seamlessly integrates exponential quantization with the ETC mechanism. Second, it replaces the conventional linear combination of sign and power-law terms with a unified exponential term $e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1^w}$. This achieves simultaneous improvements in structural compactness, convergence speed, and resource utilization efficiency.

Remark 4.

The non-separation approach is adopted in this work, as opposed to the decomposition-based Lyapunov method [5,10,11], for three key reasons: First, decomposing OVNNs into real-valued subsystems disrupts the intrinsic non-commutative and non-associative nature of the octonion algebra. Second, this approach avoids the computational complexity incurred by handling eight equivalent real-valued subsystems, thereby yielding more concise stability criteria and controller designs. Finally, from a practical perspective, the non-separation method provides a more direct framework that is straightforward to implement, as it processes the complete OVNN directly without component separation.

Remark 5.

A novel dynamic ETC algorithm (6)–(9) is developed to achieve enhanced control efficiency and communication resource conservation. Unlike conventional static triggering schemes [28,29,30,34,35], dynamic auxiliary variables ${\Gamma }_{1p}(t)\ge 0$ and ${\Gamma }_{2r}(t)\ge 0$ are incorporated into the proposed methodology, enhancing adaptability and overall performance in resource-constrained environments.

In the quantized controller (6) and the dynamic ETC mechanism (7)–(9), if ${\mathcal{T}}_{pre}={\mathcal{T}}_{max}$, Theorem 1 degenerates to the following result.

Corollary 1.

Under Assumptions 1–3, the quantized control protocol (6) and the dynamic ETC mechanism (7)–(9), if ${\mathcal{T}}_{pre}={\mathcal{T}}_{max}$ and $k_1<min\{k_2,k_3{4}^{1-2w}\}$, systems (1) and (2) can attain FDT projective lag synchronization, and the ST is reckoned as ${\mathcal{T}}_{max}$, where ${\mathcal{T}}_{max}$ is given in (6).

Next, we prove that the Zeno phenomenon is eliminated in the quantized controller (6) with the dynamic ETC mechanism (7)–(9).

Theorem 2.

By adopting the quantized controller (6) with the dynamic ETC mechanism (7)–(9), the Zeno behavior can be excluded.

Proof. 

Assume Zeno behavior occurs, i.e., there is a moment $\breve{t}$ that satisfies ${lim}_{\sigma \to \infty }{t}_{\sigma }=\breve{t}<+\infty$. According to the definition of limits, for any $\tilde{e}>0$, there is $N(\tilde{e})$ such that $t_{\sigma }\in (\breve{t}-\tilde{e},\breve{t}+\tilde{e})$ for any $k\ge N(\tilde{e})$, which implies that $t_{N(\tilde{e})+1}-t_{N(\tilde{e})}<2\tilde{e}$. For $t\in [t_{\sigma },t_{\sigma +1})$, by Lemma 3, one derives

$$\begin{array}{c}\hfill D^{+}|{\mathcal{E}}_{1p}{(t)|}_1\le |{\dot{\mathcal{E}}}_{1p}{(t)|}_1\le {\nabla }_{\sigma }^{1p}{|{\dot{\xi }}_{1p}(t)|}_1,\end{array}$$

(19)

where ${\nabla }_{\sigma }^{1p}=8{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}{\beta }_{1p}w{(1+ϵ)}^w{e}^{{\beta }_{1p}{|Q({\xi }_{1p}(t))|}_1^w}{|{\xi }_{1p}(t)|}_1^{w-1}$. Since $D^{+}{V}_1(t)\le 0$, one can get

$$\begin{array}{c}\hfill |{\xi }_{1p}{(t)|}_1\le \sum _{p=1}^{n_2}{|{\xi }_{1p}(t)|}_1\le V_1(t)\le V_1(0).\end{array}$$

(20)

According to systems (1) and (2), then

$$\begin{array}{cc}\hfill |{\dot{\xi }}_{1p}{(t)|}_1\le & d_{1p}{V}_1(0)+\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_1(1+|\rho |){\widehat{\daleth }}_{r1}+\sum _{r=1}^{n_1}|a_{pr}{|}_1(1+|\rho |){\widehat{\daleth }}_{r1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +(1+|\rho |)|W_{1p}{|}_1+8{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}|e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1^w}|.\hfill \end{array}$$

(21)

Substituting (20) and (21) into (19), one can obtain

$$\begin{array}{cc}\hfill D^{+}{|{\mathcal{E}}_{1p}(t)|}_1\le & {\nabla }_{\sigma }^{1p}(d_{1p}{V}_1(0)+\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_1(1+|\rho |){\widehat{\daleth }}_{r1}+\sum _{r=1}^{n_1}|a_{pr}{|}_1(1+|\rho |){\widehat{\daleth }}_{r1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & + (1+|\rho |)|W_{1p}{|}_1+8{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}|e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1^w}|)\triangleq {\Psi }_1.\hfill \end{array}$$

(22)

Analogously,

$$\begin{array}{cc}\hfill D^{+}{|{\mathcal{E}}_{2r}(t)|}_1\le & {\nabla }_{\sigma }^{2r}(d_{2r}{V}_3(0)+\sum _{p=1}^{n_2}|e_{rp}{|}_1(1+|\rho |){\widehat{F}}_{p1}+\sum _{p=1}^{n_2}|l_{rp}{|}_1(1+|\rho |){\widehat{F}}_{p1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & + 8{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}|e^{{\beta }_{2r}{|Q({\xi }_{2r}(t_{\sigma }))|}_1^w}|)\triangleq {\Psi }_2,\hfill \end{array}$$

(23)

where ${\nabla }_{\sigma }^{2r}=8{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}{\beta }_{2r}w{(1+ϵ)}^w{e}^{{\beta }_{2r}{|Q({\xi }_{2r}(t))|}_1^w}{|{\xi }_{2r}(t)|}_1^{w-1}$. Combining (22) and (23),

$$\begin{array}{c}\hfill D^{+}|{\mathcal{E}}_{1p}{(t)|}_1+D^{+}{|{\mathcal{E}}_{2r}(t)|}_1\le {\Psi }_1+{\Psi }_2.\end{array}$$

Thus, for $t\in [t_{\sigma },t_{\sigma +1})$, one gets $|{\mathcal{E}}_{1p}{(t)|}_1+{|{\mathcal{E}}_{2r}(t)|}_1\le ({\Psi }_1+{\Psi }_2)(t-t_{\sigma }).$ If $t=t_{\sigma +1}^{-}$,

$$\begin{array}{c}\hfill |{\mathcal{E}}_{1p}(t_{\sigma +1}^{-}){|}_1+{|{\mathcal{E}}_{2r}(t_{\sigma +1}^{-})|}_1\le ({\Psi }_1+{\Psi }_2)(t_{\sigma +1}-t_{\sigma }).\end{array}$$

(24)

In addition, by utilizing the triggering condition (7), one has

$$\begin{array}{c}\hfill |{\mathcal{E}}_{1p}(t_{\sigma +1}^{-}){|}_1+{|{\mathcal{E}}_{2r}(t_{\sigma +1}^{-})|}_1>{\delta }_{1p}{\Gamma }_{1p}(t)+{\delta }_{2r}{\Gamma }_{2r}(t).\end{array}$$

(25)

By dynamic variables (9) and Lemma 7, for $t\in [t_{\sigma },t_{\sigma +1})$, one has

$$\begin{array}{cc}\hfill {\Gamma }_{1p}(t)=& e^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_{t_{\sigma }}^t(-{\gamma }_{1p}{\Gamma }_{1p}^{w-1}(t)-{\mu }_{1p}{\Gamma }_{1p}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{1p}(t_{\sigma })\hfill \\ \hfill =& e^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_{t_{\sigma -1}}^t(-{\gamma }_{1p}{\Gamma }_{1p}^{w-1}(t)-{\mu }_{1p}{\Gamma }_{1p}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{1p}(t_{\sigma -1})\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& \dots =e^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_0^t(-{\gamma }_{1p}{\Gamma }_{1p}^{w-1}(t)-{\mu }_{1p}{\Gamma }_{1p}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{1p}(0)>0\hfill \end{array}$$

(26)

and

$$\begin{array}{cc}\hfill {\Gamma }_{2r}(t)=& e^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_{t_{\sigma }}^t(-{\gamma }_{2r}{\Gamma }_{2r}^{w-1}(t)-{\mu }_{2r}{\Gamma }_{2r}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{2r}(t_{\sigma })\hfill \\ \hfill =& e^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_{t_{\sigma -1}}^t(-{\gamma }_{2r}{\Gamma }_{2r}^{w-1}(t)-{\mu }_{2r}{\Gamma }_{2r}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{2r}(t_{\sigma -1})\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& \dots =e^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_0^t(-{\gamma }_{2r}{\Gamma }_{2r}^{w-1}(t)-{\mu }_{2r}{\Gamma }_{2r}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{2r}(0)>0.\hfill \end{array}$$

(27)

Substitute (26) and (27) into (25),

$$\begin{array}{cc}\hfill & |{\mathcal{E}}_{1p}(t_{\sigma +1}^{-}){|}_1+{|{\mathcal{E}}_{2r}(t_{\sigma +1}^{-})|}_1\hfill \\ \hfill >& {\delta }_{1p}{e}^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_0^t(-{\gamma }_{1p}{\Gamma }_{1p}^{w-1}(t)-{\mu }_{1p}{\Gamma }_{1p}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{1p}(0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\delta }_{2r}{e}^{{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\int }_0^t(-{\gamma }_{2r}{\Gamma }_{2r}^{w-1}(t)-{\mu }_{2r}{\Gamma }_{2r}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{2r}(0)>0.\hfill \end{array}$$

(28)

Ulteriorly, it can be seen from (24) and (28),

$$\begin{array}{cc}\hfill t_{\sigma +1}-t_{\sigma }\ge & {\displaystyle \frac{{\delta }_{1p}{e}^{\frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}{\int }_0^t(-{\gamma }_{1p}{\Gamma }_{1p}^{w-1}(t)-{\mu }_{1p}{\Gamma }_{1p}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{1p}(0)}{{\Psi }_1+{\Psi }_2}}\hfill \\ \hfill & +{\displaystyle \frac{{\delta }_{2r}{e}^{\frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}{\int }_0^t(-{\gamma }_{2r}{\Gamma }_{2r}^{w-1}(t)-{\mu }_{2r}{\Gamma }_{2r}^{2w-1}(t))\mathrm{dv}}·{\Gamma }_{2r}(0)}{{\Psi }_1+{\Psi }_2}}>0.\hfill \end{array}$$

Then it can be further deduced that $t_{\sigma +1}-t_{\sigma }\ge 2\tilde{e}$, which contradicts with $t_{\sigma +1}-t_{\sigma }<2\tilde{e}$. Consequently, the Zeno behavior can be precluded. □

In quantized ETC strategy (6), if $w=1$, the controller becomes the following form

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{U}_{1p}(t)=\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1},\hfill \\ U_{2r}(t)=\hfill & -{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma }))e^{{\beta }_{2r}{|Q({\xi }_{2r}(t_{\sigma }))|}_1},\hfill \end{array}\right.\end{array}$$

(29)

where ${\mathcal{T}}_{max}$ is defined as

$${\mathcal{T}}_{max}=\left\{\begin{array}{cc}{\mathcal{T}}_{max}^{*}=\hfill & {\displaystyle \frac{\pi }{k_3}}{({\displaystyle \frac{k_3}{{\Phi }_1}})}^{{\displaystyle \frac{1}{2}}},k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2\le 0,\hfill \\ {\mathcal{T}}_{max}^{**}=\hfill & {\displaystyle \frac{2\pi }{k_3}}{({\displaystyle \frac{\frac{1}{4}{k}_3}{{\Phi }_1-k_1+\frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}{k}_2}})}^{{\displaystyle \frac{1}{2}}}\mathbf{I}({\displaystyle \frac{\frac{1}{4}{k}_3}{{\Phi }_1+\frac{1}{4}{k}_3-k_1+\frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}{k}_2}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})\hfill \\ \hfill & +{\displaystyle \frac{\pi }{2{\Phi }_1}}{({\displaystyle \frac{{\Phi }_1}{\frac{1}{4}{k}_3-k_1+\frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}{k}_2}})}^{{\displaystyle \frac{1}{2}}}\mathbf{I}({\displaystyle \frac{k_2}{{\Phi }_1+\frac{1}{4}{k}_3-k_1+\frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}{k}_2}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}),\hfill \\ \hfill & 0<k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2<min\{{\displaystyle \frac{1}{4}}{k}_3,{\Phi }_1\},\hfill \end{array}\right.$$

and the remaining undefined parameters are consistent with those in (6).

