Impulsive Destabilization Effect on Novel Existence of Solution and Global μ-Stability for MNNs in Quaternion Field

Abstract

In this paper, a novel memristor-based non-delay Hopfield neural network with impulsive effects is designed in a quaternion field. Some special inequalities, differential inclusion, Hamilton rules and impulsive system theories are utilized in this manuscript to investigate potential solutions and obtain some sufficient criteria. In addition, through choosing proper $\mu \left(t\right)$ and impulsive points, the global $\mu$-stability of the solution is discussed and some sufficient criteria are presented by special technologies. Then, from the obtained sufficient criteria of global $\mu$-stability, other stability criteria including exponential stability and power stability can be easily derived. Finally, one numerical example is given to illustrate the feasibility and validity of the derived conclusions.

1. Introduction

In the past decade or so, the memristor [1,2] has attracted the attention of numerous scholars because of its special characteristics related to brain-inspired computing. Among all the studies, the study of memristor-based neural networks (MNNs) is an important research direction including properties of the solution, algorithms and so on. Due to the hysteresis loop characteristic of the memristor, when it is used as a connection weight, the neural networks become a differential inclusion system. Therefore, based on Filippov’s solution [3], some significant dynamic behaviors of MNNs were examined [4,5,6,7,8,9,10,11,12]. Up to now, the dynamic research of MNNs is still an open hot topic field.

In the process of studying MNNs, it can be found that its connection weights have a switching jump with the change of threshold voltage. Furthermore, when the systems are running, sudden noises or other unexpected abrupt perturbations may burst in and affect the stability of systems. Thus, abrupt changes phenomena, named impulsive effects, should be considered in the construction of any system model. In recent years, some typical results for the dynamic behaviors of impulsive systems have been published [13,14,15,16,17,18,19,20,21]. For example, the authors in [13] discussed the periodic solution of an impulsive system. By impulsive controller and other techniques, the authors investigated the quasi-synchronization [16,17] and input-to-state stability [18,19]. In addition, the authors in [20,21] researched the stability of quaternion-valued NNs (QVNNs) with impulsive disturbance in a quaternion field, in which a novel NN model–QVNN was provided and investigated. QVNNs have attracted more and more attention in recent years.

A quaternion field is a set of quaternions, which was constructed by Hamilton in 1843. Recently, with the development of technology, quaternions have been used to solve some multidimensional problems [22,23,24,25,26]. Under such circumstances, as the general form of real and complex field NNs, QVNNs are naturally constructed and investigated by some scholars. In the past several years, many scholars have been paying attention to the QVNNs system and have obtained some significant conclusions [7,20,27,28,29,30]. In [27,28], the stability criteria of the LMI form for delayed QVNNs were given by the Lyapunov–Krasovskii functional. In [20,29], mixed delayed QVNNs were studied to obtain their existence and stability conditions. Then, differing from these normal methods, the authors in [30] provided the stability of QVNNs by introducing $\{\xi ,\infty \}$-norm [31], which is an effective meaningful technique for the dynamical behaviors investigation of nonlinear systems.

Recently, a novel model-QVMNNs based on MNNs and QVNNs was constructed and studied by some scholars [7,9,11,12]. Since its states, activation functions, external input and memristive connection weights are QVs, it is more complicated but also more effective than RVNNs and CVNNs to some extent. Therefore, some special technologies should be introduced to study the dynamical behaviors of QVMNNs. Furthermore, due to the complexity of impulsive systems, the dynamical behavior analysis of QVMNNs with impulsive disturbances is still an important open problem, which motivates us to research the $\mu$-stability of QVMNNs with impulsive destabilization by special techniques.

Among all the dynamical behaviors, stability and synchronization [32,33,34,35] are the most important properties. Therefore, as a generalization stability, $\mu$-stability, including power and exponential stability, has attracted many scholars’ attentions [30,31,36,37]. These papers provided several methods for the $\mu$-stability of different neural network models with or without impulsive effects. In [37], the $\mu$-stability of mixed delayed NNs with impulsive effects was discussed by constructing a special Lyapunov–Krasovskii functional. The solution existence and its $\mu$-stability criteria for the mixed delayed neutral-type impulsive CVNNs were investigated by constructing proper Lyapunov–Krasovskii functionals in [36]. The generalized $\{\xi ,\infty \}$-norm was, respectively, used in [30,31] to discuss the $\mu$-stable conditions of QVNNs and RVNNs without impulsive effects. According to the methods of these papers and the $\{\xi ,\infty \}$-norm, the research of $\mu$-stability for the QVMNNs with impulsive destabilization is a meaningful and possible objective.

Based on these analyses, this paper mainly aims to study the solution existence and $\mu$-stability conditions of impulsive QVMNNs by the $\{\xi ,\infty \}$-norm and other techniques. The main contributions are as follows:

(1)

A novel more complex multidimensional nonlinear dynamical system QVMNNs mathematical model with impulsive destabilization effects is established. Hamilton rules and differential inclusion theories are used to work out the noncommutativity of quaternion multiplication and switching memristive connection weight.

(2)

Compared with the existing results, the existence sufficient criteria of an equilibrium point of complex impulsive QVMNNs are firstly obtained by an M-matrix, differential inclusion theories and a homeomorphism map.

(3)

The $\mu$-stability criteria of impulsive QVMNNs are obtained by the concept of a generalized $\{\xi ,\infty \}$-norm, which is different from some published conclusions by constructing Lyapunov functions. Taking advantage of the M-matrix, $\{\xi ,\infty \}$-norm and impulse analysis method, the proving process of globally $\mu$-stable conditions for impulsive QVMNNs is simpler and more concise than some existing results. At the same time, exponential stability criteria and power stability criteria can also be concluded from $\mu$-stability criteria, which promotes some published theories.

2. Models and Preliminaries

The next QVMNNs with impulsive destabilization will be constructed in the quaternion field, which is a more complicated system. Since a quaternion has some different properties from real and complex numbers, its basic knowledge can be found in [27,28], which are omitted in this paper.

Based on MNNs and QVNNs, the following impulsive QVMNN without delays is constructed by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_p\left(t\right)=-a_p{x}_p\left(t\right)+\sum _{q=1}^n{b}_{pq}\left(x_p\left(t\right)\right)g_q\left(x_q\left(t\right)\right)+{\theta }_p,t\ne {\tau }_{\kappa },\\ ▵x_p\left({\tau }_k\right)=x_p\left({\tau }_k\right)-x_p\left({\tau }_k^{-}\right)={\mathcal{J}}_{pk}{x}_p\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+},p=1,2,\cdots ,n,\end{array}\right.\end{array}$$

(1)

or, equivalently

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}\left(t\right)=-\mathcal{A}x\left(t\right)+\mathcal{B}\left(x\left(t\right)\right)g\left(x\left(t\right)\right)+\Theta ,t\ne {\tau }_k,\\ ▵x\left({\tau }_k\right)=x\left({\tau }_k\right)-x\left({\tau }_k^{-}\right)={\mathcal{J}}_{k}x\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+},\end{array}\right.\end{array}$$

