Finite-Time Synchronization and Mittag–Leffler Synchronization for Uncertain Fractional-Order Delayed Cellular Neural Networks with Fuzzy Operators via Nonlinear Adaptive Control

Abstract

This paper investigates a class of uncertain fractional-order delayed cellular neural networks (UFODCNNs) with fuzzy operators and nonlinear activations. Both fuzzy AND and fuzzy OR are considered, which help to improve the robustness of the model when dealing with various uncertain problems. To achieve the finite-time (FT) synchronization and Mittag–Leffler synchronization of the concerned neural networks (NNs), a nonlinear adaptive controller consisting of three information feedback modules is devised, and each submodule performs its function based on current or delayed historical information. Based on the fractional-order comparison theorem, the Lyapunov function, and the adaptive control scheme, new FT synchronization and Mittag–Leffler synchronization criteria for the UFODCNNs are derived. Unlike previous feedback controllers, the control strategy proposed in this article can adaptively adjust the strength of the information feedback, and partial parameters only need to satisfy inequality constraints within a local time interval, which shows our control mechanism has a significant advantage in conservatism. The experimental results show that our mean synchronization time and variance are $11.397\%$ and $12.5\%$ lower than the second-ranked controllers, respectively.

1. Introduction

In recent years, people’s work and lives have been inextricably linked to various complex networks and NNs [1,2]. For example, urban public transportation networks, energy and power systems, information transmission networks, online social networks, and deep learning networks have dramatically changed all aspects of people [3,4]. NNs of varying distinctiveness have been widely utilized in signal processing, cooperative control, system stability, dynamics analysis, and other fields [5,6,7,8]. Synchronization, as an important aggregation behavior of NNs, inevitably becomes a cutting-edge direction for researchers [9]. Synchronized neuronal firing plays an important role in cognitive function by enhancing the processing of specific sensory information, thereby improving concentration. Fireflies flashing in nature, birds flying in flocks, and crickets chirping in unison show the importance of synchronized behavior to the life of biota [10,11]. Up to the present, various synchronization patterns have emerged, including complete synchronization [12], impulsive synchronization [13], exponential synchronization [14], Mittag–Leffler synchronization [15], and finite-time synchronization [16].

Few NNs can achieve synchronization consistency without the aid of external control. To finish various synchronization tasks for NNs, there exist many control strategies, such as PID control [17,18,19,20,21], sampled-data control [22], event-triggered control [23], feedback control [24], and adaptive control [25]. In [26], the global synchronization of NNs was investigated through feedback control of a few key nodes. The authors of [27] conducted the asymptotic synchronization of delayed networks with uncertainties under feedback control at partial discrete instants. By concise control schemes and solution formulas, Hua et al. [28] derived the sufficient synchronization conditions of NNs with multi-link frameworks and multiple delays. In [29], Wan et al. studied the master–slave synchronization of delayed network systems by using the Barbalat lemma and adaptive control. An adaptive feedback controller was designed for the global synchronization of quaternion NNs in [30] to reduce the control cost and conservatism.

Compared to integer-order calculus, fractional-order calculus can more accurately describe the dynamic changes of actual systems due to its infinite memory and genetic characterizations [31,32]. The advantages of fractional calculus make it successfully applied in various engineering and biophysical fields [33]. For instance, the stress relaxation and creep behavior of materials such as soil and asphalt can be more accurately captured by fractional-order differential constitutive models, which are more concise and effective than integer-order models [34]. Many scholars have introduced fractional calculus into network modeling and studied the synchronization control of fractional NNs [35,36,37]. In [38], Xiao et al. solved the FT synchronization of fractional delayed BAM NNs by using a linear feedback controller. In [39], Popa et al. analyzed the Mittag–Leffler master–slave synchronization problem for fractional NNs with hybrid delays via state feedback schemes. Using feedback control methods, the authors derived the flexible synchronization criteria and the Mittag–Leffler synchronization of fractional fuzzy NNs [40,41].

The dynamic behavior of NNs often manifests as strong nonlinearity, time variation, and inaccuracy measurements [42]. For a biological neural network, a neuron receives input signals from thousands of synapses. It does not simply add up all the input values linearly. The neuron’s state response is more akin to a nonlinear threshold decision-making process [43]. Fuzzy logic operators provide just the right mathematical framework to express this nonlinear integration [44]. For example, Li and Cao et al. [45] aimed to probe a class of memristive NNs with different fuzzy operators. In [46], the synchronization of discontinuous delayed neural networks with fuzzy AND and fuzzy OR was discussed using adaptive control schemes. Based on nonlinear feedback control methods, the global synchronization and the FT synchronization of fuzzy network systems with delays were discussed in [47,48].

In addition to fuzziness, the dynamic behavior evolution of NNs is often accompanied by uncertainties at the same time. In [49], Ansari et al. investigated the FT synchronization for fractional quaternion-valued NNs by the desired sliding motion. In [50], Chen et al. obtained the synchronization conditions of discontinuous T-S fuzzy NNs with system uncertainties under a fuzzy feedback controller. The FT synchronization of fuzzy cellular NNs with uncertainties and the FT Mittag–Leffler synchronization of Caputo fractional memristive systems were considered in [51,52]. However, few works have investigated both FT synchronization and FT Mittag–Leffler synchronization of fractional fuzzy NNs with time delays and system uncertainties through nonlinear adaptive control and feedback control schemes. Compared to the PID control [17,18] and the fixed gain feedback control [16,50,51,52], we adopt adaptive gain feedback control, which reduces control time by adjusting its control strength in real time, ensuring the robustness of the controller under uncertain parameters. The design of the control law and parameter update law works together to ensure the negative definiteness of the Lyapunov function. The experimental results show that our controller saves at least $11.397\%$ control time compared with the related controllers. The primary challenge of this paper is to design energy-efficient adaptive controllers for synchronization tasks when parameters are only within a local time interval.

Sparked by the above discussions, we study the adaptive FT synchronization and Mittag–Leffler synchronization for fuzzy UFODCNNs. The main contributions of this article lie in three points. First, we give a general fractional-order fuzzy model, which includes uncertainties and delays, and nonlinear activation functions. Second, neuroendocrine PID controllers [20] and BELBIC PID controllers [21] focus on set-point regulation or non-delayed systems, lacking direct compensation mechanisms for time-delayed systems. The fixed-gain feedback controllers in [16,50,51,52] do not have adaptive adjustment mechanisms. Compared with these existing control schemes, we design an adaptive nonlinear feedback controller consisting of three information feedback modules, and each submodule performs its function based on current and delayed historical information. Third, novel FT synchronization and FT Mittag–Leffler synchronization conditions for fuzzy UFODCNNs can be achieved based on the fractional comparison theorem and the adaptive controller. Different from the parameters in [44,45,46,47] satisfying inequality constraints from the initial moment to the current moment, the partial system parameters of our work only need to satisfy inequality constraints within a local time interval $[\underline{t},\overline{t}]$. Our control method can be used for synchronization analysis of fractional-order delayed memristive NNs and can also be applied to the FT consistency control of multi-agent systems. The FT synchronization results obtained in this article can be considered for the image information encryption and decryption within the scheduled time.

