Matrix measure-based exponential stability and synchronization of Markovian jumping QVNNs with time-varying delays and delayed impulses

1.   Introduction

Over the last couple of decades, famous neural networks have garnered special attention and have been extensively researched owing to their myriad of applications across various domains, as evidenced by numerous studies ^[1,2,3,4,5]. These applications encompass a wide spectrum, including but not limited to signal processing, image processing, and engineering optimization. Moreover, neural networks, including real-valued and complex-valued ones, have led to the presentation of numerous groundbreaking results in the field, as detailed in ^[6]. These results have significantly contributed to the advancement of neural network technologies and their practical applications. Furthermore, the quaternion, discovered by the British mathematician W. R. Hamilton, has found its way into various practical applications. It has been utilized in array processing, which involves the manipulation and analysis of data arrays in various contexts such as color image processing, which leverages the quaternion representation to capture and manipulate color information more accurately; and modeling of 3-D wind signals, which employs quaternion algebra to represent and analyze complex wind patterns in three-dimensional space. These applications, as referenced in ^[7,8,9,10], demonstrate the versatility and power of quaternion-based methods in addressing real-world problems.

It is worth noting that quaternion-valued neural networks (QVNNs) can be viewed as an extension of complex-valued neural networks (CVNNs), which exhibit far more intricate properties compared to CVNNs due to the non-commutative property of quaternion multiplication. These properties encompass quaternion-valued states, quaternion-valued connection weights, and quaternion-valued activation functions. In the realm of complex numbers, according to Liouville's theorem ^[11], every bounded entire function must be constant. From this point of view, choosing an appropriate activation function is very important. Fortunately, to date, numerous authors have delved into the analyticity problem ^[12,13]. For instance, the existence issue and stability problem of stochastic delayed QVNNs have been addressed in ^[14]. Finite-time synchronization of fractional-order delayed QVNNs has been given in ^[15]. By resorting to the lexicographical order method, the exponential synchronization and state estimation problems of inertial quaternion-valued Cohen-Grossberg neural networks have been researched in ^[16]. Therefore, addressing the dynamics of neural networks with delays has risen as a vital issue.

It is well-established that numerous prior studies have primarily concentrated on deterministic neural models ^[17,18,19]. However, random phenomenons are ubiquitous in social life because of fluctuations in the environment. Therefore, they should be modeled using stochastic systems to accurately represent real-world situations ^[20]. With the help of the stochastic analysis technique, dissipativity results of delayed Markovian jumping CVNNs were investigated in ^[21]. Meanwhile, lots of dynamical works on stochastic Markovian switching models was reported ^{[22,23,24,25,26,27,28]}. For example, the asynchronous output feedback control has been addressed in ^[29] for semi-Markov systems with random delays. Additionally, time-varying gain controller synthesis was examined in ^[30] for nonhomogeneous semi-Markovian switching linear systems. The self-triggered control problem of Markovian jumping nonlinear systems with stochastic disturbances was researched in ^[31]. When talking about global exponential stability and synchronization in delayed Markovian jumping QVNNs, little research attention has been paid to these issues. This scarcity of research serves as the primary motivation behind our ongoing research efforts.

Apart from stochastic perturbations, impulsive effects and delayed impulses are also prevalent in neural systems ^[32,33,34]. Furthermore, impulse is also an effective method to implement in practical scenarios. For instance, combining the average impulsive interval with the Lyapunov method, $p$-th moment exponential stability was addressed in ^[35] of impulsive random delayed nonlinear systems with average-delay impulses. In ^[36], the almost sure exponential stability issue was demonstrated for the impulsive stochastic differential delay equation with bounded variable delays. By jointly introducing the matrix measure protocol, the global and exponential leader-following consensus issue was concentrated in ^[37] for nonlinear multi-agent systems with mixed delays. Despite the research on stability and synchronization of impulsive systems, there is limited literature focusing on Markovian jumping QVNNs under delayed impulses. Therefore, with the help of the matrix measure method, our primary objective aims to propose the stability and synchronization results of delayed Markovian jumping QVNNs with delayed impulses.

Inspired by the preceding statements, this paper delves into exponential stability and synchronization of delayed Markovian jumping QVNNs with delayed impulses. The key highlights can be presented as follows. (1) With the consideration of a Markov chain, the investigated model becomes significantly more realistic in practical applications. Our findings reveal that impulses have a profound effect on the stability and synchronization of QVNNs. (2) By resorting to the advantage of the matrix measure approach, i.e., the measure value of the matrix can be positive or negative, the desired results can be derived. Meanwhile, the achieved criteria verify that it is a more practical and effective tool for dealing with the considered model. (3) This paper first takes into account not only time-varying delays but also delayed impulses on dynamical phenomenons of the proposed QVNNs.