The triggering condition is exploited as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{t}_{\sigma +1}=inf\{t>t_{\sigma }|{\widehat{B}}_{1p}(t)\wedge {\widehat{B}}_{2r}(t)\ge 0\},\hfill \\ {\widehat{B}}_{1p}(t)={\upsilon }_{1p}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}+{|{\mathcal{E}}_{1p}(t)|}_1-{\delta }_{1p}{\Gamma }_{1p}(t)+{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\Phi ,\hfill \\ {\widehat{B}}_{2r}(t)={\upsilon }_{2r}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}+{|{\mathcal{E}}_{2r}(t)|}_1-{\delta }_{2r}{\Gamma }_{2r}(t)+{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\Phi ,\hfill \end{array}\right.\end{array}$$

(30)

where ${\delta }_{1p},{\delta }_{2r}>0$, ${\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}>{\upsilon }_{1p}+{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\Phi$, ${\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}>{\upsilon }_{2r}+{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}\Phi$, ${\mathcal{E}}_{1p}(t)$ and ${\mathcal{E}}_{2r}(t)$ are described as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{E}}_{1p}(t)={\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_1}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{1p}\mathbf{Sign}({\xi }_{1p}(t))e^{{\beta }_{1p}{|Q({\xi }_{1p}(t))|}_1},\hfill \\ {\mathcal{E}}_{2r}(t)={\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma }))e^{{\beta }_{2r}{|Q({\xi }_{2r}(t_{\sigma }))|}_1}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\alpha }_{2r}\mathbf{Sign}({\xi }_{2r}(t))e^{{\beta }_{2r}{|Q({\xi }_{2r}(t))|}_1},\hfill \end{array}\right.\end{array}$$

(31)

for $t\in [t_{\sigma },t_{\sigma +1}]$. Furthermore, dynamic variables ${\Gamma }_{1p}(t)$ and ${\Gamma }_{2r}(t)$ are designed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\Gamma }}_{1p}(t)=-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\gamma }_{1p}{\Gamma }_{1p}(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\mu }_{1p}{\Gamma }_{1p}^2(t),\hfill \\ {\dot{\Gamma }}_{2r}(t)=-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\gamma }_{2r}{\Gamma }_{2r}(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\mu }_{2r}{\Gamma }_{2r}^2(t),\hfill \end{array}\right.\end{array}$$

(32)

where ${\Gamma }_{1p}(0),{\Gamma }_{2r}(0)>0$, ${\gamma }_{1p},{\mu }_{1p},{\gamma }_{2r}$ and ${\mu }_{2r}>0$.

Theorem 3.

Under Assumptions 1–3, the quantized control protocol (29) and the dynamic ETC mechanism (30)–(32), if $k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2<min\{{\displaystyle \frac{1}{4}}{k}_3,{\Phi }_1\}$, the OV-BAMNNs (1) and (2) can achieve PDT projective lag synchronization in ${\mathcal{T}}_{pre}$, where ${\mathcal{T}}_{pre}\le {\mathcal{T}}_{max}$ and ${\mathcal{T}}_{max}$ is given in (29).

Proof. 

Analogous to the proof of Theorem 1, it is not difficult to obtain

$$\begin{array}{cc}\hfill D^{+}V(t)\le & (k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2)V(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{4{\mathcal{T}}_{pre}}}{k}_3{V}^2(t)-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{\Phi }_1\hfill \\ \hfill \le & \left\{\begin{array}{cc}-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}({\displaystyle \frac{k_3}{4}}{V}^2(t)+{\Phi }_1),\hfill & k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2\le 0,\hfill \\ {\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}((k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2)V(t)-{\displaystyle \frac{k_3}{4}}{V}^2(t)-{\Phi }_1),\hfill & 0<k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2<min\{{\displaystyle \frac{1}{4}}{k}_3,{\Phi }_1\}.\hfill \end{array}\right.\hfill \end{array}$$

From Lemma 5, the QV-BAMNNs (1) and (2) can achieve PDT projective lag synchronization in ${\mathcal{T}}_{pre}$ by the quantized control scheme (29) and the dynamic ETC mechanism (30)–(32). □

Corollary 2.

Under Assumptions 1–3, the quantized control strategy (29) and the dynamic ETC mechanism (30)–(32), if ${\mathcal{T}}_{pre}={\mathcal{T}}_{max}$ and $k_1-{\displaystyle \frac{{\mathcal{T}}_{max}}{{\mathcal{T}}_{pre}}}{k}_2<min\{{\displaystyle \frac{1}{4}}{k}_3,{\Phi }_1\}$, the OV-BAMNNs (1) and (2) can achieve FDT projective lag synchronization, and the ST is reckoned as ${\mathcal{T}}_{max}$, where ${\mathcal{T}}_{max}$ is given in (29).

Corollary 3.

By adopting the quantized controller (29) with the dynamic ETC mechanism (30)–(32), the Zeno behavior can be excluded.

To further strengthen the practicability of the results in this article, the PDT projective lag synchronization of OV-BAMNNs will be investigated by employing the 2-norm.

For convince, the related parameters are provided.

$$\begin{array}{cc}\hfill & {\tilde{d}}_1=\underset{p\in \tilde{\mathbf{p}}}{max}\{-2d_{1p}+\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_2{\daleth }_{r2}+\sum _{r=1}^{n_1}|e_{rp}{|}_2{F}_{p2}+1\},\delta =\underset{p\in \tilde{\mathbf{p}},r\in \tilde{\mathbf{r}}}{max}\{{\tilde{\delta }}_{1p},{\tilde{\delta }}_{2r}\},\hfill \\ \hfill & {\tilde{d}}_2=\underset{r\in \tilde{\mathbf{r}}}{max}\{-2d_{2r}+\sum _{p=1}^{n_2}|e_{rp}{|}_2{F}_{p2}+\sum _{p=1}^{n_2}|{\omega }_{pr}{|}_2{\daleth }_{r2}+1\},\tilde{d}=max\{{\tilde{d}}_1,{\tilde{d}}_2\},\hfill \\ \hfill & {\tilde{\upsilon }}_{1p}=2\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_2({\tilde{\daleth }}_{r2}+{\widehat{\daleth }}_{r2}(1+|\rho |))+2\sum _{r=1}^{n_1}|{\alpha }_{pr}{|}_2{\widehat{\daleth }}_{r2}(1+|\rho |)+2(1+|\rho |)|W_{1p}{|}_2,\hfill \\ \hfill & {\tilde{\upsilon }}_{2r}=2\sum _{p=1}^{n_2}|e_{rp}{|}_2({\tilde{F}}_{p2}+{\widehat{F}}_{p2}(1+|\rho |))+2\sum _{p=1}^{n_2}|l_{rp}{|}_2{\widehat{F}}_{r2}(3+|\rho |)+2(1+|\rho |)|W_{2r}{|}_2,\hfill \\ \hfill & {\tilde{\Phi }}_1=n_2\tilde{\Phi }+n_1\tilde{\Phi },{\tilde{k}}_1=max\{\tilde{d},\tilde{\delta }\},{\tilde{k}}_2=min\{{\check{k}}_2,{\widehat{k}}_2\},{\tilde{k}}_3=min\{n_2^{{\displaystyle \frac{1}{2}}-w}{\check{k}}_3,n_1^{{\displaystyle \frac{1}{2}}-w}{\widehat{k}}_3\},\hfill \\ \hfill & {\check{k}}_2=\underset{p\in \tilde{\mathbf{p}}}{min}\{2{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}{(1-\widetilde{ϵ})}^{\tilde{w}},2{\tilde{\gamma }}_{1p}\},{\widehat{k}}_2=\underset{r\in \tilde{\mathbf{l}}}{min}\{2{\tilde{\alpha }}_{2r}{\tilde{\beta }}_{2r}{(1-\widetilde{ϵ})}^{\tilde{w}},2{\tilde{\gamma }}_{2r}\},\hfill \\ \hfill & {\check{k}}_3=\underset{p\in \tilde{\mathbf{p}}}{min}\{{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}^2{(1-\widetilde{ϵ})}^{2\tilde{w}},2{\tilde{\mu }}_{1p}\},{\widehat{k}}_3=\underset{r\in \tilde{\mathbf{l}}}{min}\{{\tilde{\alpha }}_{2r}{\tilde{\beta }}_{2r}^2{(1-\widetilde{ϵ})}^{2\tilde{w}},2{\tilde{\mu }}_{2r}\}.\hfill \end{array}$$

Next, the exponential quantized ETC scheme based on 2-norm for $\tilde{t}\in [{\tilde{t}}_{\sigma },{\tilde{t}}_{\sigma +1})$ can be designed as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{U}_{1p}(t)=\hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))e^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_2^{\tilde{w}}},\hfill \\ U_{2r}(t)=\hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma }))e^{{\tilde{\beta }}_{2r}{|Q({\xi }_{2r}(t_{\sigma }))|}_2^{\tilde{w}}},\hfill \end{array}\right.\end{array}$$

(33)

where ${\tilde{\alpha }}_{1p},{\tilde{\beta }}_{1p},{\tilde{\alpha }}_{2r},{\tilde{\beta }}_{2r}>0$, ${\displaystyle \frac{1}{2}}<\tilde{w}<1$, $p\in \tilde{\mathbf{p}}$, $r\in \tilde{\mathbf{r}}$, ${\left\{{\tilde{t}}_{\sigma }\right\}}_{\sigma \in \mathbf{Z}}$ denotes an increasing sequence of release instants and ${\tilde{t}}_0=0$, ${\tilde{\mathcal{T}}}_{pre}$ is a preassigned-time and

$$\begin{array}{cc}\hfill {\tilde{\mathcal{T}}}_{max}={\tilde{\mathcal{T}}}_{max}^{**}=& {\displaystyle \frac{2\pi csc(\hslash \pi )}{{\tilde{k}}_3{4}^{\frac{1}{2}-w}w}}{({\displaystyle \frac{{\tilde{k}}_3{4}^{\frac{1}{2}-w}}{{\tilde{k}}_2-{\tilde{k}}_1}})}^{1-\hslash }\mathbf{I}({\displaystyle \frac{{\tilde{k}}_3{4}^{\frac{1}{2}-w}}{{\tilde{k}}_2+{\tilde{k}}_3{4}^{\frac{1}{2}-w}-{\tilde{k}}_1}},\hslash ,1-\hslash )\hfill \\ \hfill & +{\displaystyle \frac{2\pi csc(\hslash \pi )}{{\tilde{k}}_{2}w}}{({\displaystyle \frac{{\tilde{k}}_2}{{\tilde{k}}_3{4}^{\frac{1}{2}-w}-{\tilde{k}}_1}})}^{\hslash }\mathbf{I}({\displaystyle \frac{{\tilde{k}}_2}{{\tilde{k}}_2+{\tilde{k}}_3{4}^{\frac{1}{2}-w}-{\tilde{k}}_1}},1-\hslash ,\hslash ),\hfill \end{array}$$

and $\hslash ={\displaystyle \frac{1-w}{w}}$. $Q({\xi }_{1p}(t))={\sum }_{j=0}^7\widehat{Q}({\xi }_{1p}^{(j)}(t))$, $Q({\xi }_{2r}(t))={\sum }_{j=0}^7\widehat{Q}({\xi }_{2r}^{(j)}(t))$ and $\widehat{Q}(\diamond )$ is a quantization function, whose definition is consistent with that in Theorem 1.

