Novel Synchronization Conditions for the Unified System of Multi-Dimension-Valued Neural Networks

Abstract

This paper discusses the novel synchronization conditions about the unified system of multi-dimension-valued neural networks (USOMDVNN). First of all, the general model of USOMDVNN is successfully set up, mainly on the basis of multidimensional algebra, Kirchhoff current law, and neuronal property. Then, the concise Lyapunov–Krasovskii functional (LKF) and switching controllers are constructed for the USOMDVNN. Moreover, the new inequalities, whose variables, together with some parameters, are employed in a concise and unified form whose variables can be translated into special ones, such as real, complex, and quaternion. It is worth mentioning that the useful parameters really make some contributions to the construction of the concise LKF, the design of the general controllers, and the acquisition of flexible criteria. Further, we acquire the newer criteria mainly by employing Lyapunov analysis, constructing new LKF, applying two unified inequalities, and designing nonlinear controllers. Particularly, the value of the fixed time is less than the other ones in some existing results, owing to the adjustable parameters. Finally, three multidimensional simulations are presented, to demonstrate the availability and progress of the achieved acquisitions.

1. Introduction

Since the 1980s, neural networks have been playing a key role in more and more areas such as image compression, signal processing, artificial intelligence, and so on [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. However, as application demand increases, the usual neural networks based on lower algebra inevitably show their weaknesses or even powerlessness in some complicated and advanced fields. Fortunately, the application areas of neural networks develop rapidly because of the progress made by higher algebra, and concepts such as complex or quaternion are introduced to construct USOMDVNN [4,8,9,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43].

In the real synchronous control system, the signal transmission sometimes needs to depend on the general communication network. For example, these include secure communication in wireless networks, robot cooperation in harsh environments, and multi-agent distributed formation control. The system of neural networks based on multi-dimension is employed to the networked synchronous control system for modeling and analysis. More applications can be realized through the remote control and remote operation of the control system. In addition, more and more researchers have employed the synchronization of complex dynamic networks in information science, power systems, and biological systems, and so on [2,3,4,5,6,7,17,18]. Both the structure and function of the brain’s nervous system can be simulated by more complex dynamic networks with multi-dimension at different levels. Meanwhile, some researchers have applied more complex dynamic networks with multi-dimension in planning the robotic path, collaboratively controlling unmanned aerial vehicles, and searching satellite formation. Naturally, there is an innovative idea for combining neural networks with higher algebra to build complex neural networks that can simultaneously have more in-depth improvement.

Recently, some researchers have focused on various dynamics of neural networks with real algebra [13,16,44], neural networks in the area of the complex field [8,32,33], or neural networks in the quaternion field [4,9,19,20,21,22,23,24,25,26]. Among the various dynamical behaviors, stability is the basic one that receives close attention from more scholars [4,19,20]. Particularly, owing to the definition of master–slave synchronization, the research of the master system’s synchronization with the slave one is equal to the discussion about the stability of the related error system with the master–slave systems.Therefore, many researchers have investigated the different kinds of synchronization criteria on the various systems of USOMDVNN [22,23,24,25,26]. Asymptotic synchronization and exponential synchronization are both considered as infinite-time synchronization, whose essence indicates that synchronization can just be finished, as long as the time verges to infinity [19,20,21,29,30,31]. Different with the above two kinds of synchronization, finite-time synchronization [23] owns absolute advantages in practical engineering applications. For finite-time synchronization, synchronization could be realized in a certain amount of time, which depends on the initial value. However, the certain amount of time can not be estimated without the accurate initial value. In 2012, Polyakov, in [11], put forward the thought of fixed-time synchronization, which indicates that the master systems can synchronize with the slave ones in a finite time without the precise initial value. That is, the finite time has the fixed upper bound for the discussed system under any initial conditions. Very recently, on one hand, many researchers have devoted themselves to deriving the fixed time with the tighter upper bound, by establishing some formulas [12,13,45]. On the other hand, because the basic commutative law can not be directly applied in the quaternion multiplication, many researchers have focused on studying different methods such as separation or nonseparation in solving the unavoidable difficulty [4,5,19,20]. The research about different dynamics in the system of USOMDVNN with many methods is still meaningful and worthy of much attention, as Liu et al. in [19] has said. Focusing on the previous results on the special systems of RVNN [13,16,44], CVNN [8,29,35,36], and QVNN [21,22,23], it is not difficult to find out that some analyses, especially those based on the separation of the systems of CVNN and QVNN, may inevitably increase the amount of calculation. Therefore, we need the search for the concise LKF and switching controllers for the novel synchronization criteria, including both asymptotic synchronization and the fixed-time one about the unified system of USOMDVNN.

Motivated by the above investigation and research, we put forward a unified analysis on globally asymptotic synchronization and fixed-time synchronization for USOMDVNN in this paper. First of all, the general model of USOMDVNN is successfully constructed, mainly on the basis of multidimensional algebra, Kirchhoff current law, and neuronal feature. Then, the new inequalities with some parameters are applied in a unified form whose variable can be translated into its special one, such as real, complex, and quaternion. It is worth mentioning that the useful parameters really make some contributions to the novel structure of the concise LKF, the new construction of the switching controllers, and the easy acquisition of the flexible criteria. Further, we acquire the newer criteria mainly by employing Lyapunov analysis, constructing new LKF, applying two unified inequalities, and designing novel controllers. It is worth mentioning that the concrete value of the fixed time is less than the other ones in the existing results, owing to the adjustable parameters. The following points are the the main contributions of this paper.

(1) The unified model of the investigated USOMDVNN is generally set up.

(2) Owing to the new establishment of the extended derivative of the absolute value function and the new application of the generalized Cauchy–Schwarz inequality, the concise LKF is successfully constructed and is used to solve the problem of non-commutativity for multidimensional multiplication.

(3) Inspired by the previous work [25,26], both the newly constructed LKF and the derived criteria contain the flexible parameters ${\breve{\varpi }}_p$, and the novelly designed controllers involve the adjustable parameters ${\breve{\varpi }}_p$. By adjusting the values of ${\breve{\varpi }}_p$ properly, the LKF can become multiple, the relevant conditions both in Theorems 1 and 2 can become flexible, and the controllers in Theorem 2 can reach the smaller synchronization fixed time ${\breve{T}}_{\tau }$.

Notations: The sets of all $p\times q$-dimensional matrices that own multidimensional algebra are denoted as ${\mathfrak{M}}^{p\times q}$. Define the multi-dimension-valued variable $\breve{z}\left(t\right)$ as follows.

$$\begin{array}{c}\hfill \breve{z}\left(t\right)=\left\{\begin{array}{cccc}& {\breve{z}}_R\left(t\right)+\overrightarrow{i}{\breve{z}}_I\left(t\right)+\overrightarrow{j}{\breve{z}}_J\left(t\right)+\overrightarrow{k}{\breve{z}}_K\left(t\right)),\hfill & & \breve{z}\left(t\right)\in \mathfrak{Q},\hfill \\ & {\breve{z}}_R\left(t\right)+\overrightarrow{i}{\breve{z}}_I\left(t\right),\hfill & & \breve{z}\left(t\right)\in \mathfrak{C},\hfill \\ & {\breve{z}}_R\left(t\right),\hfill & & \breve{z}\left(t\right)\in \mathfrak{R}.\hfill \end{array}\right.\end{array}$$

Correspondingly, the derivative of variable $\breve{z}\left(t\right)$ in the multi-dimension-valued form is regarded to be multi-dimension formed by the derivatives of every element ${\breve{z}}_M\left(t\right)$ (M can be equal to be $R,I,J$ or K; M can be equal to be R or I; or M can be equal to be R) of the multi-dimension variable $\breve{z}\left(t\right)$ with respect to t:

$$\begin{array}{c}\hfill \dot{\breve{z}}\left(t\right)=\left\{\begin{array}{cccc}& {\dot{\breve{z}}}_R\left(t\right)+\overrightarrow{i}{\dot{\breve{z}}}_I\left(t\right)+\overrightarrow{j}{\dot{\breve{z}}}_J\left(t\right)+\overrightarrow{k}{\dot{\breve{z}}}_K\left(t\right),\hfill & & \breve{z}\left(t\right)\in \mathfrak{Q},\hfill \\ & {\dot{\breve{z}}}_R\left(t\right)+\overrightarrow{i}{\dot{\breve{z}}}_I\left(t\right),\hfill & & \breve{z}\left(t\right)\in \mathfrak{C},\hfill \\ & {\dot{\breve{z}}}_R\left(t\right),\hfill & & \breve{z}\left(t\right)\in \mathfrak{R}.\hfill \end{array}\right.\end{array}$$

More specific descriptions can be found in the previous works [25,26].