(2)

where $x_p\left(t\right)=x_p^R\left(t\right)+ıx_p^I\left(t\right)+ȷx_p^J\left(t\right)+\kappa x_p^K\left(t\right)\in \mathbb{Q}$ is the quaternion state vector (ı, ȷ, $\kappa$ are the imaginary units), $g_q(x_q\left(t\right)$ is the quaternion activation function, ${\tau }_k$ is the impulse time sequence, which satisfies $0\le {\tau }_0<{\tau }_1<{\tau }_2\cdots <{\tau }_k<\cdots$, and ${lim}_{k\to +\infty }{\tau }_k=+\infty$, $x_p\left({\tau }_k^{-}\right)={lim}_{t\to {\tau }_k^{-}}{x}_p\left(t\right)$, $x_p\left({\tau }_k^{+}\right)={lim}_{t\to {\tau }_k^{+}}{x}_p\left(t\right)=x_p\left({\tau }_k\right)$, ${\mathcal{J}}_{pk}={\mathcal{J}}_{pk}^R+ı{\mathcal{J}}_{pk}^I+ȷ{\mathcal{J}}_{pk}^J+\kappa {\mathcal{J}}_{pk}^K\in \mathbb{Q}$ denotes the impulsive jump operator at ${\tau }_k$. Here, ${\mathcal{J}}_{pk}{x}_p\left({\tau }_k^{-}\right)$ is denoted by ${\mathcal{J}}_{pk}^R{x}_p^R\left(t\right)+ı{\mathcal{J}}_{pk}^I{x}_p^I\left(t\right)+ȷ{\mathcal{J}}_{pk}^J{x}_p^J\left(t\right)+\kappa {\mathcal{J}}_{pk}^K{x}_p^K\left(t\right)$ for ease of calculation. $x\left(t\right)={(x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))}^T\in {\mathbb{Q}}^n$, $g\left(x\left(t\right)\right)={(g_1\left(x_1\left(t\right)\right),g_2\left(x_2\left(t\right)\right),\cdots ,g_n\left(x_n\left(t\right)\right))}^T$, ${\mathcal{J}}_k={({\mathcal{J}}_{1k},{\mathcal{J}}_{2k},\cdots ,{\mathcal{J}}_{nk})}^T$. The self-inhibition matrix is $\mathcal{A}=\mathrm{diag}\{a_1,a_2,\cdots ,a_n\}\in {\mathbb{R}}^{n\times n}$ with $a_p>0$, the memristor-based Quaternion connection weight matrix is $\mathcal{B}\left(x\left(t\right)\right)={\left[b_{pq}\left(x_p\left(t\right)\right)\right]}_{n\times n}\in {\mathbb{Q}}^{n\times n}$, $\Theta ={({\theta }_1,{\theta }_2,\cdots ,{\theta }_n)}^T\in {\mathbb{Q}}^n$ is the quaternion bias vector or external input, where ${\mathbb{R}}^{n\times n}$, $\mathbb{Q}$, ${\mathbb{Q}}^n$, and ${\mathbb{Q}}^{n\times n}$ are, respectively, the $n\times n$ real matrices set, the quaternion number set, the n-dimensional quaternion space, and the $n\times n$ quaternion matrices set. In the next part, the following notations may be used: $P>0$ and $P^T$ denote, respectively, the positive definite and the transpose of $P\in {\mathbb{R}}^{n\times n}$, E denotes the unit matrix, $co\left\{\varpi \right\}$ is the closure of the convex hull $\varpi \in \mathbb{Q}$.

Due to the memristor properties, the memristor-based connection weight $b_{pq}\left(x_p\left(t\right)\right)=b_{pq}^R\left(x_p\right)+ıb_{pq}^I\left(x_p\right)+ȷb_{pq}^J\left(x_p\right)+\kappa b_{pq}^K\left(x_p\right)$ can be defined by

$$\begin{array}{ccc}\hfill & & b_{pq}\left(x_p\left(t\right)\right)=\left\{\begin{array}{cc}{\widehat{b}}_{pq}={\widehat{b}}_{pq}^R+ı{\widehat{b}}_{pq}^I+ȷ{\widehat{b}}_{pq}^J+\kappa {\widehat{b}}_{pq}^K,& |x_p|\le {\mathcal{T}}_p\\ {\check{b}}_{pq}={\check{b}}_{pq}^R+ı{\check{b}}_{pq}^I+ȷ{\check{b}}_{pq}^J+\kappa {\check{b}}_{pq}^K,& |x_p|\ge {\mathcal{T}}_p,\end{array}\right.\hfill \end{array}$$

where ${\widehat{b}}_{pq}^i,{\check{b}}_{pq}^i$ are real constants with $i\in \{R,I,J,K\}\triangleq \mathbb{M}$, and ${\mathcal{T}}_p$ is the switching jump. Let $\widehat{\mathcal{B}}={\left[{\widehat{b}}_{pq}\right]}_{n\times n}$, $\check{\mathcal{B}}={\left[{\check{b}}_{pq}\right]}_{n\times n}$; then, from (1) and (2), the following forms can be obtained:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_p\left(t\right)\in -a_p{x}_p\left(t\right)+\sum _{q=1}^{n}co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}{g}_q\left(x_q\left(t\right)\right)+{\theta }_p,t\ne {\tau }_k,\\ ▵x_p\left({\tau }_k\right)={\mathcal{J}}_{pk}{x}_p\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+},\end{array}\right.\end{array}$$

(3)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}\left(t\right)\in -\mathcal{A}x\left(t\right)+co\{\widehat{\mathcal{B}},\check{\mathcal{B}}\}g\left(x\left(t\right)\right)+\Theta ,t\ne {\tau }_k,\\ ▵x\left({\tau }_k\right)={\mathcal{J}}_{k}x\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+}.\end{array}\right.\end{array}$$

(4)

The following assumptions for the quaternion activation function and differential inclusion are given as follows:

(H1)

$g_q\left(x_q\right)$ can be written as $g_q\left(x_q\right)=g_q^R(x_q^R,x_q^I,x_q^J,x_q^K)$ + $ıg_q^I(x_q^R,x_q^I,x_q^J,x_q^K)$ + $ȷg_q^J(x_q^R,x_q^I,x_q^J,x_q^K)$ + $\kappa g_q^K(x_q^R,x_q^I,x_q^J,x_q^K),$ where $g_q^i(·,·,·,·)\triangleq g_q^i:{\mathbb{R}}^4\to \mathbb{R}$ with $g_q^i(0,0,0,0)=0$ satisfies

(1)

Every partial derivative of $g_q^i$ corresponding to $x_q^j$: $\partial g_q^i/\partial x_q^j$ exists and is continuous;

(2)

For any $i,j\in \mathbb{M}$, there exists a real positive number ${\lambda }_q^{ij}$, such that $|\partial g_q^i/\partial x_q^j|\le {\lambda }_q^{ij}$ holds.