The triggering condition is designed to

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tilde{t}}_{\sigma +1}=inf\{\tilde{t}>{\tilde{t}}_{\sigma }|{\tilde{B}}_{1p}(t)\wedge {\tilde{B}}_{2r}(t)\ge 0\},\hfill \\ {\tilde{B}}_{1p}(t)=({\tilde{\upsilon }}_{1p}-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p})|{\xi }_{1p}(t){|}_2+{|{\tilde{\mathcal{E}}}_{1p}(t)|}_2^2-2{\tilde{\delta }}_{1p}{\tilde{\Gamma }}_{1p}^2(t),\hfill \\ {\tilde{B}}_{2r}(t)=({\tilde{\upsilon }}_{2r}-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r})|{\xi }_{2r}(t){|}_2+{|{\tilde{\mathcal{E}}}_{2r}(t)|}_2^2-2{\tilde{\delta }}_{2r}{\tilde{\Gamma }}_{2r}^2(t),\hfill \end{array}\right.\end{array}$$

(34)

where ${\tilde{\delta }}_{1p},{\tilde{\delta }}_{2r}>0$, $2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}>{\tilde{\upsilon }}_{1p}$, $2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}>{\tilde{\upsilon }}_{2r}$, ${\tilde{\mathcal{E}}}_{1p}(t)$ and ${\tilde{\mathcal{E}}}_{2r}(t)$ are measurable errors, which are described as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tilde{\mathcal{E}}}_{1p}(t)={\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\mathbf{Sign}({\xi }_{1p}({\tilde{t}}_{\sigma }))e^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}({\tilde{t}}_{\sigma }))|}_2^{\tilde{w}}}-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\mathbf{Sign}({\xi }_{1p}(t))e^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}(t))|}_2^w},\hfill \\ {\tilde{\mathcal{E}}}_{2r}(t)={\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}\mathbf{Sign}({\xi }_{2r}({\tilde{t}}_{\sigma }))e^{{\tilde{\beta }}_{2r}{|Q({\xi }_{2r}({\tilde{t}}_{\sigma }))|}_2^{\tilde{w}}}-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}\mathbf{Sign}({\xi }_{2r}(t))e^{{\tilde{\beta }}_{2r}{|Q({\xi }_{2r}(t))|}_2^{\tilde{w}}}\hfill \end{array}\right.\end{array}$$

(35)

for $\tilde{t}\in [{\tilde{t}}_{\sigma },{\tilde{t}}_{\sigma +1}]$. Furthermore, dynamic variables ${\tilde{\Gamma }}_{1p}(t)$ and ${\tilde{\Gamma }}_{2r}(t)$ are designed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\tilde{\Gamma }}}_{1p}(t)=-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\gamma }}_{1p}{\tilde{\Gamma }}_{1p}^{\tilde{w}}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\mu }}_{1p}{\tilde{\Gamma }}_{1p}^{2\tilde{w}}(t),\hfill \\ {\dot{\tilde{\Gamma }}}_{2r}(t)=-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\mathcal{T}}_{pre}}}{\tilde{\gamma }}_{2r}{\tilde{\Gamma }}_{2r}^{\tilde{w}}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\mu }}_{2r}{\tilde{\Gamma }}_{2r}^{2\tilde{w}}(t),\hfill \end{array}\right.\end{array}$$

(36)

where ${\tilde{\Gamma }}_{1p}(0),{\tilde{\Gamma }}_{2r}(0)>0$, ${\tilde{\gamma }}_{1p},{\tilde{\mu }}_{1p},{\tilde{\gamma }}_{2r}$ and ${\tilde{\mu }}_{2r}>0$.

Theorem 4.

Under Assumptions 1–3, the quantized control protocol (33) and the dynamic ETC mechanism (34)–(36), if ${\tilde{k}}_1<min\{{\tilde{k}}_2,{\tilde{k}}_3{4}^{{\displaystyle \frac{1}{2}}-\tilde{w}}\}$, the OV-BAMNNs (1) and (2) can achieve PDT projective lag synchronization in ${\tilde{\mathcal{T}}}_{pre}$, where ${\tilde{\mathcal{T}}}_{pre}\le {\tilde{\mathcal{T}}}_{max}$ and ${\tilde{\mathcal{T}}}_{max}$ is given in (33).

Proof. 

Construct Lyapunov function

$$\begin{array}{cc}\hfill \tilde{V}(t)=& V_5(t)+V_6(t)+V_7(t)+V_8(t),\hfill \\ \hfill V_5(t)=& \sum _{p=1}^{n_2}{|{\xi }_{1p}(t)|}_2^2,V_6(t)=\sum _{p=1}^{n_2}{\tilde{\Gamma }}_{1p}^2(t),\hfill \\ \hfill V_7(t)=& \sum _{r=1}^{n_1}{|{\xi }_{2r}(t)|}_2^2,V_8(t)=\sum _{r=1}^{n_1}{\tilde{\Gamma }}_{2r}^2(t).\hfill \end{array}$$

For $\tilde{t}\in [{\tilde{t}}_{\sigma },{\tilde{t}}_{\sigma +1})$, calculating the upper right Dini derivative $V_5(t)$ along the error trajectories of (5), one derives

$$\begin{array}{cc}\hfill D^{+}{V}_5(t)=& \sum _{p=1}^{n_2}((-d_{1p}\overline{{\xi }_{1p}(t)}+\sum _{r=1}^{n_1}\overline{G_r({\xi }_{2r}(t))}\overline{{\omega }_{pr}}+\overline{U_{1p}(t)}\hfill \\ \hfill & +\sum _{r=1}^{n_1}\overline{F_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}+\sum _{r=1}^{n_1}\overline{L_r({\xi }_{2r}(t))}\overline{{\omega }_{pr}}\hfill \\ & +\sum _{r=1}^{n_1}\overline{S_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}+(1-\rho )\overline{W_{1p}}){\xi }_{1p}(t)\hfill \\ \hfill & +\overline{{\xi }_{1p}(t)}(-d_{1p}{\xi }_{1p}(t)+\sum _{r=1}^{n_1}{\omega }_{pr}{G}_r({\xi }_{2r}(t))+U_{1p}(t)\hfill \\ \hfill & +\sum _{r=1}^{n_1}{a}_{pr}{F}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))+\sum _{r=1}^{n_1}{\omega }_{pr}{L}_r({\xi }_{2r}(t))\hfill \end{array}$$

$$\begin{array}{cc}& +\sum _{r=1}^{n_1}{a}_{pr}{S}_r({\xi }_{2r}(t-{\Im }_{2r}(t)))+(1-\rho )W_{1p})).\hfill \end{array}$$

(37)

From Assumptions 2 and 3, and by means of Lemma 1, one gets that

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{G_r({\xi }_{2r}(t))}\overline{{\omega }_{pr}}{\xi }_{1p}(t)\hfill \\ \hfill & +\overline{{\xi }_{1p}(t)}{\omega }_{pr}{G}_r({\xi }_{2r}(t)))\hfill \\ \hfill \le & 2\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_2{|{\xi }_{1p}(t)|}_2\left({\daleth }_{r2}{|{\xi }_{2r}(t)|}_2+{\tilde{\daleth }}_{r2}\right)\hfill \\ \hfill =& \sum _{r=1}^{n_1}{|{\omega }_{pr}|}_2{\daleth }_{r2}(\overline{{\xi }_{1p}(t)}{\xi }_{1p}(t)+\overline{{\xi }_{2r}(t)}{\xi }_{2r}(t))\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +2\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_2{\tilde{\daleth }}_{r2}{|{\xi }_{1p}(t)|}_2\hfill \end{array}$$

(38)

and

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{F_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}{\xi }_{1p}(t)\hfill \\ \hfill & +\overline{{\xi }_{1p}(t)}{a}_{pr}{F}_r({\xi }_{2r}(t-{\Im }_{2r}(t))))\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 4\sum _{r=1}^{n_1}|a_{pr}{|}_2{\widehat{\daleth }}_{r2}{|{\xi }_{1p}(t)|}_2\hfill \end{array}$$

(39)

and

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{L_r({\xi }_{2r}(t))}\overline{{\omega }_{pr}}{\xi }_{1p}(t)\hfill \\ \hfill & +\overline{{\xi }_{1p}(t)}{\omega }_{pr}{L}_{2r}({\xi }_{2r}(t)))\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 2\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_2{\widehat{\daleth }}_{r2}|{\xi }_{1p}{(t)|}_2(1+|\rho |)\hfill \end{array}$$

(40)

and

$$\begin{array}{cc}\hfill & \sum _{r=1}^{n_1}(\overline{S_r({\xi }_{2r}(t-{\Im }_{2r}(t)))}\overline{a_{pr}}{\xi }_{1p}(t)\hfill \\ \hfill & +a_{pr}{S}_r({\xi }_{2r}(t-{\Im }_{2r}(t))){\xi }_{1p}(t))\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 2\sum _{r=1}^{n_1}|a_{pr}{|}_2{\widehat{\daleth }}_{r2}|{\xi }_{1p}{(t)|}_2(1+|\rho |)\hfill \end{array}$$

(41)

and

$$\begin{array}{c}\hfill (1-\rho )\overline{W_{1p}}{\xi }_{1p}(t)+\overline{{\xi }_{1p}(t)}(1-\rho )W_{1p}\le 2(1+|\rho |)|{\xi }_{1p}{(t)|}_2{|W_{1p}|}_2.\end{array}$$

(42)

