Finite-Time Adaptive Cluster Synchronization of Heterogeneous Fractional-Order Dynamic Networks with Community Structure and Co-Competition Interactions

Abstract

This paper is devoted to investigating the problem of finite-time (FnT) adaptive cluster synchronization for heterogeneous fractional-order dynamic networks (FODNs) with community structure and co-competition interactions. By designing a suitable adaptive controller and using reduction to absurdity, some sufficient conditions are derived to ensure the considered heterogeneous FODNs can achieve cluster synchronization over a FnT interval. Meanwhile, the cluster-synchronized setting times (CSSTs) are evaluated effectively by means of the monotonicity of the Mittag-Leffler function. It is indicated that the estimated CSSTs are associated with the order of the derivation and the control parameters. Finally, numerical simulations are carried out to validate the effectiveness of our theoretical results.

1. Introduction

Community structure, as a kind of interaction pattern among network nodes, is a distinctive feature shared by many real-world dynamic networks (DNs), such as social networks, technological networks and biological networks [1,2,3]. A DN with community structure is constituted by a certain number of groups (or clusters), where nodes within each group connect to each other relatively densely but nodes across different groups sparsely [1]. Currently, the examination of DNs with community structure has attracted extensive attention because of their plentiful applications in biology, sociology and informatics [1,4,5,6].

Synchronization is an interesting and important collective behavior of DNs, which is indispensable to the proper functioning of a variety of natural and engineered systems, ranging from electric power grids to neuronal and biological networks [7]. In DNs with community structure, nodes within the same group generally evolve in unison (or are completely synchronized), while those from different groups are unsynchronized [8]. This phenomenon is known as cluster or partial synchronization [8,9], which has been an active research issue in various disciplines due to its significant to the theoretical investigation on brain science and biological science [10,11,12]. There have been several studies on this topic, such as [13,14,15,16,17,18,19,20,21]. Notably, all of these studies were exclusively concerned with integer-order DNs. However, considering that the non-locality and memory effect of many materials and processes, numerous real-world systems may more accurately be described as fractional-order (FO) systems, such as neural networks, viscoelastic systems, biological systems and financial systems [22,23,24,25]. Consequently, it is more meaningful to explore the issue of cluster synchronization in networks of coupled FO systems, termed as FODNs. In recent years, several interesting results in this direction have emerged [26,27,28,29,30]. For example, Wang et al. [26] proposed an adaptive pinning control approach to achieve cluster synchronization of FODNs. In [27], an intermittent pinning controller was designed to steer directed FODNs towards cluster synchronization. Moreover, a re-arranging nodes’ order method was developed in [28] to study the problem of pinning cluster synchronization for multi-weighted FODNs.

It is noted that all the above achievements have focused on asymptotical cluster synchronization, which may be impracticable for applications because the setting time of cluster synchronization tends towards infinity. From the perspective of practical applications, it is always expected that cluster synchronization can be reached over a finite time (FnT) interval in the consideration of the security and economic costs [30,31]. In recent years, in view of some excellent performances including good robustness and anti-interference exhibited in FnT synchronization schemes [31,32,33,34], the research on FnT cluster synchronization of FODNs has become a hot topic, and some outstanding works have been conducted. For instance, FnT cluster synchronization of coupled FO delayed neural networks with discontinuous feedback control was investigated in [30]. Based on the non-decomposition approach, FnT cluster synchronization of nonlinearly coupled FO complex-valued neural networks was discussed in [35]. However, the self-dynamics of uncoupled nodes are assumed to be identical in [35], i.e., only homogeneous FODNs are considered therein, which is not always realistic. Actually, in many real-world DNs, nodes belonging to different groups are essentially different due to their distinct functions [6,36]. Moreover, Ref. [35] also assumed the connection weights between nodes are non-negative, i.e., only cooperative interactions exist between nodes, which is scarce in practice. In fact, co-competition interactions are ubiquitous in the real world. For example, the exciting and inhibiting effects between neurons exist simultaneously in neural networks, and the friendly and hostile relationships between people are coexisting in social networks [2,28]. Therefore, it is of vital importance to explore FnT cluster synchronization of heterogeneous FODNs with community structure and co-competition interactions, which has not been studied yet. This is one motivation of our research work.

On the other hand, in the traditional state feedback control [27,28,29,30,35], the control gains (or feedback strengths) are fixed and their values required to achieve cluster synchronization are usually much larger than what are actually needed. This is mainly because of the conservatism of theoretical results. In practice, small control gains mean small control effort, and thus the control gains are highly desired to be as small as possible. An effective approach to prevent the occurrence of high control gains is to adaptively adjust the control gains. It has been revealed that the adaptive control technique is a good way to obtain optimal control gains [18,37], by which the control parameters can update themselves using appropriate rules. Furthermore, the adaptive control technique is also effective in overcoming disturbances, uncertainties and unknown parameters of the addressed systems [16,38]. Hence, adaptive control schemes are more practical and cost-effective to achieve synchronization of FODNs. In [39], FnT synchronization of multi-weighted complex-valued FODNs was realized via adaptive control. In [40], adaptive synchronization of FO memristor neural networks with leakage delay was considered. Unfortunately, as far as the authors know, there are few studies on FnT adaptive cluster synchronization of heterogeneous FODNs, providing another motivation of our research work.

This paper aims to address the issue of FnT adaptive cluster synchronization in heterogeneous FODNs with community structure and co-competition interactions. The main contributions are summarized as follows:

(1) A general FODN is formulated, where node dynamics in different groups are allowed to be nonidentical, and moreover, the cooperative and competitive interactions can coexist between nodes within the same groups or across different groups. Compared with previous relevant studies, such as [30,35,39], our presented model is more realistic.

(2) An effective and simple adaptive controller is designed to achieve FnT cluster synchronization of the considered heterogeneous FODN. In contrast to the delayed feedback control in [30] and nonlinear feedback control in [35], the proposed adaptive control can significantly reduce the control cost.

(3) According to the monotonicity of the Mittag-Leffler function, the cluster-synchronized settling time (CSST) is estimated reliably. The estimated CSST is found to be closely related to the order of the derivative and the control parameters.

The rest of this work is constructed as follows. In Section 2, some necessary preliminaries are given and a general FODN is formulated. Section 3 presents the main results, an adaptive control policy is developed to achieve FnT cluster synchronization of the constructed heterogeneous FODN. A numerical example is provided in Section 4. Finally, Section 5 concludes this work.

Notations. Let $\mathbb{R}$, $\mathbb{C}$, ${\mathbb{R}}^m$ and ${\mathbb{R}}^{m\times r}$ represent the sets of real numbers, complex numbers, m-dimensional real vectors and $m\times r$ real matrices, respectively. The superscript ‘⊤’ denotes the transpose and ⊗ denotes the Kronecker product. For a symmetrical matrix $Q\in {\mathbb{R}}^{m\times m}$, ${\lambda }_{max}\left(Q\right)$ denotes the largest eigenvalue of Q, and $Q<0$ $(>0)$ means that it is negative definite (positive definite). For a vector $\xi ={\left({\xi }_1,{\xi }_2,\dots ,{\xi }_m\right)}^T\in {\mathbb{R}}^m$, ${\left|\right|\xi \left|\right|}_1$ = ${\sum }_{i=1}^m\left|{\xi }_i\right|$, $\left|\right|\xi \left|\right|$ = $\sqrt{{\sum }_{i=1}^m{\xi }_i^2}$, and sgn$\left(\xi \right)$ = (sgn$\left({\xi }_1\right)$, sgn$\left({\xi }_2\right)$,…, sgn$\left({\xi }_m\right)$)${\phantom{­}}^T$, where sgn(·) stands for the sign function. $I_m\in {\mathbb{R}}^{m\times m}$ represents the m-dimensional identity matrix, and $\mathrm{diag}({\eta }_1,{\eta }_2,\dots ,{\eta }_m)\in {\mathbb{R}}^{m\times m}$ denotes the diagonal matrix with diagonal entries ${\eta }_i$. ${\mathcal{C}}^1([t_0,+\infty ),\mathbb{R})$ signifies the set comprised by all continuous-differential functions from $[t_0,+\infty )$ into $\mathbb{R}$.

2. Preliminaries and Model Description

2.1. Preliminaries

Before presenting the network model, some definitions and lemmas concerning fractional calculus are introduced.

Definition 1 

([22]). For a function $\mathcal{F}\left(t\right)\in {\mathcal{C}}^1([t_0,+\infty ),\mathbb{R})$, its Caputo fractional derivative of order $0<q<1$ is given by

$$\begin{array}{c}\hfill \phantom{­}{\phantom{­}}_{t_0}^C{D}_t^q\mathcal{F}\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-q)}}{\int }_{t_0}^t{\displaystyle \frac{\mathcal{F}{\phantom{­}}^{\prime }\left(s\right)}{{(t-s)}^q}}\phantom{­}\mathrm{d}s,t\ge t_0,\end{array}$$

where $\mathsf{\Gamma }(·)$ is the Gamma function, i.e., $\mathsf{\Gamma }(1-q)={\int }_0^{+\infty }{s}^{-q}{\mathrm{e}}^{-s}\mathrm{d}s.$

Definition 2 

([22]). For the Mittag-Leffler function, its one-parameter and two-parameter forms are respectively expressed as

$$\begin{array}{c}\hfill \phantom{­}{E}_{\eta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(k\eta +1)}}\mathrm{and}{E}_{\eta ,\phantom{­}\theta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(k\eta +\theta )}},\end{array}$$

where $\eta >0$, $\theta >0$, and $z\in \mathbb{C}$. Specially, $E_{\eta }\left(0\right)=1$ and $E_1\left(z\right)={\mathrm{e}}^z$.

Lemma 1. 

Let $0<q<1$, $\vartheta >0$, and $t\ge t_0$, then the function $E_q\left(-\vartheta {(t-t_0)}^q\right)$ is strictly decreasing with respect to t and $0\le E_q\left(-\vartheta {(t-t_0)}^q\right)\le 1$.

Proof. 