Then, for any $x_q=x_q^R+ıx_q^I+ȷx_q^J+\kappa x_q^K$, $y_q=y_q^R+ıy_q^I+ȷy_q^J+\kappa y_q^K\in \mathbb{Q}$, we can obtain that

$$\begin{array}{c}\hfill |g_q^i(x_q^R,x_q^I,x_q^J,x_q^K)-g_q^i(y_q^R,y_q^I,y_q^J,y_q^K)|\le \sum _{j\in \mathbb{M}}{\lambda }_q^{ij}|x_q^j-y_q^j|.\end{array}$$

(H2)

Suppose that $x_q$, $y_q\in \mathbb{Q}$ are any two solutions of impulsive QVMNN (3), ${\phi }_q$, ${\psi }_q\in \mathbb{Q}$ are their initial conditions, respectively, the condition $co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}{g}_q\left(x_q\right)-co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}{g}_q\left(y_q\right)$⊆$co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}(g_q\left(x_q\right)-g_q\left(y_q\right))$ holds.

By (H1), Equation (3) can be separated into the following four general RVNNs forms:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_p^{\mathcal{L}}\left(t\right)\in -a_p{x}_p^{\mathcal{L}}\left(t\right)+\sum _{q=1}^n{\left(co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}{g}_q\left(x_q\left(t\right)\right)\right)}^{\mathcal{L}}+{\theta }_p^{\mathcal{L}},t\ne {\tau }_k,\\ ▵x_p^{\mathcal{L}}\left({\tau }_k\right)={\mathcal{J}}_{pk}^{\mathcal{L}}{x}_p^{\mathcal{L}}\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+},\end{array}\right.\end{array}$$

(5)

or there exists $b_{pq}\left(t\right)\in co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}$, such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_p^{\mathcal{L}}\left(t\right)=-a_p{x}_p^{\mathcal{L}}\left(t\right)+\sum _{q=1}^n{\left(b_{pq}\left(t\right)g_q\left(x_q\left(t\right)\right)\right)}^{\mathcal{L}}+{\theta }_p^{\mathcal{L}},t\ne {\tau }_k,\\ ▵x_p^{\mathcal{L}}\left({\tau }_k\right)={\mathcal{J}}_{pk}^{\mathcal{L}}{x}_p^{\mathcal{L}}\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+}.\end{array}\right.\end{array}$$

(6)

In matrix format, they can be rewritten as

$$\left\{\begin{array}{c}{\dot{x}}^{\mathcal{L}}\left(t\right)\in -\mathcal{A}{x}^{\mathcal{L}}\left(t\right)+{\left(co\{\widehat{\mathcal{B}},\check{\mathcal{B}}\}g\left(x\left(t\right)\right)\right)}^{\mathcal{L}}+{\Theta }^{\mathcal{L}},t\ne {\tau }_k,\\ ▵x^{\mathcal{L}}\left({\tau }_k\right)={\mathcal{J}}_k^{\mathcal{L}}{x}^{\mathcal{L}}\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+},\end{array}\right.$$

or

$$\left\{\begin{array}{c}{\dot{x}}^{\mathcal{L}}\left(t\right)=-\mathcal{A}{x}^{\mathcal{L}}\left(t\right)+{\left(\mathcal{B}\left(t\right)g\left(x\right(t\left)\right)\right)}^{\mathcal{L}}+{\Theta }^{\mathcal{L}},t\ne {\tau }_k,\\ ▵x^{\mathcal{L}}\left({\tau }_k\right)={\mathcal{J}}_k{x}^{\mathcal{L}}\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+}\end{array}\right.$$

, where $\mathcal{L}\in \mathbb{M}$, $\mathcal{B}\left(t\right)\in co\{\widehat{\mathcal{B}},\check{\mathcal{B}}\}$, ${\theta }_p^L\left({\Theta }^L\right)$ is the real part or one imaginary part of ${\theta }_p(\Theta )$.

For any $i,j\in \mathbb{M}$, let ${\mathcal{B}}^{ij}={\mathcal{B}}^{ij}\left(t\right)\in co\{{\widehat{\mathcal{B}}}^i,{\check{\mathcal{B}}}^i\}$ with ${\widehat{\mathcal{B}}}^i={\left[{\widehat{b}}_{pq}^i\right]}_{n\times n}$ and ${\check{\mathcal{B}}}^i={\left[{\check{b}}_{pq}^i\right]}_{n\times n}$, $\mathcal{Z}\left(t\right)={(x^R,x^I,x^J,x^K)}^T$, $\Xi ={({\Theta }^R,{\Theta }^I,{\Theta }^J,{\Theta }^K)}^T$, $G^i\left(\mathcal{X}\left(t\right)\right)={\left(g^i,g^i,g^i,g^i\right)}^T$, $\Gamma =\mathrm{diag}\{\mathcal{A},\mathcal{A},\mathcal{A},\mathcal{A}\}$, $A_R=\mathrm{diag}\{{\mathcal{B}}^{RR},{\mathcal{B}}^{II},{\mathcal{B}}^{JJ},{\mathcal{B}}^{KK}\}$, $A_I=\mathrm{diag}\{-{\mathcal{B}}^{IR},{\mathcal{B}}^{RI},{\mathcal{B}}^{KJ},-{\mathcal{B}}^{JK}\}$, $A_J=\mathrm{diag}\{-{\mathcal{B}}^{JR},-{\mathcal{B}}^{KI},{\mathcal{B}}^{RJ},{\mathcal{B}}^{IK}\}$, $A_K=\mathrm{diag}\{-{\mathcal{B}}^{KR},{\mathcal{B}}^{JI},-{\mathcal{B}}^{IJ},{\mathcal{B}}^{RK}\}$, and

$$\begin{array}{c}\hfill \Omega =\left(\begin{array}{cccc}{\mathcal{B}}^R& {\mathcal{B}}^I& {\mathcal{B}}^J& {\mathcal{B}}^K\\ {\mathcal{B}}^I& {\mathcal{B}}^R& {\mathcal{B}}^K& {\mathcal{B}}^J\\ {\mathcal{B}}^J& {\mathcal{B}}^K& {\mathcal{B}}^R& {\mathcal{B}}^I\\ {\mathcal{B}}^K& {\mathcal{B}}^J& {\mathcal{B}}^I& {\mathcal{B}}^R\end{array}\right),\Lambda =\left(\begin{array}{cccc}{\lambda }^{RR}& {\lambda }^{RI}& {\lambda }^{RJ}& {\lambda }^{RK}\\ {\lambda }^{IR}& {\lambda }^{II}& {\lambda }^{IJ}& {\lambda }^{IK}\\ {\lambda }^{JR}& {\lambda }^{JI}& {\lambda }^{JJ}& {\lambda }^{JK}\\ {\lambda }^{KR}& {\lambda }^{KI}& {\lambda }^{KJ}& {\lambda }^{KK}\end{array}\right)\end{array}$$

where ${\mathcal{B}}^i={\left[b_{pq}^i\right]}_{n\times n}$ with $b_{pq}^i=max\left\{\right|{\widehat{b}}_{pq}^i|,|{\check{b}}_{pq}^i\left|\right\}$, ${\lambda }^{ij}=\mathrm{diag}\{{\lambda }_1^{ij},{\lambda }_2^{ij},\cdots ,{\lambda }_n^{ij}\}$, then, from the equations of ${\dot{x}}^i\left(t\right)$, it can be obtained that

$$\begin{array}{ccc}\hfill \dot{\mathcal{Z}}\left(t\right)& =& -\Gamma \mathcal{Z}\left(t\right)+A_R{G}^R\left(\mathcal{Z}\left(t\right)\right)+A_I{G}^I\left(\mathcal{Z}\left(t\right)\right)+A_J{G}^J\left(\mathcal{Z}\left(t\right)\right)+A_K{G}^K\left(\mathcal{Z}\left(t\right)\right)+\Xi \hfill \\ & =& -\Gamma \mathcal{Z}\left(t\right)+\sum _{i\in \mathbb{M}}{A}_i{G}^i\left(\mathcal{Z}\left(t\right)\right)+\Xi .\hfill \end{array}$$

(7)

Remark 1.