Notations: The notations $\mathbb{R}^n$ and $\mathbb{Q}^{n}$ denote $n$-dimensional real vectors and $n$-dimensional quaternion value vectors, respectively. ${\mathbb{R}}^{m\times n}$ and $\mathbb{Q}^{m\times n}$ refer to $m\times n$-dimensional real matrices as well as $m\times n$-dimensional quaternion value matrices, respectively. The symbol "$\mathcal{I}$'' denotes identity matrix, with its dimension determined by the context. $\tilde{A}^T$' signifies the transpose of matrix $\tilde{A}$, and "$\mathrm{diag}$'' is used to represent a diagonal matrix. The notation "$*$'' is employed to indicate an element that is symmetrically implied in a given context, while $|x|$ refers to the magnitude of vector $x$. $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}\}_{t\geq 0}, \hat{\mathcal{P}})$ denotes the complete probability space. Moreover, $\lambda_{\max}(\tilde{A})$ ($\lambda_{\min}\; (\tilde{A})$) means the largest (smallest) eigenvalue of involved matrix $\tilde{A}$. $\mathbb{E}\{\cdot\}$ denotes mathematical expectation, and $\min\{a, b\}$ means the minimum value between $a$ and $b$.

2.   Problem formulation and preliminaries

Consider the delayed Markovian jumping QVNNs with delayed impulses below:

in which $k\in \mathbb{N}_{+} = \{1, 2, \ldots\}$, $\hbar(t) = (\hbar_1(t), \hbar_2(t), \ldots, \hbar_n(t))^T\in \mathbb{Q}^n$ refers to the state value of QVNN (2.1). $\mathcal{R}_{\varsigma(t)} = {\rm diag}\{\tilde{c}_{1\varsigma(t)(t)}, \tilde{c}_{2\varsigma(t)(t)}, \ldots, \tilde{c}_{n\varsigma(t)(t)}\} \in \mathbb{R}^{n\times n}$ with $\tilde{c}_{\iota \varsigma(t)(t)} > 0$ denotes the self-feedback matrix, $\iota = 1, 2, \ldots, n$. $\mathcal{S}_{r(t)} = (a_{\iota\upsilon \varsigma(t)})\in \mathbb{Q}^{n\times n}$ and $\mathcal{T}_{\varsigma(t)} = (b_{\iota\upsilon \varsigma(t)})\in \mathbb{Q}^{n\times n}$ refer to connection matrix without delays and the delayed one, respectively. $\mathfrak{f}(\hbar(t))$ and $\mathfrak{g}(\hbar(t-\Im(t)))$ denote the quaternion-valued nonlinear activation functions without and with time delays, respectively. $\Im(t)$ stands for time-varying delay subject to $0 < \Im(t) < \vec{\Im}$, and $0 < \vec{\omega}_k\leq\vec{\Im}$ means the impulsive delay. $t_k$ is the impulsive point satisfying $t_1 < t_2 < \ldots$ and ${\rm lim}_{k\rightarrow +\infty}t_k = +\infty$. $\Delta \hbar(t_k)\triangleq \hbar(t^{+}_k)-\hbar(t^{-}_k)$ means impulsive difference. Set $\hbar(t_k) = \hbar(t^{+}_k)\triangleq {\rm lim}_{t\rightarrow t^{+}_k}\hbar(t)$ and $\hbar(t^{-}_k)\triangleq{\rm lim}_{t\rightarrow t^{-}_k}\hbar(t)$. $\mathcal{L}_k$ and $\mathcal{M}_k$ refer to the strength of impulses. $\varsigma(t)$ denotes a Markov process, which is specified on space $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}\}_{t\geq 0}, \hat{\mathcal{P}})$ and takes its precise value in a set $\hat{\Upsilon} = \{1, 2, \ldots, \mathrm{N}\}$. Moreover, the corresponding generator is $\Theta = (\theta_{l\nu})_{\mathrm{N}\times\mathrm{N}}$, and the probability can be presented as

in which $\tilde{\Delta t} > 0$ as well as $\lim_{\tilde{\Delta t}\rightarrow0}(o(\tilde{\Delta t})/\tilde{\Delta t}) = 0$. In addition, $\vec{\theta}_{l\nu} > 0\; (l\neq \nu)$ refers to the jumping rate and $\vec{\theta}_{ll}\triangleq-\sum_{\nu = 1, \nu\neq l}^N \vec{\theta}_{l\nu}$.

In order to achieve the desired outcomes, all involved nonlinear functions are presumed to meet the constraints below.