2. Model Description and Preliminaries

First of all, we give the general master system of USOMDVNN as follows.

$$\begin{array}{ccc}\hfill {\dot{\breve{x}}}_m\left(t\right)& =& -{\breve{c}}_m{\breve{x}}_m\left(t\right)+\sum _{n=1}^q{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{x}}_n\left(t\right)\right)+{\breve{I}}_m\left(t\right),\hfill \end{array}$$

(1)

where $t\ge 0,m,n=1,2,\cdots ,q$, $\breve{x}\left(t\right)={({\breve{x}}_1\left(t\right),{\breve{x}}_2\left(t\right),\cdots ,{\breve{x}}_q\left(t\right))}^T\in {\mathfrak{M}}^q$, ${\breve{x}}_m\left(t\right)$ stands for the state variable of the m-th neuron; ${\breve{c}}_m>0$, ${\breve{c}}_m\in \mathfrak{R}$ ($m=1,2,\cdots ,q$) describes the decay matrix; the external input vector is denoted as $\breve{I}\left(t\right)={({\breve{I}}_1,{\breve{I}}_2,\cdots ,{\breve{I}}_q)}^T\in {\mathfrak{M}}^q$; the corresponding multi-dimension-valued activation functions are expressed by $\breve{f}\left(\breve{x}\left(t\right)\right)={({\breve{f}}_1\left({\breve{x}}_1\left(t\right)\right),{\breve{f}}_2\left({\breve{x}}_2\left(t\right)\right),\cdots ,{\breve{f}}_q\left({\breve{x}}_q\left(t\right)\right))}^T\in {\mathfrak{M}}^q$.

The relevant slave system of USOMDVNN can be similarly established as follows,

$$\begin{array}{ccc}\hfill {\dot{\breve{y}}}_m\left(t\right)& =& -{\breve{c}}_m{\breve{y}}_m\left(t\right)+\sum _{n=1}^q{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{y}}_n\left(t\right)\right)+{\breve{I}}_m\left(t\right)+{\breve{J}}_m\left(t\right),\hfill \end{array}$$

(2)

where, $t\ge 0,m=1,2,\cdots ,q,n=1,2,\cdots ,q$; $\breve{y}\left(t\right)={({\breve{y}}_1\left(t\right),{\breve{y}}_2\left(t\right),\cdots ,{\breve{y}}_q\left(t\right))}^T\in {\mathfrak{M}}^q$ stands for the state variable of the m-th neuron; ${\breve{c}}_m>0$, ${\breve{c}}_m\in \mathfrak{R}$ ($m=1,2,\cdots ,p$) describes the corresponding decay matrix; ${\breve{J}}_m\left(t\right)$ denotes the subsequently designed controllers.

Generally, design the switching controllers as follows,

$$\begin{array}{c}\hfill {\breve{J}}_m\left(t\right)=\left\{\begin{array}{cccc}& -|{\breve{k}}_m|{\breve{z}}_m\left(t\right),\hfill & & some{\breve{z}}_m^{*}\left(t\right)=0,\hfill \\ & -|{\breve{k}}_m|{\breve{z}}_m\left(t\right)\hfill \\ & -|{\breve{l}}_m|{\left({\breve{\omega }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\beta }}\hfill \\ & -|{\breve{m}}_m|{\left({\breve{\omega }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\gamma }},\hfill & & any{\breve{z}}_m^{*}\left(t\right)\ne 0.\hfill \end{array}\right.\end{array}$$

where, $m=1,2,\dots ,q$, $0<\breve{\beta }<1$, $\breve{\gamma }>1$, ${\breve{k}}_m$, ${\breve{l}}_m$, ${\breve{m}}_m$; ${\breve{\omega }}_m$ stand for the gains of the above controllers, and all can be any nonzero constants.

Denote ${\breve{z}}_m\left(t\right)={\breve{y}}_m\left(t\right)-{\breve{x}}_m\left(t\right)$. According to the above master system and slave one, we can obtain the following general error one:

$$\begin{array}{ccc}\hfill {\dot{\breve{z}}}_m\left(t\right)& =& -{\breve{c}}_m{\breve{z}}_m\left(t\right)+\sum _{n=1}^q{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)+{\breve{J}}_m\left(t\right),\hfill \end{array}$$

(3)

where, $\breve{z}\left(t\right)={({\breve{z}}_1\left(t\right),{\breve{z}}_2\left(t\right),\cdots ,{\breve{z}}_q\left(t\right))}^T\in {\mathfrak{M}}^q$, ${\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)={\breve{f}}_n\left({\breve{y}}_n\left(t\right)\right)-{\breve{f}}_n\left({\breve{x}}_n\left(t\right)\right)$.

Definition 1

([19]). The error system of USOMDVNN (3) is defined to achieve the globally asymptotical stability if $li{m}_{t\to +\infty }{\parallel \breve{z}\parallel }_1=0$.

Definition 2

([23]). The error system of USOMDVNN (3) is considered to achieve the finite-time stability if $li{m}_{t\to T\left(\breve{z}\left(0\right)\right)}{\parallel \breve{z}\parallel }_1=0$ and $\parallel \breve{z}{\parallel }_1=0$ for $\forall t>\breve{T}\left(\breve{z}\left(0\right)\right)$, where, $\breve{T}\left(\breve{z}\left(0\right)\right)$ is a constant and $\breve{T}\left(\breve{z}\left(0\right)\right)>0$. Moreover, the finite setting time is denoted as $\breve{T}\left(\breve{z}\left(0\right)\right)$.

Definition 3

([13]). The error system of USOMDVNN (3) is regarded to achieve the fixed-time stability, if the following conditions hold: (1) The error system of USOMDVNN (3) can achieve the finite-time stability; (2) Let ${\breve{T}}_{\tau }$ is a positive and fixed constant. Then, there exists $\breve{T}\left(\breve{z}\left(0\right)\right)<{\breve{T}}_{\tau }$, for any $\breve{z}\left(0\right)$. Moreover, ${\breve{T}}_{\tau }$ is defined to be the fixed time.

Assumption 1.

Assume that the activation functions ${\breve{f}}_n(·)$ could satisfy the basic inequality $\parallel {\breve{f}}_n\left(\breve{y}\right)-{\breve{f}}_n\left(\breve{x}\right)\parallel \le {\breve{\lambda }}_n\parallel \breve{y}-\breve{x}\parallel$, where $n=1,2,\cdots ,q$, ${\breve{\lambda }}_n>0$.

Lemma 1

([11]). Suppose $\breve{V}(·):{\mathfrak{R}}^q⟶{\mathfrak{R}}_{+}\cup 0$ to be a radially continuous and unbounded function, and satisfy: (1) $\breve{V}\left(\breve{z}\left(t\right)\right)=0$⟺$\breve{z}\left(t\right)=0$; (2) Any solution $\breve{z}\left(t\right)$ of the error system of USOMDVNN (3) meets $D^{+}\breve{V}\left(\breve{z}\left(t\right)\right)\le -\breve{\vartheta }{\breve{V}}^{\breve{\beta }}\left(\breve{z}\left(t\right)\right)-\breve{\upsilon }{\breve{V}}^{\breve{\gamma }}\left(\breve{z}\left(t\right)\right)$, for $\breve{\vartheta }>0$, $\breve{\upsilon }>0$, $0<\breve{\beta }<1$ and $\breve{\gamma }>1$, where $D^{+}\breve{V}\left(\breve{z}\left(t\right)\right)$ stands for the upper right-hand Dini differential of $\breve{V}\left(\breve{z}\left(t\right)\right)$. Then, the error system of USOMDVNN (3) can achieve the fixed-time stability and the concrete fixed time ${\breve{T}}_{\tau }={(\breve{\vartheta }(1-\breve{\beta }))}^{-1}+{\left(\breve{\upsilon }(\breve{\gamma }-1)\right)}^{-1}$.

