Fixed/Preassigned-Time Stabilization for Complex-Valued Inertial Neural Networks with Distributed Delays: A Non-Separation Approach

Abstract

This article explores complex-valued inertial neural networks (CVINNs) with distributed delays (DDs). By constructing two new feedback controllers, some novel results on fixed-time stabilization (FTS) and preassigned-time stabilization (PTS) of CVINNs are established. Unlike most of the previous works, FTS and PTS obtained here are explored without dividing the original complex-valued system into two separate real valued subsystems. Eventually, to verify the effectiveness and reliability of the results of this article, we provide several numerical examples. The FTS and PTS of CVINNs are successfully implemented at T = 6, 5.5, and 5, and the settling time is not affected by system parameters and initial values.

1. Introduction

Neural networks (NNs) have wide-ranging applications in signal processing, combinatorial optimisation, automatic control, and deep learning, which have received substantial attention in the last few decades. During the applications of NNs, of which dynamic behaviours are the key points. Among them, stabilization is the important one that has a wide range of applications in bioengineering, information processing, and other fields. As is well known, the settling time (ST) of finite-time is linked to the system’s initial conditions. Furthermore, a well-known fact is that the initial conditions of NNs are difficult to obtain in some cases. Therefore, the results of finite-time are conservative. In 2012, Polyakov [1] proposed the fixed-time theory, in which the estimation of the ST is independent of the initial state. In comparison to finite time stabilization, fixed-time stabilization (FTS) has better prospects for application and have come to be the focus of scientists and researchers. In [2], Liu and Zhang studied the FTS of fuzzy NNs by means of the theory of differential inclusion. In [3], the authors investigated the FTS problems for delayed memristive neural networks. In [4], Liu et al. have researched FTS of fuzzy inertial neural networks. However, FTS also has its disadvantages, of which ST is fixed and cannot be adjusted to meet some engineering applications. Fortunately, ST of preassigned-time stabilization (PTS) and synchronization discussed in [5,6] can be adjusted according to practical need, which has more meaningful in practical engineering applications.

Babcock and Westervelt first presented the concept of inertial neural networks (INNs) in 1986 by introducing inductance into neural circuits [7]. INNs are usually described by second-order differential equations. Compared with the traditional first-order NNs, INNs show more complex dynamic characteristics [8], such as instability, bifurcation, chaos, etc., and INNs helps to realize a chaotic search of memory in the application [9]. At the same time, adding inductors into NNs has broad biological background, which can simulate the membrane characteristics of the protruding pigeon vestibular hair cells of squid through inductance [10]. INNs have important applications in information processing; therefore, studies about the dynamics and control of INNs is of great importance. In [11,12,13,14], the theoretical results of global exponential stability, finite time stability, FTS, and preassigned-time synchronization (PTS) for INNs were obtained by employing the method of inserting intermediate variables and designing suitable controllers. Refs. [11,12,15] provide the application of INNs in the production of pseudo-random numbers and image encryption. In 2019, refs. [16,17] research the stability of the INNs by variable replacement. Many academics have conducted in-depth research on INNs and achieved some good theoretical results because INNs can solve many practical problems. This encourages us to further study INNs.

At the same time, because the limitations of physical components, which lead to the information transferring of NNs have always time delays [18]. Time delay can have an influence on the stability of NNs, which often leads to unstable oscillations and chaos [19,20]. Several classes of delay, such as constant delay, time-varying delay and distributed delay (DDs) have been, respectively, studied in relevant papers [21,22,23,24]. In [25,26], the authors discussed the passivity of fractional order NNs and quaternions NNs with time-varying delays, respectively. The stability of NNs with time-varying delays was investigated and some effective conditions were obtained in the work by Chen et al. [27]. Li et al. [28] studied the finite-time synchronization of quaternion-valued NNs with mixed-delays and some effective synchronisation criteria are reached by constructing novel feedback strategies. However, most of the above results are discussed in finite-time, and the NNs are also considered without inertial terms. So, the investigation of FTS and PTS of INNs with DDs are meaningful and necessary.

On the other hand, the above discussions are all based on real-valued NNs, in fact, in some practical applications, there often exist complex signals, such as information, flow of speech synthesis, and optoelectronics [29,30,31,32,33]. Compared to real-valued NNs, complex-valued neural networks (CVNNs) have superior characteristics, such as faster computing speed, greater processing capacity, and can solve problems that the real-valued system cannot, such as the well-known XOR problem [34]. As we know, the research method of complex-valued system is different from a real-valued system. In most cases, the complex-valued systems are split into two systems with real and imaginary parts, then real-valued approach can be used. There are a lot of literatures, e.g., see [35,36,37,38,39,40,41,42], which use separation method to study asymptotic synchronization, exponential synchronization and FTS. Although the separation approach is feasible, two issues will arise, firstly, dimension of the systems will increase after decomposition, and the processes of theoretical analysis will become more cumbersome. Secondly, it is difficult to decompose complex valued systems into corresponding real part systems and imaginary part systems in practical applications. Hence, it is profoundly meaningful to deal with complex-valued systems with a non-separation approach.

Based on the above discussions, to ensure that the results of complex-valued systems are more rigorous and scientific, in this article, without dividing complex-valued INNs into the real and imaginary parts, some effective results are derived to realize the FTS and PTS for complex-valued inertial neural networks (CVINNs) with DDs. The following contents are the major contributions of this article:

1. Unlike the NNs considered in previous works, refs. [43,44,45,46,47,48] only defined in the real number field, ref. [5] without inertial term, refs. [2,3,9,13] do not consider the DDs. The model of this article is with inertial term, DDs and defined in complex number domain, which is more general and meaningful.

2. In contrast to split the complex system into real and imaginary parts [35,36,37,38,39,40,41,42], this article proposes a suitable control strategy to implement FTS of CIVNNs with DDs by using a non-separation approach and some new criteria are obtained.

3. PTS were not investigated in [2,3,9,13,47,48]. By using a non-separation approach, we obtain some new results on PTS of CIVNNs with DDs, and the ST of PTS do not depend on any initial values and parameters of the CIVNNs.

Symbol Description: In this article, $\aleph =\{1,2,3\dots ,n\}$$\mathbb{R},\mathbb{C},{\mathbb{C}}^n$ stand for set of real numbers, set of complex numbers and n-dimensional complex-valued vectors, respectively. In addition, m represents the imaginary unit, for any $\vartheta =a+bi\in \mathbb{C}$, the conjugate expression of $\vartheta$ is $\overline{\vartheta }=a-bi,\left|\vartheta \right|=\sqrt{a\overline{a}}$, $\left[\vartheta \right]=sign\left(a\right)+isign\left(b\right)$, $m,\ell \in \aleph$. In addition, the $sign\left(a\right)$ represents the sign function of any $a\in \mathbb{R}$.