Assumption 2.1. Set $\hbar = \hbar^R+\vec{i}\hbar^I+\vec{j}\hbar^J+\vec{k}\hbar^K$ with $\hbar^R, \hbar^I, \hbar^J, \hbar^K\in\mathbb{R}$, and the considered $\mathfrak{f}_\iota(\hbar)$ and $\mathfrak{g}_\iota(\hbar)$ can be expressed as follows:

in which $\mathfrak{f}_\iota^r(\cdot), \mathfrak{g}_\iota^r(\cdot):\mathbb{R}\rightarrow \mathbb{R}\; (\iota = 1, 2, \cdots, n), r="R", "I", "J", "K"$ and satisfy

in which $\mathfrak{m}_\iota^r$ and $\mathfrak{q}_\iota^r$ are known positive scalars, and $\tilde{\hbar}$, $\bar{\hbar}$ are scalars in $\mathbb{R}$. Moreover, $\mathfrak{f}_\iota(0) = \mathfrak{g}_\iota(0) = 0$.

Assumption 2.2. For $\hbar = \hbar^R+\vec{i}\hbar^I+\vec{j}\hbar^J+\vec{k}\hbar^K$ with $\hbar^R, \hbar^I, \hbar^J, \hbar^K\in\mathbb{R}$, $\mathfrak{f}_\iota(\hbar)$ and $\mathfrak{g}_\iota(\hbar)$ can be expressed as follows:

where $r="R", "I", "J", "K"$. Moreover, when $\tilde{\hbar}\neq\bar{\hbar}$, $\mathfrak{f}_\iota^r$ and $\mathfrak{g}_\iota^r$ satisfy the inequalities below:

Moreover, $\mathfrak{f}_\iota(0) = \mathfrak{g}_\iota(0) = 0$.

Next, set $\hbar(t) = \hbar^R(t)+\vec{i}\hbar^I(t)+\vec{j}\hbar^J(t)+\vec{k}\hbar^K(t)$, and then model (2.1) can also be converted to

in which $\aleph(t) = ((\hbar^R(t))^T, (\hbar^I(t))^T, (\hbar^J(t))^T, (\hbar^K(t))^T)^T$, and other symbols are showed as follows:

Next, some important lemmas and definition are presented.

Definition 2.1. ^[38] For a matrix $\tilde{\mathfrak{A}} = (\tilde{\mathfrak{a}}_{\theta\vartheta})_{n\times n}$, the homologous matrix measure value is set as

in which $\delta$ takes its value in $\{1, 2, \infty\}$. The symbol $\|\cdot\|_\delta$ refers to the matrix $\delta$-norm defined as

and the corresponding matrix measurements can be acquired as outlined below:

Lemma 2.1. ^[38] The involved matrix measure $\mu_\delta(\cdot)$ and it owns several properties: for a positive constant $\mathfrak{\zeta}$, matrices $\tilde{\mathfrak{A}}$, $\tilde{\mathfrak{B}}\in \mathbb{R}^{n\times n}$ (1): $-\|\tilde{\mathfrak{A}}\|_\delta \leq \mu_\delta(\tilde{\mathfrak{A}}) \leq \|\tilde{\mathfrak{A}}\|_\delta$; (2): $\mu_\delta(\mathfrak{\zeta} \tilde{\mathfrak{A}}) = \zeta\mu_\delta(\tilde{\mathfrak{A}})$; (3): $\mu_\delta(\tilde{\mathfrak{A}}+\tilde{\mathfrak{B}})\leq\mu_\delta(\tilde{\mathfrak{A}})+\mu_\delta(\tilde{\mathfrak{B}})$. From these properties, the matrix measure can be negative.

Lemma 2.2. ^[39] Take the differential inequalities as

in which $\vec{\mathcal{H}}(t)$ is nonnegative and continuous in $[t_0-\vec{\Im}, t_0]$. Then, when $\vec{\chi}_1 > \vec{\chi}_2\geq0$, there owns a positive scalar $\vec{\varpi} > 1$ such that $t_k-t_{k-1} > \vec{\varpi}\vec{\Im}$, and then

in which $t\in[t_k, t_{k+1}]$ and $\tilde{\eta}_\gamma = \max\{1, \vec{a}_i+\vec{b}_i e^{\vec{\zeta}\vec{\Im}}\}$ with $\gamma = 1, 2, \ldots, k+1$. Particularly, when $\tilde{\xi}\triangleq\sup_{k\in\mathbb{N}_+}\{1, \vec{a}_k+\vec{b}_k e^{\vec{\zeta}\vec{\Im}}\}$, one yields

where $\vec{\zeta}$ refers to the unique positive root of equation $\vec{\zeta} = \vec{\chi}_1-\vec{\chi}_2 e^{\vec{\zeta}\vec{\Im}}$.