Lemma 2

([27] (Extended Cauchy-Schwarz Inequality)). For any $\breve{X},\breve{Y}\in \mathfrak{M}$, $\breve{\mu }\breve{Y}\breve{\nu }\breve{X}+{(\breve{\nu }\breve{X})}^{*}{(\breve{\mu }\breve{Y})}^{*}\le \breve{\xi }{\breve{\mu }}^2\breve{Y}{\breve{Y}}^{*}+\frac{1}{\breve{\xi }}{\breve{\nu }}^2{\breve{X}}^{*}\breve{X}$, where, $\breve{\mu },\breve{\nu }\in \mathfrak{R}$, $\breve{\xi }\in \mathfrak{R}$ ($\breve{\xi }>0$) are all constants.

Lemma 3

([26] (Extended Differential of Absolute Value Function)). If $\breve{z}\left(t\right)\in C^1([t_0,+\infty ),\mathfrak{M})$, $c\in \mathfrak{R}$, then $\frac{d|a\breve{z}\left(t\right)|}{dt}=|c|sign\left(\breve{z}\left(t\right)\right)·\dot{\breve{z}}\left(t\right)$, $t\ge t_0$, where the symbol · describes the dot product between the two vectors.

Lemma 4

([28]). If ${\breve{\epsilon }}_m\ge 0$ $(m=1,2,\dots ,p)$, $0<\breve{\iota }\le 1$, $\breve{\kappa }>1$, then ${\sum }_{m=1}^p{\breve{\epsilon }}_m^{\breve{\iota }}\ge {\left({\sum }_{m=1}^p{\breve{\epsilon }}_m\right)}^{\breve{\iota }}$, ${\sum }_{m=1}^p{\breve{\epsilon }}_m^{\breve{\kappa }}\ge n^{1-\breve{\kappa }}{\left({\sum }_{m=1}^p{\breve{\epsilon }}_m\right)}^{\breve{\kappa }}$.

3. Main Results

Generally, design the switching controllers as follows.

$$\begin{array}{c}\hfill {\breve{J}}_m\left(t\right)=\left\{\begin{array}{cccc}& -|{\breve{k}}_m|{\breve{z}}_m\left(t\right),\hfill & & some{\breve{z}}_m^{*}\left(t\right)=0,\hfill \\ & -|{\breve{k}}_m|{\breve{z}}_m\left(t\right)\hfill \\ & -|{\breve{l}}_m|{\left({\breve{\omega }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\beta }}\hfill \\ & -|{\breve{m}}_m|{\left({\breve{\omega }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\gamma }},\hfill & & any{\breve{z}}_m^{*}\left(t\right)\ne 0.\hfill \end{array}\right.\end{array}$$

(4)

where, $m=1,2,\dots ,q$, $0<\breve{\beta }<1$, $\breve{\gamma }>1$, ${\breve{k}}_m$, ${\breve{l}}_m$, ${\breve{m}}_m$; ${\breve{\omega }}_m$ stand for the gains of the above controllers, and all can be any nonzero constants.

When some ${\breve{z}}_m^{*}\left(t\right)=0$, $m=1$ or $m=2$, …, or $m=q$, we just design the linear controllers as follows.

$$\begin{array}{ccc}\hfill {\breve{J}}_m\left(t\right)& =& -|{\breve{k}}_m|{\breve{z}}_m\left(t\right),\hfill \end{array}$$

where, ${\breve{k}}_m$ ($m=1,2,\dots ,q$) describe the corresponding gains of the above linear controllers and ${\breve{k}}_m\ne 0$.

When any ${\breve{z}}_m^{*}\left(t\right)\ne 0$, we design

$$\begin{array}{ccc}\hfill {\breve{J}}_m\left(t\right)& =& -|{\breve{k}}_m|{\breve{z}}_m\left(t\right)\hfill \\ \hfill & & -|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\beta }}\hfill \\ \hfill & & -|{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\gamma }},\hfill \end{array}$$

where, $m=1,2,\dots ,q$, $0<\breve{\beta }<1$, $\breve{\gamma }>1$, ${\breve{k}}_m$, ${\breve{l}}_m$, ${\breve{m}}_m$, and ${\breve{\varpi }}_m$ describe the gains of the nonlinear controllers, and all can be any nonzero constants.

Remark 1.

Compared to the precious work in [25], the above switching controllers not only contain linear ones, but also include nonlinear ones. The key contribution of the controllers can solve the singularity, which may generate in the similar ones of [25]. In fact, it is worth noting that, as the condition $0<\breve{\beta }<1$, the variable ${\breve{z}}_m^{*}\left(t\right)$ contains the power ($\breve{\beta }-1$), and a singularity may exist when the variable is equal to 0. However, as long as the variable is not equal to 0, there will be no singularity. Therefore, the newly designed controllers can help realize both the globally asymptotical synchronization and the fixed-time one, which will be discussed in the next theorems and shown in the simulations.

Next, we discuss two kinds of the synchronization according to the above values of the switching controllers (4).

3.1. Globally Asymptotical Synchronization of USOMDVNN

According to the switching controllers (4), when some ${\breve{z}}_m^{*}\left(t\right)=0$, $m=1$ or $m=2$, …, or $m=q$, we just design the linear controllers as follows.

$$\begin{array}{ccc}\hfill {\breve{J}}_m\left(t\right)& =& -|{\breve{k}}_m|{\breve{z}}_m\left(t\right),\hfill \end{array}$$

(5)

where, ${\breve{k}}_m$ ($m=1,2,\dots ,q$) describe the corresponding gains of the above linear controllers and ${\breve{k}}_m\ne 0$.

Theorem 1.

Under Assumption 1, if there exist constants ${\breve{\varpi }}_m\ne 0$, ${\breve{k}}_m\ne 0$, ${\breve{l}}_m\ne 0$, ${\breve{m}}_m\ne 0$, $\breve{\rho }>0$, $0<\breve{\beta }<1$, $\breve{\gamma }>1$, ${\breve{\lambda }}_m>0$, and $\breve{\sigma }>0$, such that

$$\begin{array}{c}\hfill -\breve{\sigma }=\underset{1\le m\le p}{max}\{-2{\breve{c}}_m-2|{\breve{k}}_m|+n\breve{\rho }|{\breve{\varpi }}_m|+\frac{{{\breve{\lambda }}_m}^2}{\breve{\rho }{\left({\breve{\varpi }}_m\right)}^2}\sum _{n=1}^q|{\breve{\varpi }}_n|{\breve{a}}_{nm}^{*}{\breve{a}}_{nm}\}<0,\\ \hfill m=1,2,\dots ,q,n=1,2,\dots ,q,\end{array}$$

(6)

then the master system of USOMDVNN (1) can reach globally asymptotical synchronization with the slave system of USOMDVNN (3) under the linear controllers (5).

Proof. 