By the measure error ${\tilde{\mathcal{E}}}_{1p}(t)$, one has

$$\begin{array}{c}\hfill U_{1p}(t)=-{\tilde{\mathcal{E}}}_{1p}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\mathbf{Sign}({\xi }_{1p}(t))e^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}(t))|}_2^{\tilde{w}}}.\end{array}$$

According to Lemmas 2 and 6,

$$\begin{array}{cc}\hfill & \overline{U_{1p}(t)}{\xi }_{1p}(t)+\overline{{\xi }_{1p}(t)}{U}_{1p}(t)\hfill \\ \hfill =& -\left(\overline{{\tilde{\mathcal{E}}}_{1p}(t)}{\xi }_{1p}(t)+\overline{{\xi }_{1p}(t)}{\tilde{\mathcal{E}}}_{1p}(t)\right)\hfill \\ & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}{e}^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}(t))|}_2^{\tilde{w}}}\left(\overline{{{\rm Y}}_{1p}(t)}{\xi }_{1p}(t)+\overline{{\xi }_{1p}(t)}{{\rm Y}}_{1p}(t)\right)\hfill \\ \hfill \le & 2|{\tilde{\mathcal{E}}}_{1p}{(t)|}_2|{\xi }_{1p}{(t)|}_2-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}{e}^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}(t))|}_2^{\tilde{w}}}{|{\xi }_{1p}(t)|}_2\hfill \\ \hfill \le & |{\tilde{\mathcal{E}}}_{1p}{(t)|}_2^2+|{\xi }_{1p}{(t)|}_2^2-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\left(1+{\tilde{\beta }}_{1p}|Q({\xi }_{1p}(t)){|}_2^{\tilde{w}}+{\displaystyle \frac{1}{2}}{\tilde{\beta }}_{1p}^2{|Q({\xi }_{1p}(t))|}_2^{2\tilde{w}}\right){|{\xi }_{1p}(t)|}_2\hfill \\ \hfill \le & |{\tilde{\mathcal{E}}}_{1p}{(t)|}_2^2+|{\xi }_{1p}{(t)|}_2^2-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}|{\xi }_{1p}{(t)|}_2-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}{(1-\widetilde{ϵ})}^{\tilde{w}}{|{\xi }_{1p}(t)|}_2^{\tilde{w}+1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}^2{(1-\widetilde{ϵ})}^{2\tilde{w}}{|{\xi }_{1p}(t)|}_2^{2\tilde{w}+1},\hfill \end{array}$$

(43)

where ${{\rm Y}}_{1p}(t)\in \overline{\mathrm{co}}(\mathbf{Sign}({\xi }_{1p}(t)))$.

Substituting (38)–(43) into (37), and using the dynamic ETC mechanism under 1-norm (34)–(36), one can conclude that

$$\begin{array}{cc}\hfill D^{+}{V}_5(t)\le & \sum _{p=1}^{n_2}(-2d_{1p}+\sum _{r=1}^{n_1}{|{\omega }_{pr}|}_2{\daleth }_{r2}+1){|{\xi }_{1p}(t)|}_2^2+\sum _{p=1}^{n_2}(({\tilde{\upsilon }}_{1p}-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p})\hfill \\ \hfill & ·|{\xi }_{1p}{(t)|}_2+|{\tilde{\mathcal{E}}}_{1p}{(t)|}_2^2)-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}{(1-\widetilde{ϵ})}^{\tilde{w}}{|{\xi }_{1p}(t)|}_2^{\tilde{w}+1}\hfill \\ \hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}^2{(1-\widetilde{ϵ})}^{2\tilde{w}}|{\xi }_{1p}(t){|}_1^{2\tilde{w}+1}+\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_2{\daleth }_{r2}{|{\xi }_{2r}(t)|}_2^2\hfill \\ \hfill \le & \sum _{p=1}^{n_2}(-2d_{1p}+\sum _{r=1}^{n_1}{|{\omega }_{pr}|}_2{\daleth }_{r2}+1)|{\xi }_{1p}{(t)|}_2^2+\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}|{\omega }_{pr}{|}_2{\daleth }_{r2}{|{\xi }_{2r}(t)|}_2^2\hfill \\ \hfill & +\sum _{p=1}^{n_2}{\tilde{\delta }}_{1p}{\tilde{\Gamma }}_{1p}^2(t)-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}{(1-\widetilde{ϵ})}^{\tilde{w}}{|{\xi }_{1p}(t)|}_2^{\tilde{w}+1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}^2{(1-\widetilde{ϵ})}^{2\tilde{w}}{|{\xi }_{1p}(t)|}_1^{2\tilde{w}+1}.\hfill \end{array}$$

(44)

Analogously,

$$\begin{array}{cc}\hfill D^{+}{V}_7(t)\le & \sum _{r=1}^{n_1}(-2d_{2r}+\sum _{p=1}^{n_2}{|e_{rp}|}_2{F}_{p2}+1){|{\xi }_{2r}(t)|}_2^2+\sum _{r=1}^{n_1}(({\tilde{\upsilon }}_{2r}-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r})\hfill \\ \hfill & ·|{\xi }_{2r}{(t)|}_2+|{\tilde{\mathcal{E}}}_{2r}{(t)|}_2^2)-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{1p}{\tilde{\beta }}_{1p}{(1-\widetilde{ϵ})}^{\tilde{w}}{|{\xi }_{2r}(t)|}_2^{\tilde{w}+1}\hfill \\ \hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{2r}{\tilde{\beta }}_{2r}^2{(1-\widetilde{ϵ})}^{2w}|{\xi }_{2r}(t){|}_1^{2w+1}+\sum _{r=1}^{n_1}\sum _{p=1}^{n_2}|e_{rp}{|}_2{\daleth }_{p2}{|{\xi }_{1p}(t)|}_2^2\hfill \\ \hfill \le & \sum _{r=1}^{n_1}(-2d_{2r}+\sum _{p=1}^{n_2}{|e_{rp}|}_2{F}_{p2}+1)|{\xi }_{2r}{(t)|}_2^2+\sum _{r=1}^{n_1}\sum _{p=1}^{n_2}|e_{rp}{|}_2{F}_{r2}{|{\xi }_{1p}(t)|}_2^2\hfill \\ \hfill & +\sum _{r=1}^{n_1}{\tilde{\delta }}_{2r}{\tilde{\Gamma }}_{2r}^2(t)-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{2r}{\tilde{\beta }}_{2r}{(1-\widetilde{ϵ})}^{\tilde{w}}{|{\xi }_{2r}(t)|}_2^{\tilde{w}+1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\sum _{p=1}^{n_2}\sum _{r=1}^{n_1}{\tilde{\alpha }}_{2r}{\tilde{\beta }}_{2r}^2{(1-\widetilde{ϵ})}^{2\tilde{w}}{|{\xi }_{2r}(t)|}_2^{2\tilde{w}+1}.\hfill \end{array}$$

(45)

Combining (44), (45) and adaptive law (36),

$$\begin{array}{cc}\hfill D^{+}\tilde{V}(t)\le & {\tilde{d}}_1\sum _{p=1}^{n_2}|{\xi }_{1p}{(t)|}_2^2+\tilde{\delta }\sum _{p=1}^{n_2}{\tilde{\Gamma }}_{1p}^2(t)+{\tilde{d}}_2\sum _{r=1}^{n_1}{|{\xi }_{2r}(t)|}_2^2+\tilde{\delta }\sum _{r=1}^{n_1}{\tilde{\Gamma }}_{2r}^2(t)\hfill \\ \hfill & -{\check{k}}_2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{V}_5^{{\displaystyle \frac{\tilde{w}+1}{2}}}(t)-{\check{k}}_2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{V}_6^{{\displaystyle \frac{\tilde{w}+1}{2}}}(t)-{\widehat{k}}_2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{V}_7^{{\displaystyle \frac{\tilde{w}+1}{2}}}(t)\hfill \\ \hfill & -{\widehat{k}}_2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{V}_8^{{\displaystyle \frac{\tilde{w}+1}{2}}}(t)-{\check{k}}_3{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{n}_2^{{\displaystyle \frac{1}{2}}-\tilde{w}}{V}_5^{\tilde{w}+{\displaystyle \frac{1}{2}}}(t)-{\check{k}}_3{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{n}_2^{{\displaystyle \frac{1}{2}}-\tilde{w}}{V}_6^{\tilde{w}+{\displaystyle \frac{1}{2}}}(t)\hfill \\ \hfill & -{\widehat{k}}_3{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{n}_2^{{\displaystyle \frac{1}{2}}-\tilde{w}}{V}_7^{\tilde{w}+{\displaystyle \frac{1}{2}}}(t)-{\widehat{k}}_3{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{n}_2^{{\displaystyle \frac{1}{2}}-\tilde{w}}{V}_8^{\tilde{w}+{\displaystyle \frac{1}{2}}}(t)\hfill \\ \hfill \le & {\tilde{k}}_1\tilde{V}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2{\tilde{V}}^{{\displaystyle \frac{\tilde{w}+1}{2}}}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_3{4}^{{\displaystyle \frac{1}{2}}-\tilde{w}}{\tilde{V}}^{\tilde{w}+{\displaystyle \frac{1}{2}}}(t).\hfill \end{array}$$

Since ${\tilde{k}}_1=max\{\tilde{d},\tilde{\delta }\}>0$ and ${\tilde{\mathcal{T}}}_{pre}<{\tilde{\mathcal{T}}}_{max}$,

$$\begin{array}{c}\hfill D^{+}\tilde{V}(t)\le {\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}({\tilde{k}}_1\tilde{V}(t)-{\tilde{k}}_2{\tilde{V}}^{{\displaystyle \frac{\tilde{w}+1}{2}}}(t)-{\tilde{k}}_3{4}^{{\displaystyle \frac{1}{2}}-w}{\tilde{V}}^{\tilde{w}+{\displaystyle \frac{1}{2}}}(t)).\end{array}$$

Therefore, by virtue of Lemma 5, if ${\tilde{k}}_1<min\{{\tilde{k}}_2,{\tilde{k}}_3{4}^{{\displaystyle \frac{1}{2}}-\tilde{w}}\}$, the networks (1) and (2) can attain PDT projective lag synchronization within ${\tilde{\mathcal{T}}}_{pre}$ via the quantized controller (33) with the dynamic ETC mechanism (34)–(36). □

Corollary 4.

Under Assumptions 1–3, the quantized control protocol (33) and the dynamic ETC mechanism (34)–(36), if ${\tilde{\mathcal{T}}}_{pre}={\tilde{\mathcal{T}}}_{max}$ and ${\tilde{k}}_1<min\{{\tilde{k}}_2,{\tilde{k}}_3{4}^{{\displaystyle \frac{1}{2}}-\tilde{w}}\}$, the OV-BAMNNs (1) and (2) can achieve FDT projective lag synchronization, and the ST is reckoned as ${\tilde{\mathcal{T}}}_{max}$.

Corollary 5.

By adopting the quantized controller (33) with the dynamic ETC mechanism (34)–(36), the Zeno behavior can be excluded.