Remark 2.1. Compared to the existing relevant results ^[40,41], they utilize the matrix norm method to handle the introduced matrix, resulting in a positive value. On the other hand, this paper employs the matrix measure method to deal with the introduced matrix, which can yield a negative value because matrix measures offer greater flexibility when applied to matrices. Therefore, the derived results in this paper, which are expressed in the format of a matrix measure, are more accurate than those achieved using the matrix norm method.

3.   Main results

In this section, by utilizing the matrix measure strategy, exponential stability criteria can be derived.

Theorem 3.1. Under the premise of Assumption 2.1, system (2.3) can attain the provided global exponential stability that two constraints hold below for $\delta$:

$¶_1$ $\omega_1 > \omega_2\geq 0$, in which

with $\delta$ belonging to $\{1, 2, \infty\}$, and

$¶_2$ There exists a constant $\vec{\beta} > \frac{{\rm ln}(\sigma e^{\vec{\varrho}\vec{\Im}})}{\vec{\varrho}\vec{\Im}}$, and one gets

in which $\sigma \triangleq \sup_{ k\in N_+}\{1, a_k+b_ke^{\vec{\varrho}\vec{\Im}}\}$, $a_k \triangleq\|\tilde{\mathcal{L}}_k+\mathcal{I}\|_\delta$, $b_k \triangleq\|\tilde{\mathcal{M}}_k\|_\delta$, and $\vec{\varrho}$ denotes the unique positive root of $\vec{\varrho} = \omega_1-\omega_2e^{\vec{\varrho}\vec{\Im}}$.

Proof. In the case of $t\neq t_k$, take the differential of $\|\aleph(t)\|_\delta$, and we get

in which

It follows from the expression of $\mathfrak{f}_1(\aleph(t))$ and Assumption 2.1 that

Utilizing the similar method in (3.2), we have

and the inequalities are satisfied below:

Next, by substituting the inequalities from (3.2) to (3.4) into (3.1), we can derive that

which further verifies that the following inequality is valid:

In addition, when $t = t_k$, one gets

Then, combine (3.6), (3.7), the conditions $¶_1$ and $¶_2$ in Theorem 3.1, as well as Lemma 2.2, and one can infer

in which $\vec{\varrho}$ denotes the unique positive root of $\vec{\varrho} = \omega_1-\omega_2e^{\vec{\varrho}\vec{\Im}}$, which further insures global exponential stability of system (2.3) under Assumption 2.1. The proof is now complete.

Theorem 3.2. Under the premise of Assumption 2.2, considered system (2.3) can attain the provided global exponential stability of the given constraints below are valid for $\delta = 1$ or $\infty$:

$¶_1$ $\omega_1 > \omega_2\geq 0$, in which

where $\delta\in \{1, 2, \infty\}$, $\mathcal{D} = {\rm diag}\left\{\mathfrak{m}_1^R, \cdots, \mathfrak{m}_n^R, \mathfrak{m}_1^I, \cdots, \mathfrak{m}_n^I, \mathfrak{m}_1^J, \cdots, \mathfrak{m}_n^J, \mathfrak{m}_1^K, \cdots, \mathfrak{m}_n^K\right\}$, $\tilde{\mathfrak{m}} = \mathfrak{m}^R+\mathfrak{m}^I+\mathfrak{m}^J+\mathfrak{m}^K$, $\mathcal{S}_1^l = (\bar{a}^l_{\iota\upsilon})_{4n\times4n}$, $\mathcal{S}_2^l = (\tilde{a}^l_{\iota\upsilon})_{4n\times4n}$, $\mathcal{S}_3^l = (\hat{a}^l_{\iota\upsilon})_{4n\times4n}$, $\mathcal{S}_4^l = (\check{a}^l_{\iota\upsilon})_{4n\times4n}$ and $\tilde{\mathcal{S}_l}^* = (a^l_{\iota\upsilon})_{4n\times4n}$ with $a^l_{\iota\upsilon} = \max\{0, \bar{a}^l_{\iota\upsilon}+\tilde{a}^l_{\iota\upsilon}+\hat{a}^l_{\iota\upsilon}+\check{a}^l_{\iota\upsilon}\}$ while $\iota = \upsilon$. Otherwise $a^l_{\iota\upsilon} = \bar{a}^l_{\iota\upsilon}+\tilde{a}^l_{\iota\upsilon}+\hat{a}^l_{\iota\upsilon}+\check{a}^l_{\iota\upsilon}$, and all other symbols retain their definitions as stated in Theorem 3.1.