The concise LKF in vector form can be constructed as follows

$$\breve{V}\left(t\right)=|{\breve{\varpi }}^{*}{\breve{Z}}^{*}|·|\breve{\varpi }\breve{Z}|,$$

where,

$$\breve{\varpi }=({\breve{\varpi }}_1,{\breve{\varpi }}_2,\dots ,{\breve{\varpi }}_q)\in {\mathfrak{R}}^q$$

$$\breve{Z}={({\breve{z}}_1\left(t\right),{\breve{z}}_2\left(t\right),\dots ,{\breve{z}}_q\left(t\right))}^T\in {\mathfrak{M}}^q$$

$${\breve{Z}}^{*}={({\breve{z}}_1^{*}\left(t\right),{\breve{z}}_2^{*}\left(t\right),\dots ,{\breve{z}}_q^{*}\left(t\right))}^T\in {\mathfrak{M}}^q.$$

According to the basic differential of the absolute value function, we have

$$D^{+}\breve{V}\left(t\right)=(\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m{\left(t\right)|)}^{\prime }=\sum _{m=1}^p(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right){|)}^{\prime }$$

$$=\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}{\left(t\right)|}^{\prime }|{\varpi }_m{\breve{z}}_m\left(t\right)|+\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m{\left(t\right)|}^{\prime }.$$

Based on new Lemma 3, it follows

$$D^{+}\breve{V}\left(t\right)=\sum _{m=1}^q|{\breve{\varpi }}_m^{*}|sgn\left({\breve{z}}_m^{*}\left(t\right)\right){\dot{\breve{z}}}_m^{*}\left(t\right)|{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|+\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m|sgn\left({\breve{z}}_m\left(t\right)\right){\dot{\breve{z}}}_m\left(t\right).$$

According to the error system (3), it can hold

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & \sum _{m=1}^q|{\breve{\varpi }}_m^{*}|sgn\left({\breve{z}}_m^{*}\left(t\right)\right)[-{\breve{c}}_m{\breve{z}}_m^{*}\left(t\right)\hfill \\ \hfill & & +\sum _{n=1}^q{\breve{f}}_n^{*}\left({\breve{z}}_n^{*}\left(t\right)\right){\breve{a}}_{mn}^{*}-|{\breve{k}}_m|{\breve{z}}_m^{*}\left(t\right)]|{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m|sgn\left({\breve{z}}_m\left(t\right)\right)[-{\breve{c}}_m{\breve{z}}_m\left(t\right)\hfill \\ \hfill & & +\sum _{n=1}^q{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)-|{\breve{k}}_m|{\breve{z}}_m\left(t\right)].\hfill \end{array}$$

(7)

That is,

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & \sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|[-2{\breve{c}}_m-2|{\breve{k}}_m|]\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}|\sum _{n=1}^q|{\breve{f}}_n^{*}\left({\breve{z}}_n^{*}\left(t\right)\right){\breve{a}}_{mn}^{*}||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m|\sum _{n=1}^q|{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)|]\hfill \end{array}$$

(8)

By employing Lemma 2, we can see there exist $\breve{\rho }>0$, such that

$$\begin{array}{c}\hfill {\breve{f}}_n^{*}({\breve{z^{*}}}_n\left(t\right)){\breve{a}}_{mn}^{*}{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)+{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right){\breve{a}}_{mn}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)\le \breve{\rho }{\left({\breve{\varpi }}_m\right)}^2{\breve{z}}_m^{*}\left(t\right){\breve{z}}_m\left(t\right)+\frac{1}{\breve{\rho }}{\breve{f}}_n^{*}({\breve{z^{*}}}_n\left(t\right)){\breve{a}}_{mn}{\breve{a}}_{mn}^{*}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right).\end{array}$$

Then, it goes

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & \sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|[-2{\breve{c}}_m-2|{\breve{k}}_m|]\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}|\sum _{n=1}^q[\breve{\rho }{\left({\breve{\varpi }}_m\right)}^2{\breve{z}}_m^{*}\left(t\right){\breve{z}}_m\left(t\right)\hfill \\ \hfill & & +\frac{1}{\breve{\rho }}{\breve{f}}_n^{*}({\breve{z^{*}}}_n\left(t\right)){\breve{a}}_{mn}{\breve{a}}_{mn}^{*}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)]\hfill \\ \hfill & =& \sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|[-2{\breve{c}}_m-2|{\breve{k}}_m|+n\breve{\rho }|{\breve{\varpi }}_m|\hfill \\ \hfill & & +\frac{{{\lambda }_m}^2}{\breve{\rho }{\left({\breve{\varpi }}_m\right)}^2}\sum _{n=1}^q|{\breve{\varpi }}_n|a_{qp}^{*}{a}_{qp}]\hfill \end{array}$$

(9)

Based on the conditions of Theorem 1, it can be easily obtained that

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & -\breve{\sigma }\breve{V}\left(t\right).\hfill \end{array}$$

(10)

Based on Definition 1 and Lemma 1, if and only if every component of ${\breve{z}}_m\left(t\right)=0$ ($m=1,2,\dots ,q$), then $\breve{Z}={({\breve{z}}_1\left(t\right),{\breve{z}}_2\left(t\right),\dots ,{\breve{z}}_q\left(t\right))}^T\in {\mathfrak{M}}^q=0$ and $\breve{V}\left(t\right)=0$. If $\breve{Z}\ne 0$, then $D^{+}\breve{V}\left(t\right)<0$. It can be concluded that the error system of USOMDVNN (3) can reach globally asymptotical stability with the linear controllers (5). □

3.2. Fixed-Time Synchronization of USOMDVNN

According to the switching controllers (4), when any ${\breve{z}}_m^{*}\left(t\right)\ne 0$, we design

$$\begin{array}{ccc}\hfill {\breve{J}}_m\left(t\right)& =& -|{\breve{k}}_m|{\breve{z}}_m\left(t\right)\hfill \\ \hfill & & -|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\beta }}\hfill \\ \hfill & & -|{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\gamma }},\hfill \end{array}$$

(11)

where, $m=1,2,\dots ,q$, $0<\breve{\beta }<1$, $\breve{\gamma }>1$, ${\breve{k}}_m$, ${\breve{l}}_m$, ${\breve{m}}_m$, and ${\breve{\varpi }}_m$ describe the gains of the nonlinear controllers and all can be any nonzero constants.

Theorem 2.

Under Assumption 1, if there exist constants ${\breve{\varpi }}_m\ne 0$, ${\breve{k}}_m\ne 0$, ${\breve{l}}_m\ne 0$, ${\breve{m}}_m\ne 0$, $\breve{\rho }>0$, $0<\breve{\beta }<1$, $\breve{\gamma }>1$, ${\breve{\lambda }}_m>0$, and $\breve{\sigma }>0$, such that

$$\begin{array}{c}\hfill -\breve{\sigma }=\underset{1\le m\le p}{max}\{-2{\breve{c}}_m-2|{\breve{k}}_m|+n\breve{\rho }|{\breve{\varpi }}_m|+\frac{{{\breve{\lambda }}_m}^2}{\breve{\rho }{\left({\breve{\varpi }}_m\right)}^2}\sum _{n=1}^q|{\breve{\varpi }}_n|{\breve{a}}_{nm}^{*}{\breve{a}}_{nm}\}<0,\\ \hfill m=1,2,\dots ,q,n=1,2,\dots ,q,\end{array}$$

(12)

then under the nonlinear controllers (11), the master system of USOMDVNN (1) and the slave system of USOMDVNN (3) can achieve fixed-time synchronization with each other. In addition, the concrete fixed time is

$$\begin{array}{c}\hfill {\breve{T}}_{\tau }={(\breve{\vartheta }(1-\breve{\beta }))}^{-1}+{\left(\breve{\upsilon }(\breve{\gamma }-1)\right)}^{-1},\end{array}$$

(13)

where, $\breve{\vartheta }={min}_{1\le m\le q}\{2|{\breve{\varpi }}_m{|}^{4-2\breve{\beta }}|{\breve{l}}_m|\}$, $\breve{\upsilon }={min}_{1\le m\le q}\{2|{\breve{\varpi }}_m{|}^{4-2\breve{\gamma }}|{\breve{m}}_m|\}{q}^{1-\breve{\gamma }}$.

Proof. 