We use the following notations. ${\mathbb{N}}_1^k=\{1,2,\cdots ,k\}$. $\mathbb{R}$ is the set of real numbers. ${\mathbb{C}}^1([t_0,+\infty ),\mathbb{R})$ is the set of continuous differential functions from $[t_0,+\infty )$ into $\mathbb{R}$. ${}_{t_0}{}^c\mathcal{D}_t^{\alpha }\eta \left(t\right)$ and ${}_{t_0}I_t^{\alpha }\left(t\right)$ denote the $\alpha$-order Caputo derivative and the $\alpha$-order integral for a function $\eta \left(t\right)$, respectively. $\Gamma (·)$ denotes the gamma function. sign(·) is the signum function. $E_{\varphi q}\left(t\right)$ and $E_{\varphi }\left(t\right)$ denote the double-parameter and single-parameter Mittag–Leffler functions, respectively.

2. Fundamental Knowledge and Network Models

This section introduces the relevant definitions, assumptions, and lemmas. Then, we give fractional-order drive-response NNs that include multiple uncertainties and fuzzy operators.

Definition   1

([31]). The $\alpha$-order Caputo derivative of $\eta \left(t\right)$ is

$$\begin{array}{c}\hfill {}_{t_0}{}^c\mathcal{D}_t^{\alpha }\eta \left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_{t_0}^t{(t-\varpi )}^{-\alpha }{\eta }^{\prime }\left(\varpi \right)d\varpi ,t\ge t_0,\end{array}$$

(1)

where $0<\alpha <1$, and $\Gamma (·)$ denotes the gamma function.

Definition 2

([31]). The $\alpha$-order integral of $\eta \left(t\right)$ is

$$\begin{array}{c}\hfill {}_{t_0}I_t^{\alpha }\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{t_0}^t{(t-\varpi )}^{\alpha -1}\eta \left(\varpi \right)d\varpi ,t\ge t_0,\end{array}$$

(2)

where $\alpha >0$.

Consider nonlinear UFODCNNs with two types of fuzzy operators as below:

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }{\varrho }_m\left(t\right)=-(b_m+\Delta b_m\left(t\right)){\varrho }_m\left(t\right)+{\displaystyle \sum _{n=1}^k}(c_{mn}+\Delta c_{mn}\left(t\right))h_n\left({\varrho }_n\left(t\right)\right)\hfill \\ +{\displaystyle \sum _{n=1}^k}{a}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m,\hfill \\ {\varrho }_m\left(t\right)={\beta }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(3)

where $0<\alpha <1$, and $m\in {\mathbb{N}}_1^k$. ${\varrho }_m\left(t\right)$ represents the m-th neuron state. $h_n(·)$ and $g_n(·)$ represent the activation functions of the n-th neuron. $\tau >0$ signifies the constant delay. $b_m$ denotes the self-regulating coefficient. $\Delta b_m\left(t\right)$ and $\Delta c_{mn}\left(t\right)$ are the bounded uncertainties satisfying the constraints $|\Delta b_m\left(t\right)|\le {\Xi }_m$ and $|\Delta c_{mn}\left(t\right)|\le {\Pi }_{mn}$. $x_m$ and $I_m$ represent the input and bias. ⋀ and ⋁ signify the fuzzy operators AND and OR. $c_{mn}$ and $a_{mn}$ characterize elements of feedback and feed-forward templates. $u_{mn}$ and $v_{mn}$ signify elements of feedback MIN and MAX templates. ${\widehat{P}}_{mn}$ and ${\widehat{Q}}_{mn}$ signify feed-forward MIN and MAX templates.

Take fractional-order NNs (3) as the drive system, and one can get the response system as

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }{\rho }_m\left(t\right)=-(b_m+\Delta b_m\left(t\right)){\rho }_m\left(t\right)+{\displaystyle \sum _{n=1}^k}(c_{mn}+\Delta c_{mn}\left(t\right))h_n\left({\rho }_n\left(t\right)\right)\hfill \\ +{\displaystyle \sum _{n=1}^k}{a}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m+{\Theta }_m\left(t\right),\hfill \\ {\rho }_m\left(t\right)={\gamma }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(4)

where $0<\alpha <1$, and $m\in {\mathbb{N}}_1^k$. ${\rho }_m\left(t\right)$ represents the m-th response neuron state. ${\Theta }_m\left(t\right)$ signifies the feedback controller with appropriate adaptive laws. ${\gamma }_m\left(t\right)$ is the initial value of system (4). The meanings of the remaining parameters can be inferred from the fractional-order NNs (3).

We define the m-th neuron error as ${ϵ}_m\left(t\right)={\rho }_m\left(t\right)-{\varrho }_m\left(t\right)$. Based on drive-response NNs (3) and (4), one can deduce that the error variable satisfies

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }{ϵ}_m\left(t\right)=-(b_m+\Delta b_m\left(t\right)){ϵ}_m\left(t\right)+{\displaystyle \sum _{n=1}^k}(c_{mn}+\Delta c_{mn}\left(t\right))\left[h_n\left({\rho }_n\left(t\right)\right)-h_n\left({\varrho }_n\left(t\right)\right)\right]\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)-{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)-{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)+{\Theta }_m\left(t\right)\hfill \\ {ϵ}_m\left(t\right)={\phi }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(5)

where $0<\alpha <1$, and $m\in {\mathbb{N}}_1^k$.

Remark 1.

Integer-order calculus has been widely used in network system modeling and various synchronous behavior research, such as the bipartite synchronization [26], the asymptotic synchronization [27], the complete synchronization [28], and the master–slave synchronization [29]. However, these feedback control or adaptive control schemes cannot be directly utilized for fractional-order NNs because of the significant difference in properties between the two types of calculus.

Remark 2.

The FT synchronization of fractional-order NNs has attracted the attention of scholars. Refs. [38,40] considered the FT Mittag–Leffler synchronization of fractional NNs, but they ignored fuzzy characteristics and uncertainties. Refs. [34,41] studied the FT synchronization of fractional fuzzy NNs, but the model did not consider uncertainties. This article considers both uncertainties and fuzziness in the network model and simultaneously studies the FT synchronization and FT Mittag–Leffler synchronization of the network. Accordingly, the model and synchronization type in this article are more generalized than those in the literature mentioned above.

Definition   3

([31]). A double-parameter Mittag–Leffler function can be defined by

$$\begin{array}{c}\hfill E_{\alpha q}\left(t\right)=\sum _{\mu =0}^{\infty }{\displaystyle \frac{t^{\mu }}{\Gamma (\mu \alpha +q)}},\end{array}$$

(6)

where $\alpha ,q>0$. For $q=1$, the simplified single-parameter version is given by

$$\begin{array}{c}\hfill E_{\alpha }\left(t\right)=\sum _{\mu =0}^{\infty }{\displaystyle \frac{t^{\mu }}{\Gamma (\mu \alpha +1)}}.\end{array}$$

(7)

Definition   4

([41]). For $\parallel \phi \parallel ={\mathit{sup}}_{s\in [-\tau ,t_0]}\parallel \phi \left(s\right)\parallel$, UFODCNNs (3) and (4) are FT synchronized with respect to $\{\sigma ,\epsilon ,\underline{t},\overline{t}\}$ if $\parallel \phi \parallel \le \sigma$ signifies $\parallel ϵ\left(t\right)\parallel \le \epsilon$ for $t\in [\underline{t},\overline{t}]$, where $0<\sigma <\epsilon$.