2. Preliminaries

The system can be described by

$$\begin{array}{cc}\hfill \frac{d^2{\vartheta }_m\left(t\right)}{d{t}^2}=& -a_m{\vartheta }_m\left(t\right)-b_m\frac{d{\vartheta }_m\left(t\right)}{dt}+\sum _{\ell =1}^n{\alpha }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(t\right)\right)+\sum _{\ell =1}^n{\beta }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))\right)\hfill \\ \hfill & +\sum _{\ell =1}^n{\psi }_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(s\right)\right)ds,t\ge 0,m\in \aleph \hfill \end{array}$$

(1)

and the initial values are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\vartheta }_m\left(\theta \right)={\varphi }_m\left(\theta \right),\hfill \\ \\ {\dot{\vartheta }}_m\left(\theta \right)={\phi }_m\left(\theta \right),\theta \in [-L,0],m\in \aleph \hfill \end{array}\right.\end{array}$$

(2)

where $m\in \aleph$, n represents number of neurons, ${\vartheta }_m\left(t\right)\in \mathbb{C}$ is the state of the $m\mathrm{th}$ neuron at time t; $a_m>0,b_m>0$; ${\alpha }_{m\ell }\in \mathbb{C},{\beta }_{m\ell }\in \mathbb{C},{\psi }_{m\ell }\in \mathbb{C}$ depict the neuron connection weights. ${ϝ}_{\ell }(·):\mathbb{C}\to \mathbb{C}$ denote the activation function. ${\zeta }_{\ell }\left(t\right)$ is delays, ${\mu }_{\ell }\left(t\right)$ is distributed delays. Furthermore, $0\le {\zeta }_{\ell }\left(t\right)\le {\zeta }_{\ell },0<{\mu }_{\ell }\left(t\right)\le {\mu }_{\ell }$, $\underset{\ell \in \aleph }{max}\{{\zeta }_{\ell },{\mu }_{\ell }\}=L$.

When the parameters of system (1) are inappropriate, the system (1) is unstable. To stabilize system (1), we build a controller and add it to the system (1) to obtain the system (3).

$$\begin{array}{cc}\hfill \frac{d^2{\vartheta }_m\left(t\right)}{d{t}^2}=& -a_m{\vartheta }_m\left(t\right)-b_m\frac{d{\vartheta }_m\left(t\right)}{dt}+\sum _{\ell =1}^n{\alpha }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(t\right)\right)+\sum _{\ell =1}^n{\beta }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))\right)\hfill \\ \hfill & +\sum _{\ell =1}^n{\psi }_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(s\right)\right)ds+u_m\left(t\right),t\ge 0,m\in \aleph \hfill \end{array}$$

(3)

where $u_m\left(t\right)$ is the control, which is given later.

Hypothesis 1.

For each $m=1,2,3,\cdots n$, there are real numbers ${\Im }_m>0$, such that for any $\hslash ,\hslash \in \mathbb{C}$,

$$\begin{array}{c}\hfill |{ϝ}_m(\hslash )-{ϝ}_m(\hslash )|\le {\Im }_m|\hslash -\hslash |.\end{array}$$

To further analyse the dynamical behaviour of the system (3), we give an intermediate variable ${\varpi }_m\left(t\right)=c_m\frac{d{\vartheta }_m\left(t\right)}{dt}+d_m{\vartheta }_m\left(t\right)$, where $c_m\ne 0,d_m$ are suitable constant. So, the system (3) was rewritten as

$$\begin{array}{cc}\hfill \frac{d{\vartheta }_m\left(t\right)}{dt}& =-\frac{d_m}{c_m}{\vartheta }_m\left(t\right)+\frac{1}{c_m}{\varpi }_m\left(t\right)\hfill \\ \hfill \frac{d{\varpi }_m\left(t\right)}{dt}& =-{\xi }_m{\varpi }_m\left(t\right)-{\delta }_m{\vartheta }_m\left(t\right)+\sum _{j=1}^n{c}_m{\alpha }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(t\right)\right)+\sum _{j=1}^n{c}_m{\beta }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))\right)\hfill \\ \hfill & +\sum _{\ell =1}^n{c}_m{\psi }_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(s\right)\right)ds+u_m\left(t\right),t\ge 0,m\in \aleph \hfill \end{array}$$

(4)

where ${\xi }_m=b_m-\frac{d_m}{c_m}$, ${\delta }_m=a_m{c}_m-{\xi }_m{d}_m$.

Definition 1

([6]). If for $\forall {\vartheta }_m\left(0\right),{\varpi }_m\left(0\right)\in \mathbb{C},T(\vartheta \left(0\right),\varpi \left(0\right))\ge 0$, $T(\vartheta \left(0\right),\varpi \left(0\right))\le \breve{T}$, and $\breve{T}$ is positive and irrelevant of the initial value of (4), one has ${lim}_{t\to \breve{T}}( \Vert \vartheta (t) \Vert +\Vert \varpi (t) \Vert )=0$, $\forall t\ge \breve{T}$, then, system (4) implements FTS at $\breve{T}$.

Definition 2

([49]). If system (4) is FTS, for a predefined constant ${\breve{T}}_p>0$, ST $T(\vartheta \left(0\right),\varpi \left(0\right))\le {\breve{T}}_p$, then, system (4) is called the PTS.

Lemma 1

([50]). For System (4), if $V\left(t\right)$ is a radial unbounded function, the following inequality stands

$$\begin{array}{c}\hfill \dot{V}\left(t\right)\le {\partial }_{1}V\left(t\right)-{\partial }_2{V}^{\alpha }\left(t\right)-{\partial }_3{V}^{\beta }\left(t\right)-o\end{array}$$

where ${\partial }_1\in \mathbb{R},{\partial }_2>0,{\partial }_3>0,\alpha >1,0\le \beta <1$ and ${\partial }_1<min\{{\partial }_2,{\partial }_3\}$, the system can achieve FTS, and the ST is estimated by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{T}_1=\frac{({\partial }_3^{\frac{1}{\beta }}+o^{\frac{1}{\beta }}{)}^{1-\beta }-o^{\frac{1-\beta }{\beta }}}{{\partial }_3^{\frac{1}{\beta }}(1-\beta )}+\frac{2^{\alpha -1}({\partial }_2^{\frac{1}{\alpha }}+o^{\frac{1}{\alpha }}{)}^{1-\alpha }}{{\partial }_2^{\frac{1}{\alpha }}(\alpha -1)},{\partial }_1\le 0\hfill \\ T_2=\frac{[{({\partial }_3-{\partial }_1)}^{\frac{1}{\beta }}+o^{\frac{1}{\beta }}{]}^{1-\beta }-o^{\frac{1-\beta }{\beta }}}{{({\partial }_3-{\partial }_1)}^{\frac{1}{\beta }}(1-\beta )}+\frac{2^{\alpha -1}[{({\partial }_2-{\partial }_1)}^{\frac{1}{\alpha }}+o^{\frac{1}{\alpha }}{]}^{1-\alpha }}{{({\partial }_2-{\partial }_1)}^{\frac{1}{\alpha }}(\alpha -1)},{\partial }_1>0.\hfill \end{array}\right.\end{array}$$