The same concise LKF in vector form can be constructed as follows,

$$\breve{V}\left(t\right)=|{\breve{\varpi }}^{*}{\breve{Z}}^{*}|·|\breve{\varpi }\breve{Z}|,$$

where,

$$\breve{\varpi }=({\breve{\varpi }}_1,{\breve{\varpi }}_2,\dots ,{\breve{\varpi }}_q)\in {\mathfrak{R}}^q,$$

$$\breve{Z}={({\breve{z}}_1\left(t\right),{\breve{z}}_2\left(t\right),\dots ,{\breve{z}}_q\left(t\right))}^T\in {\mathfrak{M}}^q,$$

$${\breve{Z}}^{*}={({\breve{z}}_1^{*}\left(t\right),{\breve{z}}_2^{*}\left(t\right),\dots ,{\breve{z}}_q^{*}\left(t\right))}^T\in {\mathfrak{M}}^q.$$

According to the basic differential of the absolute value function, we have

$$D^{+}\breve{V}\left(t\right)=(\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m{\left(t\right)|)}^{\prime }=\sum _{m=1}^p(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right){|)}^{\prime }$$

$$=\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}{\left(t\right)|}^{\prime }|{\varpi }_m{\breve{z}}_m\left(t\right)|+\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m{\left(t\right)|}^{\prime }.$$

Based on new Lemma 3, it follows:

$$D^{+}\breve{V}\left(t\right)=\sum _{m=1}^q|{\breve{\varpi }}_m^{*}|sgn\left({\breve{z}}_m^{*}\left(t\right)\right){\dot{\breve{z}}}_m^{*}\left(t\right)|{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|+\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m|sgn\left({\breve{z}}_m\left(t\right)\right){\dot{\breve{z}}}_m\left(t\right).$$

According to the error system (3), it can hold

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & \sum _{m=1}^q|{\breve{\varpi }}_m^{*}|sgn\left({\breve{z}}_m^{*}\left(t\right)\right)[-{\breve{c}}_m{\breve{z}}_m^{*}\left(t\right)\hfill \\ \hfill & & +\sum _{n=1}^q{\breve{f}}_n^{*}\left({\breve{z}}_n^{*}\left(t\right)\right){\breve{a}}_{mn}^{*}-|{\breve{k}}_m|{\breve{z}}_m^{*}\left(t\right)\hfill \\ \hfill & & -|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m^{*}\left(t\right)\right)|{\breve{z}}_m\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m^{*}\left(t\right)|}^{\breve{\beta }}\hfill \\ \hfill & & -|{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m^{*}\left(t\right)\right)|{\breve{z}}_m\left(t\right){|}^{\breve{\gamma }-1}|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }}]|{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m|sgn\left({\breve{z}}_m\left(t\right)\right)[-{\breve{c}}_m{\breve{z}}_m\left(t\right)\hfill \\ \hfill & & +\sum _{n=1}^q{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)-|{\breve{k}}_m|{\breve{z}}_m\left(t\right)\hfill \\ \hfill & & -|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\beta }}\hfill \\ \hfill & & -|{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}|{\breve{z}}_m\left(t\right){|}^{\breve{\gamma }}].\hfill \end{array}$$

(14)

That is,

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & \sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|[-2{\breve{c}}_m-2|{\breve{k}}_m|]\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}|\sum _{n=1}^q|{\breve{f}}_n^{*}\left({\breve{z}}_n^{*}\left(t\right)\right){\breve{a}}_{mn}^{*}||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m|\sum _{n=1}^q|{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)|\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}|[-|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^2|{\breve{z}}_m\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m^{*}\left(t\right)|}^{\breve{\beta }}\hfill \\ \hfill & & -{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^2|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}|{\breve{z}}_m\left(t\right){|}^{\breve{\gamma }}]|{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m|[-|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^2|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\beta }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\beta }}\hfill \\ \hfill & & -|{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^2|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}|{\breve{z}}_m\left(t\right){|}^{\breve{\gamma }}]\hfill \end{array}$$

(15)

By applying Lemma 2, we can obtain that there exist $\breve{\rho }>0$, such that

$$\begin{array}{c}\hfill {\breve{f}}_n^{*}({\breve{z^{*}}}_n\left(t\right)){\breve{a}}_{mn}^{*}{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)+{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right){\breve{a}}_{mn}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)\le \breve{\rho }{\left({\breve{\varpi }}_m\right)}^2{\breve{z}}_m^{*}\left(t\right){\breve{z}}_m\left(t\right)+\frac{1}{\breve{\rho }}{\breve{f}}_n^{*}({\breve{z^{*}}}_n\left(t\right)){\breve{a}}_{mn}{\breve{a}}_{mn}^{*}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right).\end{array}$$

In addition, it can easily hold that

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & \sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|[-2{\breve{c}}_m-2|{\breve{k}}_m|]\hfill \\ \hfill & & +\sum _{m=1}^q|{\breve{\varpi }}_m^{*}|\sum _{n=1}^q[\breve{\rho }{\left({\breve{\varpi }}_m\right)}^2{\breve{z}}_m^{*}\left(t\right){\breve{z}}_m\left(t\right)\hfill \\ \hfill & & +\frac{1}{\breve{\rho }}{\breve{f}}_n^{*}\left({\breve{z^{*}}}_n\left(t\right)\right){\breve{a}}_{mn}{\breve{a}}_{mn}^{*}{\breve{f}}_n\left({\breve{z}}_n\left(t\right)\right)]\hfill \\ \hfill & & +2\sum _{m=1}^p{\breve{\varpi }}_m^2[-|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^2|{\breve{z}}_m\left(t\right){|}^{\breve{\beta }}{|{\breve{z}}_m^{*}\left(t\right)|}^{\breve{\beta }}\hfill \\ \hfill & & -{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^2|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }}|{\breve{z}}_m\left(t\right){|}^{\breve{\gamma }}]\hfill \\ \hfill & =& \sum _{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|[-2{\breve{c}}_m-2|{\breve{k}}_m|+n\breve{\rho }|{\breve{\varpi }}_m|\hfill \\ \hfill & & +\frac{{{\lambda }_m}^2}{\breve{\rho }{\left({\breve{\varpi }}_m\right)}^2}\sum _{n=1}^q|{\breve{\varpi }}_n|a_{qp}^{*}{a}_{qp}]\hfill \\ \hfill & & -2\sum _{m=1}^q|{\breve{\varpi }}_m{|}^{4-2\breve{\beta }}|{\breve{l}}_m|(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right){|)}^{\breve{\beta }}\hfill \\ \hfill & & -2\sum _{m=1}^q|{\breve{\varpi }}_m{|}^{4-2\breve{\gamma }}|{\breve{m}}_m|(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right){|)}^{\breve{\gamma }}\hfill \end{array}$$

(16)

Based on the conditions of Theorem 2, it can be obtained that

$$\begin{array}{ccc}\hfill D^{+}\breve{V}\left(t\right)& \le & -\underset{1\le m\le q}{min}\{2|{\breve{\varpi }}_m{|}^{4-2\breve{\beta }}|{\breve{l}}_m|\}\sum _{m=1}^p(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right){|)}^{\breve{\beta }}\hfill \\ \hfill & & -\underset{1\le m\le q}{min}\{2|{\breve{\varpi }}_m{|}^{4-2\breve{\gamma }}|{\breve{m}}_m|\}\sum _{m=1}^p(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right){|)}^{\breve{\gamma }}.\hfill \end{array}$$

(17)

Because the conditions $0<\breve{\beta }<1$ and $\breve{\gamma }>1$ hold, it is easy to obtain that $D^{+}\breve{V}\left(t\right)\le -\breve{\vartheta }(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m{\left(t\right)|)}^{\breve{\beta }}-\breve{\upsilon }(|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right){|)}^{\breve{\gamma }}$. Based on Lemma 1 and under the nonlinear controllers (11), it can be concluded thatthe error system of USOMDVNN (3) can achieve the fixed-time stability and also can obtain the concrete fixed time ${\breve{T}}_{\tau }={(\breve{\vartheta }(1-\breve{\beta }))}^{-1}+{\left(\breve{\upsilon }(\breve{\gamma }-1)\right)}^{-1}.$ □

Remark 2.