In quantized ETC strategy (33), if $w=1$, the controller becomes the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{U}_{1p}(t)=\hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\mathbf{Sign}({\xi }_{1p}(t_{\sigma }))e^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}(t_{\sigma }))|}_2},\hfill \\ U_{2r}(t)=\hfill & -{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}\mathbf{Sign}({\xi }_{2r}(t_{\sigma }))e^{{\tilde{\beta }}_{2r}{|Q({\xi }_{2r}(t_{\sigma }))|}_2},\hfill \end{array}\right.\end{array}$$

(46)

where ${\tilde{\mathcal{T}}}_{max}$ is defined as

$${\tilde{\mathcal{T}}}_{max}=\left\{\begin{array}{cc}{\tilde{\mathcal{T}}}_{max}^{*}=\hfill & {\displaystyle \frac{8\sqrt{3}\pi }{9{\tilde{k}}_3}}{({\displaystyle \frac{{\tilde{k}}_3}{2\tilde{\Phi }}})}^{{\displaystyle \frac{2}{3}}},{\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2\le 0,\hfill \\ {\tilde{\mathcal{T}}}_{max}^{**}=\hfill & {\displaystyle \frac{8\sqrt{3}\pi }{9{\tilde{k}}_3}}{({\displaystyle \frac{\frac{1}{2}{\tilde{k}}_3}{{\tilde{\Phi }}_1-{\tilde{k}}_1+\frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}{\tilde{k}}_2}})}^{{\displaystyle \frac{1}{3}}}\mathbf{I}({\displaystyle \frac{\frac{1}{2}{\tilde{k}}_3}{{\tilde{\Phi }}_1+\frac{1}{2}{\tilde{k}}_3-{\tilde{k}}_1+\frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}{\tilde{k}}_2}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{1}{3}})\hfill \\ \hfill & +{\displaystyle \frac{4\sqrt{3}\pi }{9{\tilde{\Phi }}_1}}{({\displaystyle \frac{{\tilde{\Phi }}_1}{\frac{1}{2}{\tilde{k}}_3-{\tilde{k}}_1+\frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}{\tilde{k}}_2}})}^{{\displaystyle \frac{1}{3}}}\mathbf{I}({\displaystyle \frac{{\tilde{\Phi }}_1}{{\tilde{\Phi }}_1+\frac{1}{2}{\tilde{k}}_3-{\tilde{k}}_1+\frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}{\tilde{k}}_2}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}}),\hfill \\ \hfill & 0<{\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2<min\{{\displaystyle \frac{1}{2}}{\tilde{k}}_3,{\tilde{\Phi }}_1\},\hfill \end{array}\right.$$

and the remaining undefined parameters are consistent with those in (33).

The ETC condition is exploited as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tilde{t}}_{\sigma +1}=inf\{\tilde{t}>{\tilde{t}}_{\sigma }|{\check{B}}_{1p}(t)\wedge {\check{B}}_{2r}(t)\ge 0\},\hfill \\ {\check{B}}_{1p}(t)=({\tilde{\upsilon }}_{1p}-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p})|{\xi }_{1p}(t){|}_2+{|{\tilde{\mathcal{E}}}_{1p}(t)|}_2^2-2{\tilde{\delta }}_{1p}{\tilde{\Gamma }}_{1p}^2(t)+{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\tilde{\Phi },\hfill \\ {\check{B}}_{2r}(t)=({\tilde{\upsilon }}_{2r}-2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r})|{\xi }_{2r}(t){|}_2+{|{\tilde{\mathcal{E}}}_{2r}(t)|}_2^2-2{\tilde{\delta }}_{2r}{\tilde{\Gamma }}_{2r}^2(t)+{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}\tilde{\Phi },\hfill \end{array}\right.\end{array}$$

(47)

where ${\tilde{\delta }}_{1p},{\tilde{\delta }}_{2r}>0$, $2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}>{\tilde{\upsilon }}_{1p}$, $2{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}>{\tilde{\upsilon }}_{2r}$, ${\tilde{\mathcal{E}}}_{1p}(t)$ and ${\tilde{\mathcal{E}}}_{2r}(t)$ are described as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tilde{\mathcal{E}}}_{1p}(t)={\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\mathbf{Sign}({\xi }_{1p}({\tilde{t}}_{\sigma }))e^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}({\tilde{t}}_{\sigma }))|}_2}-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{1p}\mathbf{Sign}({\xi }_{1p}(t))e^{{\tilde{\beta }}_{1p}{|Q({\xi }_{1p}(t))|}_2},\hfill \\ {\tilde{\mathcal{E}}}_{2r}(t)={\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}\mathbf{Sign}({\xi }_{2r}({\tilde{t}}_{\sigma }))e^{{\tilde{\beta }}_{2r}{|Q({\xi }_{2r}({\tilde{t}}_{\sigma }))|}_2}-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\alpha }}_{2r}\mathbf{Sign}({\xi }_{2r}(t))e^{{\tilde{\beta }}_{2r}{|Q({\xi }_{2r}(t))|}_2}\hfill \end{array}\right.\end{array}$$

(48)

for $\tilde{t}\in [{\tilde{t}}_{\sigma },{\tilde{t}}_{\sigma +1}]$. Furthermore, dynamic variables ${\tilde{\Gamma }}_{1p}(t)$ and ${\tilde{\Gamma }}_{2r}(t)$ are designed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\tilde{\Gamma }}}_{1p}(t)=-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\gamma }}_{1p}{\tilde{\Gamma }}_{1p}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\mu }}_{1p}{\tilde{\Gamma }}_{1p}^2(t),\hfill \\ {\dot{\tilde{\Gamma }}}_{2r}(t)=-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\mathcal{T}}_{pre}}}{\tilde{\gamma }}_{2r}{\tilde{\Gamma }}_{2r}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\mu }}_{2r}{\tilde{\Gamma }}_{2r}^2(t),\hfill \end{array}\right.\end{array}$$

(49)

where ${\tilde{\Gamma }}_{1p}(0),{\tilde{\Gamma }}_{2r}(0)>0$, ${\tilde{\gamma }}_{1p},{\tilde{\mu }}_{1p},{\tilde{\gamma }}_{2r}$ and ${\tilde{\mu }}_{2r}>0$.

Theorem 5.

Under Assumptions 1–3, the quantized control protocol (46) and the dynamic ETC mechanism (47)–(49), if ${\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2<min\{{\displaystyle \frac{1}{2}}{\tilde{k}}_3,{\tilde{\Phi }}_1\}$, the OV-BAMNNs (1) and (2) can achieve PDT projective lag synchronization in ${\tilde{\mathcal{T}}}_{pre}$, where ${\tilde{\mathcal{T}}}_{pre}\le {\tilde{\mathcal{T}}}_{max}$ and ${\tilde{\mathcal{T}}}_{max}$ is given in (46).

Proof. 

Similar to the proof of Theorem 4, it is not difficult to obtain

$$\begin{array}{cc}\hfill D^{+}\tilde{V}(t)\le & ({\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2)\tilde{V}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{2{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_3{\tilde{V}}^{{\displaystyle \frac{3}{2}}}(t)-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{\Phi }}_1\hfill \\ \hfill \le & \left\{\begin{array}{cc}-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}({\displaystyle \frac{{\tilde{k}}_3}{2}}{\tilde{V}}^{{\displaystyle \frac{3}{2}}}(t)+{\tilde{\Phi }}_1),\hfill & {\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2\le 0,\hfill \\ {\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}(({\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2)\tilde{V}(t)-{\displaystyle \frac{{\tilde{k}}_3}{2}}{\tilde{V}}^{{\displaystyle \frac{3}{2}}}(t)-{\tilde{\Phi }}_1),\hfill & 0<{\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_p}}{\tilde{k}}_2<min\{{\displaystyle \frac{1}{2}}{\tilde{k}}_3,{\tilde{\Phi }}_1\}.\hfill \end{array}\right.\hfill \end{array}$$

By virtue of Lemma, if ${\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2<min\{{\displaystyle \frac{1}{2}}{\tilde{k}}_3,{\tilde{\Phi }}_1\}$, the networks (1) and (2) can attain PDT projective lag synchronization within ${\tilde{\mathcal{T}}}_{pre}$ via the quantized controller (46) with the dynamic ETC mechanism (47)–(49). □

Corollary 6.

Under Assumptions 1–3, the quantized control protocol (46) and the dynamic ETC mechanism (47)–(49), if ${\tilde{k}}_1-{\displaystyle \frac{{\tilde{\mathcal{T}}}_{max}}{{\tilde{\mathcal{T}}}_{pre}}}{\tilde{k}}_2<min\{{\displaystyle \frac{1}{2}}{\tilde{k}}_3,{\tilde{\Phi }}_1\}$, systems (1) and (2) can implement FDT projective lag synchronization, and the ST is assessed by ${\tilde{\mathcal{T}}}_{max}$, where ${\tilde{\mathcal{T}}}_{max}$ is given in (46).

Corollary 7.

By adopting the quantized controller (46) with the dynamic ETC mechanism (47)–(49), the Zeno behavior can be excluded.

Remark 6.

Under the framework of the non-decomposition method, Theorems 1 and 4 establish the PDT projective lag synchronization of OV-BAMNNs using the 1-norm and 2-norm, respectively. In particular, Theorem 1 is derived by designing an exponential quantized ETC protocol under the 1-norm, while Theorem 4 is obtained using the 2-norm. It is noteworthy that two distinct Lyapunov functions are formulated in the theoretical analysis to demonstrate synchronization under the two respective norms. Moreover, compared with conventional decomposition-based methods [5,10,11] or the previous studies limited to a single norm [7,18], the proposed approach not only yields more comprehensive theoretical results but also significantly reduces computational redundancy.

Remark 7.

The proposed PDT projective lag synchronization scheme offers a unified framework that generalizes several conventional synchronization types. As defined by the error ${\xi }_{1p}(t)={\tilde{z}}_{1p}(t)-\rho z_{1p}(t-\epsilon )$ and ${\xi }_{2r}(t)={\tilde{z}}_{2r}(t)-\rho z_{2r}(t-\epsilon )$, our results encompass: (i) PDT complete synchronization when $\rho =1,\epsilon =0$; (ii) PDT lag synchronization when $\rho =1,\epsilon \ne 0$; and (iii) PDT anti-synchronization when $\rho =-1,\epsilon =0$. While previous studies often focus on these special cases, they fail to address the combined challenges of PDT convergence [7,8], projective scaling [15], and transmission delays [38] simultaneously. Hence, the findings of the paper are more generalizable.

Remark 8.

In contrast to the finite-time and FDT synchronization, the ST ${\mathcal{T}}_{pre}$ of PDT synchronization can be predetermined depending on actual demands. Under the identical criteria, the ST of the PDT synchronization is noticeably less than that of the FDT synchronization. From this viewpoint, the PDT synchronization results in this paper are greater flexibility and practicality compared to the approaches in [28,29,30].

Remark 9.

This paper is compared with recent studies [24,25,27] to highlight its innovations. Unlike [25], which used quantized controllers for multi-layer networks, our study employs a more efficient quantized ETC scheme for OV-BAMNNs. Compared with [27], which achieved finite-time switching via intermittent control, this work solves the PDT projective lag synchronization problem, with a ST that can be preset independently of initial conditions. In contrast to [24], which developed PDT control for launch vehicles, our approach ensures PDT synchronization for NNs without requiring complex observers or performance functions.

4. Numerical Simulations

This section presents two numerical examples to demonstrate the validity and feasibility of the theoretical findings.

