On the Stability and Synchronization of Distributed-Order Coupled Delayed Neural Networks: A Novel Halanay Inequality Technique

Abstract

This paper investigates the stability and synchronization of distributed-order coupled delayed neural networks (DOCDNNs). First, an analytical solution to the distributed-order linear system is proved, thus resulting in an asymptotic stability criterion for distributed-order linear systems. This solution function is an extension of the Mittag–Leffler function. Then, a series of pivotal mathematical properties of the solution function are established, encompassing differential formula, monotonicity, weak additivity, and asymptotic property. It is further demonstrated that a novel distributed-order non-autonomous Halanay inequality can be derived from the unique properties of the solution function. Based on the proposed Halanay inequality technique, an asymptotic stability determination theorem for distributed-order nonlinear systems is derived. Moreover, the stability and synchronization of DOCNNs are analyzed using this theorem. The efficacy of the proposed method is substantiated by two numerical examples.

1. Introduction

Fractional-order calculus theory can be viewed as an extension of classical integration and differentiation, enabling the description of the dynamic behavior of complex systems [1]. Fractional-order calculus provides new insights into the analysis of systems with memory and genetic properties, characteristics not present in integer-order systems [2]. Research on fractional-order systems has made significant progress over the past few decades. Many scholars have recognized the extensive applicability of fractional-order calculus in both practical applications and theoretical analysis across various fields of science and engineering, such as signal processing [3], electromagnetic phenomena [4], neural networks [5], viscoelastic materials [6], module identification [7], and automatic control systems [8]. Additionally, the dynamic behavior analysis of fractional-order systems has garnered significant attention and research from many scholars, including stability [9], synchronization [10], and dissipativity [11].

Distributed-order calculus is an extension of fractional-order calculus developed based on the model of single fractional-order calculus [12]. It was first proposed by Caputo in 1969 [13], and further expanded and studied in 1995 [14]. When describing actual physical systems, a constant-value fractional-order system still cannot fully and accurately capture certain complex phenomena. Since distributed-order operators are defined by analytical expressions obtained through integration of power series terms, distributed-order systems with distributed-order operators can accurately capture certain complex phenomena [12]. The distributed-order system not only retains the nonlocal characteristics of fractional-order systems but also better describes the influence of order distribution in real systems, demonstrating significant advantages in aspects such as viscoelastic dynamics [15], robotics [16], non-uniform phenomena [17], fluid mechanics [18], and so on. In recent years, the stability analysis of distributed-order systems has become one of the research hotspots. The fractional-order Lyapunov direct method proposed by Li et al. [19] was extended by Taghavian & Tavazoei to the case of distributed-order systems [20]. Wu et al. established a stability criterion based on the Lyapunov method for discrete-time nabla distributed-order nonlinear systems [21]. Furthermore, Aminikhah et al. investigated the chaotic behavior and stability of distributed-order Chen systems, further demonstrating the rich dynamics exhibited by distributed-order systems in nonlinear dynamics [22]. Therefore, it is of great significance for the stability analysis of distributed-order systems.

Neural networks are artificial intelligence models inspired by the structure and function of biological neurons. Their core architecture consists of a large number of interconnected processing units, which collaborate through complex hierarchical structures and weighting mechanisms to achieve perception, learning, and abstraction of complex data patterns [23]. Coupled neural networks are complex network systems constructed by interconnecting multiple neural network structures [24]. Previous literature has studied and discussed integer-order coupled neural networks [25,26]. Since fractional-order neural networks can serve as important tools for accurately describing neurons in the human brain, and since the brain is a complex network formed by multiple regions through coupling, fractional-order coupled models can better simulate the time delays, memory accumulation, and nonlinear feedback in information transmission between regions. Numerous researchers have focused on the dynamic characteristics of fractional-order coupled neural networks. In [27], Sun et al. introduced symbolic graph theory into fractional-order coupled neural networks (FCNNs) and developed a direct error method to study the bipartite synchronization problem of leaderless FCNNs. In [28], Zhang investigated the bipartite quasi-synchronization problem of fractional symbolic networks composed of a set of antagonistic coupled neural networks. Then, in [29], Wang explored the output synchronization problem of multi-weight coupled fractional-order neural networks (MWCFONNs) with certain and uncertain parameters, and then utilized the properties of the Mittag–Leffler function and Laplace transform to established some output synchronization criteria for MWCFONNs. It is evident that current academic research focuses more on fractional-order coupled neural networks. However, in actual neural networks, different nodes or connections may have different memory characteristics. The distributed-order model allows assigning different order distribution functions $\omega \left(\alpha \right)$ to each coupled path, better reflecting the heterogeneity of biological networks. Moreover, in fields such as control systems, neural networks, and biomechanics, time delays are ubiquitous and may disrupt system stability or performance. Therefore, further study is needed on the dynamic characteristics of distributed-order coupled delayed neural networks (DOCDNNs).

The Lyapunov–Krasovskii method provides a general framework for stability analysis of delayed systems by constructing a delay-dependent Lyapunov functional [30]. However, its functional form is complex and parameter selection is challenging in practical applications. The Lyapunov–Razumikhin method simplifies the functional design [31]. However, it requires the time-delay state energy to be strictly constrained by the current state. This results in highly sensitive outcomes to time-delay magnitude, making it difficult to handle scenarios with large or time-varying delays. Therefore, the Lyapunov–Krasovskii and Lyapunov–Razumikhin methods have significant limitations when analyzing the dynamic performance of systems with time delays. To address these limitations, researchers have adopted inequality-based approaches to analyze the dynamic characteristics of systems with time delays [32,33].

Halanay inequalities play a significant role in exploring the dynamic characteristics of systems with time delays. For example, in [34], Chen et al. proposed two novel finite-time stability theorems using a generalized form of the Halanay inequality, and further derived the finite-time synchronization criteria for time-delay neural networks. In [35], Thi Thu Huong et al. established a generalized Halanay inequality with distributed time delays and derived Mittag–Leffler stability conditions of fractional-order differential equations with time delays. In [36], Yang et al. derived a distributed-order autonomous Halanay inequality by promoting the well-known Halanay inequality, and then derived a continuous lag synchronization criterion for heterogeneous distributed-order coupled neural networks with unbounded delay coupling. Since real-world systems are typically influenced by external environmental factors and internal parameter changes, non-autonomous systems are real-world entities or indispensable [37]. For the DOCDNNs, non-autonomous characteristics reflect the dependence of network dynamic behavior on time variations, which may originate from external stimuli, environmental changes, or adjustments to internal parameters. The distributed-order autonomous Halanay inequality has some limitations when analyzing the dynamic characteristics of distributed-order non-autonomous neural networks. It is evident that there is a necessity for further research to be conducted on a more effective distributed-order non-autonomous Halanay inequality.

Based on the discussion above, a novel distributed-order non-autonomous Halanay inequality and an asymptotic stability determination theorem for distributed-order nonlinear systems are derived. Furthermore, this theorem is used to analyze the stability and synchronization of DOCDNNs. The main innovations of this paper are as follows.

(1) An analytical solution to the distributed-order linear system is presented. A series of pivotal mathematical properties of solution function are proved, including differential formula, monotonicity, weak additivity and asymptotic property. In fact, the solution function can be seen as an extension of Mittag–Leffler function and exponential function.

(2) A novel distributed-order non-autonomous Halanay inequality is derived based on the mathematical properties of the solution function, which can be regarded as a generalization of classical Halanay inequality. Based on this novel Halanay inequality technique, an asymptotic stability determination theorem for distributed-order nonlinear systems is derived.

(3) The stability and synchronization of distributed-order coupled neural networks with time delays are analyzed by the asymptotic stability determination theorem for distributed-order nonlinear systems. Simulation results validate the correctness of the proposed theory.

The rest of this paper is organized as follows. Section 2 presents a series of definitions and lemmas related to distributed-order calculus. Section 3 presents an analytical solution to the distributed-order linear system. Several pivotal properties of the solution function are substantiated and a new distributed-order non-autonomous Halanay inequality is proven. Based on this inequality, an asymptotic stability determination theorem for distributed-order nonlinear systems is derived. Furthermore, stability and synchronization analysis for DOCDNNs are derived using the asymptotic stability determination theorem for distributed-order nonlinear systems. Section 4 provides empirical demonstrations of the proposed method’s efficacy through two numerical examples. Finally, Section 5 offers a comprehensive summary of this paper.

Notations: In this paper, the complex set is denoted by $\mathbb{C}$, the positive integer set by $\mathbb{N}$, the nonnegative integers set by ${\mathbb{N}}^{+}$, the real set by $\mathbb{R}$, and the positive real set by ${\mathbb{R}}^{+}$. Let ${\mathbb{R}}^n$ (${\mathbb{R}}^{n\times n}$) be the set of n-dimensional vectors ($n\times n$-dimensional matrices). For $y_k\in {\mathbb{R}}^n(k=1,2,\dots ,N)$, the notation ${\left[y_k\right]}_{Nn\times 1}$ denotes the vector $[y_1^T,y_2^T,\dots ,y_N^T]\in {\mathbb{R}}^{Nn}$. $V\succ 0$ denotes $V\in {\mathbb{R}}^{n\times n}$ being positive definite. $V\in {\mathbb{R}}_{+}^{n\times n}$ or $V\ge 0$ ($y\in {\mathbb{R}}_{+}^n$ or $y\ge 0$) denotes the elements of $V\in {\mathbb{R}}^{n\times n}$ ($y\in {\mathbb{R}}^n$) being non-negative. $\parallel ·\parallel$ denotes the Euclidean norm. The symbol ${\lambda }_{max}(·)$ is used to denote the largest eigenvalue of a matrix, while ${\lambda }_{min}(·)$ is employed to represent the minimum eigenvalue.