Definition   5

([40]). UFODCNNs (3) and (4) are FT Mittag–Leffler synchronized with respect to $\{\sigma ,\epsilon ,\underline{t},\overline{t}\}$ if $\parallel \phi \parallel \le \sigma$ signifies $\parallel ϵ\left(t\right)\parallel \le \iota \left(\sigma \right)E_{\alpha }(-\varphi {(t-t_0)}^{\alpha })\le \epsilon$ for $t\in [\underline{t},\overline{t}]$, where $t_0<\underline{t}<\overline{t}$, $0<\sigma <\epsilon ,\varphi >0,0<\alpha <1$, and $\iota \left(\sigma \right)$ denotes a nonnegative function.

Remark 3.

Currently, there exist two types of FT synchronization. The first one is where the synchronization error of the system tends to zero in a finite time. The second is where the synchronization error is not above the desired threshold during the transients. In this article, we mainly focus on the second type (Definitions 4 and 5).

Lemma 1

([32]). For a function $\Phi \left(t\right)\in {\mathbb{C}}^1([t_0,+\infty ),\mathbb{R})$, we obtain

$${}_{t_0}{}^c\mathcal{D}_t^{\alpha }|\Phi \left(t\right)|\le \mathrm{sign}{(\Phi \left(t\right))}_{t_0}^c{\mathcal{D}}_t^{\alpha }\Phi \left(t\right),0<\alpha <1.$$

(8)

Assumption 1

([51]). For $\forall {\vartheta }_1,{\vartheta }_2\in \mathbb{R}$, there exist constants ${\mathcal{L}}_n^h>0$ and ${\mathcal{L}}_n^g>0$ satisfying

$$\begin{array}{c}\hfill |h_n\left({\vartheta }_2\right)-h_n\left({\vartheta }_1\right)| \le {\mathcal{L}}_n^h|{\vartheta }_2-{\vartheta }_1|,\end{array}$$

(9)

and

$$\begin{array}{c}\hfill |g_n\left({\vartheta }_2\right)-g_n\left({\vartheta }_1\right)| \le {\mathcal{L}}_n^g|{\vartheta }_2-{\vartheta }_1|,\end{array}$$

(10)

where $n\in {\mathbb{N}}_1^k$.

Lemma 2

([15]). If ${\rho }_n$ and ${\varrho }_n$ signify different states of UFODCNNs (3) and (4), then we can obtain

$$|\underset{n=1}{\overset{k}{\bigwedge }}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)-\underset{n=1}{\overset{k}{\bigwedge }}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)|\le \sum _{n=1}^k|u_{mn}||g_n\left({\rho }_n(t-\tau )\right)-g_n\left({\varrho }_n(t-\tau )\right)|,$$

(11)

and

$$|\underset{n=1}{\overset{k}{\bigvee }}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)-\underset{n=1}{\overset{k}{\bigvee }}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)|\le \sum _{n=1}^k|v_{mn}||g_n\left({\rho }_n(t-\tau )\right)-g_n\left({\varrho }_n(t-\tau )\right)|,$$

(12)

where $m,n\in {\mathbb{N}}_1^k$.

Lemma 3

([41]). Let $0<\alpha <1,\varphi >0,\psi \le 0$, and $W_1\left(t\right)$ and $W_2\left(t\right)$ be nonnegative functions satisfying

$${}_{t_0}{}^c\mathcal{D}_t^{\alpha }(W_1\left(t\right)+W_2\left(t\right))\le -\varphi W_1\left(t\right)+\psi ,t\ge t_0.$$

(13)

Then, for $\forall \lambda >0$, there is a constant $\overline{t}$ such that

$$W_1\left(t\right)\le \left(W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }(-\varphi {(t-t_0)}^{\alpha })+{\displaystyle \frac{\psi }{\varphi }},$$

(14)

where $t\in [t_0,\overline{t}]$, and $\overline{t}$ signifies the solution of $E_{\alpha }(-\varphi {(t-t_0)}^{\alpha })-{\displaystyle \frac{W_1\left(t_0\right)+W_2\left(t_0\right)}{W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda }}=0$.

3. New Synchronization Results of UFODCNNs

To accomplish the FT synchronization mission between UFODCNNs (3) and (4), we consider an information feedback controller with adaptive laws as follows:

$${\Theta }_m\left(t\right)=\left\{\begin{array}{c}-\chi {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|}}|{ϵ}_m(t-\tau )|-{\varsigma }_m\left(t\right){ϵ}_m\left(t\right)-\delta {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|}},|{ϵ}_m\left(t\right)|\ne 0,\hfill \\ 0,|{ϵ}_m\left(t\right)|=0,\hfill \end{array}\right.$$

(15)

where ${\varsigma }_m\left(t\right)$ represents the variable feedback strength satisfying ${}_{t_0}{}^c\mathcal{D}_t^{\alpha }{\varsigma }_m\left(t\right)={\kappa }_m|{ϵ}_m\left(t\right)|$, $m\in {\mathbb{N}}_1^k$. $\chi ,\delta ,$ and ${\kappa }_m$ signify positive control parameters.

Remark 4.

Figure 1 shows the control block diagram between drive system (3) and response system (4). When the error norm does not conform to the synchronization definition, the nonlinear adaptive controller (15) composed of three stacked submodules is applied to the response system. The systems stop external control input when the synchronization goal is achieved.

Remark 5.

Different from the nonadaptive feedback control schemes in [16,50,51,52], this article devises an adaptive nonlinear feedback controller including two situations, i.e., ${ϵ}_m\left(t\right)=0$ and ${ϵ}_m\left(t\right)\ne 0$. The information feedback with three feedback modules is activated only under the situation ${ϵ}_m\left(t\right)\ne 0$. Particularly, the first module $-\chi {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|}}|{ϵ}_m(t-\tau )|$ can deal with the impact of delays on system stability. The second module $-{\varsigma }_m\left(t\right){ϵ}_m\left(t\right)$ can adaptively remove quasi-linear growth errors. The adaptive law reflects that the feedback strength automatically adjusts its size as the error changes. The rest module $-\delta {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|}}$ has the function of reducing the synchronization error between the drive–response NNs. These three functional modules have distinct functions and are indispensable for handling network synchronization problems that involve uncertainties and delays.

Remark 6.