$¶_2$ For a constant $\vec{\beta} > \frac{{\rm ln}(\sigma e^{\vec{\varrho}\vec{\Im}})}{\vec{\varrho}\vec{\Im}}$, the inequality

holds, where $\sigma \triangleq \sup_{ k\in N_+}\{1, a_k+b_ke^{\vec{\varrho}\vec{\Im}}\}$, $a_k \triangleq\|\tilde{\mathcal{L}}_k+\mathcal{I}\|_\delta$, $b_k \triangleq\|\tilde{\mathcal{M}}_k\|_\delta$, and $\vec{\varrho}$ stands for the unique positive root of $\vec{\varrho} = \omega_1-\omega_2e^{\vec{\varrho}\vec{\Im}}$.

Proof. In the case of $t\neq t_k$, analogous to the derivative process of Theorem 3.1, one can derive (3.1). Set

Based on the explanation of $\mathfrak{f}_1(\aleph(t))$, one yields

in which

Similarly, one can yield

in which

in which

in which

In addition, substituting (3.8)–(3.11) into $\|\aleph(t)+\varrho\dot{\aleph}(t)+o(\varrho)\|_\delta$, one can derive that

Hence, the result can eventually be reached that

Based on this, in the case of $t = t_k$, the conclusion is immediately available that

in which $\vec{\varrho}$ denotes the unique positive root of $\vec{\varrho} = \omega_1-\omega_2e^{\vec{\varrho}\vec{\Im}}$, and this solution further ensures the desired exponential stability under Assumption 2.2. The detailed proof is thus concluded.

In the subsequent section, we will be devoted to exploring the synchronization issue of considered system. If the drive system is (2.1), then the corresponding response system can be shown as

where $\Re(t) = \Re^R(t)+\vec{i}\Re^I(t)+\vec{j}\Re^J(t)+\vec{k}\Re^K(t)$ refers to the corresponding status value with $\Re^R(t), \Re^I(t), \Re^J(t), \Re^K(t)\in \mathbb{R}^n$, and $\mathfrak{U}_{r(t)}(t)$ stands for the designed control input as

where $\mathfrak{P}_{\varsigma(t)}\in \mathbb{R}^{n\times n}$ represents designed the gain matrix and $\wp(t)$ is given later.

Take the initial value of (3.15) as

where $\vartheta(s) = \vartheta^R(s)+\vec{i}\vartheta^I(s)+\vec{j}\vartheta^J(s)+\vec{k}\vartheta^K(s)$ with $\vartheta^R(s), \vartheta^I(s), \vartheta^J(s), \vartheta^K(s)\in\mathbf{C}([t_0-\vec{\Im}, t_0], \mathbb{R}^n)$.

Set the error system $\wp(t)\triangleq \Re(t)-\hbar(t)$ with $\wp(t) = ((\wp^R(t))^T, (\wp^I(t))^T, (\wp^J(t))^T, (\wp^K(t))^T)^T$ with $\wp^R(t) = \Re^R(t)-\hbar^R(t), \wp^J(t) = \Re^J(t)-\hbar^J(t), \wp^I(t) = \Re^I(t)-\hbar^I(t), \wp^K(t) = \Re^K(t)-\hbar^K(t)$, and it follows from systems (2.1) and (3.15), as well as controller (3.16), that one obtains

in which $\tilde{\mathfrak{f}}^R(\wp^R(t)) = \mathfrak{f}^R(\Re^R(t))-\mathfrak{f}^R(\hbar^R(t))$, $\tilde{\mathfrak{g}}^R(\wp^R(t-\Im(t))) = \mathfrak{g}^R(\Re^R(t-\Im(t)))-\mathfrak{g}^R(\hbar^R(t-\Im(t)))$, $\tilde{\mathfrak{f}}^I(\wp^I(t)) = \mathfrak{f}^I(\Re^I(t))-\mathfrak{f}^I(\hbar^I(t))$, $\tilde{\mathfrak{g}}^I(\wp^I(t-\Im(t))) = \mathfrak{g}^I(\Re^I(t-\Im(t)))-\mathfrak{g}^I(\hbar^I(t-\Im(t)))$, $\tilde{\mathfrak{f}}^J(\wp^J(t)) = \mathfrak{f}^J(\Re^J(t))-\mathfrak{f}^J(\hbar^J(t))$, $\tilde{\mathfrak{f}}^K(\wp^K(t)) = \mathfrak{f}^K(\Re^K(t))-\mathfrak{f}^K(\hbar^K(t))$, $\tilde{\mathfrak{g}}^J(\wp^J(t-\Im(t))) = \mathfrak{g}^J(\Re^J(t-\Im(t)))-\mathfrak{g}^J(\hbar^J(t-\Im(t)))$, $\tilde{\mathfrak{g}}^K(\wp^K(t-\Im(t))) = \mathfrak{g}^K(\Re^K(t-\Im(t)))-\mathfrak{g}^K(\hbar^K(t-\Im(t)))$. To facilitate the subsequent discussions, one useful definition can be introduced.