Since $E_q\left(-\vartheta {(t-t_0)}^q\right)$ is an entire function [23], utilizing the recursion formula $\mathsf{\Gamma }(z+1)=z\mathsf{\Gamma }\left(z\right)$ [22], it can be deduced from Definition 2 that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(E_q\left(-\vartheta {(t-t_0)}^q\right)\right)& =& {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-\vartheta {(t-t_0)}^q\right)}^k}{\mathsf{\Gamma }(kq+1)}}\right)\hfill \\ & =& -\vartheta {(t-t_0)}^{q-1}\sum _{k=1}^{\infty }\left({\displaystyle \frac{{\left(-\vartheta {(t-t_0)}^q\right)}^{k-1})}{\mathsf{\Gamma }\left(kq\right)}}\right)\hfill \\ & =& -\vartheta {(t-t_0)}^{q-1}\sum _{k^{\prime }=0}^{\infty }\left({\displaystyle \frac{{\left(-\vartheta {(t-t_0)}^q\right)}^{k^{\prime }})}{\mathsf{\Gamma }(k^{\prime }q+q)}}\right)\hfill \\ & =& -\vartheta {(t-t_0)}^{q-1}{E}_{q,\phantom{­}q}\left(-\vartheta {(t-t_0)}^q\right).\hfill \end{array}$$

In the light of the positivity of $E_{q,\phantom{­}q}(-w)$ on $[0,+\infty )$ (see Corollary 3.3 in [23]), one has ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(E_q\left(-\vartheta {(t-t_0)}^q\right)\right)<0$ for all $t>t_0$. This means that $E_q\left(-\vartheta {(t-t_0)}^q\right)$ is strictly decreasing with respect to t on $[t_0,+\infty )$, suggesting that $E_q\left(-\vartheta {(t-t_0)}^q\right)<E_q\left(-\vartheta {(t_0-t_0)}^q\right)=E_q\left(0\right)=1$ for $t>t_0$. In addition, it is implied from Theorem 3.5 in [23] that $E_q(-w)\ge 0$ for all $w\ge 0$, which indicates that $E_q\left(-\vartheta {(t-t_0)}^q\right)\ge 0$ for all $t\ge t_0$. The proof is thus completed. □

Lemma 2 

([41]). Let $\mathcal{Q}\left(t\right)\in {\mathbb{R}}^n$ be a differentiable vector-valued function. Then, for any $t\ge t_0$,

$$\begin{array}{c}\hfill {\phantom{­}}_{t_0}^C\phantom{­}{D}_t^q\left({\mathcal{Q}}^{\top }\left(t\right)\mathcal{M}\mathcal{Q}\left(t\right)\right)\le 2{\mathcal{Q}}^{\top }\left(t\right)\mathcal{M}{\phantom{­}}_{t_0}^C{D}_t^q\mathcal{Q}\left(t\right),\end{array}$$

where $\mathcal{M}>0$ and $0<q\le 1$.

Lemma 3 

([42]). Suppose that $h\left(t\right)$: $[t_0,+\infty )\to \mathbb{R}$ is a continuous-differential function, then for any constant $h^{*}\in \mathbb{R}$ and $t\in [t_0,+\infty )$,

$$\begin{array}{c}\hfill \phantom{­}{\phantom{­}}_{t_0}^C{D}_t^q{\left(h\left(t\right)-h^{*}\right)}^2\le 2\left(h\left(t\right)-h^{*}\right){\phantom{­}}_{t_0}^C{D}_t^{q}h\left(t\right),\end{array}$$

where $0<q\le 1$. 

Lemma 4 

([43]). Suppose that $\mu \left(t\right)$∈ ${\mathcal{C}}^1([t_0,+\infty ),\mathbb{R})$ and satisfies

$$\begin{array}{c}\hfill {\phantom{­}}_{t_0}^C{D}_t^q\mu \left(t\right)\le \lambda \mu \left(t\right),\end{array}$$

where $0<q<1$ and $\lambda \in \mathbb{R}$. Then

$$\begin{array}{c}\hfill \mu \left(t\right)\le \mu \left(t_0\right)E_q\left(\lambda {(t-t_0)}^q\right),t\ge t_0.\end{array}$$

2.2. Model Description

In this subsection, a heterogeneous FODN with community structure and co-competition interactions is considered, which is composed of N nodes and m groups with $1\le m<N$. For convenience, we denote by ${\mathcal{G}}_{\kappa }$ the $\kappa$-th group and let ${\mathcal{G}}_{\kappa }=\left\{{\sum }_{j=0}^{\kappa -1}{r}_j+1,{\sum }_{j=0}^{\kappa -1}{r}_j+2,\dots ,{\sum }_{j=0}^{\kappa }{r}_j\right\}$, where $\kappa \in \mathfrak{K}$ = $\{1,2,\dots ,m\}$, $r_0=0$, 1<$r_j$≤N, and ${\sum }_{j=0}^m{r}_j=N$. The dynamics of the ℓ-th node in ${\mathcal{G}}_{\kappa }$ are characterized by:

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{x}_{\ell }\left(t\right)=f_{\kappa }(t,x_{\ell }\left(t\right))+\sigma \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p,j\ne \ell }{b}_{\ell j}\mathsf{\Gamma }(x_j\left(t\right)-x_{\ell }\left(t\right)),t\ge 0,\end{array}$$

(1)

where $\ell \in {\mathcal{G}}_{\kappa }$, $\kappa \in \mathfrak{K}$, $0<q<1$, $x_{\ell }\left(t\right)={(x_{\ell 1}\left(t\right),x_{\ell 2}\left(t\right),\dots ,x_{\ell n}\left(t\right))}^{\top }\in {\mathbb{R}}^n$ is the state vector of the ℓ-th node in ${\mathcal{G}}_{\kappa }$, $f_{\kappa }(t,x_{\ell }\left(t\right))$:$[0,+\infty )\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is a continuous vector-valued function describing the self-dynamics of node ℓ in ${\mathcal{G}}_{\kappa }$, $\sigma >0$ is the overall coupling strength, $\mathsf{\Gamma }=\mathrm{diag}({\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_n)\in {\mathbb{R}}^{n\times n}$ with ${\gamma }_i\ge 0$ denotes the inner-connecting matrix, and $B={\left(b_{\ell j}\right)}_{N\times N}$ is the outer-coupling matrix, the elements of which are defined as follows: if there exists a directed edge from nodes j to ℓ$(\ell \ne j)$, then $b_{\ell j}\ne 0$; otherwise, $b_{\ell j}=0$. Additionally, the diagonal elements of matrix B are specified as $b_{\ell \ell }=-{\sum }_{j=1,j\ne \ell }^N{b}_{\ell j}$ for $\ell \in {\mathcal{G}}_{\kappa }$ and $\kappa \in \mathfrak{K}$. Note that, here, node dynamics in different groups can be nonidentical and the corresponding graph represented by matrix B can be directed and disconnected (i.e., B can be asymmetric and reducible). Obviously, our network model (1) is a generalization of those discussed in [14,17,31].

For the convenience of subsequent analysis, we rewrite matrix B as below:

$$\begin{array}{c}\hfill B=\left(\begin{array}{cccc}{B}_{11}& \phantom{­}\phantom{­}\phantom{­}{B}_{12}& \phantom{­}\phantom{­}\phantom{­}\cdots & \phantom{­}\phantom{­}\phantom{­}{B}_{1m}\\ B_{21}& \phantom{­}\phantom{­}\phantom{­}{B}_{22}& \phantom{­}\phantom{­}\phantom{­}\cdots & \phantom{­}\phantom{­}\phantom{­}{B}_{2m}\\ ⋮& \phantom{­}\phantom{­}\phantom{­}⋮& \phantom{­}\phantom{­}\phantom{­}\ddots & \phantom{­}\phantom{­}\phantom{­}⋮\\ B_{m1}& \phantom{­}\phantom{­}\phantom{­}{B}_{m2}& \phantom{­}\phantom{­}\phantom{­}\cdots & \phantom{­}\phantom{­}\phantom{­}{B}_{mm}\end{array}\right),\end{array}$$

(2)

where per diagonal sub-matrix $B_{vv}$∈${\mathbb{R}}^{r_v\times r_v}(v\in \mathfrak{K})$ denotes the intra-connections in the community ${\mathcal{G}}_v$ and per non-diagonal sub-matrix $B_{v\mu }\in {\mathbb{R}}^{r_v\times r_{\mu }}(v\ne \mu \in \mathfrak{K})$ represents the inter-connections between the communities ${\mathcal{G}}_v$ and ${\mathcal{G}}_{\mu }$.

The objective of this work is to investigate the issue of FnT cluster synchronization of the heterogeneous FODN (1) via adaptive control. For clarity, the following definition of FnT cluster synchronization is put forward for system (1).

Definition 3 

([17,35]). The heterogeneous FODN (1) is said to reach FnT cluster synchronization if there is a constant ${\mathcal{T}}^{*}\left(e\left(0\right)\right)$> 0, such that for any $\ell \in {\mathcal{G}}_{\kappa }$ and $\kappa \in \mathfrak{K}$,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill \phantom{­}& \underset{t\to {\mathcal{T}}^{*}\left(e\left(0\right)\right)}{lim}\left|\right|x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\left|\right|=0\mathrm{and}\left|\right|x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\left|\right|=0,\forall \phantom{­}t\ge {\mathcal{T}}^{*}\left(e\left(0\right)\right),\hfill \\ & \underset{t\to {\mathcal{T}}^{*}\left(e\left(0\right)\right)}{lim}\left|\right|{\pi }_{{\kappa }_1}\left(t\right)-{\pi }_{{\kappa }_2}\left(t\right)\left|\right|\ne 0,\forall \phantom{­}{\kappa }_1\ne {\kappa }_2\in \mathfrak{K},\hfill \end{array}\right.\end{array}$$

where $e\left(t\right)$ = ${\left(e_1^{\top }\left(t\right),\dots ,e_{r_1}^{\top }\left(t\right),e_{r_1+1}^{\top }\left(t\right),\dots ,e_{r_1+r_2}^{\top }\left(t\right),\dots ,e_{r_1+r_2\dots +r_{m-1}+1}^{\top }\left(t\right),\dots ,e_N^{\top }\left(t\right)\right)}^{\top }$, $e_{\ell }\left(t\right)$ = $x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)$ with $\ell \in {\mathcal{G}}_{\kappa }$, and ${\pi }_{\kappa }\left(t\right)$ is a reference trajectory satisfying ${\phantom{­}}_0^C{D}_t^q{\pi }_{\kappa }\left(t\right)=f_{\kappa }\left(t,{\pi }_{\kappa }\left(t\right)\right)$ called the target synchronization state of the κ-th group. In addition,

$$\begin{array}{c}\hfill \mathcal{T}=inf\left\{{\mathcal{T}}^{*}\left(e\left(0\right)\right)\ge 0:\left|\right|x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\left|\right|=0\phantom{­}\phantom{­}\mathrm{for}\phantom{­}\phantom{­}\mathrm{any}\phantom{­}\phantom{­}t\ge {\mathcal{T}}^{*}\left(e\left(0\right)\right)\phantom{­}\phantom{­}\mathrm{and}\phantom{­}\phantom{­}\ell \in {\mathcal{G}}_{\kappa },\kappa \in \mathfrak{K}\right\}\end{array}$$

is termed as the cluster-synchronized settling time (CSST).

Assumption 1. 

For each $\kappa \in \mathfrak{K}$, there exists a constant matrix $P_{\kappa }=\mathrm{diag}\left(p_1^{\kappa },p_2^{\kappa },\dots ,p_n^{\kappa }\right)$ with $p_j^{\kappa }\ge 0$ such that

$$\begin{array}{ccc}& & {\left(z_1\left(t\right)-z_2\left(t\right)\right)}^{\top }\left(f_{\kappa }(t,z_1\left(t\right))-f_{\kappa }(t,z_2\left(t\right))\right)\le {\left(z_1\left(t\right)-z_2\left(t\right)\right)}^{\top }{P}_{\kappa }\left(z_1\left(t\right)-z_2\left(t\right)\right)\hfill \end{array}$$

holds for any $z_1\left(t\right),z_2\left(t\right)\in {\mathbb{R}}^n$.

Remark 1. 

One can easily verify that all linear and piecewise-linear FO systems and many FO chaotic systems (for example, FO neural networks and FO Chua’s oscillator) fulfill Assumption 1, see [27,36] and the references therein.