Various conventional PID controllers [17,18] and improved versions [19,20,21] have demonstrated excellent performance in handling integer-order linear and nonlinear systems. Compared to these PID controllers, our adaptive nonlinear controller is designed for the specific structure of fractional-order UFODCNNs with fuzzy operators. First, the network model in this article is a fractional-order nonlinear delayed system. Our controller features a dedicated time-delayed feedback module that enables direct compensation of the system’s time-delayed state. In contrast, the neuroendocrine PID controller [20] and the BELBIC PID controller [21] focus on set-point regulation or non-delayed systems, lacking direct compensation mechanisms for historical states (time-delayed states). Second, the network model in this article has uncertain factors caused by parameter uncertainties and fuzzy rules. Our controller includes an adaptive feedback module, and the feedback strength is adaptively adjusted by the real-time system error, which can better address the impact of uncertain factors on the system’s stability. However, the feedback gains of the conventional PID controller [17,18] and the Sigmoid PID controller [19] are often predefined and do not have a real-time adaptive processing mechanism for uncertain information.

3.1. FT Synchronization Criteria

Theorem 1.

Under Assumption 1 and adaptive feedback controller (15), UFODCNNs (3) are FT synchronized with UFODCNNs (4) with respect to $\{\sigma ,\epsilon ,\underline{t},\overline{t}\}$ if

$$\begin{array}{c}\hfill \left(\sigma +W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)+{\displaystyle \frac{\psi }{\varphi }}\le \epsilon ,t\in [\underline{t},\overline{t}],\end{array}$$

(16)

$$\begin{array}{c}\hfill {\varsigma }^{*}>{\mathit{max}}_{1\le m\le k}\left[-b_m+{\Xi }_m+\sum _{n=1}^k\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h\right],\end{array}$$

(17)

$$\begin{array}{c}\hfill \chi >{\mathit{max}}_{1\le m\le k}\left[\sum _{n=1}^k\left(|u_{nm}|+|v_{nm}|\right){\mathcal{L}}_m^g\right],\end{array}$$

(18)

where $\varphi ={\mathit{min}}_{1\le m\le k}\left[b_m-{\Xi }_m-{\displaystyle \sum _{n=1}^k}\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h+{\varsigma }^{*}\right]>0$ and $\psi =-k\delta$. $\overline{t}$ and $\underline{t}$ denote the solutions of $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{W_1\left(t_0\right)+W_2\left(t_0\right)}{W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda }}=0$ and $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\epsilon -\frac{\psi }{\varphi }}{\sigma +W_2\left(t_0\right)+\lambda -\frac{\psi }{\varphi }}}=0$, respectively. $W_1\left(t_0\right)={\displaystyle \sum _{m=1}^k}|{ϵ}_m\left(t_0\right)|$ and $W_2\left(t_0\right)={\displaystyle \sum _{m=1}^k}{\displaystyle \frac{{({\varsigma }_m\left(t_0\right)-{\varsigma }^{*})}^2}{2{\kappa }_m}}$.

Proof. 

Consider an auxiliary function

$$\begin{array}{c}\hfill W\left(t\right)=\stackrel{W_1\left(t\right)}{\overbrace{\sum _{m=1}^k|{ϵ}_m\left(t\right)|}}+\stackrel{W_2\left(t\right)}{\overbrace{\sum _{m=1}^k{\displaystyle \frac{{({\varsigma }_m\left(t\right)-{\varsigma }^{*})}^2}{2{\kappa }_m}}}}.\end{array}$$

(19)

Based on Lemma 1, computing the $\alpha$-order derivative of $W\left(t\right)$ gives

$$\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }W\left(t\right)\hfill \\ ={}_{t_0}{}^c\mathcal{D}_t^{\alpha }\left[\sum _{m=1}^k|{ϵ}_m\left(t\right)|+\sum _{m=1}^k{\displaystyle \frac{{({\varsigma }_m\left(t\right)-{\varsigma }^{*})}^2}{2{\kappa }_m}}\right]\hfill \\ \le \sum _{m=1}^k\mathrm{sign}\left({ϵ}_m\left(t\right)\right){}_{t_0}{}^c\mathcal{D}_t^{\alpha }{ϵ}_m\left(t\right)+\sum _{m=1}^k{\displaystyle \frac{{\varsigma }_m\left(t\right)-{\varsigma }^{*}}{{\kappa }_m}}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }{\varsigma }_m\left(t\right)\hfill \\ =\sum _{m=1}^k\mathrm{sign}\left({ϵ}_m\left(t\right)\right)\{-(b_m+\Delta b_m\left(t\right)){ϵ}_m\left(t\right)+\sum _{n=1}^k(c_{mn}+\Delta c_{mn}\left(t\right))\left[h_n\left({\rho }_n\left(t\right)\right)-h_n\left({\varrho }_n\left(t\right)\right)\right]\hfill \\ +\underset{n=1}{\overset{k}{\bigwedge }}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)-\underset{n=1}{\overset{k}{\bigwedge }}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)+\underset{n=1}{\overset{k}{\bigvee }}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)\hfill \\ -\underset{n=1}{\overset{k}{\bigvee }}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)-\chi {\displaystyle \frac{{ϵ}_m\left(t\right)|{ϵ}_m(t-\tau )|}{|{ϵ}_m\left(t\right)|}}-{\varsigma }_m\left(t\right){ϵ}_m\left(t\right)-\delta {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|}}\}\hfill \\ +\sum _{m=1}^k({\varsigma }_m\left(t\right)-{\varsigma }^{*})|{ϵ}_m\left(t\right)|.\hfill \end{array}$$

(20)

Considering the boundedness of uncertainties and the Lipschitz condition of the function $h(·)$, one can get

$$\begin{array}{c}\hfill \sum _{m=1}^k\mathrm{sign}\left({ϵ}_m\left(t\right)\right)\left[-(b_m+\Delta b_m\left(t\right))\right]{ϵ}_m\left(t\right)\le \sum _{m=1}^k(-b_m+{\Xi }_m)|{ϵ}_m\left(t\right)|,\end{array}$$

(21)

and

$$\begin{array}{cc}\hfill \sum _{m=1}^k& \mathrm{sign}\left({ϵ}_m\left(t\right)\right)\sum _{n=1}^k(c_{mn}+\Delta c_{mn}\left(t\right))\left[h_n\left({\rho }_n\left(t\right)\right)-h_n\left({\varrho }_n\left(t\right)\right)\right]\hfill \\ \hfill & \le \sum _{m=1}^k\sum _{n=1}^k(|c_{mn}|+{\Pi }_{mn})|h_n\left({\rho }_n\left(t\right)\right)-h_n\left({\varrho }_n\left(t\right)\right)|\hfill \\ \hfill & \le \sum _{m=1}^k\sum _{n=1}^k(|c_{mn}|+{\Pi }_{mn}){\mathcal{L}}_n^h|{ϵ}_n\left(t\right)|.\hfill \end{array}$$

(22)