Example 1.

Consider the following OV-BAMNN

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}X-layer:{\dot{z}}_{1p}(t)=\hfill & -d_{1p}{z}_{1p}(t)+{\displaystyle \sum _{r=1}^2}{\omega }_{pr}{h}_r(z_{2r}(t))+W_{1p}\hfill \\ \hfill & +{\displaystyle \sum _{r=1}^2}{a}_{pr}{h}_r(z_{2r}(t-{\Im }_{2r}(t))),p=1,2,\hfill \\ Y-layer:{\dot{z}}_{2r}(t)=\hfill & -d_{2r}{z}_{2r}(t)+{\displaystyle \sum _{p=1}^2}{e}_{rp}{m}_p(z_{1p}(t))+W_{2r}\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^2}{l}_{rp}{m}_p(z_{1p}(t-{\Im }_{1p}(t))),r=1,2,\hfill \end{array}\right.\end{array}$$

(50)

where $z_{1p}(t)$, $z_{2q}(t)\in \mathbf{O}$, $d_{11}=d_{12}=0.9$, $d_{21}=d_{22}=1.1$, ${\Im }_{2r}(t)={\displaystyle \frac{e^t}{1+e^t}}$, ${\Im }_{1p}(t)=0.6-0.4sin(t)$, and $W_{1p}=W_{2r}=0$. $h_r(z_{2r}(t))={\sum }_{j=0}^7\left(0.6tanh(z_{2r}^{(j)}(t))+0.01\mathrm{sign}(z_{2r}^{(j)}(t))\right){\mathbf{u}}_j$, and $m_p(z_{1p}(t))={\sum }_{j=0}^7\left(0.6tanh(z_{1p}^{(j)}(t))+0.03\mathrm{sign}(z_{1p}^{(j)}(t))\right){\mathbf{u}}_j$. $W_{1p}(t)=W_{2r}(t)=0$. Apparently, Assumptions A2 and A3 are met with ${\daleth }_{r𝚥}=F_{p𝚥}=0.6$, $3{\tilde{\daleth }}_{r𝚥}={\tilde{F}}_{p𝚥}=0.06$, ${\widehat{\daleth }}_{r𝚥}=0.61$ and ${\widehat{F}}_{r𝚥}=0.63$ $(𝚥=1,2)$. The diagrams of the activation functions $h_r(z)$ and $m_p(z)$ are presented in Figure 1.

The connection weights of the network (50) are selected as

$$\begin{array}{cc}\hfill & {\omega }_{11}=0.3{\mathbf{u}}_0-0.4{\mathbf{u}}_1+0.3{\mathbf{u}}_2-0.4{\mathbf{u}}_3+0.5{\mathbf{u}}_4-0.8{\mathbf{u}}_5+0.5{\mathbf{u}}_6-0.3{\mathbf{u}}_7,\hfill \\ \hfill & {\omega }_{12}=0.5{\mathbf{u}}_0-0.7{\mathbf{u}}_1+0.5{\mathbf{u}}_2-0.6{\mathbf{u}}_3+0.5{\mathbf{u}}_4-0.4{\mathbf{u}}_5+0.2{\mathbf{u}}_6-0.2{\mathbf{u}}_7,\hfill \\ \hfill & {\omega }_{21}=0.6{\mathbf{u}}_0-0.6{\mathbf{u}}_1+0.2{\mathbf{u}}_2-0.3{\mathbf{u}}_3+0.4{\mathbf{u}}_4-1.0{\mathbf{u}}_5+0.6{\mathbf{u}}_6-0.6{\mathbf{u}}_7,\hfill \\ \hfill & {\omega }_{22}=0.25{\mathbf{u}}_0+1.0{\mathbf{u}}_1+0.3{\mathbf{u}}_2+0.2{\mathbf{u}}_3+0.2{\mathbf{u}}_4-0.1{\mathbf{u}}_5+0.7{\mathbf{u}}_6+0.3{\mathbf{u}}_7,\hfill \\ \hfill & a_{11}=0.8{\mathbf{u}}_0-0.4{\mathbf{u}}_1-0.5{\mathbf{u}}_2-1.1{\mathbf{u}}_3+0.5{\mathbf{u}}_4+0.2{\mathbf{u}}_5-0.1{\mathbf{u}}_6-1.0{\mathbf{u}}_7,\hfill \\ \hfill & a_{12}=0.2{\mathbf{u}}_0+0.3{\mathbf{u}}_1-0.2{\mathbf{u}}_2+0.6{\mathbf{u}}_3-0.4{\mathbf{u}}_4+0.3{\mathbf{u}}_5+0.3{\mathbf{u}}_6+0.2{\mathbf{u}}_7,\hfill \\ \hfill & a_{21}=-0.3{\mathbf{u}}_0+0.5{\mathbf{u}}_1+0.7{\mathbf{u}}_2+0.1{\mathbf{u}}_3-0.9{\mathbf{u}}_4-0.1{\mathbf{u}}_5+0.4{\mathbf{u}}_6-0.6{\mathbf{u}}_7,\hfill \\ \hfill & a_{22}=0.5{\mathbf{u}}_0+0.7{\mathbf{u}}_1+0.9{\mathbf{u}}_2+0.2{\mathbf{u}}_3+0.2{\mathbf{u}}_4-0.1{\mathbf{u}}_5+0.7{\mathbf{u}}_6+0.3{\mathbf{u}}_7,\hfill \\ \hfill & e_{11}=0.3{\mathbf{u}}_0-0.4{\mathbf{u}}_1+0.3{\mathbf{u}}_2-0.4{\mathbf{u}}_3+0.5{\mathbf{u}}_4-0.8{\mathbf{u}}_5+0.5{\mathbf{u}}_6-0.3{\mathbf{u}}_7,\hfill \\ \hfill & e_{12}=0.5{\mathbf{u}}_0-0.7{\mathbf{u}}_1+0.5{\mathbf{u}}_2-0.6{\mathbf{u}}_3+0.5{\mathbf{u}}_4-0.4{\mathbf{u}}_5+0.2{\mathbf{u}}_6-0.2{\mathbf{u}}_7,\hfill \\ \hfill & e_{21}=0.6{\mathbf{u}}_0-0.6{\mathbf{u}}_1+0.2{\mathbf{u}}_2-0.3{\mathbf{u}}_3+0.4{\mathbf{u}}_4-1.0{\mathbf{u}}_5+0.6{\mathbf{u}}_6-0.6{\mathbf{u}}_7,\hfill \\ \hfill & e_{22}=0.25{\mathbf{u}}_0+1.0{\mathbf{u}}_1+0.3{\mathbf{u}}_2+0.7{\mathbf{u}}_3+0.3{\mathbf{u}}_4+0.5{\mathbf{u}}_5+0.6{\mathbf{u}}_6+0.1{\mathbf{u}}_7,\hfill \\ \hfill & l_{11}=0.3{\mathbf{u}}_0-0.4{\mathbf{u}}_1+0.5{\mathbf{u}}_2-0.8{\mathbf{u}}_3+0.6{\mathbf{u}}_4-0.6{\mathbf{u}}_5+0.5{\mathbf{u}}_6-0.6{\mathbf{u}}_7,\hfill \\ \hfill & l_{12}=0.5{\mathbf{u}}_0-0.6{\mathbf{u}}_1-0.5{\mathbf{u}}_2-0.7{\mathbf{u}}_3+0.2{\mathbf{u}}_4-0.2{\mathbf{u}}_5+0.2{\mathbf{u}}_6+0.3{\mathbf{u}}_7,\hfill \\ \hfill & l_{21}=0.6{\mathbf{u}}_0-1.6{\mathbf{u}}_1+0.3{\mathbf{u}}_2-1.6{\mathbf{u}}_3+0.7{\mathbf{u}}_4-0.1{\mathbf{u}}_5+0.5{\mathbf{u}}_6-0.3{\mathbf{u}}_7,\hfill \\ \hfill & l_{22}=-0.2{\mathbf{u}}_0+1.0{\mathbf{u}}_1+0.7{\mathbf{u}}_2+0.5{\mathbf{u}}_3+0.4{\mathbf{u}}_4+0.5{\mathbf{u}}_5+0.8{\mathbf{u}}_6+0.7{\mathbf{u}}_7.\hfill \end{array}$$

Under initial conditions $z_{11}(𝚤)=1.4{\mathbf{u}}_0+1.2{\mathbf{u}}_1+0.5{\mathbf{u}}_2+1.1{\mathbf{u}}_3+1.4{\mathbf{u}}_4+1.2{\mathbf{u}}_5+1.0{\mathbf{u}}_6+1.1{\mathbf{u}}_7$, $z_{12}(𝚤)=1.3{\mathbf{u}}_0-0.2{\mathbf{u}}_1-0.7{\mathbf{u}}_2+0.7{\mathbf{u}}_3+1.3{\mathbf{u}}_4-0.2{\mathbf{u}}_5-0.7{\mathbf{u}}_6+0.7{\mathbf{u}}_7$, $z_{21}(𝚤)=1.1{\mathbf{u}}_0-1.0{\mathbf{u}}_1+0.5{\mathbf{u}}_2+0.3{\mathbf{u}}_3-1.1{\mathbf{u}}_4+0.2{\mathbf{u}}_5+0.5{\mathbf{u}}_6+0.3{\mathbf{u}}_7$ and $z_{22}(𝚤)=-0.4{\mathbf{u}}_0+1.2{\mathbf{u}}_1+0.8{\mathbf{u}}_2+0.8{\mathbf{u}}_3-0.4{\mathbf{u}}_4+0.3{\mathbf{u}}_5+1.8{\mathbf{u}}_6+0.8{\mathbf{u}}_7$, $𝚤\in [-1,0]$, the dynamic behaviors of network (50) are displayed in Figure 2.

The slave network is

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}X-layer:{\dot{\tilde{z}}}_{1p}(t)=\hfill & -d_{1p}{\tilde{z}}_{1p}(t)+{\displaystyle \sum _{r=1}^2}{\omega }_{pr}{h}_r({\tilde{z}}_{2r}(t))+W_{1p}(t)\hfill \\ \hfill & +{\displaystyle \sum _{r=1}^2}{a}_{pr}{h}_r({\tilde{z}}_{2r}(t-{\Im }_{2r}(t)))+U_{1p}(t),p=1,2,\hfill \\ Y-layer:{\dot{\tilde{z}}}_{2r}(t)=\hfill & -d_{2r}{\tilde{z}}_{2r}(t)+{\displaystyle \sum _{p=1}^2}{e}_{rp}{m}_p({\tilde{z}}_{1p}(t))+W_{2r}(t)\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^2}{l}_{rp}{m}_p({\tilde{z}}_{1p}(t-{\Im }_{1p}(t)))+U_{2r}(t),r=1,2,\hfill \end{array}\right.\end{array}$$

(51)

where ${\tilde{z}}_{11}(𝚤)=-0.2{\mathbf{u}}_0+0.2{\mathbf{u}}_1+0.5{\mathbf{u}}_2-0.5{\mathbf{u}}_3-0.4{\mathbf{u}}_4+0.2{\mathbf{u}}_5-1.0{\mathbf{u}}_6+0.1{\mathbf{u}}_7$, ${\tilde{z}}_{12}(𝚤)=-0.3{\mathbf{u}}_0+0.8{\mathbf{u}}_1-0.2{\mathbf{u}}_2+0.6{\mathbf{u}}_3-0.3{\mathbf{u}}_4+1.5{\mathbf{u}}_5+0.8{\mathbf{u}}_6-0.1{\mathbf{u}}_7$, ${\tilde{z}}_{21}(𝚤)=1.0{\mathbf{u}}_0+1.0{\mathbf{u}}_1-0.1{\mathbf{u}}_2-0.7{\mathbf{u}}_3+1.1{\mathbf{u}}_4+1.0{\mathbf{u}}_5-0.3{\mathbf{u}}_6-0.3{\mathbf{u}}_7$ and ${\tilde{z}}_{22}(𝚤)=0.5{\mathbf{u}}_0-0.6{\mathbf{u}}_1-0.6{\mathbf{u}}_2-0.3{\mathbf{u}}_3+0.6{\mathbf{u}}_4+1.7{\mathbf{u}}_5-1.2{\mathbf{u}}_6-0.5{\mathbf{u}}_7$, where $𝚤\in [-1,0]$.