Since quaternion multiplication is noncommutative, it is necessary to decompose the impulsive QVMNNs into four impulsive real-valued MNNs (RVMNNs) by Hamilton rules to investigate the solution existence and its stability. Obviously, the impulsive RVMNNs system (7) is the decomposed form of impulsive QVMNNs system (1) or (2), which implies that it is only necessary to consider solution existence problem and global μ-stability of system (7). That is, if the existence conditions and global μ-stability of (7) are derived, then the relative conditions of system (1) are concluded.

Definition 1

([31]). $\{\xi ,\infty \}$-norm: for any $u\left(t\right)={(u_1\left(t\right),u_2\left(t\right),\cdots ,u_n\left(t\right))}^T\in {\mathbb{R}}^{n\times 1}$, ${\parallel u\left(t\right)\parallel }_{(\xi ,\infty )}={max}_{1\le i\le n}\left|{\xi }_i^{-1}{u}_i\left(t\right)\right|$, here, ${\xi }_i>0$ is a constant.

Remark 2.

In Definition 1, $\{\xi ,\infty \}$-norm is defined in $\mathbb{R}$. Similarly, $\{\xi ,\infty \}$-norm in the quaternion field can be defined as ${\parallel u\left(t\right)\parallel }_{(\xi ,\infty )}={max}_{\mathcal{L}\in \mathbb{M}}{max}_{1\le i\le n}\left|{\left({\xi }_i^{\mathcal{L}}\right)}^{-1}{u}_i^{\mathcal{L}}\left(t\right)\right|$ with ${\xi }_i^{\mathcal{L}}>0$. When $\xi =1$, $\{\xi ,\infty \}$-norm is the general ∞-norm, which is a generalized norm.

Definition 2.

The equilibrium point $x^{\ast }$ of impulsive QVMNN (1) is globally μ-stable if for any solution $x\left(t\right)$, ${\parallel x\left(t\right)-x^{\ast }\parallel }_{(\xi ,\infty )}=O\left({\mu }^{-1}\left(t\right)\right)$ holds, where $\mu \left(t\right)\ge 0$ is a continuous function with ${lim}_{t\to \infty }\mu \left(t\right)=\infty$, and $O(·)$ means infinitesimal of the same order.

Lemma 1

([36]). If continuous function $\mathcal{H}\left(t\right):{\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfies (1) $\mathcal{H}\left(x\right)$ is injective on ${\mathbb{R}}^n$; (2) ${lim}_{\parallel t\parallel \to \infty }\parallel \mathcal{H}\left(t\right)\parallel =\infty$, then $\mathcal{H}\left(t\right)$ is said to be a homeomorphism in ${\mathbb{R}}^n$.

Lemma 2

([38]). Suppose $A={\left[a_{pq}\right]}_{n\times n}\in {\mathbb{R}}^{n\times n}$ is a nonsingular matrix with $a_{pq}\le 0(p\ne q)$, then A is an M-matrix if one of the next two propositions holds:

(I) 

There exists a symmetric matrix $Q>0$ so as to $AQ+Q{A}^T>0$;

(II) 

There exists $\alpha ={({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)}^T$ with ${\alpha }_p>0$ so as to $a_{pp}{\alpha }_p>{\sum }_{q\ne p}\left|a_{pq}\right|{\alpha }_q$ holds for $p=1,2,\cdots ,n$, which can also be written as ${\sum }_{q=1}^n\left|a_{pq}\right|{\alpha }_q>0$.

3. Main Results

For the impulsive QVMNNs (1), this section mainly aims to formulate its sufficient criteria of solution existence and then discuss its global $\mu$-stability by a homeomorphism map, M-matrix, $\{\xi ,\infty \}$-norm and special analytical methods of impulsive differential systems.

Theorem 1.

Suppose assumptions (H1) and (H2) hold, if $\Gamma -\Omega \Lambda$ is a nonsingular M-matrix, then system (1) or (2) has an unique equilibrium point $x^{\ast }\left(t\right)$, where Γ, Ω and Λ are defined in the above section.

Proof. 

According to (H1) and (7), we can define $\mathcal{H}\left(\mathcal{Z}\right(t\left)\right)$ as

$$\begin{array}{c}\hfill \mathcal{H}\left(\mathcal{Z}\left(t\right)\right)=-\Gamma \mathcal{Z}\left(t\right)+\sum _{i\in \mathbb{M}}{A}_i{G}^i\left(\mathcal{Z}\left(t\right)\right)+\Xi .\end{array}$$

Suppose that there exist two different values ${\mathcal{X}}_1\left(t\right)={(x_1^R,x_1^I,x_1^J,x_1^K)}^T$ and ${\mathcal{X}}_2\left(t\right)={(x_2^R,x_2^I,x_2^J,x_2^K)}^T$ such that $\mathcal{H}\left({\mathcal{X}}_1\right)=\mathcal{H}\left({\mathcal{X}}_2\right)$, we have

$$\begin{array}{c}\hfill -\Gamma ({\mathcal{X}}_1-{\mathcal{X}}_2)+\sum _{i\in \mathbb{M}}(A_i^1{G}^i\left({\mathcal{X}}_1\right)-A_i^2{G}^i\left({\mathcal{X}}_2\right))=0,\end{array}$$

(8)

where $A_i^1$ and $A_i^2$ denote, respectively, the connect weights matrix at different values ${\mathcal{X}}_1\left(t\right)$ and ${\mathcal{X}}_2\left(t\right)$. According to (H2), we have