Lemma 2

([6]). For System (4), if $V\left(t\right)$ is a radial unbounded function, the following inequality holds

$$\begin{array}{c}\hfill \dot{V}\left(t\right)\le \frac{T_f}{T_p}\left({\partial }_{1}V\left(t\right)-{\partial }_2{V}^{\alpha }\left(t\right)-{\partial }_3{V}^{\beta }\left(t\right)-o\right)\end{array}$$

where ${\partial }_1\in \mathbb{R},{\partial }_2>0,{\partial }_3>0,\alpha >1,0\le \beta <1$ and ${\partial }_1<min\left({\partial }_2,{\partial }_3\right),T_p>0,$ the system can achieve PTS within prespecified time $T_p$, and

$$\begin{array}{c}\hfill T_f=\left\{\begin{array}{c}{T}_1,{\partial }_1\le 0,\hfill \\ \\ T_2,{\partial }_1>0.\hfill \end{array}\right.\end{array}$$

Lemma 3

([51]). Let ${\vartheta }_m\ge 0,m=1,2,3\cdots n;0<\beta \le 1,$ and $\alpha >1,$ one has

$$\begin{array}{c}\hfill \sum _{m=1}^n{\vartheta }_m^{\beta }\ge (\sum _{m=1}^n{\vartheta }_m{)}^{\beta };\sum _{m=1}^n{\vartheta }_m^{\alpha }\ge n^{1-\alpha }(\sum _{m=1}^n{\vartheta }_m{)}^{\alpha }\end{array}$$

Lemma 4

([52]). For any $w\left(t\right):\mathbb{R}\to \mathbb{C}$, and any measurable selection $\kappa \left(t\right)\in \overline{co}\left(\left[w\right(t\left)\right]\right)$, in which

$$\begin{array}{c}\hfill \overline{co}\left(\left[w\right(t\left)\right]\right)=\overline{co}\left(sign\left(Re\left(w\right(t\left)\right)\right)\right)+\overline{co}\left(sign\left(Im\left(w\right(t\left)\right)\right)\right)\end{array}$$

The formula below is correct.

$$\begin{array}{c}\hfill \overline{w\left(t\right)}\kappa \left(t\right)+\overline{\kappa \left(t\right)}w\left(t\right)\ge 2\left|w\left(t\right)\right|.\end{array}$$

3. Main Results

In this part, we analyse the FTS and PTS of system (4) via novel control strategy.

3.1. FTS of System (4)

The system (4) can achieve FTS through the following control strategy:

$$\begin{array}{c}\hfill u_{1m}\left(t\right)=\left\{\begin{array}{c}-\frac{1}{\overline{{\varpi }_m\left(t\right)}}\left(k_m|{\vartheta }_m{\left(t\right)|}^2+{\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{2m}{\left|{\vartheta }_m\left(t\right)\right|}^{2\beta }+o\right)\hfill \\ -\left[{\varpi }_m\left(t\right)\right]\left(r_m|{\varpi }_m\left(t\right)|+{\eta }_{1m}|{\varpi }_m{\left(t\right)|}^{2\alpha -1}+{\eta }_{2m}{\left|{\varpi }_m\left(t\right)\right|}^{2\beta -1}\right)\hfill \\ -\left[{\varpi }_m\left(t\right)\right]{\displaystyle \sum _{\ell =1}^n}{\rho }_{m\ell }\left|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)\right|-{\displaystyle \sum _{\ell =1}^n}{h}_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds\hfill \\ {\varpi }_m\left(t\right)\ne 0,\hfill \\ \\ 0{\varpi }_m\left(t\right)=0.\hfill \end{array}\right.\end{array}$$

(5)

where $k_m,r_m\in \mathbb{R},{\eta }_{1m},{\eta }_{2m},{\rho }_{m\ell },h_{m\ell },o>0$ for $m,\ell =1,2,3...,n$, $\alpha >1,0<\beta <1$, and the framework of above designed controller is given in Figure 1.

Theorem 1.

Under Hypothesis 1, and the following inequality holds

$$\begin{array}{c}\hfill -h_{m\ell }+c_m|{\psi }_{m\ell }|{\Im }_{\ell }\le 0,-{\rho }_{m\ell }+c_m\left|{\beta }_{m\ell }\right|{\Im }_{\ell }\le 0.\end{array}$$

(6)

then, the system (4) can achieve FTS, and the ST is

$$\begin{array}{c}\hfill T_f=\left\{\begin{array}{c}\frac{({\partial }_3^{\frac{1}{\beta }}+o^{\frac{1}{\beta }}{)}^{1-\beta }-o^{\frac{1-\beta }{\beta }}}{{\partial }_3^{\frac{1}{\beta }}(1-\beta )}+\frac{2^{\alpha -1}({\partial }_2^{\frac{1}{\alpha }}+o^{\frac{1}{\alpha }}{)}^{1-\alpha }}{{\partial }_2^{\frac{1}{\alpha }}(\alpha -1)},{\partial }_1\le 0\hfill \\ \frac{[{({\partial }_3-{\partial }_1)}^{\frac{1}{\beta }}+o^{\frac{1}{\beta }}{]}^{1-\beta }-o^{\frac{1-\beta }{\beta }}}{{({\partial }_3-{\partial }_1)}^{\frac{1}{\beta }}(1-\beta )}+\frac{2^{\alpha -1}[{({\partial }_2-{\partial }_1)}^{\frac{1}{\alpha }}+o^{\frac{1}{\alpha }}{]}^{1-\alpha }}{{({\partial }_2-{\partial }_1)}^{\frac{1}{\alpha }}(\alpha -1)},{\partial }_1>0\hfill \end{array}\right.\end{array}$$

(7)

where ${\partial }_1=max\{ı,j\}$; $ı=2(-k_m-\frac{d_m}{c_m}+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_{\ell }|{\alpha }_{\ell m}|{\Im }_m)$; $j=2(-r_m-{\xi }_m+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell })$; ${\partial }_2=min\{{\eta }_{1m}\}{n}^{1-\alpha }{\left(\frac{1}{2}\right)}^{-\alpha }; {\partial }_3=min\{{\eta }_{2m}\}{\left(\frac{1}{2}\right)}^{-\beta }$.