Assumption 2. 

For each sub-matrix $B_{v\mu }$$(v,\mu \in \mathfrak{K})$, the sum of the elements in each row is zero.

Remark 2. 

In order to ensure the cluster-synchronized manifold of system (1), i.e., $\mathcal{M}$= $\{{\left(x_1^{\top }\left(t\right),x_2^{\top }\left(t\right),\dots ,x_N^{\top }\left(t\right)\right)}^{\top }$$\in {\mathbb{R}}^{nN}$|$x_{\ell }\left(t\right)$ = $x_j\left(t\right)$, $\ell ,j\in {\mathcal{G}}_{\kappa },\kappa \in \mathfrak{K}\}$, should be an invariant manifold, similar to [9,14,18], Assumption 2 is adopted here. Additionally, as indicated in [14,17], $b_{\ell j}>0$ ($b_{\ell j}<0$) means the connection from nodes j to ℓ ($\ell \ne j$) is cooperative (competitive). With regard to system (1), Assumption 2 suggests both cooperative and competitive interactions are included within the same group or across different groups. Actually, co-competition interactions are ubiquitous in real-world DNs [2,28]. Therefore, our proposed network model (1) is more consistent with reality compared with the ones introduced in [35,39], where only cooperative interactions are involved.

3. Main Results

In this part, an adaptive control strategy is proposed to steer the heterogeneous FODN (1) towards cluster synchronization over a FnT interval. By means of Lemmas 1 and 4 and reduction to absurdity, some sufficient criteria guaranteeing the FnT cluster synchronization are derived and the CSSTs are also reckoned reliably.

To economize the cost of control, an adaptive controller $u_{\ell }\left(t\right)$ is added to the ℓ-th node in ${\mathcal{G}}_{\kappa }$, which is designed as follows:

$$\begin{array}{c}\hfill u_{\ell }\left(t\right)=\left\{\begin{array}{cc}-d_{\ell }\left(t\right)\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right)}{\left|\right|x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right){\left|\right|}_1}},& {\parallel x_{\ell }\left(t\right)-{\pi }_{\kappa }\parallel }_1\left(t\right)\ne 0,\hfill \\ 0,& \parallel x_{\ell }\left(t\right)-{\pi }_{\kappa }{\left(t\right)\parallel }_1=0,\hfill \end{array}\right.\end{array}$$

(3)

with

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{d}_{\ell }\left(t\right)={\rho }_{\ell }{\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right)}^{\top }\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right)-{\displaystyle \frac{{\eta }_{\ell }}{2}}\phantom{­}\mathrm{sgn}\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)\phantom{­}|d_{\ell }\left(t\right)-d_{\ell }^{*}|,\end{array}$$

(4)

where ${\alpha }_{\ell },{\eta }_{\ell }$> 0 are positive constants, ${\rho }_{\ell },d_{\ell }^{*}\ge 0$ are tunable constants, $d_{\ell }\left(0\right)\ge 0$, and $\ell \in {\mathcal{G}}_{\kappa }$, $\kappa \in \mathfrak{K}$.

Remark 3. 

Denote ${\mathcal{H}}_{\ell }\left(t\right)=d_{\ell }\left(t\right)-d_{\ell }^{*}$ and ${\hslash }_{\ell }\left(t\right)={\rho }_{\ell }{\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right)}^{\top }\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right)$, $t\ge 0$, where $\ell \in {\mathcal{G}}_{\kappa }$ and $\kappa \in \mathfrak{K}$. Then, it is evident that ${\hslash }_{\ell }\left(t\right)\ge 0$ for all $t\ge 0$. Moreover, according to the linearity of fractional derivative and the fact that the Caputo fractional derivative for a constant is zero [22], one can obtain from Equation (4) that

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{\mathcal{H}}_{\ell }\left(t\right)=-{\displaystyle \frac{{\eta }_{\ell }}{2}}{\mathcal{H}}_{\ell }\left(t\right)+{\hslash }_{\ell }\left(t\right).\end{array}$$

(5)

Since the solution to the initial value problem ${\phantom{­}}_{t_0}^C{D}_t^{q}y\left(t\right)=-\lambda y\left(t\right)+\tilde{h}\left(t\right)$ with $0<q<1$ and $\lambda \ge 0$ has the form: $y\left(t\right)=y\left(t_0\right)E_q\left(-\lambda {(t-t_0)}^q\right)+{\int }_{t_0}^t{(t-s)}^{q-1}{E}_{q,\phantom{­}q}\left(-\lambda {(t-s)}^q\right)\tilde{h}\left(s\right)\mathrm{d}s$ [30], it follows from Equation (5) and Lemma 1 that ${\mathcal{H}}_{\ell }\left(t\right)\ge {\mathcal{H}}_{\ell }\left(0\right)E_q\left(-{\displaystyle \frac{{\eta }_{\ell }}{2}}{t}^q\right)$ for $t\ge 0$ due to the non-negative of ${\hslash }_{\ell }\left(t\right)$ and $E_{q,\phantom{­}q}(-w)>0$ on $[0,+\infty )$ [23]. Therefore, ${\mathcal{H}}_{\ell }\left(t\right)\ge 0$ holds for all $t\ge 0$ if ${\mathcal{H}}_{\ell }\left(0\right)\ge 0$. In the following, we will set $d_{\ell }\left(0\right)\ge d_{\ell }^{*}$ so that the adaptive control gain $d_{\ell }\left(t\right)\ge 0$ for all $t\ge 0$.

Theorem 1. 

Suppose Assumptions 1 and 2 hold. Then, the heterogeneous FODN (1) can reach FnT cluster synchronization under the adaptive controller (3) and (4), if there exists a positive scalar ${ϵ}_0>0$ such that, for all $\kappa \in \mathfrak{K}$,

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{\kappa }=\left(I_{r_{\kappa }}\otimes P_{\kappa }+{\varpi }_0{I}_{r_{\kappa }}\otimes I_n-D_{\kappa }\otimes I_n+{\displaystyle \frac{\sigma }{2}}\left(B_{\kappa \kappa }+B_{\kappa \kappa }^{\top }\right)\otimes \mathsf{\Gamma }\right)<0,\end{array}$$

(6)

where ${\varpi }_0$ = ${\displaystyle \frac{(m-1)\sigma }{2}}\left({\displaystyle {ϵ}_0\underset{1\le \kappa \ne p\le m,\phantom{­}\kappa \ne p}{max}\left\{{\lambda }_{max}\left(B_{\kappa p}{B}_{\kappa p}^{\top }\right){\lambda }_{max}\left(\mathsf{\Gamma }{\mathsf{\Gamma }}^{\top }\right)\right\}+{ϵ}_0^{-1}}\right)$ and $D_{\kappa }=\mathrm{diag}\left(d_{l_{k-1}+1}^{*},d_{l_{\kappa -1}+2}^{*},\dots ,d_{l_{\kappa }}^{*}\right)$ with $l_{\kappa -1}$ = ${\sum }_{i=0}^{\kappa -1}{r}_i$. Moreover, the CSST is reckoned by $\mathcal{T}\le {\mathcal{T}}_1$ with ${\mathcal{T}}_1>0$ being the unique solution of the equation below:

$${\displaystyle E_q\left(-\mathsf{\Theta }{t}^{\phantom{­}q}\right)-\frac{{\alpha }^{*}}{\mathsf{\Theta }\mathcal{V}\left(0\right)+{\alpha }^{*}}=0,}$$

in which Θ=${\displaystyle min\left\{-\underset{1\le \phantom{­}\kappa \le m}{min}\{2{\lambda }_{min}\left({\mathsf{\Omega }}_{\kappa }\right)\},{\eta }^{*}\right\}}$ with ${\displaystyle {\eta }^{*}=\underset{1\le \ell \le N}{min}\left\{{\eta }_{\ell }\right\}}$, ${\displaystyle {\alpha }^{*}=\underset{1\le \ell \le N}{min}\left\{{\alpha }_{\ell }\right\}}$, and $\mathcal{V}\left(0\right)={\displaystyle \frac{1}{2}}{\sum }_{\kappa =1}^m{\sum }_{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\left(x_{\ell }\left(0\right)-{\pi }_{\kappa }\left(0\right)\right)}^{\top }\left(x_{\ell }\left(0\right)-{\pi }_{\kappa }\left(0\right)\right)+{\sum }_{\kappa =1}^m{\sum }_{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(0\right)-d_{\ell }^{*}\right)}^2$.

Proof. 

Based on Assumption 2, it is easy to observe that, for any $\ell \in {\mathcal{G}}_{\kappa }$ and $k\in \mathfrak{K}$,

$$\sum _{j\phantom{­}\in {\mathcal{G}}_p,j\ne \ell }{b}_{\ell j}\mathsf{\Gamma }\left(x_j\left(t\right)-x_{\ell }\left(t\right)\right)=\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\mathsf{\Gamma }{x}_j\left(t\right)\phantom{­}\phantom{­}\phantom{­}\mathrm{and}\phantom{­}\phantom{­}\phantom{­}\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\mathsf{\Gamma }{\pi }_p\left(t\right)={\mathbf{0}}_n,\forall p\in \mathfrak{K},$$

where ${\mathbf{0}}_n$ represents the n-dimensional vector of zeros. Recalling that $e_{\ell }\left(t\right)=x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)$, with $\ell \in {\mathcal{G}}_{\kappa }$, represents the synchronized error of the $\kappa$-th group, we can derive the following error system from Equations (1), (3) and (4):

$$\begin{array}{c}\hfill {\phantom{­}}_0^{\phantom{­}C}{D}_t^q{e}_{\ell }\left(t\right)={\tilde{f}}_{\kappa }(t,x_{\ell },{\pi }_{\kappa })+\sigma \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\mathsf{\Gamma }{e}_j\left(t\right)-d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{{\sum }_{i=1}^n\left|e_{\ell i}\left(t\right)\right|}},t\ge 0,\end{array}$$

(7)

where $\left|\right|e_{\ell }\left(t\right){\left|\right|}_1\ne 0$ and ${\tilde{f}}_{\kappa }(t,x_{\ell },{\pi }_{\kappa })=f_{\kappa }(t,x_{\ell }\left(t\right))-f_{\kappa }(t,{\pi }_{\kappa }\left(t\right))$. Clearly, the FnT cluster synchronization problem herein is equivalent to the FnT stabilization problem of the error system (7) in the sense of Definition 3.