Take $\rho =0.9$ and $\epsilon =1$. When $U_{1p}(t)=U_{2r}(t)=0$, the state trajectories of the OV-BAMNNs (50) and (51) are depicted in Figure 3, which demonstrates that the OV-BAMNNs (50) and (51) cannot realize synchronization without control inputs.

First, the PDT projective lag synchronization of systems (50) and (51) will be verified via the quantized control scheme (6) and the dynamic ETC mechanism (7)–(9). By simple calculation, ${\upsilon }_{11}=27.672$, ${\upsilon }_{12}=36.246$, ${\upsilon }_{21}=29.936$ and ${\upsilon }_{22}=36.07$. Choose the quantizer density $ϵ=0.7$. The parameters in (6) are selected as ${\alpha }_{11}=50$, ${\alpha }_{12}=60$, ${\alpha }_{21}=55$, ${\alpha }_{22}=60$, ${\beta }_{11}={\beta }_{12}={\beta }_{21}={\beta }_{22}=0.6$, $w=0.6$. The parameters in (7) are selected as ${\delta }_{11}={\delta }_{12}={\delta }_{21}={\delta }_{22}=4$. The parameters in (9) are selected as ${\gamma }_{11}={\gamma }_{12}={\gamma }_{21}={\gamma }_{22}=14$, ${\mu }_{11}={\mu }_{12}={\mu }_{21}={\mu }_{22}=10$, ${\Gamma }_{11}(0)=10$, ${\Gamma }_{12}(0)=30$, ${\Gamma }_{21}(0)=50$ and ${\Gamma }_{22}(0)=70$. According to the selection of the above parameters, it is not difficult to calculate $k_1=4$, $k_2=14$ and $k_3=6.21$. Obviously, the inequality $0<k_1<min\{k_2,k_3{4}^{1-2w}\}$ holds and ${\mathcal{T}}_{max}={\mathcal{T}}_{max}^{**}=3.7$. So, we take ${\mathcal{T}}_{pre}=2.5$, the OV-BAMNNs (50) and (51) can achieve PDT projective lag synchronization in ${\mathcal{T}}_{pre}=2.5$ by Theorem 1. The trajectories of the synchronization error ${\xi }_{1p}(t)$ and ${\xi }_{2r}(t)$ are plotted in Figure 4a. The triggering moments of each neuron in systems (50) and (51) are depicted in Figure 4b. The trajectories of dynamic variables ${\Gamma }_{1p}(t)$ and ${\Gamma }_{2r}(t)$ are showed in Figure 4c. Besides, Figure 5 displays the state trajectories of OV-BAMNNs (50) and (51) with $\rho =0.9$ and $\epsilon =1$.

4.1. Verification with Different Activation Functions

To verify the universality of the ETC protocol (6)–(9) with respect to different activation functions, the following two representative activation functions are additionally employed for validation. Figure 6a depicts the evolution of PDT projective lag synchronization errors under the saturation function $\mathrm{Sat}(x)=sign(x)·min(|x|,1)$, while Figure 6b illustrates the synchronization error evolution under the piecewise linear function $\mathrm{PWL}(x)=0.5·(|x+1|-|x-1|)$. The results demonstrate that the synchronization errors converge within ${\mathcal{T}}_{pre}$ under both activation functions, confirming the generality of our control strategy.

4.2. Performance Evaluation of ETC Scheme

To evaluate the performance of the ETC protocol (6)–(9) and rule out Zeno behavior, we conducted comprehensive simulations comparing it with traditional time-triggered control (TTC).

Table 3. Average inter-event time of dynamic ETC Scheme.

Neuron	Total Events	Simulation Time (s)	Average Inter-Event Time (s)
X-layer neuron 1	28	3.0	0.1071
X-layer neuron 2	22	3.0	0.1364
Y-layer neuron 1	24	3.0	0.1250
Y-layer neuron 2	34	3.0	0.0882
Overall	108	3.0	0.0278

displays the average inter-event times for the four neurons in the OV-BAMNNs. The results show that all neurons exhibit positive and finite inter-event intervals, ranging from 0.0882 to 0.1364 seconds, providing empirical evidence for the exclusion of Zeno behavior.

Furthermore,

Table 4. Comparisons between ETC and TTC.

Control Scheme	X-Layer		Y-Layer
	Neuron 1	Neuron 2	Neuron 1	Neuron 2
ETC Count	28	22	24	34
TTC Count	60	60	60	60
Reduction	53.3%	63.3%	60.0%	43.3%
Total reduction: 55.0% (108 vs. 240)

presents a comparative analysis between the proposed ETC scheme (6)–(9) and traditional TTC with a fixed triggering interval of 0.05 seconds. The comparison results demonstrate that our ETC scheme achieves a significant 55% reduction in total triggering events (108 vs. 240) while maintaining the desired synchronization performance. This substantial reduction in communication frequency translates into considerable savings in network bandwidth utilization and energy consumption, highlighting the practical advantages of the proposed ETC mechanism over traditional TTC methods.

4.3. Comparative Experments

To validate the effectiveness of the proposed dynamic ETC scheme (6)–(9), comparative experiments are conducted in this section. The proposed method is compared with the static ETC mechanism in [35] and the dynamic ETC mechanism in [36]. The experimental results can be found in Figure 7 and

Table 5. Statistics of the triggering times.

Neuron	In [35] (Static)	In [36] (Dynamic)	Proposed Method
X-layer neuron 1	474	29	28
X-layer neuron 2	477	26	22
Y-layer neuron 1	332	48	24
Y-layer neuron 2	324	39	34
Total	1607	142	108

. Figure 7 presents a comparison of the triggering instants under the same experimental conditions for the three methods. Table 5 details the statistical results of the triggering times for the three methods. It can be observed that while maintaining the system’s synchronization performance, the number of triggering times of the proposed dynamic ETC mechanism is significantly less than that of the static ETC mechanism in [35]. Compared with the dynamic ETC mechanism in [36], the proposed method exhibits a more uniform distribution of triggering instants and fewer triggering times.

Example 2.

To test the correctness of the theoretical result of Theorem 4, OV-BAMNNs (50) and (51) in Example 1 will continue to be considered. By simple calculation, ${\tilde{\upsilon }}_{11}=9.794$, ${\tilde{\upsilon }}_{12}=15.526$, ${\tilde{\upsilon }}_{21}=15.344$ and ${\tilde{\upsilon }}_{22}=21.68$. Choose the quantizer density $\widetilde{ϵ}=0.7$. The parameters in (33) are selected as ${\tilde{\alpha }}_{11}=25$, ${\tilde{\alpha }}_{12}=25$, ${\tilde{\alpha }}_{21}=25$, ${\tilde{\alpha }}_{22}=30$, ${\tilde{\beta }}_{11}={\tilde{\beta }}_{12}={\tilde{\beta }}_{21}={\tilde{\beta }}_{22}=0.6$, $\tilde{w}=0.6$. The parameters in (34) are selected as ${\tilde{\delta }}_{11}={\tilde{\delta }}_{12}={\tilde{\delta }}_{21}={\tilde{\delta }}_{22}=2$. The parameters in (36) are selected as ${\tilde{\gamma }}_{11}={\tilde{\gamma }}_{12}={\tilde{\gamma }}_{21}={\tilde{\gamma }}_{22}=8$, ${\tilde{\mu }}_{11}={\tilde{\mu }}_{12}={\tilde{\mu }}_{21}={\tilde{\mu }}_{22}=5$, ${\tilde{\Gamma }}_{11}(0)=10$, ${\tilde{\Gamma }}_{12}(0)=30$, ${\tilde{\Gamma }}_{21}(0)=50$ and ${\tilde{\Gamma }}_{22}(0)=70$. According to the selection of the above parameters, it is straightforward to calculate that ${\tilde{k}}_1=2.88$, ${\tilde{k}}_2=16$ and ${\tilde{k}}_3=6.65$. Obviously, the inequality $0<{\tilde{k}}_1<min\{{\tilde{k}}_2,{\tilde{k}}_3{4}^{{\displaystyle \frac{1}{2}}-w}\}$ holds and ${\tilde{\mathcal{T}}}_{max}={\tilde{\mathcal{T}}}_{max}^{**}=2.6$. So, we take ${\tilde{\mathcal{T}}}_p=1$. According to Theorem 1, systems (50) and (51) can achieve PDT projective lag synchronization in ${\tilde{\mathcal{T}}}_{pre}=1$, where $\rho =0.9$ and $\epsilon =1$. The trajectories of synchronization errors ${\xi }_{1p}(t)$ and ${\xi }_{2r}(t)$ are presented in Figure 8a. The triggering moments of each neuron in systems (50) and (51) are plotted in Figure 8b. The trajectories of dynamic variables ${\tilde{\Gamma }}_{1p}(t)$ and ${\tilde{\Gamma }}_{2r}(t)$ are presented in Figure 8c.

5. Application to Image Encryption

As a practical example, an image encryption algorithm is proposed in accordance with the PDT projective lag synchronization results in Example 1 of Numerical simulations. The architecture for image encryption and decryption is depicted in Figure 9.

5.1. Image Encryption Algorithm

The encryption algorithm is as stated below:

Step 1: Image pre-processing. Three images, namely baboon, flower and retina [39], are selected, and the following pre-processing is performed on each respectively to convert them into two-dimensional matrices. Specifically, each input color image $W_1(x_1,y_1,r)$ with dimensions $\mathbb{S}\times \mathbb{Z}\times 3$ is reshaped into a 2D matrix $W_2(x_2,y_2)$ of size $3\mathbb{S}\times \mathbb{Z}$ through channel concatenation:

$$W_2(x_2,y_2)=\left\{\begin{array}{cc}{W}_1(x_2,y_2,1),\hfill & x_2\in \{1,\dots ,\mathbb{S}\},\hfill \\ W_1(x_2,y_2,2),\hfill & x_2\in \{\mathbb{S}+1,\dots ,2\mathbb{S}\},\hfill \\ W_1(x_2,y_2,3),\hfill & x_2\in \{2\mathbb{S}+1,\dots ,3\mathbb{S}\},\hfill \end{array}\right.$$

for $y_2\in \{1,\dots ,\mathbb{Z}\}$.