Proof. 

When ${\varpi }_m\left(t\right)\ne 0$, because the controller (5) is discontinuous, there is

$$\begin{array}{cc}\hfill u_{1m}\left(t\right)\in & -\frac{1}{\overline{{\varpi }_m\left(t\right)}}\left(k_m|{\vartheta }_m{\left(t\right)|}^2+{\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{2m}{|{\vartheta }_m\left(t\right)|}^{2\beta }+o\right)\hfill \\ \hfill & -\overline{co}\left(\left[{\varpi }_m\left(t\right)\right]\right)\left(r_m|{\varpi }_m\left(t\right)|+{\eta }_{1m}|{\varpi }_m{\left(t\right)|}^{2\alpha -1}+{\eta }_{2m}{|{\varpi }_m\left(t\right)|}^{2\beta -1}\right)\hfill \\ \hfill & -\overline{co}\left(\left[{\varpi }_m\left(t\right)\right]\right)\sum _{\ell =1}^n{\rho }_{m\ell }|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)|-\sum _{\ell =1}^n{h}_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds.\hfill \end{array}$$

(8)

By virtue of the measurable selection theorem, and because ${\kappa }_m\left(t\right)\in \overline{co}\left(\left[{\varpi }_m\left(t\right)\right]\right)$, such that

$$\begin{array}{cc}\hfill u_{1m}\left(t\right)=& -\frac{1}{\overline{{\varpi }_m\left(t\right)}}\left(k_m|{\vartheta }_m{\left(t\right)|}^2+{\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{2m}{|{\vartheta }_m\left(t\right)|}^{2\beta }+o\right)\hfill \\ \hfill & -{\kappa }_m\left(t\right)\left(r_m|{\varpi }_m\left(t\right)|+{\eta }_{1m}|{\varpi }_m{\left(t\right)|}^{2\alpha -1}+{\eta }_{2m}{|{\varpi }_m\left(t\right)|}^{2\beta -1}\right)\hfill \\ \hfill & -{\kappa }_m\left(t\right)\sum _{\ell =1}^n{\rho }_{m\ell }|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)|-\sum _{\ell =1}^n{h}_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds.\hfill \end{array}$$

(9)

Now, we build the Lyapunov function as follows:

$$\begin{array}{cc}\hfill V_1\left(t\right)& =\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right).\hfill \end{array}$$

(10)

We calculate the derivative of $V_1\left(t\right)$ along the solution of system (4), then, one has

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& \frac{1}{2}\sum _{m=1}^n\left(\frac{d{\vartheta }_m\left(t\right)}{dt}\overline{{\vartheta }_m\left(t\right)}+{\vartheta }_m\left(t\right)\overline{\frac{d{\vartheta }_m\left(t\right)}{dt}}+\frac{d{\varpi }_m\left(t\right)}{dt}\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{\frac{d{\varpi }_m\left(t\right)}{dt}}\right)\hfill \\ \hfill =& \sum _{m=1}^n\left(Re(\frac{d{\vartheta }_m\left(t\right)}{dt}\overline{{\vartheta }_m\left(t\right)})+Re(\frac{d{\varpi }_m\left(t\right)}{dt}\overline{{\varpi }_m\left(t\right)})\right)\hfill \\ \hfill =& \sum _{m=1}^n\left(-\frac{d_m}{c_m}{\vartheta }_m\left(t\right)\overline{{\vartheta }_m\left(t\right)}+\frac{1}{c_m}Re({\varpi }_m\left(t\right)\overline{{\vartheta }_m\left(t\right)})\right)\hfill \\ \hfill +& \sum _{m=1}^n\left(-{\xi }_m{\varpi }_m\left(t\right)\overline{{\varpi }_m\left(t\right)}-{\delta }_{m}Re(\overline{{\vartheta }_m\left(t\right)}{\varpi }_m\left(t\right))\right)\hfill \\ \hfill +& \sum _{m=1}^n\sum _{\ell =1}^n{c}_{m}Re\left({\alpha }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(t\right)\right)\overline{{\varpi }_m\left(t\right)}\right)\hfill \\ \hfill +& \sum _{m=1}^n\sum _{\ell =1}^n{c}_{m}Re\left({\beta }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))\right)\overline{{\varpi }_m\left(t\right)}\right)\hfill \\ \hfill +& \sum _{m=1}^n\sum _{\ell =1}^n{c}_{m}Re\left({\psi }_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{ϝ}_{\ell }({\vartheta }_{\ell }\left(s\right)ds\overline{{\varpi }_m\left(t\right)}\right)\hfill \\ \hfill +& \frac{1}{2}\sum _{m=1}^n\left(u_{1m}\left(t\right)\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{u_{1m}\left(t\right)}\right).\hfill \end{array}$$

(11)

□

According to the properties of norm of complex-valued number and Assumption 1,

$$\begin{array}{cc}\hfill & \sum _{m=1}^n(\frac{1}{c_m}-{\delta }_m)Re\left(\overline{{\vartheta }_m\left(t\right)}{\varpi }_m\left(t\right)\right)\hfill \\ \hfill & \le \sum _{m=1}^n|\frac{1}{c_m}-{\delta }_m\left|\right|\overline{{\vartheta }_m\left(t\right)}{\varpi }_m\left(t\right)|\hfill \\ \hfill & \le \frac{1}{2}\sum _{m=1}^n|\frac{1}{c_m}-{\delta }_m|\left(|{\vartheta }_m{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \sum _{m=1}^n\sum _{\ell =1}^n{c}_{m}Re\left({\alpha }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(t\right)\right)\overline{{\varpi }_m\left(t\right)}\right)\hfill \\ \hfill & \le \sum _{m=1}^n\sum _{\ell =1}^n{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell }|{\vartheta }_{\ell }\left(t\right)||\overline{{\varpi }_m\left(t\right)}|\hfill \\ \hfill & \le \frac{1}{2}\sum _{m=1}^n\sum _{\ell =1}^n{c}_m\left|{\alpha }_{m\ell }\right|{\Im }_{\ell }\left(|{\vartheta }_{\ell }{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right).\hfill \end{array}$$

(13)

Similar to above:

$$\begin{array}{cc}\hfill & \sum _{m=1}^n\sum _{\ell =1}^n{c}_{m}Re\left({\beta }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))\right)\overline{{\varpi }_m\left(t\right)}\right)\hfill \\ \hfill & \le \sum _{m=1}^n\sum _{\ell =1}^n{c}_m|{\beta }_{m\ell }|{\Im }_{\ell }|{\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))||\overline{{\varpi }_m\left(t\right)}|.\hfill \end{array}$$