Let ${\widehat{Z}}_{\kappa }\left(t\right)={\left(e_{l_{k-1}+1}^{\top }\left(t\right),e_{l_{k-1}+2}^{\top }\left(t\right),\dots ,e_{l_k}^{\top }\left(t\right)\right)}^{\top }$ with $\kappa \in \mathfrak{K}$, and consider a Lyapunov function having the following form:

$$\begin{array}{c}\hfill \phantom{­}\mathcal{V}\left(t\right)={\displaystyle \frac{1}{2}}\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)+\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2\phantom{­}.\end{array}$$

(8)

By Lemmas 2 and 3, when $\left|\right|e_{\ell }\left(t\right){\left|\right|}_1\ne 0$, the q-order Caputo derivative of $\mathcal{V}\left(t\right)$ along the trajectory of system (7) follows

$$\begin{array}{ccc}\hfill {\phantom{­}}_0^C{D}_t^q\mathcal{V}\left(t\right)& =& {\displaystyle \frac{1}{2}}\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\phantom{­}}_0^C{D}_t^q\left(e_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)\right)+\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\phantom{­}}_0^C{D}_t^q\left({\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2\right)\hfill \\ & \le & \sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{e}_{\ell }^{\top }\left(t\right){\phantom{­}}_0^C{D}_t^q{e}_{\ell }\left(t\right)\phantom{­}+\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{{\rho }_{\ell }}}\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right){\phantom{­}}_0^C{D}_t^q{d}_{\ell }\left(t\right)\hfill \\ & =& \sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{e}_{\ell }^{\top }\left(t\right)\left({\tilde{f}}_{\kappa }(t,x_{\ell },{\pi }_{\kappa })+\sigma \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\mathsf{\Gamma }{e}_j\left(t\right)-d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{{\sum }_{i=1}^n\left|e_{\ell i}\left(t\right)\right|}}\right)\hfill \\ & & \phantom{­}\phantom{­}+\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{{\rho }_{\ell }}}\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)\left({\rho }_{\ell }{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)-{\displaystyle \frac{{\eta }_{\ell }}{2}}\phantom{­}\mathrm{sgn}\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)\phantom{­}|d_{\ell }\left(t\right)-d_{\ell }^{*}|\right)\hfill \\ & =& \sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{e}_{\ell }^{\top }\left(t\right){\tilde{f}}_{\kappa }(t,x_{\ell },{\pi }_{\kappa })\phantom{­}+\sigma \sum _{\kappa =1}^m\sum _{p=1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\phantom{­}{e}_{\ell }^{\top }\left(t\right)\mathsf{\Gamma }{e}_j\left(t\right)-\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\alpha }_{\ell }\Delta \left(e_{\ell }\left(t\right)\right)\hfill \\ & & \phantom{­}\phantom{­}-\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{d}_{\ell }^{*}{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)-\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2,\hfill \end{array}$$

(9)

where $\Delta \left(e_{\ell }\left(t\right)\right)={\displaystyle \frac{e_{\ell }^{\top }\left(t\right)\mathrm{sgn}\left(e_l\left(t\right)\right)}{{\sum }_{i=1}^n\left|e_{\ell i}\left(t\right)\right|}}$.

From Assumption 1, we have

$$\begin{array}{c}\hfill \sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{e}_{\ell }^{\top }\left(t\right){\tilde{f}}_{\kappa }(t,x_{\ell },{\pi }_{\kappa })\le \sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{e}_{\ell }^{\top }\left(t\right)P_{\kappa }{e}_{\ell }\left(t\right)=\sum _{\kappa =1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(I_{r_{\kappa }}\otimes P_{\kappa }\right){\widehat{Z}}_{\kappa }\left(t\right).\end{array}$$

(10)

By the properties of Kronecker product of two matrices, one has

$$\begin{array}{ccc}& & \sigma \sum _{\kappa =1}^m\sum _{p=1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\phantom{­}{e}_{\ell }^{\top }\left(t\right)\mathsf{\Gamma }{e}_j\left(t\right)\hfill \\ & & =\sigma \sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_k}\sum _{j\phantom{­}\in {\mathcal{G}}_{\kappa }}{b}_{\ell j}\phantom{­}{e}_{\ell }^{\top }\left(t\right)\mathsf{\Gamma }{e}_j\left(t\right)+\sigma \sum _{\kappa =1}^m\sum _{p=1,p\ne \kappa }^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\phantom{­}{e}_{\ell }^{\top }\left(t\right)\mathsf{\Gamma }{e}_j\left(t\right)\hfill \\ & & =\sigma \sum _{\kappa =1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(B_{\kappa \kappa }\otimes \mathsf{\Gamma }\right){\widehat{Z}}_{\kappa }\left(t\right)+\sigma \sum _{\kappa =1}^m\sum _{p=1,p\ne \kappa }^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(B_{\kappa p}\otimes \mathsf{\Gamma }\right){\widehat{Z}}_p\left(t\right)\hfill \\ & & \le \sigma \sum _{\kappa =1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(B_{\kappa \kappa }\otimes \mathsf{\Gamma }\right){\widehat{Z}}_{\kappa }\left(t\right)+{\displaystyle \frac{\sigma }{2}}\sum _{\kappa =1}^m\sum _{p=1,p\ne \kappa }^m[{ϵ}_0{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(B_{\kappa p}{B}_{\kappa p}^{\top }\otimes \mathsf{\Gamma }{\mathsf{\Gamma }}^{\top }\right){\widehat{Z}}_{\kappa }\left(t\right)\hfill \\ & & +{ϵ}_0^{-1}{\widehat{Z}}_p^{\top }\left(t\right)\left(I_{r_p}\otimes I_n\right){\widehat{Z}}_p\left(t\right)]\hfill \\ & & \le {\displaystyle \frac{(m-1)\sigma }{2}}\left({ϵ}_0\underset{1\le \kappa ,p\le m,\phantom{­}\kappa \ne p}{max}\left\{{\lambda }_{max}\left(B_{\kappa p}{B}_{\kappa p}^{\top }\right){\lambda }_{max}\left(\mathsf{\Gamma }{\mathsf{\Gamma }}^{\top }\right)\right\}+{ϵ}_0^{-1}\right)\sum _{k=1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(I_{r_{\kappa }}\otimes I_n\right){\widehat{Z}}_{\kappa }\left(t\right)\hfill \\ & & +\sigma \sum _{\kappa =1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(B_{\kappa \kappa }\otimes \mathsf{\Gamma }\right){\widehat{Z}}_{\kappa }\left(t\right)\hfill \\ & & ={\varpi }_0\sum _{\kappa =1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(I_{r_{\kappa }}\otimes I_n\right){\widehat{Z}}_{\kappa }\left(t\right)+{\displaystyle \frac{\sigma }{2}}\sum _{\kappa =1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(\left(B_{\kappa \kappa }+B_{\kappa \kappa }^{\top }\right)\otimes \mathsf{\Gamma }\right){\widehat{Z}}_{\kappa }\left(t\right).\hfill \end{array}$$

(11)

Substituting Equations (10) and (11) into Equation (9) yields

$$\begin{array}{ccc}\hfill {\phantom{­}}_0^C{D}_t^q\mathcal{V}\left(t\right)& \le & \sum _{\kappa =1}^m{\widehat{Z}}_{\kappa }^{\top }\left(t\right)\left(I_{r_{\kappa }}\otimes P_{\kappa }+{\varpi }_0{I}_{r_{\kappa }}\otimes I_n-D_{\kappa }\otimes I_n+{\displaystyle \frac{\sigma }{2}}\left(B_{\kappa \kappa }+B_{\kappa \kappa }^{\top }\right)\otimes \mathsf{\Gamma }\right){\widehat{Z}}_{\kappa }\left(t\right)\hfill \\ & & -\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\alpha }_{\ell }\Delta \left(e_{\ell }\left(t\right)\right)-\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2\hfill \\ & \le & \sum _{\kappa =1}^m{\lambda }_{min}\left({\mathsf{\Omega }}_{\kappa }\right){\widehat{Z}}_{\kappa }^{\top }\left(t\right){\widehat{Z}}_{\kappa }\left(t\right)-{\eta }^{*}\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2-\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\alpha }_{\ell }\Delta \left(e_{\ell }\left(t\right)\right)\hfill \\ & \le & \underset{1\le \kappa \le m}{min}\{2{\lambda }_{min}\left({\mathsf{\Omega }}_{\kappa }\right)\}\sum _{\kappa =1}^m{\displaystyle \frac{1}{2}}{\widehat{Z}}_{\kappa }^{\top }\left(t\right){\widehat{Z}}_{\kappa }\left(t\right)-{\eta }^{*}\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2\hfill \\ & & -\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\alpha }_{\ell }\Delta \left(e_{\ell }\left(t\right)\right)\hfill \\ & \le & -\mathsf{\Theta }\left({\displaystyle \frac{1}{2}}\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)+\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2\right)-\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\alpha }_{\ell }\Delta \left(e_{\ell }\left(t\right)\right)\hfill \\ & =& -\mathsf{\Theta }\mathcal{V}\left(t\right)-\sum _{\kappa =1}^m\sum _{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\alpha }_{\ell }\Delta \left(e_{\ell }\left(t\right)\right).\hfill \end{array}$$

(12)

On the other hand, it can be found that

$$\begin{array}{c}\hfill \Delta \left(e_{\ell }\left(t\right)\right)=\left\{\begin{array}{cc}& \phantom{­}\phantom{­}1,||e_{\ell }\left(t\right){||}_1\ne 0,\hfill \\ & \phantom{­}\phantom{­}0,||e_{\ell }\left(t\right){||}_1=0.\hfill \end{array}\right.\end{array}$$

Therefore, if there exists a $\ell \in \{1,2,\dots ,N\}$ such that $e_{\ell }\left(t\right)\ne 0$ for all $t\ge 0$, then ${\sum }_{\kappa =1}^m{\sum }_{\ell \phantom{­}\in {\mathcal{G}}_{\kappa }}{\alpha }_{\ell }\phantom{­}\Delta \left(e_{\ell }\left(t\right)\right)\ge {\alpha }_{\ell }$. Combining this with Equation (12) gives

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q\mathcal{V}\left(t\right)\le -\mathsf{\Theta }\mathcal{V}\left(t\right)-{\alpha }_{\ell }\le -\mathsf{\Theta }\mathcal{V}\left(t\right)-{\alpha }^{*},t\ge 0,\end{array}$$

(13)

where ${\alpha }^{*}={min}_{1\le \ell \le N}\left\{{\alpha }_{\ell }\right\}$.

Denote ${\displaystyle \widehat{\mathcal{V}}\left(t\right)=\mathcal{V}\left(t\right)+\frac{{\alpha }^{*}}{\mathsf{\Theta }}}$, it follows from Lemma 4 and Equation (13) that

$$\begin{array}{c}\hfill \widehat{\mathcal{V}}\left(t\right)\le \widehat{\mathcal{V}}\left(0\right)E_q\left(-\phantom{­}\mathsf{\Theta }{t}^{\phantom{­}q}\right),t\ge 0,\end{array}$$

which shows that

$$\begin{array}{c}\hfill \mathcal{V}\left(t\right)\le \left(\mathcal{V}\left(0\right)+{\displaystyle \frac{{\alpha }^{*}}{\mathsf{\Theta }}}\right)E_q\left(-\mathsf{\Theta }{t}^{\phantom{­}q}\right)-{\displaystyle \frac{{\alpha }^{*}}{\mathsf{\Theta }}},t\ge 0.\end{array}$$

(14)

Let ${\displaystyle g\left(t\right)=E_q\left(-\mathsf{\Theta }{t}^{\phantom{­}q}\right)-\frac{{\alpha }^{*}}{\mathsf{\Theta }\mathcal{V}\left(0\right)+{\alpha }^{*}}}$, $t\ge 0$. Since $\mathsf{\Theta }>0$, by virtue of the analyticity of the Mittag-Leffler function [23] and Lemma 1, it is deduced that the function $g\left(t\right)$ is a continuous and monotonically decreasing function of t, and

$${\displaystyle g\left(0\right)=1-\frac{{\alpha }^{*}}{\mathsf{\Theta }\mathcal{V}\left(0\right)+{\alpha }^{*}}>0,\underset{t\to +\infty }{lim}g\left(t\right)=0-\frac{{\alpha }^{*}}{\mathsf{\Theta }\mathcal{V}\left(0\right)+{\alpha }^{*}}<0.}$$

It follows that the equation ${\displaystyle g\left(t\right)=E_q\left(-\mathsf{\Theta }{t}^{\phantom{­}q}\right)-\frac{{\alpha }^{*}}{\mathsf{\Theta }\mathcal{V}\left(0\right)+{\alpha }^{*}}=0}$ has an unique solution ${\mathcal{T}}_1>0$ (i.e., $g\left({\mathcal{T}}_1\right)=0$) and $g\left(t\right)>0$ for any $t\in [0,{\mathcal{T}}_1)$.