$$\begin{array}{ccc}\hfill & & a_p(x_{1p}^R\left(t\right)-x_{2p}^R\left(t\right))\in \sum _{q=1}^{n}co\{{\widehat{b}}_{pq}^R,{\check{b}}_{pq}^R\}\left(g_q^R(x_{1q}^R\left(t\right),x_{1q}^I\left(t\right),x_{1q}^J\left(t\right),x_{1q}^K\left(t\right))-g_q^R(x_{2q}^R\left(t\right),x_{2q}^I\left(t\right),x_{2q}^J\left(t\right),x_{2q}^K\left(t\right))\right)\hfill \\ & & -\sum _{q=1}^{n}co\{{\widehat{b}}_{pq}^I,{\check{b}}_{pq}^I\}\left(g_q^I(x_{1q}^R\left(t\right),x_{1q}^I\left(t\right),x_{1q}^J\left(t\right),x_{1q}^K\left(t\right))-g_q^I(x_{2q}^R\left(t\right),x_{2q}^I\left(t\right),x_{2q}^J\left(t\right),x_{2q}^K\left(t\right))\right)\hfill \\ & & -\sum _{q=1}^{n}co\{{\widehat{b}}_{pq}^J,{\check{b}}_{pq}^J\}\left(g_q^J(x_{1q}^R\left(t\right),x_{1q}^I\left(t\right),x_{1q}^J\left(t\right),x_{1q}^K\left(t\right))g_q^J(x_{2q}^R\left(t\right),x_{2q}^I\left(t\right),x_{2q}^J\left(t\right),x_{2q}^K\left(t\right))\right)\hfill \\ & & -\sum _{q=1}^{n}co\{{\widehat{b}}_{pq}^K,{\check{b}}_{pq}^K\}\left(g_q^K(x_{1q}^R\left(t\right),x_{1q}^I\left(t\right),x_{1q}^J\left(t\right),x_{1q}^K\left(t\right))-g_q^K(x_{2q}^R\left(t\right),x_{2q}^I\left(t\right),x_{2q}^J\left(t\right),x_{2q}^K\left(t\right))\right).\hfill \end{array}$$

Furthermore, by (H1), we have

$$\begin{array}{ccc}& & \mathcal{A}|x_1^R\left(t\right)-x_2^R\left(t\right)|\le ({\mathcal{B}}^R{\lambda }^{RR}+{\mathcal{B}}^I{\lambda }^{IR}+{\mathcal{B}}^J{\lambda }^{JR}+{\mathcal{B}}^K{\lambda }^{KR})|x_1^R\left(t\right)-x_2^R\left(t\right)|+({\mathcal{B}}^R{\lambda }^{RI}+{\mathcal{B}}^I{\lambda }^{II}+\hfill \\ & & {\mathcal{B}}^J{\lambda }^{JI}+{\mathcal{B}}^K{\lambda }^{KI}\left)\right|x_1^I\left(t\right)-x_2^I\left(t\right)|+({\mathcal{B}}^R{\lambda }^{RJ}+{\mathcal{B}}^I{\lambda }^{IJ}+{\mathcal{B}}^J{\lambda }^{JJ}+{\mathcal{B}}^K{\lambda }^{KJ})\hfill \\ & & |x_1^J\left(t\right)-x_2^J\left(t\right)|+({\mathcal{B}}^R{\lambda }^{RK}+{\mathcal{B}}^I{\lambda }^{IK}+{\mathcal{B}}^J{\lambda }^{JK}+{\mathcal{B}}^K{\lambda }^{KK})|x_1^K\left(t\right)-x_2^K\left(t\right)|.\hfill \end{array}$$

By means of a similar way, we can obtain other inequalities for $\mathcal{A}|x_1^i\left(t\right)-x_2^i\left(t\right)|,i\in \{I,J,K\}$, and the details are omitted.

Let $|{\mathcal{X}}_1-{\mathcal{X}}_2|=(|x_1^R-x_2^R|,|x_1^I-x_2^I|,|x_1^J-x_2^J|,|x_1^K-x_2^K{\left|\right)}^T$, then we can derive

$$\begin{array}{c}\hfill (\Gamma -\Omega \Lambda )|{\mathcal{X}}_1-{\mathcal{X}}_2|\le 0.\end{array}$$

(9)

However, from the given condition, $\Gamma -\Omega \Lambda$ is a nonsingular M-matrix, and it can be obtained that

$$\begin{array}{c}\hfill (\Gamma -\Omega \Lambda )|{\mathcal{X}}_1-{\mathcal{X}}_2|>0.\end{array}$$

(10)

Obviously, (9) and (10) are contradictory. That is to say, $\mathcal{H}\left(\mathcal{Z}\right(t\left)\right)$ is injective.

On the other hand, according to M-matrix $\Gamma -\Omega \Lambda$ and Lemma 1, we can find a constant $ϵ>0$ and a matrix ${\rm Y}>0$, such that ${\rm Y}(\Omega \Lambda -\Gamma )+{(\Omega \Lambda -\Gamma )}^T{\rm Y}\le -ϵE_{4n}<0$. Then, by (H1), we can obtain $\mathcal{H}\left(t\right)\triangleq \mathcal{H}\left(\mathcal{Z}\right)-\mathcal{H}\left(0\right)=-\Gamma \mathcal{Z}+{\sum }_{i\in \mathbb{M}}(A_i{G}^i\left(\mathcal{Z}\right)-A_i{G}^i\left(0\right))$. As a result,

$$\begin{array}{ccc}& & {\mathcal{Z}}^T\left(t\right){\rm Y}\mathcal{H}\left(t\right)+{\mathcal{H}}^T\left(t\right){\rm Y}\mathcal{Z}\left(t\right)\le {\left|\mathcal{Z}\left(t\right)\right|}^T{\rm Y}(\Omega \Lambda -\Gamma )\left|\mathcal{Z}\left(t\right)\right|+{\left|\mathcal{Z}\left(t\right)\right|}^T{(\Omega \Lambda -\Gamma )}^T{\rm Y}\left|\mathcal{Z}\left(t\right)\right|\le -{ϵ\parallel \mathcal{Z}\left(t\right)\parallel }^2.\hfill \end{array}$$

From that, it can be obtained ${ϵ\parallel \mathcal{Z}\left(t\right)\parallel }^2\le 2\parallel {\rm Y}\parallel \parallel \mathcal{H}\left(t\right)\parallel \parallel \mathcal{Z}\left(t\right)\parallel ,$ which means that ${lim}_{\parallel \mathcal{Z}\left(t\right)\parallel \to \infty }\parallel \mathcal{H}\left(\mathcal{Z}\left(t\right)\right)\parallel =\infty$.

Based on the above analyses, the conditions of the homeomorphism map are satisfied. Therefore, $\mathcal{H}\left(\mathcal{Z}\right(t\left)\right)$ is injective and surjective in ${\mathbb{R}}^{4n}$, which implies that QVMNNs system (1) or (2) has an unique equilibrium point. □

Remark 3.

For this theorem, it is proved that the equilibrium point of QVMNNs system (1) or (2) exists and is unique, which can also be used to study the delayed QVMNNs. Based on the study of Theorem 1, the global μ-stability of this system will be discussed by the generalized $\{\xi ,\infty \}$-norm definition and other techniques.

Theorem 2.

Suppose (H1) and (H2) hold; then, the impulsive QVMNNs system (1) or (2) is globally μ-stable if there exist constants $\varsigma ,\sigma >0$ and continuous differential function $\mu \left(t\right)\ge 0$, such that $\Gamma -\Omega \Lambda -(\sigma +\varsigma )E_{4n}$ is a nonsingular M-matrix,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\mu \left(t\right)=\infty ,\underset{t\to \infty }{lim}\frac{\dot{\mu }\left(t\right)}{\mu \left(t\right)}=\varsigma ,0<\frac{ln(1+{\mathcal{J}}_{pk}^{\mathcal{L}})}{{\tau }_k-{\tau }_{k-1}}\le \sigma ,\end{array}$$

Proof. 