Definition 3.1. Systems (2.1) and (3.15) are said to realize exponential synchronization if the impulsive switched error system (3.17) reaches the desired exponential stability, i.e., there exist positive scalars $\zeta$ and $\xi$, and one can derive

Theorem 3.3. Under the premise of Assumption 2.1, the considered system (2.1) can attain globally exponential synchronization with system (3.15) if the given constraints below are valid for $\delta$:

$¶_1$ $\omega_1 > \omega_2\geq 0$, in which

in which $\delta\in\{1, 2, \infty\}$, and $\widetilde{\mathfrak{m}} = \max\{\mathfrak{m}^R, \mathfrak{m}^I, \mathfrak{m}^J, \mathfrak{m}^K\}$, $\mathfrak{m}^R = \max_{ 1\le \iota\le n}\{\mathfrak{m}_\iota^R\}$, $\mathfrak{m}^I = \max_{ 1\le \iota\le n}\{\mathfrak{m}_\iota^I\}$, $\mathfrak{m}^J = \max_{ 1\le \iota\le n}\{\mathfrak{m}_\iota^J\}$, $\mathfrak{m}^K = \max_{ 1\le \iota\le n}\{\mathfrak{m}_\iota^K\}$, $\widetilde{\mathfrak{q}} = \max\{\mathfrak{q}^R, \mathfrak{q}^I, \mathfrak{q}^J, \mathfrak{q}^K\}$, $\mathfrak{q}^R = \max_{ 1\le \iota\le n}\{\mathfrak{q}_\iota^R\}$, $\mathfrak{q}^I = \max_{ 1\le \iota\le n}\{\mathfrak{q}_\iota^I\}$, $\mathfrak{q}^J = \max_{ 1\le \iota\le n}\{\mathfrak{q}_\iota^J\}$, and $\mathfrak{q}^K = \max_{ 1\le \iota\le n}\{\mathfrak{q}_\iota^K\}$.

$¶_2$ There exists a constant $\vec{\beta} > \frac{{\rm ln}(\sigma e^{\vec{\varrho}\vec{\Im}})}{\vec{\varrho}\vec{\Im}}$, where

in which $\sigma \triangleq \sup_{ k\in N_+}\{1, a_k+b_ke^{\vec{\varrho}\vec{\Im}}\}$, $a_k \triangleq\|\mathcal{L}_k+\mathcal{I}\|_\delta$, $b_k \triangleq\|\mathcal{M}_k\|_\delta$, and $\vec{\varrho}$ stands for the unique positive root of $\vec{\varrho} = \omega_1-\omega_2e^{\vec{\varrho}\vec{\Im}}$.

Proof. For the case of $t\neq t_k$, choose the Lyapunov functional factor as

and then, take the differential of $V(\wp(t))$, and one gets

Combining the expression of $\|\tilde{\mathfrak{f}}^R(\wp^R(t))\|_\delta$ with Assumption 2.1, one gets

Using a similar approach, the following inequalities can be derived:

and one also obtains

By resorting to the inequalities (3.21)–(3.27), one can calculate

In addition, by the similar method as in (3.28), one also has

By substituting equations (3.28) to (3.31) into Eq (3.19), one can have

Likewise, when $t = t_k$, one gets

Combine (3.32) and (3.33) with Lemma 2.2 as well as conditions $¶_1$ and $¶_2$ stated in this theorem, one can yield

is right, in which $\vec{\varrho}$ represents the unique positive root of $\vec{\varrho} = \omega_1-\omega_2e^{\vec{\varrho}\vec{\Im}}$. Consequently, in accordance with Definition 3.1, we can deduce that systems (2.1) and (3.15) achieve global exponential synchronization. Thus, the proof has been fully established.

Theorem 3.4. Under the premise of Assumption 2.2, the system (2.1) can attain global exponential synchronization with the system (3.15) if the given constraints below are valid for $\delta$:

$¶_1$ $\bar{\omega}_1 > \bar{\omega}_2\geq 0$, in which

with $\mathcal{D} = {\rm diag}\left\{\mathfrak{m}_1^R, \cdots, \mathfrak{m}_n^R, \mathfrak{m}_1^I, \cdots, \mathfrak{m}_n^I, \mathfrak{m}_1^J, \cdots, \mathfrak{m}_n^J, \mathfrak{m}_1^K, \cdots, \mathfrak{m}_n^K\right\}$.

$¶_2$ There exists a constant $\vec{\beta} > \frac{{\rm ln}(\sigma e^{\vec{\varrho}\vec{\Im}})}{\vec{\varrho}\vec{\Im}}$, where

in which $\delta = 1$ or $\infty$, $\sigma \triangleq \sup_{ k\in N_+}\{1, a_k+b_ke^{\vec{\varrho}\vec{\Im}}\}$, $a_k \triangleq\|\mathcal{I} +\mathcal{L}_k\|_\delta$, $b_k \triangleq\|\mathcal{M}_k\|_\delta$, and $\vec{\varrho}$ denotes unique positive root of $\vec{\varrho} = \bar{\omega}_1-\bar{\omega}_2e^{\vec{\varrho}\tau}$. The remaining symbols retain their definitions as stated in Theorem 3.2.