Denote

$$\mathsf{\Phi }\left(t\right)=\left(\mathcal{V}\left(0\right)+{\displaystyle \frac{{\alpha }^{*}}{\mathsf{\Theta }}}\right)E_q\left(-\mathsf{\Theta }{t}^{\phantom{­}q}\right)-{\displaystyle \frac{{\alpha }^{*}}{\mathsf{\Theta }}},t\ge 0.$$

Next, we prove that

$$\begin{array}{c}\hfill \mathcal{V}\left(t\right)\equiv 0,\phantom{­}\phantom{­}\mathrm{for}\phantom{­}\phantom{­}\mathrm{all}\phantom{­}\phantom{­}\phantom{­}t\ge {\mathcal{T}}_1.\end{array}$$

(15)

If it does not hold, by the non-negativity of function $\mathcal{V}\left(t\right)$, then there exists a moment $\breve{t}\ge {\mathcal{T}}_1$ such that $\mathcal{V}\left(\breve{t}\right)>0$. This, together with Lemma 1, implies that

$$\begin{array}{c}\hfill \mathsf{\Phi }\left(\breve{t}\right)\le \left(\mathcal{V}\left(0\right)+{\displaystyle \frac{{\alpha }^{*}}{\mathsf{\Theta }}}\right)E_q\left(-\mathsf{\Theta }{\left({\mathcal{T}}_1\right)}^{\phantom{­}q}\right)-{\displaystyle \frac{\alpha }{\mathsf{\Theta }}}=\left(\mathcal{V}\left(0\right)+{\displaystyle \frac{{\alpha }^{*}}{\mathsf{\Theta }}}\right)g\left({\mathcal{T}}_1\right)=0.\end{array}$$

(16)

However, we can infer from Equation (14) that $\mathsf{\Phi }\left(\breve{t}\right)\ge \mathcal{V}\left(\breve{t}\right)>0$, which contradicts the inequality (16). Therefore, $\mathcal{V}\left(t\right)\equiv 0$ holds for all $t\ge {\mathcal{T}}_1$, meaning that $\left|\right|e_{\ell }\left(t\right)\left|\right|\equiv 0$ for all $t\ge {\mathcal{T}}_1$ and $\ell =1,2,\dots ,N.$ According to Definition 3, the FnT cluster synchronization of the heterogeneous FODN (1) with the adaptive controller (3) and (4) is achieved no later than ${\mathcal{T}}_1$. The proof is completed. □

When ${\alpha }_{\ell }=0$ and ${\eta }_{\ell }=0$ for $\ell =1,2,\dots ,N$, the adaptive controller (3) and (4) is simplified as the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& u_{\ell }\left(t\right)=-d_{\ell }\left(t\right)\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right),\hfill \\ & {\phantom{­}}_0^C{D}_t^q{d}_{\ell }\left(t\right)={\rho }_{\ell }{\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right)}^{\top }\left(x_{\ell }\left(t\right)-{\pi }_{\kappa }\left(t\right)\right),\hfill \end{array}\right.\end{array}$$

(17)

where $\ell \in {\mathcal{G}}_{\kappa }$ and $\kappa \in \mathfrak{K}$. For the control scheme (17), similar to the analysis in [40,44], one can obtain the following result.

Corollary 1. 

Suppose Assumptions 1 and 2 hold. Then, the heterogeneous FODN (1) can achieve Mittag-Leffler cluster synchronization under the adaptive controller (17), if there exists a positive scalar ${ϵ}_0>0$ such that, for all $\kappa \in \mathfrak{K}$,

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{\kappa }=\left(I_{r_{\kappa }}\otimes P_{\kappa }+{\varpi }_0{I}_{r_{\kappa }}\otimes I_n-D_{\kappa }\otimes I_n+{\displaystyle \frac{\sigma }{2}}\left(B_{\kappa \kappa }+B_{\kappa \kappa }^{\top }\right)\otimes \mathsf{\Gamma }\right)<0,\end{array}$$

(18)

where ${\varpi }_0$ and $D_{\kappa }$ are defined in Theorem 1.

Remark 4. 

From Theorem 1 and Corollary 1, one can discover that the linear part $-d_{\ell }\left(t\right)e_{\ell }\left(t\right)$ in the controller (3) ensures system (1) to realize Mittag-Leffler cluster synchronization and the other term $-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{{\sum }_{i=1}^n\left|e_{\ell i}\left(t\right)\right|}}$ guarantees the cluster synchronization would be reached within a FnT time estimated by ${\mathcal{T}}_1$.

Remark 5. 

Apparently, when $m=1$, i.e., the heterogeneous FODN (1) contains only one group, FnT complete synchronization will happen. That is to say, our proposed adaptive control approach can also be applied to achieve FnT complete synchronization of heterogeneous FODNs with co-competition interactions.

Remark 6. 

Theorem 1 indicates that the estimate ${\mathcal{T}}_1$ is relevant to the parameters q, Θ and ${\alpha }^{*}$. In practical applications, one may adjust these three parameters properly to realize cluster synchronization within a specified time. The relationship between each of the three parameters and the CSST will be revealed through numerical simulations in the next section.

If only cooperative interactions are involved in the heterogeneous FODN (1), then the corresponding heterogeneous FODN with cooperative couplings can be written as

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{x}_{\ell }\left(t\right)=f_{\kappa }(t,x_{\ell }\left(t\right))+\sigma \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p}{\tilde{b}}_{\ell j}\phantom{­}\mathsf{\Gamma }{x}_j\left(t\right),\ell \in {\mathcal{G}}_{\kappa }\phantom{­}\phantom{­}\mathrm{and}\phantom{­}\phantom{­}\kappa \in \mathfrak{K},\end{array}$$

(19)

where $t\ge 0$, ${\tilde{b}}_{\ell j}\ge 0\phantom{­}(j\ne \ell )$ for all $\ell ,j=1,2,\dots ,N$, and ${\tilde{b}}_{\ell \ell }=-{\sum }_{j=1,j\ne \ell }^N{\tilde{b}}_{\ell j}$ for $\ell =1,2,\dots ,N$. For the heterogeneous FODN (19), Assumption 2 is no longer needed and, due to the fact that ${\sum }_{j=1}^N{\tilde{b}}_{\ell j}=0$ for all $\ell =1,2,\dots ,N$, we have

$$\begin{array}{c}\hfill \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p}{\tilde{b}}_{\ell j}\mathsf{\Gamma }{\pi }_p\left(t\right)={\mathbf{0}}_n,\forall \ell \in {\mathcal{G}}_{\kappa }\backslash {\breve{\mathcal{G}}}_{\kappa },\end{array}$$

(20)

where $\kappa \in \mathfrak{K}$ and ${\breve{\mathcal{G}}}_{\kappa }$ is a subset of ${\mathcal{G}}_{\kappa }$ composed of all nodes belonging to the κ-th group that have directional edges connecting the nodes in other groups. In view of (20), inspired by Refs. [6,35], one can design the following adaptive controller

$$\begin{array}{c}\hfill u_{\ell }\left(t\right)=\left\{\begin{array}{cc}& -d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{\left|\right|e_{\ell }\left(t\right){\left|\right|}_1}}-\sigma \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p}{\tilde{b}}_{\ell j}\mathsf{\Gamma }{\pi }_p\left(t\right),\ell \in {\breve{\mathcal{G}}}_{\kappa }\phantom{­}\phantom{­}\mathrm{and}\phantom{­}\phantom{­}{\parallel e_{\ell }\left(t\right)\parallel }_1\left(t\right)\ne 0,\hfill \\ & -d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{\left|\right|e_{\ell }\left(t\right){\left|\right|}_1}},\phantom{­}\phantom{­}\ell \in {\mathcal{G}}_{\kappa }\setminus {\breve{\mathcal{G}}}_{\kappa }\phantom{­}\phantom{­}\mathrm{and}\phantom{­}\phantom{­}{\parallel e_{\ell }\left(t\right)\parallel }_1\left(t\right)\ne 0,\hfill \\ & 0,\parallel e_{\ell }{\left(t\right)\parallel }_1=0,\hfill \end{array}\right.\end{array}$$

(21)

with

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{d}_{\ell }\left(t\right)={\rho }_{\ell }{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)-{\displaystyle \frac{{\eta }_{\ell }}{2}}\phantom{­}\mathrm{sgn}\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)\phantom{­}|d_{\ell }\left(t\right)-d_{\ell }^{*}|,\end{array}$$

(22)

to steer the heterogeneous FODN (19) towards FnT cluster synchronization.

Remark 7. 

Intuitively, the role of the term $-\sigma {\sum }_{p=1}^m{\sum }_{j\phantom{­}\in {\mathcal{G}}_p}{\tilde{b}}_{\ell j}\mathsf{\Gamma }{\pi }_p\left(t\right)$ in Equation (21) is to eliminate the interplays among groups at the intersection nodes. In fact, without this term, the cluster synchronization will never occur. The remaining two terms in Equation (21) are to finite-timely synchronize all nodes within the same group.

In order to analyze conveniently, for ℓ∈${\mathcal{G}}_{\kappa }$, we let ${\mu }_{\ell }$ = κ, then the function $f_{\kappa }(t,·)$ in system (19) can be represented by $f_{{\mu }_{\ell }}(t,·)$. Certainly, for each pair of nodes ℓ and j belonging to different groups (i.e., ${\mu }_{\ell }$ ≠ ${\mu }_j$), then $f_{{\mu }_{\ell }}(·)$ ≠ $f_{{\mu }_j}(·)$. With the aforesaid statement, when $\parallel e_{\ell }{\left(t\right)\parallel }_1\left(t\right)$ ≠ 0, one can deduce from Equations (19)–(21) that

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{e}_{\ell }\left(t\right)=f_{{\mu }_{\ell }}(t,x_{\ell }\left(t\right))-f_{{\mu }_{\ell }}(t,{\pi }_{{\mu }_{\ell }}\left(t\right))+\sigma \sum _{j=1}^N{\tilde{b}}_{\ell j}\phantom{­}\mathsf{\Gamma }{e}_j\left(t\right)-d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{{\sum }_{i=1}^n\left|e_{\ell i}\left(t\right)\right|}},\end{array}$$

(23)

where $t\ge 0$ and $\ell =1,2,\dots ,N$.