Suppose $x^{\ast }=x^{\ast R}+ıx^{\ast I}+ȷx^{\ast J}+\kappa x^{\ast K}$ is the equilibrium point of the impulsive QVMNNs system (1). Let $\overline{\omega }\left(t\right)=x\left(t\right)-x^{\ast }=x^R-x^{\ast R}+ı(x^I-x^{\ast I})+ȷ(x^J-x^{\ast J})+\kappa (x^K-x^{\ast K})$; then from (3) and (5), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\overline{\omega }}}_p\left(t\right)\in -a_p{\overline{\omega }}_p\left(t\right)+\sum _{q=1}^{n}co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}{\overline{g}}_q\left({\overline{\omega }}_q\left(t\right)\right),t\ne {\tau }_k,\\ ▵{\overline{\omega }}_p\left({\tau }_k\right)={\mathcal{J}}_{pk}{\overline{\omega }}_p\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+}.\end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\overline{\omega }}}_p^{\mathcal{L}}\left(t\right)\in -a_p{\overline{\omega }}_p^{\mathcal{L}}\left(t\right)+\sum _{q=1}^n{\left(co\{{\widehat{b}}_{pq},{\check{b}}_{pq}\}{\overline{g}}_q\left({\overline{\omega }}_q\left(t\right)\right)\right)}^{\mathcal{L}},t\ne {\tau }_k,\\ ▵{\overline{\omega }}_p^{\mathcal{L}}\left({\tau }_k\right)={\mathcal{J}}_{pk}^{\mathcal{L}}{\overline{\omega }}_p^{\mathcal{L}}\left({\tau }_k^{-}\right),k\in {\mathbb{Z}}^{+},\end{array}\right.\end{array}$$

where ${\overline{g}}_q\left({\overline{\omega }}_q\left(t\right)\right)=g_q({\overline{\omega }}_q\left(t\right)+x_q^{\ast })-g_q\left(x_q^{\ast }\right)$.

According to the nonsingular M-matrix $\Gamma -\Omega \Lambda -(\sigma +\varsigma )E_{4n}$ and condition (II) of Lemma 1, there exist a series of real constants ${\xi }_p^{\mathcal{L}}>0$, such that

$$\begin{array}{c}\hfill {\xi }_p^{\mathcal{L}}(a_p-(\sigma +\varsigma ))-\sum _{q=1}^n\sum _{(i,j)\in {\mathbb{M}}^{\mathcal{L}}}\sum _{r\in \mathbb{M}}{b}_{pq}^i{\lambda }_q^{jr}{\xi }_q^r>0,\end{array}$$

holds, where ${\mathbb{M}}^{\mathcal{L}}\in \{{\mathbb{M}}^R,{\mathbb{M}}^I,{\mathbb{M}}^J,{\mathbb{M}}^K\}$, where ${\mathbb{M}}^R$ is the set $\left\{\right(R,R),(I,I),(J,J),(K,K\left)\right\}$, ${\mathbb{M}}^I$ is the set $\left\{\right(R,I),(I,R),(J,K),(K,J\left)\right\}$, ${\mathbb{M}}^J$ is the set $\left\{\right(R,J),(I,K),(J,R),(K,I\left)\right\}$, and ${\mathbb{M}}^K$ is the set $\left\{\right(R,K),(I,J),(J,I),(K,R\left)\right\}$. That is, there exists a large enough constant number $\mathcal{T}>0$, such that

$$\begin{array}{c}\hfill {\xi }_p^{\mathcal{L}}\left(a_p-\frac{{\left(e^{\sigma t}\mu \left(t\right)\right)}^{\prime }}{e^{\sigma t}\mu \left(t\right)}\right)-\sum _{q=1}^n\sum _{(i,j)\in {\mathbb{M}}^{\mathcal{L}}}\sum _{r\in \mathbb{M}}{b}_{pq}^i{\lambda }_q^{jr}{\xi }_q^r>0.\end{array}$$

Define $u^{\mathcal{L}}\left(t\right)={(u_1^{\mathcal{L}}\left(t\right),u_2^{\mathcal{L}}\left(t\right),\cdots ,u_n^{\mathcal{L}}\left(t\right))}^T$ with $u_p^{\mathcal{L}}\left(t\right)=e^{\sigma t}\mu \left(t\right){\overline{x}}_p^{\mathcal{L}}\left(t\right)$ and $\mathcal{U}\left(t\right)={max}_{\mathcal{L}\in \mathbb{M}}{\mathcal{U}}^{\mathcal{L}}\left(t\right)$ with ${\mathcal{U}}^{\mathcal{L}}\left(t\right)={sup}_{s\le t}\{\parallel u^{\mathcal{L}}\left(t\right){\parallel }_{\{\xi ,\infty \}}\}$; then, we can derive that $\mathcal{U}\left(t\right)$ is bounded.

When ${\zeta }_0>\mathcal{T}$ and ${\zeta }_0\ne {\tau }_k$, that is, ${\zeta }_0\in ({\tau }_{k-1},{\tau }_k)\cap (\mathcal{T},+\infty )$, let $p_0^{\mathcal{L}}=p_0^{\mathcal{L}}\left({\zeta }_0\right)$ be such an index that ${\parallel {\overline{\omega }}^{\mathcal{L}}\left({\zeta }_0\right)\parallel }_{\{\xi ,\infty \}}=\left|{\left({\xi }_{p_0^{\mathcal{L}}}^{\mathcal{L}}\right)}^{-1}{\overline{\omega }}_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left(t_0\right)\right|$, then