To derive the criteria for the two kinds of synchronization problem for USOMDVNN (3), there are main mathematical difficulties: (1) the establishment of the unified system; (2) the construction of the new LKF; (3) the act of the varied coefficients; (4) the new design of the switching controllers; (5) the different simulations of the USOMDVNN, as the algebra is different.

Remark 3.

When there may be some variable ${\breve{z}}_m\left(t\right)=0$ of the error system of USOMDVNN, (3), the common linear controllers (5) are applied to realize the globally asymptotical synchronization of the error system of USOMDVNN (3); When there is any variable ${\breve{z}}_m\left(t\right)\ne 0$, the new nonlinear controllers (11) are employed to achieve the fixed-time synchronization of the error system of USOMDVNN (3). In fact, the variable ${\breve{z}}_m\left(t\right)$ is almost not equal to 0. Therefore, the accomplished synchronization is almost the fixed-time one. In addition, compared to the conditions in Theorems 1 and 2, it is easy to find that the basic criteria are the same. That is to say, whether the criteria for the globally asymptotical synchronization or for the fixed-time synchronization, it is newer. Particularly, the criteria for the fixed-time synchronization not only contains the foundation condition of synchronization, but also includes the concrete time of synchronization.

Remark 4.

Compared with the recently constructed LKF in [21,22,23], the novelty of the new LKF $\breve{V}\left(t\right)=|{\breve{\varpi }}^{*}{\breve{Z}}^{*}|·|\breve{\varpi }\breve{Z}|$ lies in the following aspects: (1) the vector form makes the LKF more concise and general; (2) the number of the useful coefficients can be reduced; (3) the method to solve the added coefficients becomes more extensive. Mainly employing Lemmas 2 and 3, the useful parameters such as $\breve{\varpi }$ and $\breve{\rho }$ can be used to establish more proper criteria. Moreover, owing to the application of the above new lemmas, the useful parameters can be properly adjusted in order to make the derived criteria more multiple, more flexible, less computative, and less conservative. Compared to the designed controllers in [21,22,23], the controllers in this paper are more general and more useful because of the adjustable coefficients ${\breve{\varpi }}_m$. If the value of $\breve{\varpi }$ is zero, the controllers reduce to be the same ones as in [21,22,23]. Moreover, by adjusting the values of ${\breve{\varpi }}_m$, ${\breve{k}}_m$, ${\breve{l}}_m$, and ${\breve{m}}_m$ comprehensively, we can derive the the final tighter fixed time of synchronization while the same one can not be obtained in [21,22,23].

Remark 5.

Compared to the results in [5,8,13], there are the following advantages in our paper: (1) The generalization of the unified model of USOMDVNN (1) lies in that it contains RVNN, CVNN, and QVNN, while the model in [5] just contains the system of QVNN; the system in [8] is just considered to be CVNN and the model in [13] only includes the system of RVNN; (2) To solve the difficulty in quaternion multiplication, in this paper we employ the direct method, which can reduce the unnecessary computation burden that can be caused in the separation of QVNN in [5]; (3) The designed controllers in our paper are newer, more flexible, and less complicated than the ones in [5,8]. Moreover, by designing the controllers with some adjustable parameters properly, we can obtain the results in less fixed time than the results in [8,13].

Remark 6.

Inspired by the new methods in [23,25], we have extended the above results and have successfully established the inner relationship between the novel LKF, the new controllers, and the final criteria in this paper. We have desined the common coefficients ${\breve{\varpi }}_m$ in the specific LKF $\breve{V}\left(t\right)={\sum }_{m=1}^q|{\breve{\varpi }}_m^{*}{\breve{z}}_m^{*}\left(t\right)||{\breve{\varpi }}_m{\breve{z}}_m\left(t\right)|$, the nonlinear controllers ${\breve{J}}_m\left(t\right)=-|{\breve{k}}_m|{\breve{z}}_m\left(t\right)-|{\breve{l}}_m|{\left({\breve{\varpi }}_m\right)}^2$ $sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}{\left(t\right)|}^{\breve{\beta }-1}|{\breve{z}}_m{\left(t\right)|}^{\breve{\beta }}-|{\breve{m}}_m|{\left({\breve{\varpi }}_m\right)}^{2}sign\left({\breve{z}}_m\left(t\right)\right)|{\breve{z}}_m^{*}\left(t\right){|}^{\breve{\gamma }-1}{|{\breve{z}}_m\left(t\right)|}^{\breve{\gamma }}$, and the final criteria ${max}_{1\le m\le p}\{-2c_m-2|{\breve{k}}_m|+n\breve{\rho }|{\breve{\varpi }}_m|+\frac{{{\lambda }_m}^2}{\breve{\rho }{\left({\breve{\varpi }}_m\right)}^2}{\sum }_{n=1}^q|{\breve{\varpi }}_n|a_{nm}^{*}{a}_{nm}\}<0$.

4. Numerical Simulation

In this part, three numerical simulations will be supported to demonstrate the above theoretical results and compare the criteria with some existing results.

Example 1.

The following master system of QVNN with 2-neuron is considered.

$$\begin{array}{cc}\hfill {\dot{\breve{x}}}_m\left(t\right)=& -{\breve{c}}_m{\breve{x}}_m\left(t\right)+{\breve{I}}_m\left(t\right)+\sum _{n=1}^2{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{x}}_n\left(t\right)\right),\hfill \end{array}$$

(18)

where, ${\breve{x}}_m\left(t\right)\in \mathfrak{Q}$, ${\breve{c}}_1=6$, ${\breve{c}}_2=4$, ${\breve{f}}_m\left({\breve{x}}_m\left(t\right)\right)=0.1(tanh\left({\breve{x}}_m\left(t\right)\right)-1)$, ${\breve{\lambda }}_m=0.1$, ${\breve{a}}_{11}=3-4i+8j-9k$, ${\breve{a}}_{12}=-2+2i-6j+7k$, ${\breve{a}}_{21}=5-3i-5j-3k$, ${\breve{a}}_{22}=-1+i-8j+2k$, ${\breve{I}}_1\left(t\right)=[sin\left(t\right)+isin\left(t\right)+jsin\left(t\right)+ksin\left(t\right)]$, ${\breve{I}}_2\left(t\right)=[cos\left(t\right)+icos\left(t\right)+jcos\left(t\right)+kcos\left(t\right)]$.

The corresponding slave system of QVNN with 2-neuron

$$\begin{array}{cc}\hfill {\dot{\breve{y}}}_m\left(t\right)=& -{\breve{c}}_m{\breve{y}}_m\left(t\right)+{\breve{I}}_m\left(t\right)+\sum _{n=1}^2{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{y}}_n\left(t\right)\right)+{\breve{J}}_m\left(t\right),\hfill \end{array}$$

(19)

where, ${\breve{y}}_m\left(t\right)\in \mathfrak{Q}$, ${\breve{c}}_1=6$, ${\breve{c}}_2=4$, ${\breve{f}}_m\left({\breve{y}}_m\left(t\right)\right)=0.1(tanh\left({\breve{y}}_m\left(t\right)\right)-1)$, ${\breve{\lambda }}_m=0.1$, ${\breve{I}}_1\left(t\right)=[sin\left(t\right)+isin\left(t\right)+jsin\left(t\right)+ksin\left(t\right)]$, ${\breve{I}}_2\left(t\right)=[cos\left(t\right)+icos\left(t\right)+jcos\left(t\right)+kcos\left(t\right)]$, and the switching controllers are desined in (4) where, $\breve{\beta }=0.5$, $\breve{\gamma }=1.5$, $m=1,2$, ${\breve{k}}_1={\breve{k}}_2=\pm 6$, ${\breve{l}}_1={\breve{l}}_2=\pm 1$, ${\breve{m}}_1={\breve{m}}_2=\pm 1$.