Proof. In the case of $t\neq t_k$, based on the specific expression of $\tilde{\mathfrak{f}}_1(\wp(t))$, one can infer

in which

Utilizing an analogous approach employed for Eq (3.35), we can likewise derive that

in which

in which

in which

with

By incorporating the inequalities (3.35)–(3.38) into $\|\wp(t)+\varrho\dot{\wp}(t)+o(\varrho)\|_\delta$, one further derives

Hence, it follows from the presented first condition in Theorem 3.4 that one has

When $t = t_k$, combine (3.39) and (3.40) with Lemma 2.2, and one can conclude

in which $\vec{\varrho}$ denotes the unique positive root of $\vec{\varrho} = \bar{\omega}_1-\bar{\omega}_2e^{\vec{\varrho}\vec{\Im}}$. Hence, according to Definition 3.1, we obtain that systems (2.1) and (3.15) can reach global exponential synchronization. The specific proof has been completed.

Remark 3.1. The sufficient and effective criteria for achieving exponential synchronization between systems (2.1) and (3.15) are contingent upon several factors, including the time delay, network parameters, intensity of the impulses, pulse delay, and the minimum duration between impulsive intervals. Furthermore, these criteria build upon previous studies outlined in references ^{[29,30,31,32,33,34,35,36,37]}, offering new insights and expanding our understanding of synchronization in such systems.

Remark 3.2. It is crucial to acknowledge that time delays are inherently prevalent in information processing because signals travel through various links at a limited speed. Deservedly, the time delay also exists in pulse signals, which are usually called delay impulses. In comparison to the existing literatures ^{[32,33,36,39]}, this paper first addresses the exponential stability and synchronization control issues of QVNNs with delayed impulses, which is more in line with practical engineering.

Remark 3.3. The connection between a Markov process and general stochastic process can be described as follows. A Markov process is a specific subset of a stochastic process. While a stochastic process represents a broad framework for modeling random variables evolving over time, a Markov process imposes additional structure through its memoryless property, making them particularly suitable for modeling scenarios. This distinction allows Markov processes to serve as foundational models in areas such as queueing theory, population dynamics, and financial modeling ^[42,43].

4.   Numerical example

Example 4.1. Consider a two-neuron QVNN given in system (2.1) with two jumping modes as the master system, and the involved parameters can be taken as $\mathcal{R}_1 = \mathrm{diag}\{0.4, 0.5\}$, $\mathcal{R}_2 = \mathrm{diag}\{0.4, 0.6\}$, $\mathcal{L}_1 = \mathcal{L}_2 = \mathrm{diag}\{0.2, 0.4\}$, $\mathcal{M}_1 = \mathcal{M}_2 = \mathrm{diag}\{0.2, 0.3\}$, $\Im(t) = 0.2+0.2\sin2t$, $\vec{\omega}_k = 0.4$ with $\vec{\Im} = 0.4$, and

Besides, the involved nonlinear activation functions take the form

Based on this, it can be computed according to Assumption 2.1 that $\mathfrak{m}_\iota^r = \mathfrak{q}_\iota^r = 0.5$. In addition, the considered response system can be described as (3.15) with mode-dependent controller (3.16), and the controller gain matrices are set as follows:

Besides, the considered Markov process with initial modal value $\varsigma(0) = 1$ follows exponential distribution, and the corresponding jumping trajectory can be presented in Figure 1. Moreover, the infinitesimal generator matrix is set as