(14)

If the results of (12)–(14) are combined, then (11) is equivalent to:

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)\le & \sum _{m=1}^n(-\frac{d_m}{c_m}+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\sum _{\ell =1}^n\frac{1}{2}{c}_{\ell }|{\alpha }_{\ell m}|{\Im }_m){|{\vartheta }_m\left(t\right)|}^2\hfill \\ \hfill +& \sum _{m=1}^n(-{\xi }_m+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\sum _{\ell =1}^n\frac{1}{2}{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell }){|{\varpi }_m\left(t\right)|}^2\hfill \\ \hfill +& \sum _{m=1}^n\sum _{\ell =1}^n{c}_m|{\beta }_{m\ell }|{\Im }_{\ell }|{\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))||\overline{{\varpi }_m\left(t\right)}|\hfill \\ \hfill +& \sum _{m=1}^n\sum _{\ell =1}^n{c}_{m}Re({\psi }_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(s\right)\right)ds\overline{{\varpi }_m\left(t\right)})\hfill \\ \hfill +& \frac{1}{2}\sum _{m=1}^n\left(u_{1m}\left(t\right)\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{u_{1m}\left(t\right)}\right).\hfill \end{array}$$

(15)

Among,

$$\begin{array}{cc}\hfill & \frac{1}{2}\sum _{m=1}^n\left(u_{1m}\left(t\right)\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{u_{1m}\left(t\right)}\right)\hfill \\ \hfill & =-\sum _{m=1}^n\left(k_m|{\vartheta }_m{\left(t\right)|}^2+{\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{2m}{|{\vartheta }_m\left(t\right)|}^{2\beta }+o\right)\hfill \\ \hfill & -\frac{1}{2}\sum _{m=1}^n\left({\kappa }_m\left(t\right)\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{{\kappa }_m\left(t\right)}\right)(r_m|{\varpi }_m\left(t\right)|+{\eta }_{1m}{|{\varpi }_m\left(t\right)|}^{2\alpha -1}\hfill \\ \hfill & +{\eta }_{2m}|{\varpi }_m{\left(t\right)|}^{2\beta -1})-\frac{1}{2}\sum _{m=1}^n\sum _{\ell =1}^n\left({\kappa }_m\left(t\right)\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{{\kappa }_m\left(t\right)}\right){\rho }_{m\ell }|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)|\hfill \\ \hfill & -\frac{1}{2}\sum _{m=1}^n\sum _{\ell =1}^n(h_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{h_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds})\hfill \\ \hfill & \le -\sum _{m=1}^n\left(k_m|{\vartheta }_m{\left(t\right)|}^2+{\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{2m}{|{\vartheta }_m\left(t\right)|}^{2\beta }+o\right)\hfill \\ \hfill & -|{\varpi }_m\left(t\right)|\left(r_m|{\varpi }_m\left(t\right)|+{\eta }_{1m}|{\varpi }_m{\left(t\right)|}^{2\alpha -1}+{\eta }_{2m}{|{\varpi }_m\left(t\right)|}^{2\beta -1}\right)\hfill \\ \hfill & -\sum _{m=1}^n\sum _{\ell =1}^n{\rho }_{m\ell }|{\varpi }_m\left(t\right)||{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)|-\sum _{m=1}^n\sum _{\ell =1}^{n}Re(h_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds\overline{{\varpi }_m\left(t\right)})\hfill \\ \hfill & \le -\sum _{m=1}^n{k}_m|{\vartheta }_m{\left(t\right)|}^2-\sum _{m=1}^n{r}_m{|{\varpi }_m\left(t\right)|}^2-\sum _{m=1}^n\left({\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{1m}{|{\varpi }_m\left(t\right)|}^{2\alpha }\right)\hfill \\ \hfill & -\sum _{m=1}^n\left({\eta }_{2m}|{\vartheta }_m{\left(t\right)|}^{2\beta }+{\eta }_{2m}{|{\varpi }_m\left(t\right)|}^{2\beta }\right)-\sum _{m=1}^n\sum _{\ell =1}^n{\rho }_{m\ell }|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)||{\varpi }_m\left(t\right)|\hfill \\ \hfill & -\sum _{m=1}^n\sum _{\ell =1}^n|h_{m\ell }|{\int }_{t-{\mu }_{\ell }\left(t\right)}^t|{\vartheta }_{\ell }\left(s\right)|ds|\overline{{\varpi }_m\left(t\right)}|-o.\hfill \end{array}$$

(16)

According to Lemma 3, we can obtain

$$\begin{array}{cc}\hfill & -\sum _{m=1}^n\left({\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{1m}{\left|{\varpi }_m\left(t\right)\right|}^{2\alpha }\right)\hfill \\ \hfill & \le -\sum _{m=1}^n{\eta }_{1m}\left({(|{\vartheta }_m{\left(t\right)|}^2)}^{\alpha }+{\left(|{\varpi }_m{\left(t\right)|}^2\right)}^{\alpha }\right)\hfill \\ \hfill & \le -{\eta }_{1m}{n}^{1-\alpha }{\left(\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right)\right)}^{\alpha }\hfill \\ \hfill & =-{\eta }_{1m}{n}^{1-\alpha }{(\frac{1}{2})}^{-\alpha }{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right)\right)}^{\alpha }.\hfill \end{array}$$

(17)

Similarly,

$$\begin{array}{cc}\hfill & -\sum _{m=1}^n\left({\eta }_{2m}|{\vartheta }_m{\left(t\right)|}^{2\beta }+{\eta }_{2m}{\left|{\varpi }_m\left(t\right)\right|}^{2\beta }\right)\hfill \\ \hfill & \le -\sum _{m=1}^n{\eta }_{2m}\left({\left(|{\vartheta }_m{\left(t\right)|}^2\right)}^{\beta }+{\left(|{\varpi }_m{\left(t\right)|}^2\right)}^{\beta }\right)\hfill \\ \hfill & \le -{\eta }_{2m}{\left(\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right)\right)}^{\beta }\hfill \\ \hfill & =-{\eta }_{2m}{(\frac{1}{2})}^{-\beta }{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right)\right)}^{\beta }.\hfill \end{array}$$

(18)