2. Preliminaries

This section is devoted to the presentation of a series of definitions and lemmas that are related to the fractional-order calculus and distributed-order calculus.

Definition 1 

([1]). The Riemann–Liouville fractional-order integral with order $\alpha >0$ for a function f is defined as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{\mathit{RL}}I_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{f\left(s\right)}{{(t-s)}^{1-\alpha }}}ds,\end{array}\end{array}$$

(1)

where, $\mathrm{\Gamma }(·)$ represents the gamma function, which is defined by the integral $\mathrm{\Gamma }\left(\alpha \right)={\int }_0^{+\infty }{t}^{\alpha -1}{e}^{-t}dt$.

Definition 2 

([1]). The Caputo fractional-order derivative of order α is defined as follows:

$${}_0{}^{c}D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(m-\alpha )}}{\int }_0^t{\displaystyle \frac{f^{\left(m\right)}\left(s\right)}{{(t-s)}^{\alpha +1-m}}}ds,$$

(2)

where $\alpha >0$, $\alpha \in (m-1,m)$ and $m\in {\mathbb{N}}^{+}$.

The Riemann–Liouville fractional-order derivative of order α is defined as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{\mathit{RL}}D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(m-\alpha )}}{\displaystyle \frac{{\mathrm{d}}^m}{\mathrm{d}{t}^m}}{\int }_0^t{(t-s)}^{m-\alpha -1}f\left(s\right)ds,\end{array}\end{array}$$

(3)

where $\mathrm{\Gamma }(m-\alpha )={\int }_0^{+\infty }{t}^{m-\alpha -1}{\mathrm{e}}^{-t}dt.$

Definition 3 

([14]). The distributed-order derivative in the Caputo sense with respect to an order density function (weight function) $\omega \left(\alpha \right)\ge 0$ is defined as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)={\int }_{m-1}^m\omega \left(\alpha \right){}_0{}^{c}D_t^{\alpha }x\left(t\right)d\alpha ,\end{array}\end{array}$$

(4)

where $\alpha \in (m-1,m)$.

As demonstrated in [14,38], the Laplace transform of the Caputo distributed-order derivative can be calculated as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{L}\left\{{}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)\right\}\left(s\right)=W\left(s\right)\mathcal{L}\left\{x\left(t\right)\right\}\left(s\right)-{\displaystyle \frac{1}{s}}W\left(s\right)x\left(t_0\right),\end{array}\end{array}$$

(5)

where $0<\alpha <1$, $W\left(s\right)={\int }_0^1\omega \left(\alpha \right)s^{\alpha }d\alpha$.

Moreover, if $\omega \left(\alpha \right)=\chi (\alpha -\xi )$ where $0<\xi <1$ and χ is the Dirac delta function, the Caputo distributed-order derivative simplifies to the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)={\int }_0^1\chi (\alpha -\epsilon ){}_0{}^{c}D_t^{\alpha }x\left(t\right)d\alpha =D_t^{\xi }x\left(t\right),\end{array}\end{array}$$

(6)

where the integral from 0 to 1 represents the weighted sum of single-valued fractional calculus [36].

Therefore, fractional-order calculus can be regarded as a special case of distributed-order derivative [20,39].

Lemma 1 

([40]). Let $x\left(t\right)\in C^1([t_0,+\infty ),\mathbb{R})$ and $\alpha \in (0,1)$. Then for any $t>t_0$, the following relation holds:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_{t_0}{}^{c}D_t^{\alpha }x\left(t\right)={}_{t_0}{}^{\mathit{RL}}D_t^{\alpha }x\left(t\right)-{\displaystyle \frac{x\left(t_0\right)}{\mathrm{\Gamma }(1-\alpha )}}{(t-t_0)}^{-\alpha },\end{array}\end{array}$$

(7)

where ${}_{t_0}{}^{c}D_t^{\alpha }$ denotes the Caputo fractional derivative and ${}_{t_0}{}^{\mathit{RL}}D_t^{\alpha }$ denotes the Riemann–Liouville fractional derivative.

Lemma 2 

([41]). Let $x\left(t\right)\in C^1([t_0,+\infty ),\mathbb{R})$ and $\alpha \in (0,1)$. Suppose there exists a time instant $t_1>t_0$ such that $x\left(s\right)\ge 0$ for $s\in [t_0,t_1)$ and $x\left(t_1\right)=0,$ then we have the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left.\left[{}_{t_0}{}^{\mathit{RL}}D_t^{\alpha }x\left(t\right)\right]\right|}_{t=t_1}\le 0.\end{array}\end{array}$$

(8)

Lemma 3 

([36]). Let $\alpha \in (0,1)$, $\lambda >0$, $\omega \left(\alpha \right)\ge 0$ and suppose all roots of $W\left(s\right)+\lambda =0$ are located in the open left-half complex plane, where $W\left(s\right)={\int }_0^1\omega \left(\alpha \right)s^{\alpha }d\alpha$. Then, we have the following:

(1) $0\le {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)\le 1$, $\forall t\ge 0$.

(2) if ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t\right)\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$, there holds ${\int }_{t_0}^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}(t-\tau )d\tau \le {\lambda }^{-1}$.

3. Main Results

The objective of this section is to present a determination theorem regarding asymptotic stability for distributed-order nonlinear systems. Moreover, stability and synchronization analysis for DOCDNNs are derived using the asymptotic stability determination theorem.

3.1. Stability Theory of Distributed-Order Nonlinear Systems

Consider the general distributed-order time-delayed nonlinear system:

$$\left\{\begin{array}{c}{}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)=-\lambda x\left(t\right)+g(t,x_h\left(t\right)),\hfill \\ x_0=\varphi .\hfill \end{array}\right.$$

(9)

where $\alpha \in (0,1)$ and $\omega \left(\alpha \right)\ge 0$ is an order density function (weight function). $x\left(t\right)$ and $x_h\left(t\right)\in {\mathbb{R}}^N$ are the state variables. $\varphi \in C([-h,0],{\mathbb{R}}^N)$ is the initial condition. $g:\mathbb{R}\times {\mathbb{R}}^N\to {\mathbb{R}}^N$ is the nonlinear function. $\lambda$ is the coefficient. $h>0$ is the time delay. For arbitrary $\lambda >0$, ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t\right)\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ and the roots of $W\left(s\right)+\lambda =0$ lie in the open left-half complex plane.

The objective of this subsection is to establish a criterion for the asymptotic stability of distributed-order nonlinear systems (9) based on the Halanay inequality technique. In the study of integer-order Halanay inequalities, the exponential function plays a pivotal role [42]. Additionally, the Mittag–Leffler function is more general than the exponential function and has attracted significant attention in studies of fractional-order Halanay inequalities [43]. It is noteworthy that the exponential function and the Mittag–Leffler function represent solutions to integer-order linear systems and fractional-order linear systems, respectively [41,44]. So, an analytical solution to the distributed-order linear system is studied first.

When $g(t,x_h\left(t\right))=0$, distributed-order nonlinear system (9) degrades into a distributed-order linear system. Then, an analytical solution to the distributed-order linear system can be obtained as follows.

Assumption 1. 

For arbitrary $\lambda >0$, ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t\right)\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ and the roots of $W\left(s\right)+\lambda =0$ lie in the open left-half complex plane.

Theorem 1. 

Under Assumption 1, the solution to the distributed-order linear system ${}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)=-\lambda x\left(t\right)$ can be expressed as $x\left(t\right)=x\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)$. Accordingly, the following equation holds:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)=-\lambda {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right).\end{array}\end{array}$$

(10)

Proof. 

By applying the Laplace transform to both sides of ${}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)=-\lambda x\left(t\right)$, we obtain the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill W\left(s\right)\mathcal{L}\left\{x\left(t\right)\right\}\left(s\right)-{\displaystyle \frac{1}{s}}W\left(s\right)x\left(0\right)=-\lambda \mathcal{L}\left\{x\left(t\right)\right\}\left(s\right),\end{array}\end{array}$$

(11)

where $\mathcal{L}\left\{x\left(t\right)\right\}\left(s\right)={\int }_0^{+\infty }{e}^{-st}x\left(t\right)dt$ is the Laplace transform of function $x\left(t\right)$.

The organization of the Equation (11) results in

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{L}\left\{x\left(t\right)\right\}\left(s\right)={\displaystyle \frac{\frac{1}{s}W\left(s\right)x\left(0\right)}{W\left(s\right)+\lambda }}.\end{array}\end{array}$$

(12)

Next, the application of the inverse Laplace transform to both sides of Equation (12) yield the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill x\left(t\right)={\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t\right)x\left(0\right).\end{array}\end{array}$$

(13)

Then, the substitution of (13) for (10) leads to the following result:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t\right)x\left(0\right)=-\lambda {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t\right)x\left(0\right).\end{array}\end{array}$$

(14)

Because $x\left(0\right)$ is a constant, we can obtain the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)=-\lambda {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right).\end{array}\end{array}$$

(15)

The proof is completed. □

If $\omega \left(\alpha \right)=\chi (\alpha -\xi )$ where $0<\xi <1$ and $\chi$ is the Dirac delta function, the ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ will degenerate into the Mittag–Leffler function [45]. Therefore, Theorem 1 can be regarded as a generalization of fractional-order linear differential equation ${}_0{}^{c}D_t^{\xi }{E}_{\xi }(-\lambda t^{\xi })=-\lambda E_{\xi }(-\lambda t^{\xi })$ proved in [40]. Subsequently, a thorough investigation into the properties of the solution function ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ will be conducted, thereby establishing the foundation for its application in distributed-order Halanay inequality.