Combine the system parameters given above, and we can get the following pleasing results. When $\delta = 1$ and $\varsigma(t) = 1$, one can get $-\mu_1(-\mathcal{R}_{1}+\mathfrak{P}_{1})-\widetilde{\mathfrak{m}}\left(\|\mathcal{S}^R_1\|_1+\|\mathcal{S}^I_1\|_1+\|\mathcal{S}^J_1\|_1+\|\mathcal{S}^K_1\|_1\right) = 9.649$, and ${\widetilde{\mathfrak{q}}}\left(\|\mathcal{T}^R_1\|_1+\|\mathcal{T}^I_1\|_1+\|\mathcal{T}^J_1\|_1+\|\mathcal{T}^K_1\|_1\right) = 4.10$. While $r(t) = 2$, it also gets $-\mu_1(-\mathcal{R}_{2}+\mathfrak{P}_{2})-\widetilde{\mathfrak{m}}\left(\|\mathcal{S}^R_2\|_1+\|\mathcal{S}^I_2\|_1+\|\mathcal{S}^J_2\|_1+\|\mathcal{S}^K_2\|_1\right) = 9.650$, and ${\widetilde{\mathfrak{q}}}\left(\|\mathcal{T}^R_2\|_1+\|\mathcal{T}^I_2\|_1+\|\mathcal{T}^J_2\|_1+\|\mathcal{T}^K_2\|_1\right) = 6.05$. In this case, one has $\omega_1 = \mathrm{min}\{9.649, 9.650\} = 9.649 > \omega_2 = \mathrm{max}\{6.05, 4.10\} = 6.05$, which means that the first condition in Theorem 3.3 is valid. Besides, it is easy to have $a_k = 1.4$, $b_k = 0.3$, $\sigma = 1.8330$, and $\vec{\varrho} = 0.9173$. According to Theorem 3.3, it can be inferred that systems (2.1) and (3.15) can attain global exponential synchronization if there exists a scalar $\vec{\beta}$ satisfying the condition $\vec{\beta}\triangleq2.66 > 2.6515$ such that the involved impulsive interval conforms to $\inf_{ k\in N_+}\{t_k-t_{k-1}\} > \vec{\beta}\vec{\Im} = 1.0640$.

Next, in order to facilitate the simulation, set the impulsive interval to 1.08, that is, $t_{k+1}-t_k = 1.08$. In addition, let the initial values of drive-response systems (2.1) and (3.15) be $\hbar(t) = (\hbar_1(t), \hbar_2(t))^T$, $\Re(t) = (\Re_1(t), \Re_2(t))^T$ with $\hbar_1(t) = -4.4+2.6\vec{i}-2.1\vec{j}+3.4\vec{k}$, $\hbar_2(t) = -2.6-3.4\vec{i}-3.6\vec{j}-1.4\vec{k}$, $\Re_1(t) = 3.4-1.6\vec{i}+2.4\vec{j}-1.4\vec{k}$, and $\Re_2(t) = 2.1+3.2\vec{i}+2.1\vec{j}+2.5\vec{k}$ for $t\in[-0.4, 0]$. Under the influence of the mode-dependent linear feedback controller $\mathfrak{U}_{\varsigma(t)}(t)$, the response time trajectories of states $\Re^R(t) = (\Re_1^R(t), \Re_2^R(t))^T$ and $\hbar^R(t) = (\hbar_1^R(t), \hbar_2^R(t))^T$ as well as their synchronization error $\wp^R(t) = (\wp_1^R(t), \wp_2^R(t))^T$ are shown in the upper-left corner of Figure 2. Similarly, the trajectory plots for the states $\hbar^I(t)$, $\Re^I(t)$ and $\wp^I(t)$, the states $\hbar^J(t)$, $\Re^J(t)$ and their synchronization error $\wp^J(t)$, as well as states $\hbar^K(t)$, $\Re^K(t)$ and error $\wp^K(t)$, are illustrated in the remaining positions of Figure 2.

5.   Conclusions

This article delves into global exponential stability and synchronization problems for a specific type of delayed Markovian jumping QVNNs that incorporate delayed impulses. By employing the matrix measure strategy and delayed differential inequality methods that account for impulsive factors, we establish several practical and effective criteria to ascertain that the impulsive QVNNs in question can attain exponential synchronization with a given response system. Additionally, the explicit exponential convergence rate is provided. Notably, the derived criteria are straightforward to verify and can be readily applied in real-world scenarios. Finally, to underscore the precision and efficacy of our theoretical findings, we present one numerical example accompanied by an explanation. Moreover, inspired by the existing research findings in references ^[44,45], our further focus will be on the event-triggered synchronization problem of semi-Markovian jumping QVNNs with time-varying delay, where the sojourn time follows a more general distribution.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 62406005 and 61906084, in part by the Natural Science Foundation of Anhui Province, China, under Grant 2408085QA008, in part by the High-Level Talent Research Foundation of Anhui Agricultural University under Grant rc382106, in part by the Ningxia Natural Science Foundation under Grant 2024AAC03157, in part by the Innovation and Entrepreneurship Training Program for College Students at Anhui Agricultural University under Grants X202410364129 and X202410364041, in part by the Philosophy and Social Science Foundation of Universities of Anhui Province under Grant 2023AH050970.

Author contributions

Miao Zhnag: Formal analysis, and original draft preparation; Bole Li: Writing-review and editing; Weiqiang Gong: Conceptualization, resources, writing-review and editing. Shuo Ma: Writing-review and editing; Qiang Li. Investigation, methodology and supervision. All authors have read and approved the final version of the manuscript for publication

Conflict of interest

The authors declare no conflict of interest.