$$\begin{array}{ccc}\hfill \frac{d|u_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left(t\right)|}{dt}{|}_{t={\zeta }_0}& =& \mathrm{sign}\left(u_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left({\zeta }_0\right)\right)\left((\sigma e^{\sigma {\zeta }_0}\mu \left({\zeta }_0\right)+e^{\sigma {\zeta }_0}\dot{\mu }\left({\zeta }_0\right)){\overline{\omega }}_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left({\zeta }_0\right)+e^{\sigma {\zeta }_0}\mu \left(t_0\right){\dot{\overline{\omega }}}_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left({\zeta }_0\right)\right)\hfill \\ & \in & \mathrm{sign}\left(u_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left({\zeta }_0\right)\right)(\sigma e^{\sigma {\zeta }_0}\mu \left({\zeta }_0\right)+e^{\sigma {\zeta }_0}\dot{\mu }\left({\zeta }_0\right)){\overline{\omega }}_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left({\zeta }_0\right)+\mathrm{sign}\left(u_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left({\zeta }_0\right)\right)e^{\sigma {\zeta }_0}\mu \left({\zeta }_0\right)\hfill \\ & & \left(-a_{p_0^{\mathcal{L}}}{\overline{\omega }}_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left({\zeta }_0\right)+\sum _{q=1}^n{\left(co\{{\widehat{b}}_{p_0^{\mathcal{L}}q},{\check{b}}_{p_0^{\mathcal{L}}q}\}{\overline{g}}_q\left({\overline{\omega }}_q\left({\zeta }_0\right)\right)\right)}^{\mathcal{L}}\right)\hfill \\ & \le & \left(-a_{p_0^{\mathcal{L}}}+\frac{{\left(e^{\sigma {\zeta }_0}\mu \left({\zeta }_0\right)\right)}^{\prime }}{e^{\sigma {\zeta }_0}\mu \left({\zeta }_0\right)}\right)\left|u_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left(t_0\right)\right|+\sum _{q=1}^n\sum _{(i,j)\in {\mathbb{M}}^{\mathcal{L}}}{b}_{p_0^{\mathcal{L}}q}^i({\lambda }_q^{jR}\left|u_q^R\left({\zeta }_0\right)\right|\hfill \\ & & +{\lambda }_q^{jI}|u_q^I\left({\zeta }_0\right)|+{\lambda }_q^{jJ}|u_q^J\left({\zeta }_0\right)|+{\lambda }_q^{jK}\left|u_q^K\left({\zeta }_0\right)\right|)\hfill \\ & \le & \left({\xi }_{p_0^{\mathcal{L}}}^{\mathcal{L}}\left(-a_{p_0^{\mathcal{L}}}+\frac{{\left(e^{\sigma {\zeta }_0}\mu \left({\zeta }_0\right)\right)}^{\prime }}{e^{\sigma {\zeta }_0}\mu \left({\zeta }_0\right)}\right)+\sum _{q=1}^n\sum _{(i,j)\in {\mathbb{M}}^{\mathcal{L}}}\sum _{r\in \mathbb{M}}{b}_{pq}^i{\lambda }_q^{jr}{\xi }_q^r\right)M\left({\zeta }_0\right)\hfill \\ & <& 0.\hfill \end{array}$$

As a result, there can be found a positive number $\epsilon$ to make $\mathcal{U}\left(t\right)=\mathcal{U}\left({\zeta }_0\right)$ hold for every $t\in ({\zeta }_0,{\zeta }_0+\epsilon )\cap (\mathcal{T},+\infty )\cap ({\tau }_{k-1},{\tau }_k)$. Especially, if $t\in [{\tau }_{k-1},{\tau }_k)\subseteq (\mathcal{T},+\infty )$, we have $\mathcal{U}\left(t\right)=\mathcal{U}\left(t_{k-1}\right)$. In addition, if $t\in [{\tau }_{k-1},{\tau }_k)⊈(\mathcal{T},+\infty )$ and $(\mathcal{T},+\infty )\cap [{\tau }_{k-1},{\tau }_k)\ne \varnothing$, then $\mathcal{U}\left(t\right)=\mathcal{U}\left(\mathcal{T}\right)$.

Next, we will discuss the case of ${\zeta }_0={\tau }_k\in [\mathcal{T},+\infty )$. Since $▵{\overline{\omega }}_p^L\left({\tau }_k\right)={\mathcal{J}}_{pk}^{\mathcal{L}}{\overline{\omega }}_p^{\mathcal{L}}\left({\tau }_k^{-}\right)$, we have ${\overline{\omega }}_p^{\mathcal{L}}\left({\tau }_k\right)=(1+{\mathcal{J}}_{pk}^{\mathcal{L}}){\overline{\omega }}_p^{\mathcal{L}}\left({\tau }_k^{-}\right)$. Therefore,

$$\begin{array}{c}\hfill |u_p^{\mathcal{L}}\left({\tau }_k\right)|=(1+{\mathcal{J}}_{pk}^{\mathcal{L}})|u_p^{\mathcal{L}}\left({\tau }_k^{-}\right)|\le (1+{\mathcal{J}}_{pk}^{\mathcal{L}})(1+{\mathcal{J}}_{p(k-1)}^{\mathcal{L}})\cdots (1+{\mathcal{J}}_{p(N+1)}^{\mathcal{L}})\left|u_p^{\mathcal{L}}\left({\tau }_N\right)\right|,\end{array}$$

where the positive integer $N\le k$, and it satisfies $[{\tau }_{N-1},{\tau }_N)⊈(\mathcal{T},+\infty )$ and $(\mathcal{T},+\infty )\cap [{\tau }_{N-1},{\tau }_N)\ne \varnothing$.

Based on this inequality and the given conditions of this theorem, we have ${\parallel \overline{\omega }\left(t\right)\parallel }_{\{\xi ,\infty \}}\le \frac{e^{-\sigma {\tau }_N}\mathcal{U}\left(\mathcal{T}\right)}{\mu \left(t\right)}$ for any $t>\mathcal{T}$, which implies $\parallel x\left(t\right)-x^{\ast }{\parallel }_{\{\xi ,\infty \}}=O\left({\mu }^{-1}\left(t\right)\right)$. Therefore, the origin impulsive QVMNNs (1) or (2) is globally $\mu$-stable. □

Remark 4.

Comparing with published results, the generalized $\{\xi ,\infty \}$-norm is firstly used in this theorem to discuss the global μ-stable criteria of impulsive QVMNNs. Since the impulsive system is discontinuous, it is necessary to consider the impulsive points and impulsive interval, which is not easy. From the analysis process, the proving of this theorem looks simpler and more concise. As we know, Lyapunov–Krasovskii functional methods were used to investigate μ-stable sufficient criteria of different impulsive NNs in [36,37], and the sufficient criteria and proving process are more complicated. Therefore, to some extend, compared with the general form, the $\{\xi ,\infty \}$-norm technique may be an effective way to research the dynamic behaviors of NNs. Furthermore, it is worth noting that μ-stability, a general stability, includes power and exponential stability. Naturally, these two kinds of stability for impulsive QVMNNs can also be derived from the above theorem.

Corollary 1.

For Theorem 2, if we let $\mu \left(t\right)=t^{\gamma }$ with $\gamma >0$, then the impulsive QVMNNs (1) is power stable; if we let $\mu \left(t\right)=e^{\varsigma t}$, then the impulsive QVMNNs (1) is globally exponentially stable.

Remark 5.

As we all know, the impulsive effect may break the stability of a stable system, which is named destabilized impulses. In this paper, destabilized impulses are considered to study the stability of QVMNNs. From the results of the research, it can be found that the negative effects of destabilized impulses can be reduced or eliminated by choosing a proper $\mu \left(t\right)$ and impulse interval and strength, and the system can maintain its stability, which implies that this stable system has the desired robustness and disturbance rejection behavior. Therefore, the choice of proper $\mu \left(t\right)$ and impulse interval and strength is very essential in this brief. In addition, for the sake of reducing the calculation complexity, a non-delayed impulsive QVMNNs system is given and studied. In fact, the methods and techniques used in this paper may also be used to research the stability, finite-time stability and synchronization for a more complex impulsive QVMNNs system with mixed delays. Furthermore, when $\xi =1$, the $\{\xi ,\infty \}$-norm is the general ∞-norm, which is an interesting definition to discuss some nonlinear systems.

4. Illustrative Examples

Example 1.