According to (17) and (18),

$$\begin{array}{cc}\hfill & \frac{1}{2}\sum _{m=1}^n\left(u_{1m}\left(t\right)\overline{{\varpi }_m\left(t\right)}+{\varpi }_m\left(t\right)\overline{u_{1m}\left(t\right)}\right)\hfill \\ \hfill & \le -\sum _{m=1}^n{k}_m|{\vartheta }_m{\left(t\right)|}^2-\sum _{m=1}^n{r}_m{|{\varpi }_m\left(t\right)|}^2\hfill \\ \hfill & -min\left\{{\eta }_{1m}\right\}{n}^{1-\alpha }{(\frac{1}{2})}^{-\alpha }{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\alpha }\hfill \\ \hfill & -min\left\{{\eta }_{2m}\right\}{(\frac{1}{2})}^{-\beta }{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\beta }\hfill \\ \hfill & -\sum _{m=1}^n\sum _{\ell =1}^n{\rho }_{m\ell }|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)||{\varpi }_m\left(t\right)|-\sum _{m=1}^n\sum _{\ell =1}^n|h_{m\ell }|{\int }_{t-{\mu }_{\ell }\left(t\right)}^t|{\vartheta }_{\ell }\left(s\right)|ds|\overline{{\varpi }_m\left(t\right)}|-o.\hfill \end{array}$$

(19)

Therefore,

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)& \le \sum _{m=1}^n(-k_m-\frac{d_m}{c_m}+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\sum _{\ell =1}^n\frac{1}{2}{c}_{\ell }|{\alpha }_{\ell m}|{\Im }_m){|{\vartheta }_m\left(t\right)|}^2\hfill \\ \hfill & +\sum _{m=1}^n(-r_m-{\xi }_m+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}\sum _{\ell =1}^n{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell }){|{\varpi }_m\left(t\right)|}^2\hfill \\ \hfill & +\sum _{m=1}^n\sum _{\ell =1}^n(-{\rho }_{m\ell }+c_m|{\beta }_{m\ell }|{\Im }_{\ell })|{\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))||\overline{{\varpi }_m\left(t\right)}|\hfill \\ \hfill & +\sum _{m=1}^n\sum _{\ell =1}^n(-h_{m\ell }+c_m|{\psi }_{m\ell }|{\Im }_{\ell }){\int }_{t-{\mu }_{\ell }\left(t\right)}^t|{\vartheta }_{\ell }\left(s\right)|ds|\overline{{\varpi }_m\left(t\right)}|\hfill \\ \hfill & -min\left\{{\eta }_{1m}\right\}{n}^{1-\alpha }{(\frac{1}{2})}^{-\alpha }{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\alpha }\hfill \\ \hfill & -min\left\{{\eta }_{2m}\right\}{(\frac{1}{2})}^{-\beta }{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\beta }-o\hfill \\ \hfill & \le {\partial }_1\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)-{\partial }_2{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\alpha }\hfill \\ \hfill & -{\partial }_3{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\beta }-o\hfill \\ \hfill & \le {\partial }_1{V}_1\left(t\right)-{\partial }_2{V}_1^{\alpha }\left(t\right)-{\partial }_3{V}_1^{\beta }\left(t\right)-o.\hfill \end{array}$$

(20)

where ${\partial }_1=max\{ı,j\}$ $ı=2(-k_m-\frac{d_m}{c_m}+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_{\ell }|{\alpha }_{\ell m}|{\Im }_m)$; $j=2(-r_m-{\xi }_m+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell })$; ${\partial }_2=min\{{\eta }_{1m}\}{n}^{1-\alpha }{\left(\frac{1}{2}\right)}^{-\alpha };{\partial }_3=min\{{\eta }_{2m}\}{\left(\frac{1}{2}\right)}^{-\beta }$.

Remark 1.

Construct simple Lyapunov function is important in discussing the FTS and PTS of NNs. Because the NNs in this article are complex-valued and Lyapunov function request positive definite, so, Lyapunov function with 2-norm are considered in our discussion.

Remark 2.

In the existing literature [2,5,53], although fixed-time synchronization or stabilization of NNs have been discussed. However, [53] ignored the DDs, [5] discussed the NNs only with first-order, and [2] only discussed the FTS in the real number domain. Therefore, the results obtained in this article are more general and extended the previous works.

3.2. PST of System (4)

The system (4) can achieve PTS through the following control strategy:

$$\begin{array}{c}\hfill u_{2m}\left(t\right)=\left\{\begin{array}{c}-\frac{1}{\overline{{\varpi }_m\left(t\right)}}\left(({ϵ}_m+\frac{T_f}{T_p}({\tilde{k}}_m-{ϵ}_m)){|{\vartheta }_m\left(t\right)|}^2\right)\hfill \\ -\left[{\varpi }_m\left(t\right)\right]\left(({\varrho }_m+\frac{T_f}{T_p}({\tilde{r}}_m-{\varrho }_m))|{\varpi }_m\left(t\right)|\right)\hfill \\ -\frac{T_f}{T_p}\frac{1}{\overline{{\varpi }_m\left(t\right)}}\left({\tilde{\eta }}_{1m}|{\vartheta }_m{\left(t\right)|}^{2\tilde{\alpha }}+{\tilde{\eta }}_{2m}{|{\vartheta }_m\left(t\right)|}^{2\tilde{\beta }}+\tilde{o}\right)\hfill \\ -\frac{T_f}{T_p}\left[{\varpi }_m\left(t\right)\right]\left({\tilde{\eta }}_{1m}|{\varpi }_m{\left(t\right)|}^{2\tilde{\alpha }-1}+{\tilde{\eta }}_{2m}{|{\varpi }_m\left(t\right)|}^{2\tilde{\beta }-1}\right)\hfill \\ -\left[{\varpi }_m\left(t\right)\right]{\displaystyle \sum _{\ell =1}^n}{\tilde{\rho }}_{m\ell }|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)|-{\displaystyle \sum _{\ell =1}^n}{\tilde{h}}_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds,\hfill \\ {\varpi }_m\left(t\right)\ne 0;\hfill \\ 0{\varpi }_m\left(t\right)=0.\hfill \end{array}\right.\end{array}$$

(21)

where ${ϵ}_m=-\frac{d_m}{c_m}+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_{\ell }|{\alpha }_{\ell m}|{\Im }_m$, ${\varrho }_m=-{\xi }_m+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell }$.

Theorem 2.

Under Hypothesis 1, the following inequality holds

$$\begin{array}{c}\hfill -{\tilde{h}}_{m\ell }+c_m|{\psi }_{m\ell }|{\Im }_{\ell }\le 0,-{\tilde{\rho }}_{m\ell }+c_m|{\beta }_{m\ell }|{\Im }_{\ell }\le 0.\end{array}$$

(22)

the system (4) can achieve preassigned-time, and the preassigned-time is $T_p$.

Proof. 