Based on Lemma 2 and Assumption 1, one can obtain

$$\begin{array}{cc}\hfill \sum _{m=1}^k& \mathrm{sign}\left({ϵ}_m\left(t\right)\right)\left[\underset{n=1}{\overset{k}{\bigwedge }}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)-\underset{n=1}{\overset{k}{\bigwedge }}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)\right]\hfill \\ \hfill & \le \sum _{m=1}^k\sum _{n=1}^k|u_{mn}||g_n\left({\rho }_n(t-\tau )\right)-g_n\left({\varrho }_n(t-\tau )\right)|\hfill \\ \hfill & \le \sum _{m=1}^k\sum _{n=1}^k|u_{mn}|{\mathcal{L}}_n^g|{ϵ}_n(t-\tau )|,\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill \sum _{m=1}^k& \mathrm{sign}\left({ϵ}_m\left(t\right)\right)\left[\underset{n=1}{\overset{k}{\bigvee }}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)-\underset{n=1}{\overset{k}{\bigvee }}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)\right]\hfill \\ \hfill & \le \sum _{m=1}^k\sum _{n=1}^k|v_{mn}||g_n\left({\rho }_n(t-\tau )\right)-g_n\left({\varrho }_n(t-\tau )\right)|\hfill \\ \hfill & \le \sum _{m=1}^k\sum _{n=1}^k|v_{mn}|{\mathcal{L}}_n^g|{ϵ}_n(t-\tau )|.\hfill \end{array}$$

(24)

Substituting inequalities (21)–(24) into (20), one derives

$$\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }W\left(t\right)\hfill \\ \le \sum _{m=1}^k[(-b_m+{\Xi }_m)|{ϵ}_m\left(t\right)|+\sum _{n=1}^k(|c_{mn}|+{\Pi }_{mn}){\mathcal{L}}_n^h|{ϵ}_n\left(t\right)|+\sum _{n=1}^k|u_{mn}|{\mathcal{L}}_n^g|{ϵ}_n(t-\tau )|\hfill \\ +\sum _{n=1}^k|v_{mn}|{\mathcal{L}}_n^g|{ϵ}_n(t-\tau )|-\chi |{ϵ}_m(t-\tau )|-{\varsigma }_m\left(t\right)|{ϵ}_m\left(t\right)|-\delta ]+\sum _{m=1}^k({\varsigma }_m\left(t\right)-{\varsigma }^{*})|{ϵ}_m\left(t\right)|\hfill \\ \le \sum _{m=1}^k\left[-b_m+{\Xi }_m+\sum _{n=1}^k(|c_{nm}|+{\Pi }_{nm}){\mathcal{L}}_m^h-{\varsigma }^{*}\right]|{ϵ}_m\left(t\right)|\hfill \\ +\sum _{m=1}^k\left[-\chi +\sum _{n=1}^k(|u_{nm}|+|v_{nm}|){\mathcal{L}}_m^g\right]|{ϵ}_m(t-\tau )|-\sum _{m=1}^k\delta .\hfill \end{array}$$

(25)

Let $\varphi ={\min }_{1\le m\le k}\left[b_m-{\Xi }_m-{\displaystyle \sum _{n=1}^k}\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h+{\varsigma }^{*}\right]>0$ and $\psi =-k\delta$. Then, one can get

$${}_{t_0}{}^c\mathcal{D}_t^{\alpha }(W_1\left(t\right)+W_2\left(t\right))\le -\varphi W_1\left(t\right)+\psi .$$

(26)

By Lemma 3, for $\forall \lambda >0$, there exists an instant $\overline{t}$ such that

$$\begin{array}{c}\hfill \parallel ϵ\left(t\right)\parallel =W_1\left(t\right)\le \left(W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)+{\displaystyle \frac{\psi }{\varphi }},\end{array}$$

(27)

where $t\in [t_0,\overline{t}]$, and $\overline{t}$ denotes the solution of $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{W_1\left(t_0\right)+W_2\left(t_0\right)}{W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda }}=0$.

According to Definition 4 and condition (16), one can obtain

$$\begin{array}{c}\hfill \parallel ϵ\left(t\right)\parallel \le \left(\sigma +W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)+{\displaystyle \frac{\psi }{\varphi }}\le \epsilon ,\end{array}$$

(28)

where $t\in [\underline{t},\overline{t}]$, and $\underline{t}$ is the solution of $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\epsilon -\frac{\psi }{\varphi }}{\sigma +W_2\left(t_0\right)+\lambda -\frac{\psi }{\varphi }}}=0$. □

Remark 7.

Combining the relevant mathematical formulas, lemmas, and assumptions, Figure 2 shows a schematic diagram of the proof of Theorem 1. It describes the logical derivation relationship of the mathematical expressions (16)–(28). The proof idea is to first construct a positive auxiliary function (19) based on stability theory and solve for the fractional derivative of the function to obtain (20). Using the Lipschitz condition, one can obtain inequalities (21) and (22) and obtain inequalities (23) and (24) based on Lemma 2 and Assumption 1. Then, we substitute (21)–(24) into (20) for simplification and use conditions (16)–(18) and Definition 4 to obtain inequalities (26)–(28).

In view of $\varphi >0$ and $\psi \le 0$, we get ${\displaystyle \frac{\psi }{\varphi }}\le 0$. Based on the proof of Theorem 1, one can obtain the following FT Mittag–Leffler synchronization results.

3.2. FT Mittag–Leffler Synchronization Criteria

Theorem 2.

Under Assumption 1 and adaptive feedback controller (15), UFODCNNs (3) are FT Mittag–Leffler synchronized with UFODCNNs (4) with respect to $\{\sigma ,\epsilon ,\underline{t},\overline{t}\}$ if

$$\begin{array}{c}\hfill \left(\sigma +W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)\le \epsilon ,t\in [\underline{t},\overline{t}],\end{array}$$

(29)

$$\begin{array}{c}\hfill {\varsigma }^{*}>{\mathit{max}}_{1\le m\le k}\left[-b_m+{\Xi }_m+\sum _{n=1}^k\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h\right],\end{array}$$

(30)

$$\begin{array}{c}\hfill \chi >{\mathit{max}}_{1\le m\le k}\left[\sum _{n=1}^k\left(|u_{nm}|+|v_{nm}|\right){\mathcal{L}}_m^g\right],\end{array}$$

(31)

where $\varphi ={\mathit{min}}_{1\le m\le k}\left[b_m-{\Xi }_m-{\displaystyle \sum _{n=1}^k}\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h+{\varsigma }^{*}\right]>0$, and $\psi =-k\delta$. $\overline{t}$ and $\underline{t}$ denote the solutions of $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{W_1\left(t_0\right)+W_2\left(t_0\right)}{W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda }}=0$ and $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\epsilon }{\sigma +W_2\left(t_0\right)+\lambda -\frac{\psi }{\varphi }}}=0$, respectively. $W_1\left(t_0\right)={\displaystyle \sum _{m=1}^k}|{ϵ}_m\left(t_0\right)|$, and $W_2\left(t_0\right)={\displaystyle \sum _{m=1}^k}{\displaystyle \frac{{({\varsigma }_m\left(t_0\right)-{\varsigma }^{*})}^2}{2{\kappa }_m}}$.

Proof. 