Step 2: Permutation operation. After the networks achieve PDT projective lag synchronization, permutation keys are generated from the state variables of network (50):

$$\begin{array}{cc}\hfill Key1(x_2)& =({10}^{12}\times |(z_{11}^R(t_{x_2})+z_{12}^R(t_{y_2}))-\lfloor z_{11}^R(t_{x_2})+z_{12}^R(t_{x_2})\rfloor |)mod3\mathbb{S}+1,\hfill \\ \hfill Key2(y_2)& =({10}^{12}\times |(z_{12}^R(t_{y_2})+z_{13}^R(t_{y_2}))-\lfloor z_{12}^R(t_{y_2})+z_{13}^R(t_{y_2})\rfloor |)mod\mathbb{Z}+1,\hfill \end{array}$$

where $t_{x_2}\in \{t_1,\dots ,t_{3\mathbb{S}}\}$, $t_{y_2}\in \{t_1,\dots ,t_{\mathbb{Z}}\}$. The rows and columns of $W_2$ are permuted according to these keys, and the resulting matrix is converted to a sequence $P_1(a)\in {\mathbf{R}}^{1\times 3\mathbb{SZ}}$, where $a=1,\dots ,3\mathbb{S}\mathbb{Z}$.

Step 3: Diffusion operation. Diffusion keys are derived from synchronized network states:

$$Key3(a)=\left\{\begin{array}{cc}({10}^{12}\times |z_{11}^R(t_a)-\lfloor z_{11}^R(t_a)\rfloor |)mod256,\hfill & a\in \{1,\dots ,\mathbb{SZ}\},\hfill \\ ({10}^{12}\times |z_{12}^R(t_{a-\mathbb{SZ}})-\lfloor z_{12}^R(t_{a-\mathbb{SZ}})\rfloor |)mod256,\hfill & a\in \{\mathbb{SZ}+1,\dots ,2\mathbb{SZ}\},\hfill \\ ({10}^{12}\times |z_{21}^R(t_{a-2\mathbb{SZ}})-\lfloor z_{21}^R(t_{a-2\mathbb{SZ}})\rfloor |)mod256,\hfill & x\in \{2\mathbb{SZ}+1,\dots ,3\mathbb{SZ}\},\hfill \end{array}\right.$$

with $t_a\in \{t_1,\dots ,t_{3\mathbb{SZ}}\}$. The diffusion process generates a sequence $P_2(a)$:

$$\begin{array}{c}\hfill P_2(a)=P_1(a)\oplus (P_1(a)+Key3(a))\oplus P_2(a-1),\end{array}$$

where ⊕ denotes XOR, $a=1,\dots ,3\mathbb{SZ}$, and $P_2(0)$ is a prescribed constant.

Step 4: Iteration and output. Steps 2–3 are repeated $\mathbf{m}$ times to strengthen the encryption effect. After the final iteration, the resulting sequence is reshaped into a matrix $W_2^{\prime }$ of size $3\mathbb{S}\times \mathbb{Z}$, which is then reshaped to $\mathbb{S}\times \mathbb{Z}\times 3$ to obtain the final ciphertext image $W_1^{\prime }$.

The decrypted image is obtained by inversely applying the encryption algorithm, utilizing the identical keys generated by the synchronized response system (51) at the receiver end. Figure 10 shows the raw, ciphertext, and decrypted images.

5.2. Experimental Results and Security Analyses

To evaluate the security of the proposed encryption algorithm based on PDT projective lag synchronization, three images (baboon, flower, and retina) are encrypted. Histograms of the raw and ciphertext images are given in Figure 11. The uniform distribution of the ciphertext histograms confirms effective pixel randomization, indicating strong resistance to statistical attacks. Furthermore, the information entropy, key space, correlation of adjacent pixels, and resistance to differential attacks are tested to further analyze the security. To avoid redundancy, only the results for the baboon image are presented.

5.2.1. Information Entropy

Information entropy is a crucial metric for evaluating the randomness and unpredictability of encrypted images, calculated as:

$$H(m)=-\sum _{i=0}^{255}P(m_i){log}_{2}P(m_i)$$

(52)

where $P(m_i)$ represents the probability of symbol $m_i$.

Table 6. Information entropy of baboon image.

Image	R Channel	G Channel	B Channel
Raw image	7.6058	7.3581	7.6665
Ciphertext images	7.9976	7.9975	7.9971

lists the information entropy of the raw and ciphertext images for different color channels. The results show that the entropy of the ciphertext images is close to 8, demonstrating significantly improved randomness and security.

5.2.2. Correlation Coefficient

Raw images typically exhibit strong correlations between adjacent pixels, and an effective image encryption system must effectively break this correlation. The correlation coefficient is calculated as:

$$r_{xy}={\displaystyle \frac{{\sum }_{i=1}^N(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^N{(x_i-\overline{x})}^2{\sum }_{i=1}^N{(y_i-\overline{y})}^2}}}$$

(53)

where $x_i$ and $y_i$ are adjacent pixel values, $\overline{x}$ and $\overline{y}$ are their mean values, and N is the total number of pixel pairs. Using channel B as an example, we randomly select $10,000$ pairs of adjacent pixels from both the raw and ciphertext images in the horizontal, vertical, and diagonal directions for analysis. The results are shown in

Table 7. Correlation coefficient of baboon image.

Image	Channel	Horizontal	Vertical	Diagonal
Raw image	R	0.8872	0.9331	0.8693
	G	0.7938	0.8551	0.7438
	B	0.8881	0.9170	0.8587
Ciphertext image	R	0.0064	0.0035	−0.0012
	G	0.0055	0.0061	0.0025
	B	0.0011	0.0011	0.0009

and Figure 12. As shown, the correlation coefficients in the raw image are close to 1, while those in the ciphertext image are reduced to near zero. This demonstrates that the encryption system can effectively resist statistical analysis attacks.

5.2.3. Key Space

Key space is a crucial metric for evaluating the security of an image encryption algorithm. To resist brute-force attacks, the key space must be sufficiently large. According to current security standards, a secure key space should be at least on the order of $2^{100}$[39]. In our algorithm, the keys include the initial values $z_{11}^R(0)$, $z_{12}^R(0)$, and $z_{21}^R(0)$ of the driving system (50), the initial values ${\tilde{z}}_{11}^R(0)$, ${\tilde{z}}_{12}^R(0)$, and ${\tilde{z}}_{21}^R(0)$ of the response system (51), the algorithm parameter $P_2(0)$, and other system parameters. When the parameter accuracy reaches ${10}^{-14}$, the key space far exceeds the security threshold of $2^{100}$. Therefore, this algorithm has a large enough key space to effectively defend against brute force attacks.

5.2.4. Differential Attacks

Differential attacks occur when an attacker attempts to crack a cipher by analyzing changes in the ciphertext after making minor modifications to the plaintext image, such as altering a single pixel. A secure encryption algorithm must resist such attacks. This is typically evaluated using the number of pixels change rate (NPCR) and the unified average changing intensity (UACI), which measure the effect of a single-pixel change in the raw image on the ciphertext. The metrics are defined as follows:

$$\mathrm{NPCR}={\displaystyle \frac{{\sum }_{i=1}^M{\sum }_{j=1}^N{\sum }_{k=1}^{L}F(i,j,k)}{M\times N\times L}}\times 100$$

(54)

$$\mathrm{UACI}={\displaystyle \frac{{\sum }_{i=1}^M{\sum }_{j=1}^N{\sum }_{k=1}^L\frac{|E_1(i,j,k)-E_2(i,j,k)|}{255}}{M\times N\times L}}\times 100$$

(55)

where $E_1(i,j,k)$ and $E_2(i,j,k)$, represent the pixel values of two ciphertext images, and $F(i,j,k)$ is defined as:

$$F(i,j,k)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{E}_1(i,j,k)=E_2(i,j,k),\hfill \\ 1,\hfill & \mathrm{if}{E}_1(i,j,k)\ne E_2(i,j,k).\hfill \end{array}\right.$$

For the test, we use two raw images: the raw image in Figure 10a, and a modified version where the blue channel value at position $(200,200)$ is increased by 1, with all other pixels unchanged. Both images are encrypted using the same algorithm and key.

Table 8. Differential attacks of baboon image.

	$\mathsf{m}=1$	$\mathsf{m}=2$	$\mathsf{m}=3$
NPCR	0.7658	0.9971	0.9958
UACI	0.0121	0.3412	0.3355

presents the NPCR and UACI results as the number of encryption rounds $\mathbf{m}$ increases. A robust encryption algorithm should achieve NPCR = 99.61% and UACI = 33.33%. The results in Table 8 show that when $\mathbf{m}=3$, the proposed method reaches these ideal values, confirming its resistance to differential attacks.

Remark 10.

It is noted that the encryption algorithms proposed in [25,38,40] lack an analysis of differential attacks. Calculations reveal that the UACI and NPCR values for these encryption algorithms are merely $0.5984\times {10}^{-7}$ and $0.1526\times {10}^{-5}$, respectively. This indicates that these algorithms cannot effectively resist differential attacks. In contrast, the image encryption algorithm proposed in this paper, which is based on PDT projective lag synchronization, effectively resists differential attacks.

Remark 11.

PDT synchronization mechanism is the basis of the security of encryption scheme. In this framework, the drive system (50) generates a basic chaotic sequence for encryption. The core innovation is to ensure that the response system (51) and the drive system (50) are synchronized within a preassigned time. Once the PDT synchronization is achieved, the response system can replicate the same chaotic sequence originally generated by the drive system. This deterministic and timely synchronization ensures the perfect matching of highly random and complex sequences used for scrambling and diffusion between the sender and the authorized receiver, thus establishing a safe and reliable basis for the encryption process.

Remark 12.

The decryption process is the precise inverse of the encryption algorithm. Upon receiving the ciphertext, the authorized receiver utilizes its response system (51), which has achieved PDT projective lag synchronization with the drive system (50), to regenerate the identical permutation keys ($Key1$, $Key2$) and diffusion keys ($Key3$). The decryption then proceeds in reverse order: first, the inverse diffusion operation is applied to the ciphertext sequence using the regenerated $Key3$ and the known initial value $P_2(0)$; subsequently, the resulting sequence is reshaped into a matrix and undergoes inverse permutation using $Key1$ and $Key2$ to restore the original pixel positions. Finally, the matrix is restructured to recover the raw image. Correct decryption is therefore strictly contingent upon perfect synchronization between the drive and response systems.

6. Conclusions

In this study, exponential quantized ETC is employed to investigate the PDT projective lag synchronization in OV-BAMNNs incorporating discontinuous activation functions and time-varying delays. By adopting a non-separation approach and leveraging various exponential quantized ETC strategies, sufficient conditions are derived to ensure the PDT projective lag synchronization of OV-BAMNNs, while simultaneously excluding Zeno behavior. Compared with existing FDT and PDT schemes, the controller designed in this paper is more concise and resource-efficient, as it incorporates only a single exponential term and eliminates the conventional linear and power-law components.

In practical applications, communication networks are often susceptible to various external disturbances, such as denial-of-service attacks. Therefore, future research will focus on the PDT synchronization of OV-BAMNNs under such conditions.