Now, we build the Lyapunov functional as follows:

$$\begin{array}{c}\hfill V_2\left(t\right)=\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{\left|{\varpi }_m\left(t\right)\right|}^2\right).\end{array}$$

(23)

We calculate the derivative of $V_2\left(t\right)$ along the solution of system (4), similar to the proof of Theorem 1, we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(t\right)\le & \sum _{m=1}^n(-\frac{d_m}{c_m}+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\sum _{\ell =1}^n\frac{1}{2}{c}_{\ell }|{\alpha }_{\ell m}|{\Im }_m){\vartheta }_m\left(t\right)\overline{{\vartheta }_m\left(t\right))}\hfill \\ \hfill +& \sum _{m=1}^n(-{\xi }_m+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\sum _{\ell =1}^n\frac{1}{2}{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell }){\varpi }_m\left(t\right)\overline{{\varpi }_m\left(t\right)}\hfill \\ \hfill +& \sum _{m=1}^n\sum _{\ell =1}^n{c}_m|{\beta }_{m\ell }|{\Im }_{\ell }|{\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))||\overline{{\varpi }_m\left(t\right)}|\hfill \\ \hfill +& \sum _{m=1}^n\sum _{\ell =1}^n{c}_{m}Re({\psi }_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(s\right)\right)ds\overline{{\varpi }_m\left(t\right)})\hfill \\ \hfill -& \sum _{m=1}^n({ϵ}_m+\frac{T_f}{T_p}\left({\tilde{k}}_m-{ϵ}_m\right))|{\vartheta }_m{\left(t\right)|}^2-\sum _{m=1}^n({\varrho }_m+\frac{T_f}{T_p}\left({\tilde{r}}_m-{\varrho }_m\right)){|{\varpi }_m\left(t\right)|}^2\hfill \\ \hfill -& \sum _{m=1}^n\sum _{\ell =1}^n{\tilde{\rho }}_{m\ell }|{\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right)||{\varpi }_m\left(t\right)|-\sum _{m=1}^n\sum _{\ell =1}^n{\tilde{h}}_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t|{\vartheta }_{\ell }\left(s\right)|ds|\overline{{\varpi }_m\left(t\right)}|\hfill \\ \hfill -& \frac{T_f}{T_p}min\left\{{\tilde{\eta }}_{1m}\right\}{n}^{1-\tilde{\alpha }}{(\frac{1}{2})}^{-\tilde{\alpha }}{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\tilde{\alpha }}\hfill \\ \hfill -& \frac{T_f}{T_p}min\left\{{\tilde{\eta }}_{2m}\right\}{(\frac{1}{2})}^{-\tilde{\beta }}{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\tilde{\beta }}-\frac{T_f}{T_p}\tilde{o}\hfill \\ \hfill \le & 2max\{-{\tilde{k}}_m+{ϵ}_m,-{\tilde{r}}_m+{\varrho }_m\}\frac{T_f}{T_p}\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)\hfill \\ \hfill -& \frac{T_f}{T_p}min\left\{{\tilde{\eta }}_{1m}\right\}{n}^{1-\tilde{\alpha }}{(\frac{1}{2})}^{-\tilde{\alpha }}{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\tilde{\alpha }}\hfill \\ \hfill -& \frac{T_f}{T_p}min\left\{{\tilde{\eta }}_{2m}\right\}{(\frac{1}{2})}^{-\tilde{\beta }}{\left(\frac{1}{2}\sum _{m=1}^n\left(|{\vartheta }_m{\left(t\right)|}^2+{|{\varpi }_m\left(t\right)|}^2\right)\right)}^{\tilde{\beta }}-\frac{T_f}{T_p}\tilde{o}\hfill \\ \hfill =& \frac{T_f}{T_p}\left({\tilde{\partial }}_1{V}_2\left(t\right)-{\tilde{\partial }}_2{V}_2^{\tilde{\alpha }}\left(t\right)-{\tilde{\partial }}_3{V}_2^{\tilde{\beta }}\left(t\right)-\tilde{o}\right).\hfill \end{array}$$

(24)

where ${\tilde{\partial }}_1=2max\{-{\tilde{k}}_m+{ϵ}_m,-{\tilde{r}}_m+{\varrho }_m\}$, ${\tilde{\partial }}_2=min\{{\tilde{\eta }}_{1m}\}{n}^{1-\tilde{\alpha }}{\left(\frac{1}{2}\right)}^{-\tilde{\alpha }}$; ${\tilde{\partial }}_3=min\{{\tilde{\eta }}_{2m}\}{\left(\frac{1}{2}\right)}^{-\tilde{\beta }}$.

□

Remark 3.

Here, we let ${ϵ}_m=-\frac{d_m}{c_m}+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_{\ell }|{\alpha }_{\ell m}|{\Im }_m$ and ${\varrho }_m=-{\xi }_m+\frac{1}{2}|\frac{1}{c_m}-{\delta }_m|+\frac{1}{2}{\displaystyle \sum _{\ell =1}^n}{c}_m|{\alpha }_{m\ell }|{\Im }_{\ell }$, which is better to ensure stability and achieve PTS of such CVINNs.

Remark 4.

Unlike [2,54,55,56], only considered finite/fixed-time results in the real number field, the PTS of INNs with complex-valued, and the DDs are studied; the results of this article further enrich the dynamics behaviours of NNs.

Remark 5.

In order to show the advantages of this article, some comparisons between the related previous works and this article is shown in

Table 1. The comparisons between the related previous works and this paper.

References	Inertial Items	Complex Value	Distributed Delay	FTS	PTS	Non Separation Method
[2]	√	×	√	√	×	×
[3]	×	×	×	√	√	×
[4]	√	×	×	√	×	×
[12]	√	×	√	×	×	×
[13]	√	×	×	√	√	×
[42]	×	√	×	√	×	×
[43,44,45,47,48]	√	×	×	×	×	×
This paper	√	√	√	√	√	√

:

4. Numerical Example

To demonstrate the effectiveness of the FTS and PTS results, the relevant simulations and applications are as follows.

Example 1.