Let $\tilde{B}={\left({\tilde{b}}_{\ell j}\right)}_{N\times N}$, $\tilde{D}=\mathrm{diag}\left(d_1^{*},d_2^{*},\dots ,d_N^{*}\right)$, $Q=\sigma \tilde{B}-\tilde{D}$, and ${\tilde{Q}}^s={\displaystyle \frac{1}{2}}\left(Q+Q^{\top }\right)={\left({\tilde{q}}_{\ell j}\right)}_{N\times N}$. Now, on the basis of Theorem 1, we present the following result.

Theorem 2. 

Under Assumption 1 and the adaptive controller (21) and (22), if there exists a positive scalar $\tilde{\epsilon }>0$ such that

$$\begin{array}{c}\hfill {\breve{p}}_r{I}_N+{\gamma }_r{\tilde{Q}}^s+\tilde{\epsilon }{I}_N\le 0\end{array}$$

(24)

holds for all $r=1,2,\dots ,n$, where ${\displaystyle {\breve{p}}_r=\underset{1\le \ell \le N}{max}\left\{p_r^{{\mu }_{\ell }}\right\}}$, then the heterogeneous FODN (19) can reach cluster synchronization in finite time. Moreover, the CSST is reckoned by $\mathcal{T}\le {\mathcal{T}}_2$ with ${\mathcal{T}}_2>0$ being the unique solution of the equation below:

$${\displaystyle E_q\left(-\breve{\mathsf{\Theta }}{t}^{\phantom{­}q}\right)-\frac{{\alpha }^{*}}{\breve{\mathsf{\Theta }}\tilde{\mathcal{V}}\left(0\right)+{\alpha }^{*}}=0,}$$

in which $\breve{\mathsf{\Theta }}=min\left\{2\tilde{\epsilon },{\eta }^{*}\right\}$ with ${\eta }^{*}={min}_{1\le \ell \le N}\left\{{\eta }_{\ell }\right\}$, ${\displaystyle {\alpha }^{*}=\underset{1\le \ell \le N}{min}\left\{{\alpha }_{\ell }\right\}}$, and $\tilde{\mathcal{V}}\left(0\right)={\displaystyle \frac{1}{2}}{\sum }_{\ell =1}^N{\left(x_{\ell }\left(0\right)-{\pi }_{{\mu }_{\ell }}\left(0\right)\right)}^{\top }\left(x_{\ell }\left(0\right)-{\pi }_{{\mu }_{\ell }}\left(0\right)\right)+{\sum }_{\ell =1}^N{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(0\right)-d_{\ell }^{*}\right)}^2$.

Proof. 

Let ${\widehat{e}}^r\left(t\right)={\left(e_{1r}\left(t\right),e_{2r}\left(t\right),\dots ,e_{Nr}\left(t\right)\right)}^{\top }$ for $r=1,2,\dots ,n$ and introduce a Lyapunov function as below:

$$\begin{array}{c}\hfill \phantom{­}\tilde{\mathcal{V}}\left(t\right)={\displaystyle \frac{1}{2}}\sum _{\ell =1}^N{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)+\sum _{\ell =1}^N{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2.\end{array}$$

(25)

Similarly, one has (by Theorem 1)

$$\begin{array}{ccc}\hfill {\phantom{­}}_0^C{D}_t^q\tilde{\mathcal{V}}\left(t\right)& \le & \sum _{\ell =1}^N{e}_{\ell }^{\top }\left(t\right)P_{{\mu }_{\ell }}{e}_{\ell }\left(t\right)+\sigma \sum _{\ell =1}^N\sum _{j=1}^N{\tilde{b}}_{\ell j}\phantom{­}{e}_{\ell }^{\top }\left(t\right)\mathsf{\Gamma }{e}_j\left(t\right)\hfill \\ & & -\sum _{\ell =1}^N{d}_{\ell }^{*}{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)-\sum _{\ell =1}^N{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2-{\alpha }^{*}\hfill \\ & =& \sum _{\ell =1}^N{e}_{\ell }^{\top }\left(t\right)P_{{\mu }_{\ell }}{e}_{\ell }\left(t\right)+e^{\top }\left(t\right)\left((\sigma \tilde{B}-\tilde{D})\otimes \mathsf{\Gamma }\right)e\left(t\right)-\sum _{\ell =1}^N{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2-{\alpha }^{*}\hfill \\ & =& \sum _{\ell =1}^N{e}_{\ell }^{\top }\left(t\right)P_{{\mu }_{\ell }}{e}_{\ell }\left(t\right)+e^{\top }\left(t\right)\left({\tilde{Q}}^s\otimes \mathsf{\Gamma }\right)e\left(t\right)-\sum _{\ell =1}^N{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2-{\alpha }^{*},\hfill \end{array}$$

(26)

where $e\left(t\right)={\left(e_1^{\top }\left(t\right),e_2^{\top }\left(t\right),\dots ,e_N^{\top }\left(t\right)\right)}^{\top }$.

Based on the definition of ${\widehat{e}}^r\left(t\right)$, we have the following two equalities

$$\begin{array}{c}\hfill \sum _{\ell =1}^N{e}_{\ell }^{\top }\left(t\right)P_{{\mu }_{\ell }}{e}_{\ell }\left(t\right)=\sum _{\ell =1}^N\sum _{r=1}^n{p}_r^{{\mu }_{\ell }}{e}_{\ell r}^2\left(t\right)=\sum _{r=1}^n{\breve{p}}_r\sum _{\ell =1}^N{e}_{\ell r}^2\left(t\right)=\sum _{r=1}^n{\breve{p}}_r{\left({\widehat{e}}^r\left(t\right)\right)}^{\top }{I}_N{\widehat{e}}^r\left(t\right)\end{array}$$

(27)

and

$$\begin{array}{ccc}\hfill e^{\top }\left(t\right)\left({\tilde{Q}}^s\otimes \mathsf{\Gamma }\right)e\left(t\right)& =& \sum _{\ell =1}^N\sum _{j=1}^N{\tilde{q}}_{\ell j}\phantom{­}{e}_{\ell }^{\top }\left(t\right)\mathsf{\Gamma }{e}_j\left(t\right)\hfill \\ & =& \sum _{\ell =1}^N\sum _{j=1}^N{\tilde{q}}_{\ell j}\phantom{­}\sum _{r=1}^n{\gamma }_r{e}_{\ell r}\left(t\right)e_{jr}\left(t\right)\hfill \\ & =& \sum _{r=1}^n{\gamma }_r\sum _{\ell =1}^N\sum _{j=1}^N{\tilde{q}}_{\ell j}\phantom{­}{e}_{\ell r}\left(t\right)e_{jr}\left(t\right)\hfill \\ & =& \sum _{r=1}^n{\gamma }_r{\left({\widehat{e}}^r\left(t\right)\right)}^{\top }\left({\tilde{Q}}^s\right){\widehat{e}}^r\left(t\right).\hfill \end{array}$$

(28)

By substituting Equations (27) and (28) into Equation (26), one has

$$\begin{array}{ccc}\hfill {\phantom{­}}_0^C{D}_t^q\tilde{\mathcal{V}}\left(t\right)& \le & \sum _{r=1}^n{\left({\widehat{e}}^r\left(t\right)\right)}^{\top }\left({\breve{p}}_r{I}_N+{\gamma }_r{\tilde{Q}}^s\right){\widehat{e}}^r\left(t\right)-\sum _{\ell =1}^N{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2-{\alpha }^{*}.\hfill \end{array}$$

(29)

This, together with the conditions of Theorem 2, yields

$$\begin{array}{ccc}\hfill {\phantom{­}}_0^C{D}_t^q\tilde{\mathcal{V}}\left(t\right)& \le & -\sum _{r=1}^n\tilde{\epsilon }\sum _{\ell =1}^N{e}_{\ell r}^2\left(t\right)-\sum _{\ell =1}^N{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2-{\alpha }^{*}\hfill \\ & =& -\tilde{\epsilon }\sum _{\ell =1}^N{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)-\sum _{\ell =1}^N{\displaystyle \frac{{\eta }_{\ell }}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2-{\alpha }^{*}\hfill \\ & \le & -\breve{\mathsf{\Theta }}\left({\displaystyle \frac{1}{2}}\sum _{\ell =1}^N{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)+\sum _{\ell =1}^N{\displaystyle \frac{1}{2{\rho }_{\ell }}}{\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)}^2\right)-{\alpha }^{*}\hfill \\ & =& -\breve{\mathsf{\Theta }}\tilde{\mathcal{V}}\left(t\right)-{\alpha }^{*}.\hfill \end{array}$$

(30)

Using the same method in Theorem 1, we can conclude from Equation (30) that the FnT cluster synchronization of system (19) is realized and the CSST is reckoned by $\mathcal{T}\le {\mathcal{T}}_2$, where ${\mathcal{T}}_2>0$ is the unique solution of the equation ${\displaystyle E_q\left(-\breve{\mathsf{\Theta }}{t}^{\phantom{­}q}\right)-\frac{{\alpha }^{*}}{\breve{\mathsf{\Theta }}\tilde{\mathcal{V}}\left(0\right)+{\alpha }^{*}}}$ = 0. The proof is completed.

If ${\alpha }_{\ell }=0$ and ${\eta }_{\ell }=0$ for $\ell =1,2,\dots ,N$, then the controller (21) and (22) can be rewritten as:

$$\begin{array}{c}\hfill u_{\ell }\left(t\right)=\left\{\begin{array}{cc}& -d_{\ell }\left(t\right)e_{\ell }\left(t\right)-\sigma \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p}{\tilde{b}}_{\ell j}\mathsf{\Gamma }{\pi }_p\left(t\right),\ell \in {\breve{\mathcal{G}}}_{\kappa },\hfill \\ & -d_{\ell }\left(t\right)e_{\ell }\left(t\right),\ell \in {\mathcal{G}}_{\kappa }\setminus {\breve{\mathcal{G}}}_{\kappa },\hfill \end{array}\right.\end{array}$$

(31)

with

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{d}_{\ell }\left(t\right)={\rho }_{\ell }{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right).\end{array}$$

(32)

□

Corollary 2. 

Under Assumption 1 and the adaptive controller (31) and (32), if there exist a positive scalar $\tilde{\epsilon }>0$ such that

$$\begin{array}{c}\hfill {\breve{p}}_r{I}_N+{\gamma }_r{\tilde{Q}}^s+\tilde{\epsilon }{I}_N\le 0\end{array}$$

(33)

holds for all $r=1,2,\dots ,n$, where ${\displaystyle {\breve{p}}_r=\underset{1\le \ell \le N}{max}\left\{p_r^{{\mu }_{\ell }}\right\}}$, then the heterogeneous FODN (19) can reach Mittag-Leffler cluster synchronization.

Remark 8. 

Evidently, inequality (24) in Theorem 2 is easier to be verified than inequality (6) in Theorem 1. However, for the heterogeneous FODN (19) with cooperative couplings, the control design is comparatively more complex than that for system (1) with co-competition interactions because an extra control term $-\sigma {\sum }_{p=1}^m{\sum }_{j\phantom{­}\in {\mathcal{G}}_p}{\tilde{b}}_{\ell j}\mathsf{\Gamma }{\pi }_p\left(t\right)$ is required for the intersection nodes in the heterogeneous FODN (19).