Property 1. 

Under Assumption 1, for $\alpha \in (0,1)$, $\lambda >0$ and $\omega \left(\alpha \right)\ge 0$, we have that ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ is a monotonically decreasing function about $t\in [0,+\infty )$.

Proof. 

From Theorem 1, we know that $x\left(t\right)=x\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)$ is a solution of the equation ${}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)=-\lambda x\left(t\right)$. Without loss of generality, assume that $x\left(0\right)>0$. Then, an examination of the monotonicity of $x\left(t\right)$ can be conducted through three distinct cases by employing the mathematical proof by contradiction method.

Case 1. Assume that there are $t_1$ and $t_2$ such that $x^{\prime }\left(t\right)<0$ for $t\in [0,t_1)$ and $x^{\prime }\left(t\right)\ge 0$ for $t\in [t_1,t_2]$. Then, we obtain the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_2}-{}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_1}\hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right){\int }_0^{t_2}{\displaystyle \frac{{(t_2-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta d\alpha -{\int }_0^1\omega \left(\alpha \right){\int }_0^{t_1}{\displaystyle \frac{{(t_1-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta d\alpha \hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right)\left[{\int }_0^{t_1}{\displaystyle \frac{{(t_2-\theta )}^{-\alpha }-{(t_1-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta +{\int }_{t_1}^{t_2}{\displaystyle \frac{{(t_2-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta \right]d\alpha .\hfill \end{array}\end{array}$$

(16)

Because of ${(t_2-\theta )}^{-\alpha }-{(t_1-\theta )}^{-\alpha }<0$ and ${(t_2-\theta )}^{-\alpha }>0$, it follows that the following is true:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_2}-{}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_1}>0.\end{array}\end{array}$$

(17)

However, considering $\lambda >0$ and $x\left(t_1\right)\le x\left(t_2\right)$, we have the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_2}-{}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_1}=-\lambda (x\left(t_2\right)-x\left(t_1\right))\le 0,\end{array}\end{array}$$

(18)

which results in a contradiction.

Case 2. Assume that there are $t_1$ and $t_2$ such that $x^{\prime }\left(t\right)>0$ for $t\in [0,t_1)$ and $x^{\prime }\left(t\right)\le 0$ for $t\in [t_1,t_2]$. Then, we obtain the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_2}-{}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_1}\hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right){\int }_0^{t_2}{\displaystyle \frac{{(t_2-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta d\alpha -{\int }_0^1\omega \left(\alpha \right){\int }_0^{t_1}{\displaystyle \frac{{(t_1-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta d\alpha \hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right)\left[{\int }_0^{t_1}{\displaystyle \frac{{(t_2-\theta )}^{-\alpha }-{(t_1-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta +{\int }_{t_1}^{t_2}{\displaystyle \frac{{(t_2-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta \right]d\alpha .\hfill \end{array}\end{array}$$

(19)

Because of ${(t_2-\theta )}^{-\alpha }-{(t_1-\theta )}^{-\alpha }<0$ and ${(t_2-\theta )}^{-\alpha }>0$, it follows that the following is true:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_2}-{}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_1}<0.\end{array}\end{array}$$

(20)

However, considering $\lambda >0$ and $x\left(t_1\right)\ge x\left(t_2\right)$, we have the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_2}-{}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x\left(t\right)|}_{t=t_1}=-\lambda (x\left(t_2\right)-x\left(t_1\right))\ge 0,\end{array}\end{array}$$

(21)

which is a contradiction.

Case 3. Assume that the function $x\left(t\right)$ is increasing for $t\in [0,+\infty )$. Consequently, it follows that $x\left(t\right)>0$ for all $t\in [0,+\infty )$ and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)={\int }_0^1\omega \left(\alpha \right){\int }_0^t{\displaystyle \frac{{(t-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{x}^{\prime }\left(\theta \right)d\theta d\alpha \ge 0.\end{array}\end{array}$$

(22)

On the other hand,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)=-\lambda x\left(t\right)<0,\end{array}\end{array}$$

(23)

which is a contradiction.

Based on the Cases 1–3, it is proved that $x\left(t\right)=x\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)$ decreases when $t\in [0,+\infty )$. So, we have that ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)$ is a monotonically decreasing function for $t\in [0,+\infty )$.

Similarly, it can be proved that ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)$ is a monotonically decreasing function for $t\in [0,+\infty )$ when $x\left(0\right)\le 0$.

The proof is completed. □

Property 1 shows that ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ is monotonically decreasing when $\lambda >0$. It is worth noting that ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ exhibits a monotonic increase when $\lambda <0$ by employing the same methodology. The monotonicity property of ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ proved in Property 1 is an extension of the monotonicity properties of Mittag–Leffler function and exponential function [46,47].

To proceed, we will show that the weak additivity is true for the ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$.

Property 2. 

Under Assumption 1, for $\alpha \in (0,1)$, $\lambda >0$ and $\omega \left(\alpha \right)\ge 0$, it holds that the following is true:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t_1\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t_2\right)\le {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t_1+t_2\right),\end{array}\end{array}$$

(24)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)-\lambda \right)}}\right\}\left(t_1\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)-\lambda \right)}}\right\}\left(t_2\right)\ge {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)-\lambda \right)}}\right\}\left(t_1+t_2\right),\end{array}\end{array}$$

(25)

where $t_1,t_2\ge 0$. Moreover, the inequalities are strictly valid if $t_1,t_2>0$.

Proof. 

Let $v\left(t\right)={\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t_1+t\right)$ and $v_1\left(t\right)={\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)$. From Property 1, we have that $v^{\prime }\left(t\right)\le 0$. So, the following can be obtained:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}v\left(t\right)& ={\int }_0^1\omega \left(\alpha \right){\int }_0^t{\displaystyle \frac{{(t-\theta )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{v}^{\prime }\left(\theta \right)d\theta d\alpha \hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right){\int }_{t_1}^{t+t_1}{\displaystyle \frac{{(t_1+t-\sigma )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{v}^{\prime }(\sigma -t_1)d\sigma d\alpha \hfill \\ \hfill & \ge {\int }_0^1\omega \left(\alpha \right){\int }_0^{t+t_1}{\displaystyle \frac{{(t_1+t-\sigma )}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}}{v}_1^{\prime }\left(\sigma \right)d\sigma d\alpha \hfill \\ \hfill & {=}_0^c{D}_t^{\omega \left(\alpha \right)}{v}_1(t+t_1)\hfill \\ \hfill & {=}_0^c{D}_t^{\omega \left(\alpha \right)}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t_1+t\right)\hfill \\ \hfill & \stackrel{\left(10\right)}{=}-\lambda {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t+t_1\right)\hfill \\ \hfill & =-\lambda v\left(t\right),\hfill \end{array}\end{array}$$

(26)

where $0<\alpha <1$. This implies that there exists a $g\left(t\right)\ge 0$ such that ${}_0{}^{c}D_t^{\omega \left(\alpha \right)}v\left(t\right)=-\lambda v\left(t\right)+g\left(t\right)$ for all $t>0$. By the solution representation of the distributed-order nonlinear differential equation, one obtains the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v\left(t\right)& ={\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t\right)v\left(0\right)+{\int }_0^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t-\theta \right)g\left(\theta \right)d\theta \hfill \\ \hfill & \ge {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t\right)v\left(0\right)\hfill \\ \hfill & ={\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t_1\right).\hfill \end{array}\end{array}$$

(27)

That is ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t_1+t\right)\ge {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\lambda )s}}\right\}\left(t_1\right)$. So, Equation (24) holds.

In a similar manner, Equation (25) remains valid.

The proof is completed. □

The weak additivity property of Mittag–Leffler function is a crucial component in the analysis of the dynamic performance of fractional-order systems, as evidenced in the literatures [41,44,48]. The weak additivity property of ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ extends the weak additivity property that is characteristic of Mittag–Leffler function.

In the following, we demonstrate that the boundedness and asymptotic behavior of the ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ are valid.

Property 3. 

Under Assumption 1, for $\alpha \in (0,1)$, $\lambda >0$ and $\omega \left(\alpha \right)\ge 0$, define $G^{\lambda }(t;\omega \left(\alpha \right))={\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$. Then, the following identity holds:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)=1-\lambda {\int }_0^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t-\theta \right)d\theta ,\end{array}\end{array}$$

(28)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 0\le {\int }_0^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t-\theta \right)d\theta \le {\displaystyle \frac{1}{\lambda }},\end{array}\end{array}$$

(29)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)=0.\end{array}\end{array}$$

(30)

Proof. 