We consider the below reduced-order CVINNs with DDs.

$$\begin{array}{cc}\hfill \frac{d{\vartheta }_m\left(t\right)}{dt}& =-\frac{d_m}{c_m}{\vartheta }_m\left(t\right)+\frac{1}{c_m}{\varpi }_m\left(t\right)\hfill \\ \hfill \frac{d{\varpi }_m\left(t\right)}{dt}& =-{\xi }_m{\varpi }_m\left(t\right)-{\delta }_m{\vartheta }_m\left(t\right)+\sum _{j=1}^n{c}_m{\alpha }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(t\right)\right)\hfill \\ \hfill & +\sum _{j=1}^n{c}_m{\beta }_{m\ell }{ϝ}_{\ell }\left({\vartheta }_{\ell }(t-{\zeta }_{\ell }\left(t\right))\right)+\sum _{\ell =1}^n{c}_m{\psi }_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{ϝ}_{\ell }\left({\vartheta }_{\ell }\left(s\right)\right)ds+u_{1m}\left(t\right)\hfill \\ \hfill & t\ge 0,m=1,2\hfill \end{array}$$

(25)

where $m=\ell =1,2$, ${\zeta }_{\ell }\left(t\right)={\mu }_{\ell }\left(t\right)=\frac{e^t}{1+e^t}$, ${ϝ}_{\ell }(·)=tanh\left(Re(·)\right)+tanh\left(Im(·)\right)i$, $c_m=d_m=1$; ${\xi }_1=0.4$, ${\xi }_2=7.8$; ${\delta }_1=0.4$, ${\delta }_2=-7$, $u_{1m}\left(t\right)$ is control input.

$$A={\left({\alpha }_{m\ell }\right)}_{3\times 3}=\left(\begin{array}{cc}-1.8-2i& 2.0-2.3i\\ -2.2-1.6i& 1.8+2.5i\end{array}\right)$$

$$B={\left({\beta }_{m\ell }\right)}_{3\times 3}=\left(\begin{array}{cc}-0.8+1.5i& 2.6-0.2i\\ -2.5-2.3i& 2.2-2.9i\end{array}\right)$$

$$C={\left({\psi }_{m\ell }\right)}_{3\times 3}=\left(\begin{array}{cc}-2.7+2.1i& -1.2+1.5i\\ 0.1-2.8i& -2.5-0.1i\end{array}\right)$$

The initial value are ${\vartheta }_1\left(\theta \right)=2+2i,{\vartheta }_2\left(\theta \right)=-2-2i,{\varpi }_1\left(\theta \right)=-0.6+0.2i,{\varpi }_2\left(\theta \right)=-0.6-0.1i$.

The dynamic behaviour of model (25) without controller is as shown in the Figure 2 and Figure 3. Obviously, the system (25) is unstable. We therefore design a controller to stabilise the system as shown.

$$\begin{array}{c}\hfill u_{1m}\left(t\right)=\left\{\begin{array}{c}-\frac{1}{\overline{{\varpi }_m\left(t\right)}}\left(k_m|{\vartheta }_m{\left(t\right)|}^2+{\eta }_{1m}|{\vartheta }_m{\left(t\right)|}^{2\alpha }+{\eta }_{2m}{|{\vartheta }_m\left(t\right)|}^{2\beta }+o\right)\hfill \\ -\left[{\varpi }_m\left(t\right)\right]\left(r_m|{\varpi }_m\left(t\right)|+{\eta }_{1m}|{\varpi }_m{\left(t\right)|}^{2\alpha -1}+{\eta }_{2m}{|{\varpi }_m\left(t\right)|}^{2\beta -1}\right)\hfill \\ -\left[{\varpi }_m\left(t\right)\right]{\displaystyle \sum _{\ell =1}^n}{\rho }_{m\ell }|{\vartheta }_{\ell }\left(t-{\zeta }_{\ell }\left(t\right)\right)|-{\displaystyle \sum _{\ell =1}^n}{h}_{m\ell }{\int }_{t-{\mu }_{\ell }\left(t\right)}^t{\vartheta }_{\ell }\left(s\right)ds\hfill \\ {\varpi }_m\left(t\right)\ne 0;\hfill \\ 0{\varpi }_m\left(t\right)=0.\hfill \end{array}\right.\end{array}$$

(26)

In System (25), a continuous function ${ϝ}_m\left(t\right)$ satisfies Assumption 1 and ${\Im }_m={\Im }_{\ell }=1,m=\ell =1,2$. We know from straightforward computations that the conditions of Theorem 1 are all satisfied. We design the gain of the controller (26) as $\alpha =1.6,\beta =0.7,k_1=10,k_2=1.8,r_1=12.1,r_2=1.1$, ${\eta }_{11}=4.5,{\eta }_{12}=7.9,{\eta }_{21}=5.6,{\eta }_{22}=8.7$. Meanwhile, the values of ${\rho }_{m\ell }$ and $h_{m\ell }$ here are as follows:

$$\begin{array}{c}\hfill {\left({\rho }_{m\ell }\right)}_{3\times 3}=\left(\begin{array}{cc}2.2& 2.61\\ 3.4& 3.85\end{array}\right);{\left(h_{m\ell }\right)}_{3\times 3}=\left(\begin{array}{cc}3.8& 4\\ 2.9& 3.1\end{array}\right)\end{array}$$

Figure 2. The phase and states ${\vartheta }_m\left(t\right)$ trajectories of real part of CVINNs (25) with $u_{1m}=0$, m = 1, 2.

Figure 2. The phase and states ${\vartheta }_m\left(t\right)$ trajectories of real part of CVINNs (25) with $u_{1m}=0$, m = 1, 2.

Figure 3. The phase and states ${\vartheta }_m\left(t\right)$ trajectories of the image part of CVINNs (25) with $u_{1m}=0$, m = 1, 2.

Figure 3. The phase and states ${\vartheta }_m\left(t\right)$ trajectories of the image part of CVINNs (25) with $u_{1m}=0$, m = 1, 2.

CVINN with DDs (25) is stable in fixed-time under control (26) which are shown in Figure 4. Furthermore, the setting time of the system (25) is $T_{max}=7.6416$. In the same way, we change the parameters to satisfy the requirement of Theorem 2. Set $T_p=6,5.5$ and 5, respectively. From Figure 5, Figure 6 and Figure 7, we can see that the system (25) reaches stabilization at 6, 5.5, and 5, respectively.

By using results of FTS given above, the PRNG is obtained, which is represented in Figure 8. Allow $z\left(t\right)$ to be the original transmitted signal as shown in Figure 9. Then obtain the encrypted signal, $o\left(t\right)=y\left(t\right)⨂z\left(t\right)$ as illustrated in Figure 10. Due to the unstable nature of CVINNs with DDs, there is an enormous difference between encrypted signal created by PRNG and original signal.

5. Conclusions

In this article, we provide some sufficient conditions to obtain the FTS and PTS of CVINNs with DDs. In contrast to previous studies, we study the dynamic behaviour of CVINNs by using the non-separation approach, which not only reduces the computational complexity but also reduces the inconvenience. Finally, we provide numerical examples for the FTS and PTS to verify the effectiveness of the obtained results, and an application of CVINNs with DDs in pseudorandom number generation is provided. Our future research directions include (1) extending the research results to CVINNs with state-dependent switching; and (2) investigating the dynamic behaviour of CVINNs by using event-triggered control and non-reduced order approach.