Consider the following impulsive QVMNNs without delays.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_p\left(t\right)=-a_p{x}_p\left(t\right)+\sum _{q=1}^2{b}_{pq}{g}_q\left(x_q\left(t\right)\right)+{\theta }_p,t\ne {\tau }_k,\\ ▵x_p\left({\tau }_k\right)={\mathcal{J}}_{pk}{x}_p\left({\tau }_k^{-}\right),p=1,2,k\in {\mathbb{Z}}^{+},\end{array}\right.\end{array}$$

where $x_p\left(t\right)\in \mathbb{Q}$, $a_1=8$, $a_2=8$, ${\mathcal{J}}_{1k}^{\mathcal{L}}={\mathcal{J}}_{2k}^{\mathcal{L}}=0.15$, ${\tau }_k-{\tau }_{k-1}=0.1$, ${\theta }_1=-1+ı+ȷ+2\kappa$, ${\theta }_2=1-2ı+3ȷ-2\kappa$, $g_q\left(x_q\left(t\right)\right)=\frac{1-exp(-2x_q^R\left(t\right)-x_q^I\left(t\right))}{1+exp(-2x_q^R\left(t\right)-x_q^I\left(t\right))}+ı\frac{1-exp(-2x_q^I\left(t\right)-x_q^J\left(t\right))}{1+exp(-2x_q^I\left(t\right)-x_q^J\left(t\right))}+ȷ\frac{1-exp(-2x_q^J\left(t\right)-x_q^K\left(t\right))}{1+exp(-2x_q^J\left(t\right)-x_q^K\left(t\right))}+\kappa \frac{1-exp(-2x_q^K\left(t\right)-x_q^R\left(t\right))}{1+exp(-2x_q^K\left(t\right)-x_q^R\left(t\right))}$, and

$$\begin{array}{ccc}\hfill & & b_{11}=\left\{\begin{array}{cc}\frac{3}{5}-\frac{3}{10}ı-\frac{3}{5}ȷ+\frac{3}{10}\kappa ,& |x_1\left(t\right)|\le 1,\\ \frac{2}{5}-\frac{1}{5}ı-\frac{2}{5}ȷ+\frac{2}{5}\kappa ,& |x_1\left(t\right)|>1,\end{array}\right.\hfill \\ & & b_{12}=\left\{\begin{array}{cc}-\frac{3}{10}+\frac{3}{5}ı+\frac{2}{5}ȷ-\frac{1}{2}\kappa ,& |x_1\left(t\right)|\le 1,\\ -\frac{1}{5}+\frac{2}{5}ı+\frac{1}{2}ȷ-\frac{2}{5}\kappa ,& |x_1\left(t\right)|>1,\end{array}\right.\hfill \\ & & b_{21}=\left\{\begin{array}{cc}-\frac{1}{5}+\frac{3}{5}ı+\frac{2}{5}ȷ-\frac{1}{2}\kappa ,& |x_2\left(t\right)|\le 1,\\ -\frac{3}{10}+\frac{2}{5}ı+\frac{1}{2}ȷ-\frac{3}{10}\kappa ,& |x_2\left(t\right)|>1,\end{array}\right.\hfill \\ & & b_{22}=\left\{\begin{array}{cc}\frac{3}{10}-\frac{3}{10}ı-\frac{1}{2}ȷ+\frac{3}{10}\kappa ,& |x_2\left(t\right)|\le 1,\\ \frac{1}{5}-\frac{2}{5}ı-\frac{2}{5}ȷ+\frac{1}{5}\kappa ,& |x_2\left(t\right)|>1.\end{array}\right.\hfill \end{array}$$

According to the parameters of the above theorems, we have ${\mathcal{B}}^R=\left(\begin{array}{cc}\frac{3}{5}& \frac{3}{10}\\ \frac{3}{10}& \frac{3}{10}\end{array}\right)$, ${\mathcal{B}}^I=\left(\begin{array}{cc}\frac{3}{10}& \frac{3}{5}\\ \frac{3}{5}& \frac{2}{5}\end{array}\right)$, ${\mathcal{B}}^J=\left(\begin{array}{cc}\frac{3}{5}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}\end{array}\right)$, ${\mathcal{B}}^K=\left(\begin{array}{cc}\frac{2}{5}& \frac{1}{2}\\ \frac{1}{2}& \frac{3}{10}\end{array}\right)$, ${\lambda }^{RR}={\lambda }^{II}={\lambda }^{JJ}={\lambda }^{KK}=\mathrm{diag}\{1,1\}$, ${\lambda }^{RI}={\lambda }^{IJ}={\lambda }^{JK}={\lambda }^{KR}=\mathrm{diag}\{\frac{1}{2},\frac{1}{2}\}$, and ${\lambda }^S=O,\forall S\in \{RJ,RK,IR,IK,JR,JI,KI,KJ\}$, where O is a zero matrix. It can be easily calculated that M-matrix $\Gamma -\Omega \Lambda -$ is nonsingular, which implies the conditions of Theorem 1. Therefore, the impulsive QVMNNs system of this example has a unique equilibrium point. If the proper $\mu \left(t\right)$ is chosen, it can be concluded that $\Gamma -\Omega \Lambda -(\sigma +\varsigma )E_{4n}$ is a nonsingular M-matrix. Then, this system is globally $\mu$-stable. Furthermore, if we let $\mu \left(t\right)=e^{\varsigma t}$ and $\varsigma =\frac{1}{5}$, then this system is globally exponentially stable. By means of Matlab, the simulation figures of this numerical example are given and illustrated by Figure 1, Figure 2, Figure 3 and Figure 4. From these figures, it can be seen that although the impulses of this example are destabilized impulses, its stability can be maintained under proper conditions.

5. Conclusions

Although some important results on the stability of QVNNs have been given recently, little attention has been paid to the $\mu$-stability of impulsive QVMNNs. In this manuscript, a novel QVMNNs system with impulsive destabilization was firstly constructed to investigate its solution existence and $\mu$-stability by generalized $\{\xi ,\infty \}$-norm, differential inclusion, homeomorphism map and impulsive analysis theories. To solve the challenges from the noncommutativity of quaternion multiplication and the switching memristive connection weight, Hamilton rules and differential inclusion theories were considered to decompose impulsive QVMNNs into impulsive real-valued systems. Then, by differential inclusions and a homeomorphism map, sufficient criteria for the existence of the equilibrium point were discussed and proved. Next to this, the $\{\xi ,\infty \}$-norm-based method was used to investigate the boundedness of the constructed system. By combining a M-matrix with the impulsive analysis method, the global $\mu$-stable sufficient criteria of impulsive QVMNNs were obtained in a quaternion field. In addition, the exponential and power stability criteria were also derived from the $\mu$-stability criteria at last. Since the impulse effect of this paper is destabilized impulse, the stable system of this paper has the desired robustness and disturbance rejection behavior. Therefore, the choices of proper $\mu \left(t\right)$ and impulse interval and strength are essential factors for maintaining its stability of given impulsive QVMNNs. In the end, the presented conditions and conclusions of this paper were illustrated by computed and simulated results of one numerical example. Since the methods and techniques used in this paper may be used to solve other quaternion valued neural networks with discrete and distributed delays, the stability, finite-time stability and synchronization of the delayed quaternion valued nonlinear neural systems with mixed delays will be studied in the future.