Using the similar proof steps with Theorem 1, for $\forall \lambda >0$, there exists an instant $\overline{t}$ such that

$$\begin{array}{c}\hfill \parallel ϵ\left(t\right)\parallel =W_1\left(t\right)\le \left(W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)+{\displaystyle \frac{\psi }{\varphi }},\end{array}$$

(32)

where $t\in [t_0,\overline{t}]$, and $\overline{t}$ denotes the solution of $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{W_1\left(t_0\right)+W_2\left(t_0\right)}{W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda }}=0$.

According to Definition 5 and condition (29) as well as ${\displaystyle \frac{\psi }{\varphi }}\le 0$, one can get

$$\begin{array}{c}\hfill \parallel ϵ\left(t\right)\parallel \le \left(\sigma +W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)\le \epsilon ,\end{array}$$

(33)

where $t\in [\underline{t},\overline{t}]$, and $\underline{t}$ is the solution of $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\epsilon }{\sigma +W_2\left(t_0\right)+\lambda -\frac{\psi }{\varphi }}}=0$. □

3.3. FT Synchronization Corollary

If uncertainties are not taken into account, then fractional-order NNs (3) and (4) are simplified by

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }{\varrho }_m\left(t\right)=-b_m{\varrho }_m\left(t\right)+{\displaystyle \sum _{n=1}^k}{c}_{mn}{h}_n\left({\varrho }_n\left(t\right)\right)+{\displaystyle \sum _{n=1}^k}{a}_{mn}{x}_n\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m,\hfill \\ {\varrho }_m\left(t\right)={\beta }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(34)

and

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{\alpha }{\rho }_m\left(t\right)=-b_m{\rho }_m\left(t\right)+{\displaystyle \sum _{n=1}^k}{c}_{mn}{h}_n\left({\rho }_n\left(t\right)\right)+{\displaystyle \sum _{n=1}^k}{a}_{mn}{x}_n\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{k}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{k}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m+{\Theta }_m\left(t\right),\hfill \\ {\rho }_m\left(t\right)={\gamma }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(35)

where $0<\alpha <1$, and $m\in {\mathbb{N}}_1^k$. By the adaptive feedback controller (15), we obtain the following corollary utilizing a similar proof to that of Theorem 1.

Corollary 1.

Under Assumption 1 and adaptive feedback controller (15), UFODCNNs (34) are FT synchronized with UFODCNNs (35) with respect to $\{\sigma ,\epsilon ,\underline{t},\overline{t}\}$ if

$$\begin{array}{c}\hfill \left(\sigma +W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)+{\displaystyle \frac{\psi }{\varphi }}\le \epsilon ,t\in [\underline{t},\overline{t}],\end{array}$$

(36)

$$\begin{array}{c}\hfill {\varsigma }^{*}>{\mathit{max}}_{1\le m\le k}\left[-b_m+\sum _{n=1}^k|c_{nm}|{\mathcal{L}}_m^h\right],\end{array}$$

(37)

$$\begin{array}{c}\hfill \chi >{\mathit{max}}_{1\le m\le k}\left[\sum _{n=1}^k\left(|u_{nm}|+|v_{nm}|\right){\mathcal{L}}_m^g\right],\end{array}$$

(38)

where $\varphi ={\mathit{min}}_{1\le m\le k}\left[b_m-{\displaystyle \sum _{n=1}^k}|c_{nm}|{\mathcal{L}}_m^h+{\varsigma }^{*}\right]>0$, and $\psi =-k\delta$. $\overline{t}$ and $\underline{t}$ denote the solutions of $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{W_1\left(t_0\right)+W_2\left(t_0\right)}{W_1\left(t_0\right)+W_2\left(t_0\right)+\lambda }}=0$ and $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\epsilon -\frac{\psi }{\varphi }}{\sigma +W_2\left(t_0\right)+\lambda -\frac{\psi }{\varphi }}}=0$, respectively. $W_1\left(t_0\right)={\displaystyle \sum _{m=1}^k}|{ϵ}_m\left(t_0\right)|$, and $W_2\left(t_0\right)={\displaystyle \sum _{m=1}^k}{\displaystyle \frac{{({\varsigma }_m\left(t_0\right)-{\varsigma }^{*})}^2}{2{\kappa }_m}}$.

Proof. 

The overall proof approach and steps are the same as Theorem 1; so, we omit them here. □

Remark 8.

According to (34) and (35) and Corollary 1, if we replace $\left(\sigma +W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)+{\displaystyle \frac{\psi }{\varphi }}\le \epsilon ,t\in [\underline{t},\overline{t}]$ with $\left(\sigma +W_2\left(t_0\right)+\lambda -{\displaystyle \frac{\psi }{\varphi }}\right)E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)\le \epsilon ,t\in [\underline{t},\overline{t}]$, and replace $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\epsilon -\frac{\psi }{\varphi }}{\sigma +W_2\left(t_0\right)+\lambda -\frac{\psi }{\varphi }}}=0$ with $E_{\alpha }\left(-\varphi {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\epsilon }{\sigma +W_2\left(t_0\right)+\lambda -\frac{\psi }{\varphi }}}=0$, a similar FT Mittag–Leffler synchronization Corollary can be obtained,

Remark 9.

For the FT interval $[t_0,t_0+T]$ in [38,40], the left end point is unchanged, and the right end point T is updated with σ and ε. However, for the FT interval $[\underline{t},\overline{t}]$ in this article, the right end point is unchanged, and the left end point is updated with σ and ε.

4. Simulation Examples

This section presents numerical examples to verify the applicability of our theorem results.

Example 1.

Consider two-dimensional nonlinear UFODCNNs with fuzzy operators as the master system:

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{0.8}{\varrho }_m\left(t\right)=-(b_m+\Delta b_m\left(t\right)){\varrho }_m\left(t\right)+{\displaystyle \sum _{n=1}^2}(c_{mn}+\Delta c_{mn}\left(t\right))h_n\left({\varrho }_n\left(t\right)\right)\hfill \\ +{\displaystyle \sum _{n=1}^2}{a}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m,\hfill \\ {\varrho }_m\left(t\right)={\beta }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(39)

where $b_1=0.6$, $b_2=0.5$, $\Delta b_1\left(t\right)=\Delta c_{1n}\left(t\right)=0.1\sin t$, $\Delta b_2\left(t\right)=\Delta c_{2n}\left(t\right)=0.1\cos t$, $h_n\left(\varrho \right)=g_n\left(\varrho \right)=\tanh \left(\varrho \right)$, ${\left(c_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}-1.4& -3.0\\ 0.9& -1.5\end{array}\right]$, ${\left(a_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}0.3& 0.4\\ 0.2& 0.5\end{array}\right]$, ${\left({\widehat{P}}_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}0.2& 0.4\\ 0.4& 0.2\end{array}\right]$, ${\left({\widehat{Q}}_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}0.4& 0.3\\ 0.5& 0.2\end{array}\right]$, ${\left(u_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}-1.6& -0.2\\ -0.4& -2.6\end{array}\right]$, ${\left(v_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}-0.9& -0.2\\ -0.3& -1.6\end{array}\right]$, $\left(x_n\right)=\left[\begin{array}{c}0.3\\ 0.3\end{array}\right]$, and $I=\left[\begin{array}{c}0\\ 0\end{array}\right].$

According to the form of the activation functions, one can see that Assumption 1 holds for ${\mathcal{L}}_n^h={\mathcal{L}}_n^g=1$. According to fractional-order master system (39), one can get the following fractional-order slave system

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{0.8}{\rho }_m\left(t\right)=-(b_m+\Delta b_m\left(t\right)){\rho }_m\left(t\right)+{\displaystyle \sum _{n=1}^2}(c_{mn}+\Delta c_{mn}\left(t\right))h_n\left({\rho }_n\left(t\right)\right)\hfill \\ +{\displaystyle \sum _{n=1}^2}{a}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m+{\Theta }_m\left(t\right),\hfill \\ {\rho }_m\left(t\right)={\gamma }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(40)

where the network parameters are the same as (39).