Let $\breve{\rho }=1$. Because the value of ${\breve{\varpi }}_m$ varies, the acquired criteria are diverse and not just one. Here, the following two conditions of the values of ${\breve{\varpi }}_1$ and ${\breve{\varpi }}_2$ are discussed, and the final values of fixed time ${\breve{T}}_{\tau }$ are listed in Table 1 according to Theorems 1 and 2, respectively:

(1) Condition 1: ${\breve{\varpi }}_1=\pm 1$, ${\breve{\varpi }}_2=\pm 1$,

(2) Condition 2: ${\breve{\varpi }}_1=\pm 2$, ${\breve{\varpi }}_2=\pm 2$.

In addition, we simulate the corresponding state change diagrams of the different systems of USOMDVNN (18) and (18) in 30 random initial values from Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. (From Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, $x_m$ and $y_m$ denote ${\breve{x}}_m$ and ${\breve{y}}_m$, respectively.) From Figure 1, Figure 2, Figure 3 and Figure 4, the divided momentary error phase diagrams for the four elements between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) in 30 random initial values for globally asymptotical synchronization are presented. From Figure 5, Figure 6, Figure 7 and Figure 8, the divided momentary error phase diagrams for the four elements between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) in 30 random initial values for the fixed-time synchronization are presented. From Figure 9, Figure 10, Figure 11 and Figure 12, the mixed momentary error phase diagrams for the four elements between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) in 30 random initial values for the above two kinds of synchronization are presented.

Figure 1. Instantaneous error states for the R part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Figure 1. Instantaneous error states for the R part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Figure 2. Instantaneous error states for the I part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Figure 2. Instantaneous error states for the I part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Table 1. The related values under two cases.

Condition	$-\breve{\mathit{\sigma }}$	${\breve{\mathit{T}}}_{\mathit{\tau }}$
1	−16.37	2.4142
2	−15.185	0.8321

Table 1. The related values under two cases.

Condition	$-\breve{\mathit{\sigma }}$	${\breve{\mathit{T}}}_{\mathit{\tau }}$
1	−16.37	2.4142
2	−15.185	0.8321

Remark 7.

In this example, we just give two choices about the values of ${\breve{\varpi }}_m$. As the value of ${\breve{\varpi }}_m$ changes, the final criteria vary accordingly. Obviously, the two conditions are satisfied, and the two criteria have been derived and the different values of ${\breve{T}}_{\tau }$ have been acquired. Moreover, by comparison with the results in Table 1, we can judge that the value of ${\breve{T}}_{\tau }$ under condition 2 is less than the one under condition 1. This indicates that the parameter ${\breve{\varpi }}_m$ has really played an important part in adjusting the final results. That is, as we adjust the values of ${\breve{\varpi }}_m$ properly, the final value of ${\breve{T}}_{\tau }$ can be less and less, in order to satisfy the application need of engineering.

Figure 3. Instantaneous error states for the J part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Figure 3. Instantaneous error states for the J part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Remark 8.

From Figure 1, Figure 2, Figure 3 and Figure 4, we show the state trajectories of the error system under linear controllers, and the globally asymptotical synchronization has been realized. From Figure 5, Figure 6, Figure 7 and Figure 8, we demonstrate that the state trajectories of the error system under the nonlinear controllers and the fixed-time synchronization have been achieved. Moreover, we can find that the final time for asymptotical synchronization is more than the one for fixed-time synchronization. This indicates that the nonlinear controllers are both novel and effective. In general, the designed controller (4) is a newly switching one that not only plays an important part in the common synchronization problem, but also has an effect on the popular fixed-time one.

Example 2.

The following 2-neuron CVNN is considered to be the same as in Ding et al. [8],

$$\begin{array}{cc}\hfill {\dot{\breve{x}}}_m\left(t\right)=& -{\breve{c}}_m{\breve{x}}_m\left(t\right)+{\breve{I}}_m\left(t\right)+\sum _{n=1}^2{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{x}}_n\left(t\right)\right),\hfill \end{array}$$

(20)

where ${\breve{x}}_m\left(t\right)\in \mathfrak{C}$, ${\breve{c}}_1=1.2$, ${\breve{c}}_2=1.1$, ${\breve{a}}_{11}^R=2$, ${\breve{a}}_{12}^R=-1$, ${\breve{a}}_{21}^R=1$, ${\breve{a}}_{22}^R=0.8$, ${\breve{a}}_{11}^I=1.2$, ${\breve{a}}_{12}^I=-4.5$, ${\breve{a}}_{21}^I=1.7$, ${\breve{a}}_{22}^I=0.9$, ${\breve{I}}_1\left(t\right)=sint-i2cost$, ${\breve{I}}_2\left(t\right)=3cos(t+1)+isin(t-1)$, ${\breve{f}}_m\left({\breve{x}}_m\left(t\right)\right)=\frac{1-e^{-{\breve{x}}_{Rm}}}{1+e^{-{\breve{x}}_{Rm}}}+0.4sign\left({\breve{x}}_{Rm}\right)+i(\frac{1}{1+e^{-{\breve{x}}_{Im}}}+0.4sign\left({\breve{x}}_{Im}\right))$, ${\breve{\lambda }}_m=0.5$, $m=1,2.$

The corresponding 2-neuron slave system of CVNN,

$$\begin{array}{cc}\hfill {\dot{\breve{y}}}_m\left(t\right)=& -{\breve{c}}_m{\breve{y}}_m\left(t\right)+{\breve{I}}_m\left(t\right)+\sum _{q=1}^2{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{y}}_n\left(t\right)\right)+{\breve{J}}_m\left(t\right),\hfill \end{array}$$

(21)

where ${\breve{y}}_m\left(t\right)\in \mathfrak{C}$, ${\breve{c}}_1=1.2$, ${\breve{c}}_2=1.1$, ${\breve{a}}_{11}^R=2$, ${\breve{a}}_{12}^R=-1$, ${\breve{a}}_{21}^R=1$, ${\breve{a}}_{22}^R=0.8$, ${\breve{a}}_{11}^I=1.2$, ${\breve{a}}_{12}^I=-4.5$, ${\breve{a}}_{21}^I=1.7$, ${\breve{a}}_{22}^I=0.9$, ${\breve{I}}_1\left(t\right)=sint-i2cost$, ${\breve{I}}_2\left(t\right)=3cos(t+1)+isin(t-1)$, ${\breve{f}}_m\left({\breve{y}}_m\left(t\right)\right)=\frac{1-e^{-{\breve{y}}_{Rm}}}{1+e^{-{\breve{y}}_{Rm}}}+0.4sign\left({\breve{y}}_{Rm}\right)+i(\frac{1}{1+e^{-{\breve{y}}_{Im}}}+0.4sign\left({\breve{y}}_{Im}\right))$, ${\breve{\lambda }}_m=0.5$, and the switching controllers are desined in (4), where, $\breve{\beta }=0.8$, $\breve{\gamma }=1.5$, $m=1,2$, ${\breve{k}}_1=\pm 6,{\breve{k}}_2=\pm 6$, ${\breve{l}}_1=\pm 1,{\breve{l}}_2=\pm 1$, ${\breve{m}}_1=\pm 1,{\breve{m}}_2=\pm 1$.

Let $\breve{\rho }=1$ and $\breve{\rho }=2$, respectively. Because of the changed values of ${\breve{\varpi }}_m$, the acquired criteria are diverse, and not just one. Here, the following two conditions of the values of the ${\breve{\varpi }}_1$ and ${\breve{\varpi }}_2$ are discussed for Theorems 1 and 2, respectively:

(1) Condition 1: ${\breve{\varpi }}_1=\pm 1$, ${\breve{\varpi }}_2=\pm 1$, $\breve{\rho }=1$.

(2) Condition 2: ${\breve{\varpi }}_1=\pm 2$, ${\breve{\varpi }}_2=\pm 2$, $\breve{\rho }=2$.

By simple calculation, we obtain the corresponding values under the above two cases in Table 2 and Table 3. It is easily seen that we acquire the corresponding criteria and compare the fixed time ${\breve{T}}_{\tau }$ with the results in [8] under different conditions.

Figure 4. Instantaneous error states for the K part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Figure 4. Instantaneous error states for the K part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for asymptotical synchronization.