Performing the Laplace transform on ${\int }_0^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t-\theta \right)d\theta$ results in the following conclusion:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{L}\left\{{\int }_0^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t-\theta \right)d\theta \right\}={\displaystyle \frac{1}{W\left(s\right)+\lambda }}·{\displaystyle \frac{1}{s}}.\end{array}\end{array}$$

(31)

Decompose ${\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}$ into the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}={\displaystyle \frac{1}{s}}-{\displaystyle \frac{\lambda }{s(W\left(s\right)+\lambda )}}.\end{array}\end{array}$$

(32)

Taking the inverse Laplace transform of (32) yields the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}\right\}\left(t\right)=1-\lambda {\int }_0^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t-\theta \right)d\theta .\end{array}\end{array}$$

(33)

From Lemma 3, we know the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 0\le {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)\le 1.\end{array}\end{array}$$

(34)

So, substituting (33) into (34), the following can be obtained:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 0\le {\int }_0^t{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{W\left(s\right)+\lambda }}\right\}\left(t-\theta \right)d\theta \le {\displaystyle \frac{1}{\lambda }}.\end{array}\end{array}$$

(35)

Moreover, based on the final value theorem, we have the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)=\underset{s\to 0}{lim}{\displaystyle \frac{sW\left(s\right)}{s\left(W\left(s\right)+\lambda \right)}}=0.\end{array}\end{array}$$

(36)

The proof is completed. □

Based on the asymptotic behavior of the ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{s\left(W\right(s)+\lambda )}}\right\}\left(t\right)$ ascertained in Property 3, the ensuing necessary and sufficient conditions for the asymptotic stability of a distributed-order linear system can be derived.

Theorem 2. 

Under Assumption 1, for the distributed-order linear system ${}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)=-\lambda x\left(t\right)$, the solution is asymptotically stable if $\lambda >0$.

Proof. 

From (13) and (30), ${\displaystyle \underset{t\to \infty }{lim}}x\left(t\right)=0$ can be easily derived, which indicates that the distributed-order linear system ${}_0{}^{c}D_t^{\omega \left(\alpha \right)}x\left(t\right)=-\lambda x\left(t\right)$ is asymptotically stable.

The proof is completed. □

From the Properties 1–3, a novel distributed-order Halanay inequality can be established.

Theorem 3. 

Under Assumption 1 and provided that $a\left(s\right)$ and $b\left(s\right)$ satisfy $0\le b\left(s\right)\le a\left(s\right)-\delta \le M-\delta$ for some positive constants δ and M. For any nonnegative differentiable function $\mathcal{V}:[-h,+\infty )\to \mathbb{R}$, if $\mathcal{V}\left(t\right)$ satisfies the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}\mathcal{V}\left(t\right)\le -a\left(t\right)\mathcal{V}\left(t\right)+b\left(t\right)\overline{\mathcal{V}}\left(t\right),t\ge 0,\end{array}\end{array}$$

(37)

where $\overline{\mathcal{V}}\left(t\right)={sup}_{-h\le \theta \le t}\mathcal{V}\left(\theta \right)$ and h is a positive constant delay, then we have the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{V}\left(t\right)\le \overline{\mathcal{V}}\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(t\right),t\ge 0,\end{array}\end{array}$$

(38)

in which ${\zeta }_0$ represents the unique positive solution of the following equation concerning ζ.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -\zeta =-{\displaystyle \frac{\delta }{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{\left(W\right(s)+\zeta )s}\right\}\left(h\right)}}+M\left[{\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{\left(W\right(s)+\zeta )s}\right\}\left(h\right)}}-1\right].\end{array}\end{array}$$

(39)

Proof. 

First, we prove that (39) has a unique positive solution.

Construct the following auxiliary function:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \Lambda \left(t\right)=t-{\displaystyle \frac{\delta }{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{\left(W\right(s)+t)s}\right\}\left(h\right)}}+M\left[{\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{\left(W\right(s)+t)s}\right\}\left(h\right)}}-1\right].\end{array}\end{array}$$

(40)

Given $\Lambda \left(0\right)=-\delta <0$ and $\Lambda \left(M\right)={\displaystyle \frac{M-\delta }{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{\left(W\right(s)+M)s}\right\}\left(h\right)}}\ge 0$, the continuity of $\Lambda \left(t\right)$ ensures the existence of a point ${\zeta }_0\in (0,M]$ such that $\Lambda \left({\zeta }_0\right)=0$, i.e., ${\zeta }_0$ is a positive solution of Equation (39).

Because $M-\delta \ge 0$ and ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+t)s}}\right\}\left(h\right)$ is a monotone decreasing function about t, $\Lambda \left(t\right)$ is monotone increasing. So, ${\zeta }_0$ is the unique positive solution to Equation (39).

Let $\mathcal{W}\left(t\right)=c\overline{\mathcal{V}}\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+t)s}}\right\}\left(t\right)$ with $c>1$. Next we show that $\mathcal{V}\left(t\right)<\mathcal{W}\left(t\right)$ for $\forall t\ge 0$.

Without loss of generality, assume $\overline{\mathcal{V}}\left(0\right)>0$. Obviously $\mathcal{V}\left(0\right)<\mathcal{W}\left(0\right)$. For $t>0$, if $\mathcal{V}\left(t\right)<\mathcal{W}\left(t\right)$ is not true, there exists a constant $t_1>0$ such that $\mathcal{V}\left(t_1\right)=\mathcal{W}\left(t_1\right)$ and $\mathcal{V}\left(s\right)<\mathcal{W}\left(s\right)$ for $s\in (0,t_1)$. From (37), it follows that the following is true:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{}_0{}^{c}D_t^{\omega \left(\alpha \right)}\mathcal{V}\left(t\right)\right]}_{t=t_1}& \le -a\left(t_1\right)\mathcal{V}\left(t_1\right)+b\left(t_1\right)\overline{\mathcal{V}}\left(t_1\right)\hfill \\ \hfill & =-a\left(t_1\right)\mathcal{W}\left(t_1\right)+b\left(t_1\right)\overline{\mathcal{V}}\left(t_1\right).\hfill \end{array}\end{array}$$

(41)

By the definition of $\overline{\mathcal{V}}\left(t_1\right)$, there exists $\overline{\theta }\in [t_1-h,t_1]$ such that $\mathcal{V}\left(\overline{\theta }\right)=\overline{\mathcal{V}}\left(t_1\right)$.

If $t_1-h<0$, when $\overline{\theta }\in [t_1-h,0]$, it follows that $\mathcal{V}\left(\overline{\theta }\right)\le \overline{\mathcal{V}}\left(0\right)<c\overline{\mathcal{V}}\left(0\right)=\mathcal{W}\left(0\right)$. When $\overline{\theta }\in (0,t_1]$, it is true that $\mathcal{V}\left(\overline{\theta }\right)\le \mathcal{W}\left(\overline{\theta }\right)\le \mathcal{W}\left(0\right)$. If $t_1-h\ge 0$, it holds that $\mathcal{V}\left(\overline{\theta }\right)\le \mathcal{W}\left(\overline{\theta }\right)\le \mathcal{W}(t_1-h)$.

So, we have the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}\left(\overline{\theta }\right)& \le \mathcal{W}\left(0\right),\mathrm{when}{t}_1<h,\hfill \\ \hfill \mathcal{V}\left(\overline{\theta }\right)& \le \mathcal{W}(t_1-h),\mathrm{when}{t}_1\ge h.\hfill \end{array}\end{array}$$

(42)

Moreover, for $t_1<h$, we have ${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(h\right)\le {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(t_1\right)$. Therefore,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{V}\left(\overline{\theta }\right)\le {\displaystyle \frac{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\mathcal{W}\left(0\right)\le {\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\mathcal{W}\left(t_1\right),\end{array}\end{array}$$

(43)

when $t_1<h$.

For $t_1\ge h$, Property 2 indicates that the following is true:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}(t_1-h){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(h\right)\le {\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(t_1\right).\end{array}\end{array}$$

(44)

Then, $\mathcal{V}\left(\overline{\theta }\right)\le \mathcal{W}(t_1-h)=\mathcal{W}\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}(t_1-h)\le {\displaystyle \frac{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(t_1\right)}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\mathcal{W}\left(0\right)={\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\mathcal{W}\left(t_1\right)$ for $t_1\ge h$.

So, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \overline{\mathcal{V}}\left(t_1\right)\le {\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\mathcal{W}\left(t_1\right).\end{array}\end{array}$$

(45)

Substituting (45) into (41) yields the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{}_0{}^{c}D_t^{\omega \left(\alpha \right)}\mathcal{V}\left(t\right)\right]}_{t=t_1}& \le -a\left(t_1\right)\mathcal{W}\left(t_1\right)+b\left(t_1\right){\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\mathcal{W}\left(t_1\right)\hfill \\ \hfill & =\left[-a\left(t_1\right)+{\displaystyle \frac{b\left(t_1\right)}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\right]\mathcal{W}\left(t_1\right).\hfill \end{array}\end{array}$$

(46)

Based on the assumption condition, it is true that $0\le b\left(t\right)\le a\left(t\right)-\delta \le M-\delta$ and $a\left(t\right)\le M$ for all $t\ge 0$. Because ${\zeta }_0$ is the unique positive solution of (39) and $0<{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(h\right)<1$, the following can be concluded: that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{}_0{}^{c}D_t^{\omega \left(\alpha \right)}\mathcal{V}\left(t\right)\right]}_{t=t_1}& \le \left[-a\left(t_1\right)+{\displaystyle \frac{b\left(t_1\right)}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\right]\mathcal{W}\left(t_1\right)\hfill \\ \hfill & \le \left[-a\left(t_1\right)+{\displaystyle \frac{a\left(t_1\right)-\delta }{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}\right]\mathcal{W}\left(t_1\right)\hfill \\ \hfill & =\left\{-{\displaystyle \frac{\delta }{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}+a\left(t_1\right)\left[{\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}-1\right]\right\}\mathcal{W}\left(t_1\right)\hfill \\ \hfill & \le \left\{-{\displaystyle \frac{\delta }{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}+M\left[{\displaystyle \frac{1}{{\mathcal{L}}^{-1}\left\{\frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}\right\}\left(h\right)}}-1\right]\right\}\mathcal{W}\left(t_1\right)\hfill \\ \hfill & =-{\zeta }_0\mathcal{W}\left(t_1\right)\stackrel{\left(10\right)}{=}{\left[{}_0{}^{c}D_t^{\omega \left(\alpha \right)}\mathcal{W}\left(t\right)\right]}_{t=t_1}.\hfill \end{array}\end{array}$$

(47)

Then, we have ${\left[{}_0{}^{c}D_t^{\omega \left(\alpha \right)}(\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right))\right]}_{t=t_1}\le 0$.

On the other hand, because $\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right)\le 0$ for $t\in [0,t_1]$, we deduce $\left[{}_0{}^{\mathit{RL}}D_t^{\alpha }(\mathcal{V}\left(t\right)-\right.\left.\mathcal{W}\left(t\right))\right]{|}_{t=t_1}\ge 0$ from Lemmas 1 and 2. This results in the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\left[{}_0{}^{c}D_t^{\omega \left(\alpha \right)}(\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right))\right]}_{t=t_1}\hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right)\left[{}_0{}^{c}D_t^{\alpha }(\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right))\right]{|}_{t=t_1}d\alpha \hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right)\left(\left[{}_0{}^{\mathit{RL}}D_t^{\alpha }(\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right))\right]{|}_{t=t_1}-{\displaystyle \frac{\mathcal{V}\left(0\right)-\mathcal{W}\left(0\right)}{\mathrm{\Gamma }(1-\alpha )}}{\left(t_1\right)}^{-\alpha }\right)d\alpha \hfill \\ \hfill & ={\int }_0^1\omega \left(\alpha \right)\left(\left[{}_0{}^{\mathit{RL}}D_t^{\alpha }(\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right))\right]{|}_{t=t_1}-{\displaystyle \frac{\mathcal{V}\left(0\right)-c\overline{\mathcal{V}}\left(0\right)}{\mathrm{\Gamma }(1-\alpha )}}{\left(t_1\right)}^{-\alpha }\right)d\alpha \hfill \\ \hfill & \ge {\int }_0^1\omega \left(\alpha \right)\left(\left[{}_0{}^{\mathit{RL}}D_t^{\alpha }(\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right))\right]{|}_{t=t_1}+{\displaystyle \frac{(c-1)\overline{\mathcal{V}}\left(0\right)}{\mathrm{\Gamma }(1-\alpha )}}{\left(t_1\right)}^{-\alpha }\right)d\alpha \hfill \\ \hfill & \ge {\int }_0^1\omega \left(\alpha \right){\displaystyle \frac{(c-1)\overline{\mathcal{V}}\left(0\right)}{\mathrm{\Gamma }(1-\alpha )}}{\left(t_1\right)}^{-\alpha }d\alpha \hfill \\ \hfill & >0,\hfill \end{array}\end{array}$$

(48)

which contradicts ${\left[{}_0{}^{c}D_t^{\omega \left(\alpha \right)}(\mathcal{V}\left(t\right)-\mathcal{W}\left(t\right))\right]}_{t=t_1}\le 0.$

So $\mathcal{V}\left(t\right)<\mathcal{W}\left(t\right)=c\overline{\mathcal{V}}\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(t\right)$ for any $t\ge 0$ and $c>1$. As $c\to 1^{+}$, it can be deduced that $\mathcal{V}\left(t\right)\le \overline{\mathcal{V}}\left(0\right){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{(W\left(s\right)+{\zeta }_0)s}}\right\}\left(t\right)$.

The proof is completed. □

In the event that $\omega \left(\alpha \right)$ is equivalent to a specific Dirac delta function, the distributed-order Halanay inequality (37) will degrade into the fractional-order Halanay inequality proved in [41].

By the Halanay inequality technique obtained in the Theorem 3, an asymptotic stability determination theorem for the nonlinear systems (9) can be obtained.

Theorem 4. 

For the nonlinear systems (9), if there exists a non-negative function $\mathcal{V}:\mathbb{R}\times {\mathbb{R}}^N\to {\mathbb{R}}^{+}$ and positive constants $\delta ,M$ satisfying the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}\mathcal{V}(t,x\left(t\right))\le -a\left(t\right)\mathcal{V}(t,x\left(t\right))+b\left(t\right)\overline{\mathcal{V}}(t,x\left(t\right))\end{array}\end{array}$$

(49)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill b\left(s\right)\le a\left(s\right)-\delta \le M-\delta ,\\ \hfill M-\delta >0,\end{array}\end{array}$$

(50)

where $\overline{\mathcal{V}}(t,x\left(t\right))={sup}_{-h\le \theta \le t}\mathcal{V}(\theta ,x\left(\theta \right))$, then systems (9) is globally asymptotically stable.

Proof. 

Since there exist $\delta >0$ and $M>0$ such that $a\left(s\right)$ and $b\left(s\right)$ satisfy (50), by using Theorem 1, there exists a $\zeta >0$ satisfying the following inequality:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{V}(t,x\left(t\right))\le \overline{\mathcal{V}}(0,x\left(0\right)){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\zeta )s}}\right\}\left(t\right),\forall t\ge 0.\end{array}\end{array}$$

(51)

Clearly, for $\forall t\in \mathbb{R}$, ${\beta }_1{\parallel x\left(t\right)\parallel }^a\le \mathcal{V}(t,x\left(t\right))\le {\beta }_2{\parallel x\left(t\right)\parallel }^{ab}$, where ${\beta }_1$, ${\beta }_2$ are positive constants and $a>0$, $b>0$. Then, we can obtain the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \parallel x\left(t\right)\parallel \le {\beta }_1^{-{\displaystyle \frac{1}{a}}}{\mathcal{V}}^{{\displaystyle \frac{1}{a}}}(t,x\left(t\right))\le {\beta }_1^{-{\displaystyle \frac{1}{a}}}{\left[\overline{\mathcal{V}}(0,x\left(0\right)){\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\zeta )s}}\right\}\left(t\right)\right]}^{{\displaystyle \frac{1}{a}}}.\end{array}\end{array}$$

(52)

Noting the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \overline{\mathcal{V}}(0,x\left(0\right))=\underset{-h\le \theta \le 0}{sup}\mathcal{V}(\theta ,x\left(\theta \right))\le \underset{-h\le \theta \le 0}{sup}\left\{{\beta }_2{\parallel x\left(\theta \right)\parallel }^{ab}\right\}={\beta }_2{\parallel \varphi \parallel }^{ab},\end{array}\end{array}$$

(53)

we have the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \parallel x\left(t\right)\parallel \le {\left[{\displaystyle \frac{{\beta }_2}{{\beta }_1}}{\parallel \varphi \parallel }^{ab}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\zeta )s}}\right\}\left(t\right)\right]}^{{\displaystyle \frac{1}{a}}},\forall t\ge 0.\end{array}\end{array}$$

(54)

From Property 3, we can deduce that ${\displaystyle \underset{t\to +\infty }{lim}}{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{W\left(s\right)}{\left(W\right(s)+\zeta )s}}\right\}\left(t\right)=0$. By the definition of Lyapunov’s stability [49], the equilibrium of the distributed-order nonlinear system (9) is globally asymptotically stable.

The proof is completed. □

Theorem 4 provides an asymptotic stability determination method for the distributed-order nonlinear systems. It is important to note that the asymptotic stability criterion is presented in algebraic form, which facilitates its application. In contrast to the conventional Lyapunov asymptotic stability determination theorem, which utilizes a positive definite Lyapunov function, Theorem 4 employs a non-negative auxiliary function that need not be strictly positive definite.

3.2. Stability Analysis of the DOCDNNs

Consider the following DOCDNNs model:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x}_k\left(t\right)=-A{x}_k\left(t\right)+Bf\left(x_k\left(t\right)\right)+s\sum _{l=1}^N{g}_{kl}M{x}_l(t-\beta \left(t\right))+J,k\in I,\end{array}\end{array}$$

(55)

where $\omega \left(\alpha \right)>0$, $t\ge t_0$, $\alpha \in (0,1)$; $A=diag(a_1,a_2,\dots ,a_n)\succ 0$; $x_k\left(t\right)={[x_{k1}\left(t\right),x_{k2}\left(t\right),\dots ,x_{kn}\left(t\right)]}^T$ is the state vector of the k th note at time t; $f\left(x_k\left(t\right)\right)={[f_1\left(x_{k1}\left(t\right)\right),f_2\left(x_{k2}\left(t\right)\right),\dots ,f_n\left(x_{kn}\left(t\right)\right)]}^T$ is the activation function; $\beta \left(t\right)$ is a discrete delay satisfying $-\beta \le t-\beta \left(t\right)\le t$ and ${\displaystyle \underset{t\to +\infty }{lim}}(t-\beta \left(t\right))=+\infty$ for some $\beta >0$; $B:={\left[b_{ij}\right]}_{n\times n}$ denotes the connected weight matrix; $G:={\left[g_{kl}\right]}_{N\times N}$ represents the coupling strength and the outer coupling matrix, here $g_{kl}=g_{lk}=1(k\ne l)$ if there exists an edge between node k and node l, otherwise, $g_{kl}=g_{lk}=0$, and $g_{kk}=-{\sum }_{k=1,k\ne l}^N{g}_{kl}$ with $k\in I$. $J={[J_1,J_2,\dots ,J_n]}^T$ denotes the external stimulus disturbance. $s>0$ and $M=diag(m_1,m_2,\dots ,m_n)$ define the inner coupling matrix.