Let parameters $t_0=0$, $\tau =0.3$, ${\kappa }_1={\kappa }_2=1,\sigma =0.13,\delta =0.01,{\varsigma }^{*}=4.4,$ and $\chi =4.7$. Simple calculations indicate that ${\varsigma }^{*}>{\max }_{1\le m\le k}\left[-b_m+{\Xi }_m+{\displaystyle \sum _{n=1}^k}\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h\right]=4.3$, $\chi >{\max }_{1\le m\le k}\left[{\displaystyle \sum _{n=1}^k}\left(|u_{nm}|+|v_{nm}|\right){\mathcal{L}}_m^g\right]=4.6$, $\varphi ={\min }_{1\le m\le k}\left[b_m-{\Xi }_m-{\displaystyle \sum _{n=1}^k}\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h+{\varsigma }^{*}\right]=0.1$, $\psi =-0.02$, $\underline{t}=2.155,$ and $\overline{t}=33.11$. Hence, all the inequality constraints in Theorem 1 are satisfied for the above parameters.

The time evolution of the adaptive control strength ${\varsigma }_m\left(t\right)$ of the nonlinear controller (15) is given in Figure 3. The system errors of different states are shown in Figure 4. The norm of the system error can be seen in Figure 5. To quantify the variation in the error norm, the time evolution of $\parallel ϵ\left(t\right)\parallel$ between systems (39) and (40) is given in

Table 1. The time evolution of the error norm $\parallel ϵ\left(t\right)\parallel$ between UFODCNNs (39) and (40).

Time t	0.1	0.2	0.4	0.6	0.8	1.0	1.2
Error norm	0.0793	0.0418	0.0186	0.0052	0.0025	0.0010	0

. Clearly, as the control advances, the master networks (39) and slave networks (40) can reach FT synchronization.

To investigate the sensitivity of important parameters in this article, initial values were randomly selected $[-1,1]$ for each experiment. First, we scanned the uncertain parameters from 0.15sint to 0.40sint with a step size of 0.05sint. Under the same conditions, we performed independent experiments on each parameter 10 times and calculated the mean and variance of the synchronization time. As shown in

Table 2. Comparisons of mean time and variance with different uncertainties.

$\mathbf{\Delta }{\mathit{b}}_1\left(\mathit{t}\right)$	0.15 sint	0.20 sint	0.25 sint	0.30 sint	0.35 sint	0.40 sint
Mean time	1.246	1.424	1.713	2.178	2.559	2.959
Variance	0.023	0.024	0.028	0.028	0.029	0.032

, the greater the uncertainty $\Delta b_1\left(t\right)$, the larger the mean time and variance for the network to reach synchronization. Second, we scanned the time delays from 0.05 to 0.5, and the results in

Table 3. Comparisons of mean time and variance with different delays.

Delays $\mathit{\tau }$	0.05	0.1	0.2	0.3	0.4	0.5
Mean time	0.236	0.616	0.827	1.153	1.962	3.248
Variance	0.012	0.013	0.014	0.022	0.026	0.031

indicate that as the time delay increases, the network takes longer to achieve synchronization, and the variance of the synchronization time also increases. Third, we selected different control strengths $\delta$ and $\chi$ and observed their impact on the synchronization time. As shown in

Table 4. Comparisons of mean time and variance with different strengths $\delta$.

Strengths $\mathit{\delta }$	0.01	0.02	0.03	0.04	0.05	0.06
Mean time	1.152	0.993	0.784	0.635	0.493	0.346
Variance	0.021	0.019	0.018	0.018	0.017	0.016

and

Table 5. Comparisons of mean time and variance with different strengths $\chi$.

Strengths $\mathit{\chi }$	4.8	4.9	5.0	5.1	5.2	5.3
Mean time	1.143	1.092	1.014	0.957	0.865	0.732
Variance	0.020	0.019	0.019	0.018	0.018	0.017

, increasing the control strengths is beneficial for reducing the synchronization time and decreasing the time variance. Regardless of how the parameters change, as long as the theorem conditions are met, the controller proposed in this paper can achieve network synchronization under small variance. This indicates that the control method proposed in this article has good robustness.

Example 2.

Consider the following two-dimensional nonlinear UFODCNNs with fuzzy operators as the master system:

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{0.92}{\varrho }_m\left(t\right)=-(b_m+\Delta b_m\left(t\right)){\varrho }_m\left(t\right)+{\displaystyle \sum _{n=1}^2}(c_{mn}+\Delta c_{mn}\left(t\right))h_n\left({\varrho }_n\left(t\right)\right)\hfill \\ +{\displaystyle \sum _{n=1}^2}{a}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{u}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{v}_{mn}{g}_n\left({\varrho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m,\hfill \\ {\varrho }_m\left(t\right)={\beta }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(41)

where $b_1=0.2$, $b_2=0.1$, $\Delta b_1\left(t\right)=\Delta c_{1n}\left(t\right)=0.1\cos t$, $\Delta b_2\left(t\right)=\Delta c_{2n}\left(t\right)=0.1\sin t$, $h_n\left(\varrho \right)=g_n\left(\varrho \right)=\tanh \left(\varrho \right)$, ${\left(c_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}-1.2& -2.0\\ 0.8& -1.4\end{array}\right]$, ${\left(a_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}0.2& 0.3\\ 0.1& 0.4\end{array}\right]$, ${\left({\widehat{P}}_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}0.1& 0.3\\ 0.3& 0.1\end{array}\right]$, ${\left({\widehat{Q}}_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}0.3& 0.2\\ 0.4& 0.1\end{array}\right]$, ${\left(u_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}-1.5& -0.1\\ -0.3& -2.5\end{array}\right]$, ${\left(v_{mn}\right)}_{2\times 2}=\left[\begin{array}{cc}-0.8& -0.1\\ -0.2& -1.5\end{array}\right]$, $\left(x_n\right)=\left[\begin{array}{c}0.2\\ 0.2\end{array}\right]$, and $I=\left[\begin{array}{c}0\\ 0\end{array}\right].$ It is clear that Assumption 1 holds for ${\mathcal{L}}_n^h={\mathcal{L}}_n^g=1$. Based on master system (41), one can get the following slave system