Figure 5. Instantaneous error states for the R part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Figure 5. Instantaneous error states for the R part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Example 3.

Consider the same 2-neuron RVNN, as in Chen et al. [13],

$$\begin{array}{cc}\hfill {\dot{\breve{x}}}_m\left(t\right)=& -{\breve{c}}_m{\breve{x}}_m\left(t\right)+{\breve{I}}_m\left(t\right)+\sum _{n=1}^2{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{x}}_n\left(t\right)\right),\hfill \end{array}$$

(22)

where ${\breve{x}}_m\left(t\right)\in \mathfrak{R}$, ${\breve{c}}_1=0.9$, ${\breve{c}}_2=1.1$, ${\breve{a}}_{11}=2$, ${\breve{a}}_{12}=-0.1$, ${\breve{a}}_{21}=-5$, ${\breve{a}}_{22}=4.5$, ${\breve{I}}_1\left(t\right)=sint$, ${\breve{I}}_2\left(t\right)=cost$, ${\breve{f}}_m\left({\breve{x}}_m\left(t\right)\right)=0.5\left(|\right({\breve{x}}_m\left(t\right)+1|+|{\breve{x}}_m\left(t\right)-1|)$, ${\breve{\lambda }}_m=1$, $m=1,2$.

The corresponding 2-neuron slave system of RVNN:

$$\begin{array}{cc}\hfill {\dot{\breve{y}}}_m\left(t\right)=& -{\breve{c}}_m{\breve{y}}_m\left(t\right)+{\breve{I}}_m\left(t\right)+\sum _{n=1}^2{\breve{a}}_{mn}{\breve{f}}_n\left({\breve{y}}_n\left(t\right)\right)+{\breve{J}}_m\left(t\right),\hfill \end{array}$$

(23)

where ${\breve{y}}_m\left(t\right)\in \mathfrak{R}$, ${\breve{c}}_1=0.9$, ${\breve{c}}_2=1.1$, ${\breve{a}}_{11}=2$, ${\breve{a}}_{12}=-0.1$, ${\breve{a}}_{21}=-5$, ${\breve{a}}_{22}=4.5$, ${\breve{I}}_1\left(t\right)=sint$, ${\breve{I}}_2\left(t\right)=cost$, ${\breve{f}}_m\left({\breve{y}}_m\left(t\right)\right)=0.5\left(|\right({\breve{y}}_m\left(t\right)+1|+|{\breve{y}}_m\left(t\right)-1|)$, ${\breve{\lambda }}_m=1$, and the switching controllers are desined in (4), where, $\breve{\beta }=0.5$, $\breve{\gamma }=1.5$, $m=1,2$, ${\breve{k}}_1=\pm 6,{\breve{k}}_2=\pm 8$, ${\breve{l}}_1=\pm 1,{\breve{l}}_2=\pm 8$, ${\breve{m}}_1=\pm 1,{\breve{m}}_2=\pm 2$.

Let $\breve{\rho }=4$ and $\breve{\rho }=2$, respectively. Due to the different values of ${\breve{\varpi }}_m$, the acquired criteria are diverse and not just one. Here, the following two conditions of the values of ${\breve{\varpi }}_1$ and ${\breve{\varpi }}_2$ are discussed about the conditions in Theorems 1 and 2, respectively:

(1) Condition 1: ${\breve{\varpi }}_1=\pm 1$, ${\breve{\varpi }}_2=\pm 0.5$, $\breve{\rho }=4$.

(2) Condition 2: ${\breve{\varpi }}_1=\pm 2$, ${\breve{\varpi }}_2=\pm 1$, $\breve{\rho }=2$.

By simple calculation, we obtain the corresponding values under the above two cases in Table 4 and Table 5. It is easily seen that we acquire the corresponding criteria and compare the fixed time ${\breve{T}}_{\tau }$ with other ones from the above tables.

Figure 6. Instantaneous error states for the I part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Figure 6. Instantaneous error states for the I part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Table 2. The related values under two cases.

Condition	$-\breve{\mathit{\sigma }}$	${\breve{\mathit{T}}}_{\mathit{\tau }}$
1	−6.525	3.9142
2	−4.7814	1.1808

Table 2. The related values under two cases.

Condition	$-\breve{\mathit{\sigma }}$	${\breve{\mathit{T}}}_{\mathit{\tau }}$
1	−6.525	3.9142
2	−4.7814	1.1808

Table 3. The comparison of the different fixed time ${\breve{T}}_{\tau }$.

Method	${\breve{\mathit{T}}}_{\mathit{\tau }}$
[8]	4.5
condition 1	3.9142
condition 2	1.1808

Table 3. The comparison of the different fixed time ${\breve{T}}_{\tau }$.

Method	${\breve{\mathit{T}}}_{\mathit{\tau }}$
[8]	4.5
condition 1	3.9142
condition 2	1.1808

Figure 7. Instantaneous error states for the J part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Figure 7. Instantaneous error states for the J part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Table 4. The related values under two cases.

Condition	$-\breve{\mathit{\sigma }}$	${\breve{\mathit{T}}}_{\mathit{\tau }}$
1	−0.675	2.4142
2	−1.675	0.8321

Table 4. The related values under two cases.

Condition	$-\breve{\mathit{\sigma }}$	${\breve{\mathit{T}}}_{\mathit{\tau }}$
1	−0.675	2.4142
2	−1.675	0.8321

Table 5. The comparison of the different fixed time ${\breve{T}}_{\tau }$.

Method	${\breve{\mathit{T}}}_{\mathit{\tau }}$
[11]	4.3784
[45]	4.3620
[12]	3.4259
[13]	2.3897
condition 1	2.4142
condition 2	0.8321

Table 5. The comparison of the different fixed time ${\breve{T}}_{\tau }$.

Method	${\breve{\mathit{T}}}_{\mathit{\tau }}$
[11]	4.3784
[45]	4.3620
[12]	3.4259
[13]	2.3897
condition 1	2.4142
condition 2	0.8321

Figure 8. Instantaneous error states for the K part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Figure 8. Instantaneous error states for the K part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for fixed-time synchronization.

Figure 9. Instantaneous error states for the R part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

Figure 9. Instantaneous error states for the R part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

Remark 9.

From the above three examples, our acquired results have made the following progress by the comparison with the ones in [11,12,13,45]: (1) The established model of USOMDVNN (1) can be reduced the special cases in [11,12,13,45]; (2) The solution to discuss the related dynamics of different multi-dimension neural networks can be unified in this paper; (3) The controllers are switching and can be various, owing to the participation of the parameters ${\breve{\varpi }}_m$; (4) Whether the criteria for asymptotical synchronization or for a fixed-time one, the final judgements are newer; (5) The final values of the fixed time ${\breve{T}}_{\tau }$ are easy to be calculated; (6) The listed values of ${\breve{T}}_{\tau }$ in Table 5 are less than the ones in [11,12,13,45].

Figure 10. Instantaneous error states for the I part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

Figure 10. Instantaneous error states for the I part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

Figure 11. Instantaneous error states for the J part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

Figure 11. Instantaneous error states for the J part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

Figure 12. Instantaneous error states for the K part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

Figure 12. Instantaneous error states for the K part between for the variable $\breve{x}$ of (18) and $\breve{y}$ of (19) under 30 random initial values for two kinds of synchronization.

5. Conclusions

In this paper, the problems of both the asymptotical synchronization and fixed-time one have been discussed for USOMDVNN in a unified form. Mainly by applying the Lyapunov method and multi-dimension algebra, we have established the unified model and concise criteria for the researched problem. In the process of establishing the criteria, we have derived two unified inequalities with useful parameters that have been sufficiently involved in the structure of the LKF, the construction of the switching controllers, and the final tighter fixed time. Finally, we have roundly simulated the obtained results by giving three numerical examples. We will learn more and extend the novel research approach to studying the corresponding dynamics, such as dissipativity or state estimation in [37,38,39,40,41,42,43], for delayed and uncertain USOMDVNN with memristors in the future.