To facilitate the analysis, the following assumption about the activation function $f\left(x_k\left(t\right)\right)$ is made.

Assumption 2. 

The function $f_i(·)$ is bounded and Lipschitz continuous. That is, for any ξ and $\zeta \in \mathbb{R}$, there exist $F_i>0$ such that $|f_i\left(\xi \right)-f_i\left(\zeta \right)|\le F_i|\xi -\zeta |$, $i=1,2,\dots ,n$.

Theorem 5. 

Under the condition of Assumption 2, system (55) is globally asymptotically stable if ${\mu }_1\left(t\right)\ge {\mu }_2\left(t\right)$, where ${\mu }_1\left(t\right)=(2{\lambda }_{min}\left(A\right)-{\beta }_0-s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right))$, ${\mu }_2\left(t\right)=s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)$ and ${\beta }_0={\displaystyle \underset{1\le k\le n}{max}}\left\{{\sum }_{j=1}^n\left(F_j|b_{kj}|+F_k\left|b_{jk}\right|\right)\right\}$.

Proof. 

Assumption 2 ensures that there is a unique solution to (55). Without loss of generality, let $x_k^{*}={[x_{k1}^{*},x_{k2}^{*},\dots ,x_{kn}^{*}]}^T$ be the equilibrium point of (55) and $x_k\left(t\right)$ be any solution of (55).

Define $z_k\left(t\right)=x_k\left(t\right)-x_k^{*}$. The distributed-order differentiation of $z_k\left(t\right)$ can be derived as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{z}_k\left(t\right)=-A{z}_k\left(t\right)+B(f\left(x_k\left(t\right)\right)-f\left(x_k^{*}\left(t\right)\right))+s\sum _{l=1}^N{g}_{kl}M{z}_l(t-\beta \left(t\right))+J,k\in I.\end{array}\end{array}$$

(56)

Construct an auxiliary function $V\left(t\right)$ as $V\left(t\right)={\displaystyle \frac{1}{2}}{\sum }_{k=1}^N{z}_k^T\left(t\right)z_k\left(t\right)$. The application of the quadratic inequality related to distributed-order differentiation results in the following equation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\le \sum _{k=1}^N{z_k^T\left(t\right)}_0^c{D}_t^{\omega \left(\alpha \right)}{z}_k\left(t\right).\end{array}\end{array}$$

(57)

Substituting (56) into (57) yields the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\le \sum _{k=1}^N{z}_k^T\left(t\right)\left[-A{z}_k\left(t\right)+B(f\left(x_k\left(t\right)\right)-f\left(x_k^{*}\left(t\right)\right))\right]+s\sum _{l=1}^N{g}_{kl}M\left(z_l(t-\beta \left(t\right))\right).\end{array}\end{array}$$

(58)

Moreover, for the ${\sum }_{k=1}^N{z}_k^T\left(t\right)(-A)z_k\left(t\right)$, it is obvious that the following is true:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{k=1}^N{z}_k^T\left(t\right)(-A)z_k\left(t\right)& =-\sum _{k=1}^N{z}_k^T\left(t\right)A{z}_k\left(t\right)\hfill \\ \hfill & \le -{\lambda }_{min}\left(A\right)\sum _{k=1}^N{z}_k^T\left(t\right)z_k\left(t\right)\hfill \\ \hfill & =-2{\lambda }_{min}\left(A\right){\displaystyle \frac{1}{2}}\sum _{k=1}^N{z}_k^T\left(t\right)z_k\left(t\right)\hfill \\ \hfill & =-2{\lambda }_{min}\left(A\right)V\left(t\right).\hfill \end{array}\end{array}$$

(59)

For the ${\sum }_{k=1}^N{z}_k^T\left(t\right)B(f\left(x_k\left(t\right)\right)-f\left(x_k^{*}\left(t\right)\right))$, the following is evident:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{k=1}^N{z}_k^T\left(t\right)B(f\left(x_k\left(t\right)\right)-f\left(x_k^{*}\left(t\right)\right))& \le \sum _{k=1}^N\sum _{i=1}^n\sum _{j=1}^n{F}_j|b_{ij}\left|\right|z_{ki}\left(t\right)\left|\right|z_{kj}\left(t\right)|\hfill \\ \hfill & \le \sum _{k=1}^N\sum _{i=1}^n\sum _{j=1}^n{\displaystyle \frac{1}{2}}(F_j|b_{kj}|+F_k|b_{jk}\left|\right)z_{ki}^2\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{{\beta }_0}{2}}\sum _{i=1}^n{z}_i^T\left(t\right)z_i\left(t\right)\hfill \\ \hfill & ={\beta }_{0}V\left(t\right).\hfill \end{array}\end{array}$$

(60)

from Assumption 2, where ${\beta }_0={\displaystyle \underset{1\le k\le n}{max}}\left\{{\sum }_{j=1}^n\left(F_j|b_{kj}|+F_k\left|b_{jk}\right|\right)\right\}$, $z_i\left(t\right):=\left[z_{1i}\left(t\right),z_{2i}\left(t\right),\dots ,\right.{\left.z_{Ni}\left(t\right)\right]}^T$.

For the ${\sum }_{k=1}^N{z}_k^T\left(t\right)s{\sum }_{l=1}^N{g}_{kl}M\left(z_l(t-\beta \left(t\right))\right)$, it can be estimated as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \sum _{k=1}^N{z}_k^T\left(t\right)s\sum _{l=1}^N{g}_{kl}M\left(z_l(t-\beta \left(t\right))\right)\hfill \\ \hfill & =s\sum _{k=1}^N\sum _{i=1}^n\sum _{l=1}^n{g}_{kl}{m}_i{z}_{ki}^T\left(t\right)z_{li}(t-\beta \left(t\right))\hfill \\ \hfill & =s\sum _{i=1}^n{g}_{kl}{m}_i{z}_i^T\left(t\right)G{z}_i^T(t-\beta \left(t\right))\hfill \\ \hfill & \le s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{z}_i^T\left(t\right)z_i^T(t-\beta \left(t\right))\hfill \\ \hfill & \le s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{\displaystyle \frac{1}{2}}({\left(z_i^T\left(t\right)\right)}^2+{\left(z_i^T(t-\beta \left(t\right))\right)}^2)\hfill \\ \hfill & =s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{\displaystyle \frac{1}{2}}{z}_i^T\left(t\right)z_i\left(t\right)+s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{\displaystyle \frac{1}{2}}{z}_i^T(t-\beta \left(t\right))z_i(t-\beta \left(t\right))\hfill \\ \hfill & \le s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V\left(t\right)+s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V(t-\beta \left(t\right)).\hfill \end{array}\end{array}$$

(61)

By substituting (59)–(61) into (58), the following result is obtained:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\hfill \\ \hfill & \le -2{\lambda }_{min}\left(A\right)V\left(t\right)+{\beta }_{0}V\left(t\right)+s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V\left(t\right)+s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V(t-\beta \left(t\right)).\hfill \end{array}\end{array}$$

(62)

i.e.,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\hfill \\ \hfill & \le -(2{\lambda }_{min}\left(A\right)-{\beta }_0-s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right))V\left(t\right)+s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V(t-\beta \left(t\right)).\hfill \end{array}\end{array}$$

(63)

Since $-\beta \le t-\beta \left(t\right)\le t$, then $V(t-\beta \left(t\right))\le {sup}_{-\beta \le \theta \le t}V\left(\theta \right)$. Let $\overline{V}\left(t\right)={sup}_{-\beta \le \theta \le t}V\left(\theta \right)$, ${\mu }_1\left(t\right)=2{\lambda }_{min}\left(A\right)-{\beta }_0-s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)$ and ${\mu }_2\left(t\right)=s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)$. Equation (63) can be rewritten as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\le -{\mu }_1\left(t\right)V\left(t\right)+{\mu }_2\left(t\right)\overline{V}\left(t\right).\end{array}\end{array}$$

(64)

Based on the Theorem 4, it can be concluded that system (55) is globally asymptotically stable if ${\mu }_1\left(t\right)\ge {\mu }_2\left(t\right)$.

The proof is completed. □

3.3. Synchronization Analysis of DOCDNNs

In this subsection, the synchronization of DOCDNNs is analyzed by employing the asymptotic stability determination theorem for distributed-order nonlinear systems.