If $q=1$, then the heterogeneous FODN (1) becomes the following heterogeneous integer-order DN with co-competition interactions:

$$\begin{array}{c}\hfill {\dot{x}}_{\ell }\left(t\right)=f_{\kappa }(t,x_{\ell }\left(t\right))+\sigma \sum _{p=1}^m\sum _{j\phantom{­}\in {\mathcal{G}}_p,j\ne \ell }{b}_{\ell j}\mathsf{\Gamma }(x_j\left(t\right)-x_{\ell }\left(t\right)),t\ge 0,\end{array}$$

(34)

where $\ell \in {\mathcal{G}}_{\kappa }$ and $\kappa \in \mathfrak{K}$. Meanwhile, the control scheme shown in Equations (3) and (4) turns into the following one:

$$\begin{array}{c}\hfill u_{\ell }\left(t\right)=\left\{\begin{array}{cc}& -d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{\left|\right|e_{\ell }\left(t\right){\left|\right|}_1}},{\parallel e_{\ell }\left(t\right)\parallel }_1\ne 0,\hfill \\ & 0,\parallel e_{\ell }{\left(t\right)\parallel }_1=0,\hfill \end{array}\right.\end{array}$$

(35)

with

$$\begin{array}{c}\hfill {\dot{d}}_{\ell }\left(t\right)={\rho }_{\ell }{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)-{\displaystyle \frac{{\eta }_{\ell }}{2}}\phantom{­}\mathrm{sgn}\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)\phantom{­}|d_{\ell }\left(t\right)-d_{\ell }^{*}|.\end{array}$$

(36)

Regarding the heterogeneous integer-order DN (34), the following corollary can be drawn from Theorem 1 and Definition 1.

Corollary 3. 

Suppose Assumptions 1 and 2 hold. If all the conditions in Theorem 1 are met, then the heterogeneous integer-order DN (34) can reach FnT cluster synchronization under the control scheme (35) and (36) and the CSST is estimated by $\mathcal{T}\le {\mathcal{T}}_1^{♣}$ with

$${\mathcal{T}}_1^{♣}={\displaystyle \frac{1}{\mathsf{\Theta }}}ln\left({\displaystyle \frac{\mathsf{\Theta }\mathcal{V}\left(0\right)+{\alpha }^{*}}{{\alpha }^{*}}}\right),$$

where $\mathcal{V}\left(0\right)$, Θ and ${\alpha }^{*}$ are defined in Theorem 1.

Analogously, the heterogeneous integer-order DN with cooperative couplings corresponding to system (19) can be represented as

$$\begin{array}{c}\hfill {\dot{x}}_{\ell }\left(t\right)=f_{{\mu }_{\ell }}(t,x_{\ell }\left(t\right))+\sigma \sum _{j=1}^N{\tilde{b}}_{\ell j}\phantom{­}\mathsf{\Gamma }{x}_j\left(t\right),\ell =1,2,\dots ,N,\end{array}$$

(37)

where $t\ge 0$. For system (37), the adaptive controller is proposed as

$$\begin{array}{c}\hfill u_{\ell }\left(t\right)=\left\{\begin{array}{cc}& -d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{\left|\right|e_{\ell }\left(t\right){\left|\right|}_1}}-\sigma \sum _{j=1}^N{\tilde{b}}_{\ell j}\mathsf{\Gamma }{\pi }_{{\mu }_j}\left(t\right),\ell \in {\breve{\mathcal{G}}}_{\kappa }\phantom{­}\phantom{­}\mathrm{and}\phantom{­}\phantom{­}{\parallel e_{\ell }\left(t\right)\parallel }_1\left(t\right)\ne 0,\hfill \\ & -d_{\ell }\left(t\right)e_{\ell }\left(t\right)-{\alpha }_{\ell }{\displaystyle \frac{\mathrm{sgn}\left(e_{\ell }\left(t\right)\right)}{\left|\right|e_{\ell }\left(t\right){\left|\right|}_1}},\phantom{­}\phantom{­}\ell \in {\mathcal{G}}_{\kappa }\setminus {\breve{\mathcal{G}}}_{\kappa }\phantom{­}\phantom{­}\mathrm{and}\phantom{­}\phantom{­}{\parallel e_{\ell }\left(t\right)\parallel }_1\left(t\right)\ne 0,\hfill \\ & 0,\parallel e_{\ell }{\left(t\right)\parallel }_1=0,\hfill \end{array}\right.\end{array}$$

(38)

with

$$\begin{array}{c}\hfill {\dot{d}}_{\ell }\left(t\right)={\rho }_{\ell }{e}_{\ell }^{\top }\left(t\right)e_{\ell }\left(t\right)-{\displaystyle \frac{{\eta }_{\ell }}{2}}\phantom{­}\mathrm{sgn}\left(d_{\ell }\left(t\right)-d_{\ell }^{*}\right)\phantom{­}|d_{\ell }\left(t\right)-d_{\ell }^{*}|.\end{array}$$

(39)

Corollary 4. 

Under Assumption 1 and the control scheme (38) and (39), if all the conditions in Theorem 2 are met, then the heterogeneous integer-order DN (37) can reach FnT cluster synchronization and the CSST is estimated by $\mathcal{T}\le {\mathcal{T}}_2^{♣}$ with

$${\mathcal{T}}_2^{♣}={\displaystyle \frac{1}{\breve{\mathsf{\Theta }}}}ln\left({\displaystyle \frac{\breve{\mathsf{\Theta }}\tilde{\mathcal{V}}\left(0\right)+{\alpha }^{*}}{{\alpha }^{*}}}\right),$$

where $\tilde{\mathcal{V}}\left(0\right)$, $\breve{\mathsf{\Theta }}$ and ${\alpha }^{*}$ are defined in Theorem 2.

Remark 9. 

In [26,27,28,29,30], some results about cluster synchronization of FODNs have been presented. However, these results are asymptotical ones; namely, its realization can be guaranteed only when the CSST goes to infinity. In contrast, in this paper the cluster synchronization can be achieved no later than ${\mathcal{T}}_1$, which is essential in many real-world applications, such as attitude stabilization of rigid spacecrafts [45] and suppressing chaos in power systems [46]. In this sense, our theoretical results are more applicable than those in [26,27,28,29,30].

Remark 10. 

In [30,35], the FnT cluster synchronization of coupled FO delayed neural networks was studied. However, there exist two limitations in these two works. Firstly, in [30,35], only cooperative interactions exist between nodes within the same group. Moreover, Ref. [35] also impose the restriction that the interactions between nodes across different groups are collaborative. In contrast, co-competition interactions are incorporated into our network model and they are allowed to coexist within the same group or across different groups, which is obviously more realistic [2]. Secondly, the designed controllers in [30,35] are state feedback ones, which may result in overlarge control gains (or control effort) because of the conservatism caused by the theoretical analysis. Differently, in this paper an adaptive control scheme is proposed, where the control gains can be automatically adjusted based on the designed control law to obtain optimal values and thus greatly economize the cost of control. In summary, compared with [30,35], not only are our obtained results more general, but our proposed control approach is also more cost-effective.

Remark 11. 

Considering that there usually exist propagation delays in the implementation of DNs, lag synchronization of DNs has become a fascinating subject in recent years, which can be applied to avoid congestion in practice [47]. For FODNs with community structure, nodes in different groups generally reach lag synchronization with distinct propagation delays, known as cluster lag synchronization [48,49]. By consulting Refs. [48,49], our proposed adaptive control approach can be used to tackle the problem of cluster lag synchronization for the the heterogeneous FODN (1), which will be investigated in our upcoming work.

4. Numerical Simulations

In this part, a numerical example is offered to confirm the correctness of the results derived above. A heterogeneous FODN (1) with 11 nodes and two groups is considered, the topological structure of which is shown in Figure 1. Its dynamics follows

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{x}_{\ell }\left(t\right)=f_{\kappa }(t,x_{\ell }\left(t\right))+\sigma \sum _{p=1}^2\sum _{j\phantom{­}\in {\mathcal{G}}_p}{b}_{\ell j}\mathsf{\Gamma }{x}_j\left(t\right),\phantom{­}\phantom{­}\ell \in {\mathcal{G}}_{\kappa },\end{array}$$

(40)

where $\kappa =1,2,$ ${\mathcal{G}}_1=\{1,2,\dots ,5\}$, and ${\mathcal{G}}_2=\{6,7,\dots ,11\}$. The self-dynamics of isolated nodes in ${\mathcal{G}}_1$ is selected as the following FO Chua’s oscillator [50]:

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{x}_{\ell }\left(t\right)=f_1(t,x_{\ell }\left(t\right))=\left(\begin{array}{cc}& 10\left(x_{\ell 2}\left(t\right)-x_{\ell 1}\left(t\right)-{\check{g}}_1\left(x_{\ell 1}\left(t\right)\right)\right)\hfill \\ & x_{\ell 1}\left(t\right)-x_{\ell 2}\left(t\right)+x_{\ell 3}\left(t\right)\hfill \\ & \phantom{­}\phantom{­}-14.87x_{\ell 2}\left(t\right)\hfill \end{array}\right),\end{array}$$

(41)

where $q=0.98$, ${\check{g}}_1\left(\upsilon \right)=-0.68\upsilon -0.295\left(\right|\upsilon +1|-|\upsilon -1\left|\right)$, and that in ${\mathcal{G}}_2$ is chosen as the following FO neural networks [36]:

$$\begin{array}{c}\hfill {\phantom{­}}_0^C{D}_t^q{x}_{\ell }\left(t\right)=f_2(t,x_{\ell }\left(t\right))=\left(\begin{array}{cc}& -x_{\ell 1}\left(t\right)+2{\check{g}}_2\left(x_{\ell 1}\left(t\right)\right)-1.2{\check{g}}_2\left(x_{\ell 2}\left(t\right)\right)\hfill \\ & -x_{\ell 2}\left(t\right)+1.8{\check{g}}_2\left(x_{\ell 1}\left(t\right)\right)+1.7{\check{g}}_2\left(x_{l\ell 2}\left(t\right)\right)+1.15{\check{g}}_2\left(x_{\ell 3}\left(t\right)\right)\hfill \\ & -x_{\ell 3}\left(t\right)-4.75{\check{g}}_2\left(x_{l\ell 1}\left(t\right)\right)+1.0{\check{g}}_2\left(x_{\ell 3}\left(t\right)\right)\hfill \end{array}\right),\end{array}$$

(42)

where $q=0.98$ and ${\check{g}}_2\left(\upsilon \right)=tanh\left(\upsilon \right)$. Figure 2a,b depicts the phase portraits of systems (41) and (42), respectively. From Figure 2, one can observe that systems (41) and (42) have chaotic behaviors.

Figure 1. Sketch map of a FODN with 11 nodes and two groups, where distinct colored nodes represent they have different self-dynamics and arrows denote the direction of edges. The outer-coupling matrix B is defined as follows: $b_{\ell j}=1$ if there is a solid link from node j to node ℓ, while $b_{\ell j}=-1$ if there is a dashed link from node j to node ℓ.