$$\left\{\begin{array}{c}{}_{t_0}{}^c\mathcal{D}_t^{0.92}{\rho }_m\left(t\right)=-(b_m+\Delta b_m\left(t\right)){\rho }_m\left(t\right)+{\displaystyle \sum _{n=1}^2}(c_{mn}+\Delta c_{mn}\left(t\right))h_n\left({\rho }_n\left(t\right)\right)\hfill \\ +{\displaystyle \sum _{n=1}^2}{a}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{u}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{v}_{mn}{g}_n\left({\rho }_n(t-\tau )\right)\hfill \\ +{\displaystyle \underset{n=1}{\overset{2}{\bigwedge }}}{\widehat{P}}_{mn}{x}_n+{\displaystyle \underset{n=1}{\overset{2}{\bigvee }}}{\widehat{Q}}_{mn}{x}_n+I_m+{\Theta }_m\left(t\right),\hfill \\ {\rho }_m\left(t\right)={\gamma }_m\left(t\right),t\in [-\tau ,t_0],\hfill \end{array}\right.$$

(42)

where the network parameters are the same as (41).

Let parameters $t_0=0$, $\tau =0.2$, ${\kappa }_1={\kappa }_2=1,\sigma =0.12,\delta =0.01,{\varsigma }^{*}=3.7,$ and $\chi =4.3$. Simple calculations indicate that ${\varsigma }^{*}>{\max }_{1\le m\le k}\left[-b_m+{\Xi }_m+{\displaystyle \sum _{n=1}^k}\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h\right]=3.6$, $\chi >{\max }_{1\le m\le k}\left[{\displaystyle \sum _{n=1}^k}\left(|u_{nm}|+|v_{nm}|\right){\mathcal{L}}_m^g\right]=4.2$, $\varphi ={\min }_{1\le m\le k}\left[b_m-{\Xi }_m-{\displaystyle \sum _{n=1}^k}\left(|c_{nm}|+{\Pi }_{nm}\right){\mathcal{L}}_m^h+{\varsigma }^{*}\right]=0.1$, $\psi =-0.02$, $\underline{t}=1.9384,$ and $\overline{t}=19.9698$. It is not difficult to find that all the inequality constraints in Theorem 2 are satisfied for the above parameters.

By the nonlinear controller (15), the time evolution of the adaptive control strength ${\varsigma }_m\left(t\right)$ is given in Figure 6. Figure 7 shows the system errors of different states under adaptive control schemes, and Figure 8 presents the corresponding norm of the system error. To quantify the variation in the error norm, the time evolution of $\parallel ϵ\left(t\right)\parallel$ between systems (41) and (42) is given in

Table 6. The time evolution of the error norm $\parallel ϵ\left(t\right)\parallel$ between UFODCNNs (41) and (42).

Time t	0.2	0.3	0.4	0.5	0.6	0.7	0.8
Error norm	0.0372	0.0243	0.0068	0.0037	0.0018	0.0005	0

. Obviously, with the advances in control, the controlled master and slave networks (41) and (42) can achieve FT Mittag–Leffler synchronization.

To evaluate the performance of our adaptive feedback controller, we compare the differences in the mean synchronization time and variance with nonlinear feedback controllers from the existing literature [16,51,52]. To ensure fairness in comparison, four feedback controllers are compared under the same system parameters. We randomly select the initial values $[-1,1]$, and each control method is executed 10 times under the same initial conditions. As shown in

Table 7. Comparisons of mean time and variance with different control schemes.

Control Schemes	Mean Time	Variance
Nonlinear feedback control in [16]	0.857	0.021
Nonlinear feedback control in [51]	0.816	0.016
Simplified feedback control in [52]	0.945	0.018
Our adaptive feedback controller	0.723	0.014

, the performance of our adaptive controller in terms of the synchronization time and variance is superior to other controllers. From the perspective of the mean time, the cost time of our controller is $11.397\%$ less than that of the second-ranked controller, and the variance of our controller is $12.5\%$ less than that of the nonlinear feedback control in [51].

Remark 10.

Noting the memory characteristics of fractional-order systems, this paper applies the Adams–Bashforth–Moulton approach in [33] to model the error evolution of delayed fractional-order nonlinear systems by MATLAB R2020b. The approach mainly consists of two phases: prediction and correction. First, the time intervals are divided using an equally spaced grid. The product rectangular rule and the series expansion are utilized to evaluate the prediction items. Second, the product trapezoidal quadrature rule is utilized to obtain the correction term. By combining the prediction and correction, the simulation examples can be realized.

Remark 11.

In particular, the three modules in (15) show that ${lim}_{{ϵ}_m\left(t\right)\to 0}{\Theta }_m\left(t\right)=\infty$. In actual application, a bounded and smooth controller significantly reduces the control energy consumption. To avoid singularity, one can replace the $-\chi {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|}}|{ϵ}_m(t-\tau )|-{\varsigma }_m\left(t\right){ϵ}_m\left(t\right)-\delta {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|}}$ with $-\chi {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|+{ϵ}^{★}}}|{ϵ}_m(t-\tau )|-{\varsigma }_m\left(t\right){ϵ}_m\left(t\right)-\delta {\displaystyle \frac{{ϵ}_m\left(t\right)}{|{ϵ}_m\left(t\right)|+{ϵ}^{★}}}$, where ${ϵ}^{★}>0$ indicates a sufficiently small constant.

Remark 12.

The synchronization results derived in this article have potential applications in the real world. In secure communication, the drive system hides encrypted information in chaotic signals for transmission, and the response system must accurately synchronize the chaotic dynamics of the transmission end to extract the original information. The FT synchronization control of neural networks can ensure that the receiving end completes synchronization within strict time constraints, greatly improving the real-time performance and anti-interception ability of communication. In multi-robot collaborative control, one robot (leader) serves as the driving system, while other robots (followers) serve as the responding systems, requiring consistent posture, speed, or trajectory with it. FT synchronization control can ensure that all robots reach a consistent state within a finite time, thereby improving the system’s response speed and overall coordination to sudden instructions. In the smart grid distributed optimization process, FT synchronization control can force the output frequency and phase of all power nodes to achieve synchronization in a very short time. This can effectively prevent frequency fluctuations caused by slow synchronization and significantly improve the stability of the power grid in response to sudden load changes.

5. Conclusions

The FT synchronization and FT Mittag–Leffler synchronization issues have been studied for the nonlinear UFODCNNs with multiple fuzzy operators. By combining the adaptive control mechanism, the fractional comparison theorem, and the fractional stability theory, novel synchronization conditions in finite time for the UFODCNNs were obtained. All conditions were in simple scalar inequality form, eliminating the need for the linear matrix inequality toolbox and reducing the computational burden. Two experiments validated the effectiveness of the synchronization results in this article. A future direction is to study the synchronized control of fractional-order neural networks under time-varying topologies or switching connection weights to improve the model’s adaptability to real network dynamics. Another research direction is to explore the effect of variable-order derivatives on neural network dynamics to reveal the synchronization mechanism of adaptive regulation order.