In the case of isolated nodes for DOCDNNs (55), the expression is as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}\tilde{x}\left(t\right)=-A\tilde{x}\left(t\right)+Bf\left(\tilde{x}\left(t\right)\right)+J,\end{array}\end{array}$$

(65)

and the state vector $\tilde{x}\left(t\right)={({\tilde{x}}_1\left(t\right),{\tilde{x}}_2\left(t\right),\dots ,{\tilde{x}}_n\left(t\right))}^T$, denoting the target synchronization state.

In consideration of the controlled network associated with system (55), the network takes the following form:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x}_k\left(t\right)=-A{x}_k\left(t\right)+Bf\left(x_k\left(t\right)\right)+s\sum _{l=1}^N{g}_{kl}M{x}_l(t-\beta \left(t\right))+u_k\left(t\right)+J,k\in I,\end{array}\end{array}$$

(66)

where $u_k\left(t\right)={(u_{k1}\left(t\right),u_{k2}\left(t\right),\dots ,u_{kn}\left(t\right))}^T$ is a controller to be designed.

To ensure the G-synchronization, the controller $u_k\left(t\right)$ is designed as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u_k\left(t\right)=s{r}_k(\tilde{x}\left(t\right)-x_k\left(t\right)),k=1,2,\dots ,N,\end{array}\end{array}$$

(67)

where $r_k$ is the feedback gain such that $0<r_k$ for each $r_k$.

Theorem 6. 

Under the condition of Assumption 2, the system (55) can achieve synchronization through the utilization of the controller (67) if ${\psi }_1\left(t\right)\ge {\psi }_2\left(t\right)$ and ${lim}_{t\to \infty }\overline{\varsigma }\left(t\right)=0$, where ${\psi }_1\left(t\right)=2{\lambda }_{min}\left(A\right)-{\beta }_0-s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)+2s{\lambda }_{min}\left(\overline{R}\right)$, ${\psi }_2\left(t\right)=s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)$, ${\beta }_0={\displaystyle \underset{1\le k\le n}{max}}\left\{{\sum }_{j=1}^n\left(F_j|b_{kj}|+F_k\left|b_{jk}\right|\right)\right\}$.

Proof. 

Let $e_k\left(t\right)=x_k\left(t\right)-\tilde{x}\left(t\right)$ be the synchronization error. By combining (65) and (66), the following can be deduced:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{e}_k\left(t\right)=-A{e}_k\left(t\right)+B(f\left(x_k\left(t\right)\right)-f\left(\tilde{x}\left(t\right)\right))+s\sum _{l=1}^N{g}_{kl}M{e}_l(t-\beta \left(t\right))+u_k\left(t\right),k\in I.\end{array}\end{array}$$

(68)

Construct the auxiliary function $V\left(t\right)={\displaystyle \frac{1}{2}}{\sum }_{k=1}^N{e}_k^T\left(t\right)e_k\left(t\right)$. According to the quadratic inequality of the distributed-order, the following can be obtained:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\le \sum _{k=1}^N{e}_k^T\left(t\right)D_t^{\omega \left(\alpha \right)}{e}_k\left(t\right).\end{array}\end{array}$$

(69)

By substituting (68) and (69), we have the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\le & \sum _{k=1}^N{e}_k^T\left(t\right)\left[-A{e}_k\left(t\right)+B(f\left(x_k\left(t\right)\right)-f\left(\tilde{x}\left(t\right)\right))\right]\hfill \\ \hfill & +\sum _{k=1}^N{e}_k^T\left(t\right)\left[s\sum _{l=1}^N{g}_{kl}M{e}_l(t-\beta \left(t\right))+s{r}_k(\tilde{x}\left(t\right)-x_k\left(t\right))\right].\hfill \end{array}\end{array}$$

(70)

Given that $V\left(t\right)={\displaystyle \frac{1}{2}}{\sum }_{k=1}^N{e}_k^T\left(t\right)e_k\left(t\right)$, it follows that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{k=1}^N{e}_k^T\left(t\right)(-A)e_k\left(t\right)& =-\sum _{k=1}^N{e}_k^T\left(t\right)A{e}_k\left(t\right)\hfill \\ \hfill & \le -{\lambda }_{min}\left(A\right)\sum _{k=1}^N{e}_k^T\left(t\right)e_k\left(t\right)\hfill \\ \hfill & =-2{\lambda }_{min}\left(A\right){\displaystyle \frac{1}{2}}\sum _{k=1}^N{e}_k^T\left(t\right)e_k\left(t\right)\hfill \\ \hfill & =-2{\lambda }_{min}\left(A\right)V\left(t\right).\hfill \end{array}\end{array}$$

(71)

For the ${\sum }_{k=1}^N{e}_k^T\left(t\right)B(f\left(x_k\left(t\right)\right)-f\left(\tilde{x}\left(t\right)\right))$, the following can be deduced from Assumption 2:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{k=1}^N{e}_k^T\left(t\right)B(f\left(x_k\left(t\right)\right)-f\left(\tilde{x}\left(t\right)\right))& \le \sum _{k=1}^N\sum _{i=1}^n\sum _{j=1}^n{F}_j|b_{ij}\left|\right|e_{ki}\left(t\right)\left|\right|e_{kj}\left(t\right)|\hfill \\ \hfill & \le \sum _{k=1}^N\sum _{i=1}^n\sum _{j=1}^n{\displaystyle \frac{1}{2}}(F_j|b_{kj}|+F_k|b_{jk}\left|\right)e_{ki}^2\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{{\beta }_0}{2}}\sum _{i=1}^n{e}_i^T\left(t\right)e_i\left(t\right)\hfill \\ \hfill & ={\beta }_{0}V\left(t\right),\hfill \end{array}\end{array}$$

(72)

where ${\beta }_0={\displaystyle \underset{1\le k\le n}{max}}\left\{{\sum }_{j=1}^n\left(F_j|b_{kj}|+F_k\left|b_{jk}\right|\right)\right\}$, $e_i\left(t\right):={[e_{1i}\left(t\right),e_{2i}\left(t\right),e_{Ni}\left(t\right)]}^T$.

Similarly, for the ${\sum }_{k=1}^N{e}_k^T\left(t\right)s{\sum }_{l=1}^N{g}_{kl}M{e}_l(t-\beta \left(t\right))$ and ${\sum }_{k=1}^N{e}_k^T\left(t\right)s{r}_k(\tilde{x}\left(t\right)-x_k\left(t\right))$, we have the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \sum _{k=1}^N{e}_k^T\left(t\right)s\sum _{l=1}^N{g}_{kl}M{e}_l(t-\beta \left(t\right))\hfill \\ \hfill & =s\sum _{k=1}^N\sum _{i=1}^n\sum _{l=1}^n{g}_{kl}{m}_i{e}_{ki}^T\left(t\right)e_{li}(t-\beta \left(t\right))\hfill \\ \hfill & =s\sum _{i=1}^n{g}_{kl}{m}_i{e}_i^T\left(t\right)G{e}_i^T(t-\beta \left(t\right))\hfill \\ \hfill & \le s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{e}_i^T\left(t\right)e_i^T(t-\beta \left(t\right))\hfill \\ \hfill & \le s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{\displaystyle \frac{1}{2}}({\left(e_i^T\left(t\right)\right)}^2+{\left(e_i^T(t-\beta \left(t\right))\right)}^2)\hfill \\ \hfill & =s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{\displaystyle \frac{1}{2}}{e}_i^T\left(t\right)e_i\left(t\right)+s{\lambda }_{max}\left(G\right)\sum _{i=1}^n{m}_i{\displaystyle \frac{1}{2}}{e}_i^T(t-\beta \left(t\right))e_i(t-\beta \left(t\right))\hfill \\ \hfill & \le s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V\left(t\right)+s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V(t-\beta \left(t\right))\hfill \end{array}\end{array}$$

(73)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{k=1}^N{e}_k^T\left(t\right)s{r}_k(\tilde{x}\left(t\right)-x_k\left(t\right))& =-\sum _{k=1}^N{e}_k^T\left(t\right)s{r}_k{e}_k\left(t\right)\hfill \\ \hfill & \le -s{\lambda }_{min}\left(\overline{R}\right)\sum _{k=1}^N{e}_k^T\left(t\right)e_k\left(t\right)\hfill \\ \hfill & =-2s{\lambda }_{min}\left(\overline{R}\right)V\left(t\right).\hfill \end{array}\end{array}$$

(74)

Substituting (71)–(74) into (70) yields the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\hfill \\ \hfill \le & -2{\lambda }_{min}\left(A\right)V\left(t\right)+{\beta }_{0}V\left(t\right)+s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V\left(t\right)+s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)V(t-\beta \left(t\right))\hfill \\ \hfill & -2s{\lambda }_{min}\left(\overline{R}\right)V\left(t\right).\hfill \end{array}\end{array}$$

(75)

i.e.,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\le & -(2{\lambda }_{min}\left(A\right)-{\beta }_0-s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)+2s{\lambda }_{min}\left(\overline{R}\right))V\left(t\right)\hfill \\ \hfill & +\left(s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)\right)V(t-\beta \left(t\right)).\hfill \end{array}\end{array}$$

(76)

Let ${\psi }_1\left(t\right)=2{\lambda }_{min}\left(A\right)-{\beta }_0-s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)+2s{\lambda }_{min}\left(\overline{R}\right)$ and ${\psi }_2\left(t\right)=s{\lambda }_{max}\left(G\right){\lambda }_{max}\left(M\right)$. Then we have the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}V\left(t\right)\le -{\psi }_1\left(t\right)V\left(t\right)+{\psi }_2\left(t\right)V(t-\beta \left(t\right))).\end{array}\end{array}$$

(77)

Based on the Theorem 4, the system (55) is synchronized if ${\psi }_1\left(t\right)\ge {\psi }_2\left(t\right)$.

The proof is completed. □

4. Numerical Examples

In this section, the stability and synchronization theories for DOCDNNs are validated through two numerical examples.

Example 1. 

Consider the following DOCDNNs (55):

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x}_1\left(t\right)& =-\left[\begin{array}{cc}3.5& 0\\ 0& 1.5\end{array}\right]x_1\left(t\right)+\left[\begin{array}{cc}-1& 0.4\\ 0& -0.1\end{array}\right]f\left(x_1\left(t\right)\right)\hfill \\ \hfill & +s\sum _{l=1}^3{g}_{1l}\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]x_l(t-\beta \left(t\right))+J,\hfill \\ \hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x}_2\left(t\right)& =-\left[\begin{array}{cc}3.5& 0\\ 0& 1.5\end{array}\right]x_2\left(t\right)+\left[\begin{array}{cc}-1& 0.4\\ 0& -0.1\end{array}\right]f\left(x_2\left(t\right)\right)\hfill \\ \hfill & +s\sum _{l=1}^3{g}_{2l}\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]x_l(t-\beta \left(t\right))+J,\hfill \\ \hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{x}_3\left(t\right)& =-\left[\begin{array}{cc}3.5& 0\\ 0& 1.5\end{array}\right]x_3\left(t\right)+\left[\begin{array}{cc}-1& 0.4\\ 0& -0.1\end{array}\right]f\left(x_3\left(t\right)\right)\hfill \\ \hfill & +s\sum _{l=1}^3{g}_{3l}\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]x_l(t-\beta \left(t\right))+J,\hfill \end{array}\right.\end{array}$$

where $x_1\left(t\right)={[x_{11}\left(t\right),x_{12}\left(t\right)]}^T$, $x_2\left(t\right)={[x_{21}\left(t\right),x_{22}\left(t\right)]}^T$, $x_3\left(t\right)={[x_{31}\left(t\right),x_{32}\left(t\right)]}^T$, $x_l(t-\beta \left(t\right))={[x_{l1}(t-\beta \left(t\right)),x_{l2}(t-\beta \left(t\right))]}^T$.

Take $\omega \left(\alpha \right)=0.5\chi (\alpha -0.9)+0.6\chi (\alpha -0.95)$, $f_i\left(u\right)={\displaystyle \frac{1}{2}}\left(\right|u+1|-|u-1\left|\right)$, $\beta \left(t\right)=0.641|sin(t\left)\right|$, $J={[0,0]}^T$. $s=1$ and $G=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right]$.

Assumption 2 holds for $F_i=1$ for $i=1,2$ by the definition of $f_i\left(u\right)$. By calculation, we can obtain ${\mu }_1\left(t\right)$ = 0.6 and ${\mu }_2\left(t\right)=0$. It is obvious that ${\mu }_1\left(t\right)\ge {\mu }_2\left(t\right)$. Therefore, according to the Theorem 2, we can conclude that (55) is globally asymptotically stable. The states of the DOCDNNs are depicted in Figure 1 with $x_{11}\left(0\right)=0.1$, $x_{12}\left(0\right)=0.06$, $x_{21}\left(0\right)=0.03$, $x_{22}\left(0\right)=0.07$, $x_{31}\left(0\right)=0.02$, $x_{32}\left(0\right)=0.09$. It is evident that these systems ultimately converge to the equilibrium solution $x^{*}={[0,0]}^T$.

When the activation functions in system (55) are taken as $9tanh\left(u\right)$ and $2arctan\left(u\right)$, respectively, the resulting state trajectories are shown in Figure 2 and Figure 3. It is evident that under the influence of these two activation functions, the state of system (55) is globally asymptotically stable. Since the functions $9tanh\left(u\right)$ and $2arctan\left(u\right)$ satisfy Assumption 2, the simulation results are consistent with the theoretical conclusions.

Example 2. 

Consider the following DOCDNNs system as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{e}_1\left(t\right)& =-\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 4.95\end{array}\right]e_1\left(t\right)+\left[\begin{array}{ccc}1.25& -3.2& -3.2\\ -3.2& 1.1& -4.4\\ -3.2& 4.4& 1\end{array}\right]f\left(e_1\left(t\right)\right)\hfill \\ \hfill & +s\sum _{l=1}^3{g}_{1l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]e_l\left(t-\beta \left(t\right)\right)+u_1\left(t\right),\hfill \\ \hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{e}_2\left(t\right)& =-\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 4.95\end{array}\right]e_2\left(t\right)+\left[\begin{array}{ccc}1.25& -3.2& -3.2\\ -3.2& 1.1& -4.4\\ -3.2& 4.4& 1\end{array}\right]f\left(e_2\left(t\right)\right)\hfill \\ \hfill & +s\sum _{l=1}^3{g}_{2l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]e_l\left(t-\beta \left(t\right)\right)+u_2\left(t\right),\hfill \\ \hfill {}_0{}^{c}D_t^{\omega \left(\alpha \right)}{e}_3\left(t\right)& =-\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 4.95\end{array}\right]e_3\left(t\right)+\left[\begin{array}{ccc}1.25& -3.2& -3.2\\ -3.2& 1.1& -4.4\\ -3.2& 4.4& 1\end{array}\right]f\left(e_3\left(t\right)\right)\hfill \\ \hfill & +s\sum _{l=1}^3{g}_{3l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]e_l\left(t-\beta \left(t\right)\right)+u_3\left(t\right),\hfill \end{array}\right.\end{array}$$

where $e_1\left(t\right)={[e_{11}\left(t\right),e_{12}\left(t\right),e_{13}\left(t\right)]}^T$, $e_2\left(t\right)={[e_{21}\left(t\right),e_{22}\left(t\right),e_{23}\left(t\right)]}^T$, $e_3\left(t\right)={[e_{31}\left(t\right),e_{32}\left(t\right),e_{33}\left(t\right)]}^T$, $u_1\left(t\right)={[u_{11}\left(t\right),u_{12}\left(t\right),u_{13}\left(t\right)]}^T$, $u_2\left(t\right)={[u_{21}\left(t\right),u_{22}\left(t\right),u_{23}\left(t\right)]}^T$, $u_3\left(t\right)={[u_{31}\left(t\right),u_{32}\left(t\right),u_{33}\left(t\right)]}^T$, $e_l(t-\beta \left(t\right))={[e_{l1}(t-\beta \left(t\right)),e_{l2}(t-\beta \left(t\right)),e_{l3}(t-\beta \left(t\right))]}^T$.

Take $\omega \left(\alpha \right)=3\chi (\alpha -0.8)+0.8\chi (\alpha -0.7)$, $f_i\left(u\right)={\displaystyle \frac{1}{2}}\left(\right|u+1|-|u-1\left|\right)$, $\beta \left(t\right)={\displaystyle \frac{e^t}{1+e^t}}$ for $i=1,2,3$. $M=diag(1,1,1)$, $\overline{R}=diag(30,30,30)$, $s=1$ and $G=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -1& 0\\ 1& 0& -1\end{array}\right]$. Obviously the Assumption 2 holds for $F_i=1$ for $i=1,2,3$ by the definition of $f_i\left(u\right)$.

The initial values of system (65) and system (66) are respectively selected as ${\tilde{x}}_1\left(0\right)={(0.1,0.2,0.3)}^T$, ${\tilde{x}}_2\left(0\right)={(0.5,0.4,0.6)}^T$, ${\tilde{x}}_3\left(0\right)={(0.7,0.8,0.9)}^T$, and $x_1\left(0\right)={(0.3,0.1,0.45)}^T$, $x_2\left(0\right)={(0.55,0.15,0.3)}^T$, $x_3\left(0\right)={(0.8,0.65,0.7)}^T$. As illustrated in Figure 4, the state of error system (76) is oscillating in the absence of a controller. So, it is evident from Figure 4 that DOCDNNs lacking a controller are incapable of achieving synchronization.

Then, consider the error system (68) equipped with the controller (67). By calculation, we can obtain ${\psi }_1\left(t\right)$ = 52.5 and ${\psi }_2\left(t\right)=0$. It is obvious that ${\psi }_1\left(t\right)\ge {\psi }_2\left(t\right)$. According to Theorem 3, DOCDNNs (55) can achieve synchronization. As illustrated in Figure 5, the error system (68) exhibits a steady state when subjected to the controller (67).

5. Conclusions

This paper addresses the stability and synchronization problems of DOCDNNs. An analytical solution and an asymptotic stability criterion have been derived for distributed-order linear systems. A series of pivotal mathematical properties of the solution function are proven. Then, a novel distributed-order non-autonomous Halanay inequality is derived from the properties. Moreover, the Halanay inequality technique is employed to derive an asymptotic stability determination theorem applicable to distributed-order nonlinear systems. The stability and synchronization of DOCDNNs is studied based on the novel asymptotic stability determination method. The validity and practicality of the proposed method are substantiated by two numerical illustrations. The present paper provides novel analytical tools for the stability theory of distributed-order systems and establishes a theoretical foundation for future research on distributed-order neural networks and complex coupled systems.