Figure 2. (a) Phase portrait of system (41) with initial condition ${(0.2,-0.1,-0.1)}^{\top }$; (b) phase portrait of system (42) with initial condition ${(0.1,-0.2,0.3)}^{\top }.$

For the group ${\mathcal{G}}_1$, it is easy to get

$$\begin{array}{ccc}& & {\left(x_{\ell }\left(t\right)-{\pi }_1\left(t\right)\right)}^{\top }\left(f_1(t,x_{\ell }\left(t\right))-f_1(t,{\pi }_1\left(t\right))\right)\hfill \\ & & =\left(e_{\ell 1}\left(t\right),e_{\ell 2}\left(t\right),e_{\ell 3}\left(t\right)\right)\left(\begin{array}{ccc}-3.2\phantom{­}\phantom{­}& 0\phantom{­}\phantom{­}& 0\\ 0\phantom{­}\phantom{­}& -1\phantom{­}\phantom{­}& 0\\ 0\phantom{­}\phantom{­}& 0\phantom{­}\phantom{­}& 0\end{array}\right)\left(\begin{array}{c}{e}_{\ell 1}\left(t\right)\\ e_{\ell 2}\left(t\right)\\ e_{\ell 3}\left(t\right)\end{array}\right)+11e_{\ell 1}\left(t\right)e_{\ell 2}\left(t\right)-13.87e_{\ell 2}\left(t\right)e_{\ell 3}\left(t\right)\hfill \\ & & +2.95e_{\ell 1}\left(t\right)\left(\left(|x_{\ell 1}\left(t\right)+1|-|{\pi }_{11}\left(t\right)+1|\right)+\left(|{\pi }_{11}\left(t\right)-1|-|x_{\ell 1}\left(t\right)-1|\right)\right),\ell \in {\mathcal{G}}_1.\hfill \end{array}$$

 According to the absolute value inequality: $\left|\right|y_1|-|y_2\left|\right|\le |y_1-y_2|$ for all $y_1,y_2\in \mathbb{R}$, we have

$$\begin{array}{ccc}& & 2.95e_{\ell 1}\left(t\right)\left(|x_{\ell 1}\left(t\right)+1|-|{\pi }_{\ell 1}\left(t\right)+1|\right)+2.95e_{\ell 1}\left(t\right)\left(|{\pi }_{\ell 1}\left(t\right)-1|-|x_{\ell 1}\left(t\right)-1|\right)\hfill \\ & & \le 2.95|e_{\ell 1}\left(t\right)\left|\right|x_{\ell 1}\left(t\right)-{\pi }_{11}\left(t\right)|+2.95|e_{\ell 1}\left(t\right)\left|\right|{\pi }_{11}\left(t\right)-x_{\ell 1}\left(t\right)|=5.9e_{\ell 1}^2\left(t\right).\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}& & {\left(x_{\ell }\left(t\right)-{\pi }_1\left(t\right)\right)}^{\top }\left(f_1(t,x_{\ell }\left(t\right))-f_1(t,{\pi }_1\left(t\right))\right)\hfill \\ & & \le \left(e_{\ell 1}\left(t\right),e_{\ell 2}\left(t\right),e_{\ell 3}\left(t\right)\right)\left(\begin{array}{ccc}2.7\phantom{­}\phantom{­}& 0\phantom{­}\phantom{­}& 0\\ 0\phantom{­}\phantom{­}& -1\phantom{­}\phantom{­}& 0\\ 0\phantom{­}\phantom{­}& 0\phantom{­}\phantom{­}& 0\end{array}\right)\left(\begin{array}{c}{e}_{\ell 1}\left(t\right)\\ e_{\ell 2}\left(t\right)\\ e_{\ell 3}\left(t\right)\end{array}\right)+11e_{\ell 1}\left(t\right)e_{\ell 2}\left(t\right)-13.87e_{\ell 2}\left(t\right)e_{\ell 3}\left(t\right)\hfill \\ & & \le \left(e_{\ell 1}\left(t\right),e_{\ell 2}\left(t\right),e_{\ell 3}\left(t\right)\right)\left(\begin{array}{ccc}2.7\phantom{­}\phantom{­}& 0\phantom{­}\phantom{­}& 0\\ 0\phantom{­}\phantom{­}& -1\phantom{­}\phantom{­}& 0\\ 0\phantom{­}\phantom{­}& 0\phantom{­}\phantom{­}& 0\end{array}\right)\left(\begin{array}{c}{e}_{\ell 1}\left(t\right)\\ e_{\ell 2}\left(t\right)\\ e_{\ell 3}\left(t\right)\end{array}\right)+11\left({\displaystyle \frac{{ϵ}_1}{2}}{e}_{\ell 1}^2\left(t\right)+{\displaystyle \frac{1}{2{ϵ}_1}}{e}_{\ell 2}^2\left(t\right)\right)\hfill \\ & & +13.87\left({\displaystyle \frac{{ϵ}_2}{2}}{e}_{\ell 2}^2\left(t\right)+{\displaystyle \frac{1}{2{ϵ}_2}}{e}_{\ell 3}^2\left(t\right)\right),\hfill \end{array}$$

where ${ϵ}_1$ and ${ϵ}_2$ are arbitrarily positive constants. By setting ${ϵ}_1$ = $1.16$ and ${ϵ}_2$ = $0.767$, we obtain $P_1$ = $\mathrm{diag}\left(9.08,9.0605,9.0417\right)$ to satisfy Assumption 1.

For the group ${\mathcal{G}}_2$, based on the Lagrange mean value theorem, one has

$$\begin{array}{ccc}& & {\left(x_{\ell }\left(t\right)-{\pi }_2\left(t\right)\right)}^{\top }\left(f_2(t,x_{\ell }\left(t\right))-f_2(t,{\pi }_2\left(t\right))\right)\hfill \\ & & =-e_{\ell 1}^2\left(t\right)-e_{\ell 2}^2\left(t\right)-e_{\ell 3}^2\left(t\right)+2e_{\ell 1}\left(t\right)\left({\check{g}}_2\left(x_{\ell 1}\left(t\right)\right)-{\check{g}}_2\left({\pi }_{21}\left(t\right)\right)\right)\hfill \\ & & -1.2e_{\ell 1}\left(t\right)\left({\check{g}}_2\left(x_{\ell 2}\left(t\right)\right)-{\check{g}}_2\left({\pi }_{22}\left(t\right)\right)\right)+1.8e_{\ell 2}\left(t\right)\left({\check{g}}_2\left(x_{\ell 1}\left(t\right)\right)-{\check{g}}_2\left({\pi }_{21}\left(t\right)\right)\right)\hfill \\ & & +1.7e_{\ell 2}\left(t\right)\left({\check{g}}_2\left(x_{\ell 2}\left(t\right)\right)-{\check{g}}_2\left({\pi }_{22}\left(t\right)\right)\right)+1.15e_{\ell 2}\left(t\right)\left({\check{g}}_2\left(x_{\ell 3}\left(t\right)\right)-{\check{g}}_2\left({\pi }_{23}\left(t\right)\right)\right)\hfill \\ & & -4.75e_{\ell 3}\left(t\right)\left({\check{g}}_2\left(x_{\ell 1}\left(t\right)\right)-{\check{g}}_2\left({\pi }_{21}\left(t\right)\right)\right)+1.0e_{\ell 3}\left(t\right)\left({\check{g}}_2\left(x_{\ell 3}\left(t\right)\right)-{\check{g}}_2\left({\pi }_{23}\left(t\right)\right)\right)\hfill \\ & & \le e_{\ell 1}^2\left(t\right)+0.7e_{\ell 2}^2\left(t\right)+3.0|e_{\ell 1}\left(t\right)\left|\right|e_{\ell 2}\left(t\right)|+1.15|e_{\ell 2}\left(t\right)\left|\right|e_{\ell 3}\left(t\right)|+4.75|e_{\ell 1}\left(t\right)\left|\right|e_{\ell 3}\left(t\right)|\hfill \\ & & \le \left(1+1.5{ϵ}_3+2.375{ϵ}_5\right)e_{\ell 1}^2\left(t\right)+\left(0.7+{\displaystyle \frac{1.5}{{ϵ}_3}}+0.575{ϵ}_4\right)e_{\ell 2}^2\left(t\right)+\left({\displaystyle \frac{0.575}{{ϵ}_4}}+{\displaystyle \frac{2.375}{{ϵ}_5}}\right)e_{\ell 3}^2\left(t\right),\hfill \end{array}$$

where $\ell \in {\mathcal{G}}_2$ and ${ϵ}_3,{ϵ}_4,{ϵ}_5$ are arbitrarily positive constants. Taking ${ϵ}_3=0.575$ and ${ϵ}_4={ϵ}_5=0.8$, then Assumption 1 can be satisfied with $P_2$ = $\mathrm{diag}\left(3.7625,3.7687,3.6875\right)$.

Denote ${{\rm Y}}_{\kappa }={\lambda }_{max}\left(I_{r_{\kappa }}\otimes {\mathsf{\Xi }}_{\kappa }+{\varpi }_0{I}_{r_{\kappa }}\otimes I_n+{\displaystyle \frac{\sigma }{2}}\left(B_{\kappa \kappa }+B_{\kappa \kappa }^{\top }\right)\otimes \mathsf{\Gamma }\right)$, where $\kappa =1,2$. Let $\sigma =0.2$, $\mathsf{\Gamma }=\mathrm{diag}(2,1.5,1)$ and ${ϵ}_0=0.3536$, then by simple calculations, we have ${\varpi }_0=0.5657$ and ${{\rm Y}}_1=9.6904$, ${{\rm Y}}_2=4.3961$. For simplicity, we set $d_{\ell }^{*}$≡$10.6904$ for $\ell \in {\mathcal{G}}_1$ and $d_{\ell }^{*}$≡$5.3961$ for $\ell \in {\mathcal{G}}_2$, then all the conditions of Theorem 1 are satisfied. In numerical simulations, the parameters of the controller (3) and (4) are chosen as ${\alpha }_{\ell }=1$, ${\rho }_{\ell }=3$, ${\eta }_{\ell }=2$ for ℓ =$1,2,\dots ,11$, and the initial values are given by $x_{\ell }\left(0\right)={(0.5-0.2\ell ,1.2-0.25\ell ,0.15\ell )}^{\top }$ and $d_{\ell }\left(0\right)=10.7$ for $\ell \in {\mathcal{G}}_1$; $x_{\ell }\left(0\right)={(0.2+0.1\ell ,0.3-0.2\ell ,0.1+0.15\ell )}^{\top }$ and $d_{\ell }\left(0\right)=5.5$ for $\ell \in {\mathcal{G}}_2$; ${\pi }_1\left(0\right)={(0.2,-0.1,-0.1)}^{\top }$, ${\pi }_2\left(0\right)={(0.1,-0.2,0.3)}^{\top }$. With the above settings, it can be calculated that $\mathcal{V}\left(0\right)=23.4833$, $\mathsf{\Theta }=2$, ${\alpha }^{*}=1$, and the estimated CSST ${\mathcal{T}}_1=2.1636$.

Let ${\mathcal{E}}_1\left(t\right)=\sqrt{{\sum }_{\ell =1}^5\left|\right|x_{\ell }\left(t\right)-{\pi }_1\left(t\right){\left|\right|}^2}$ and ${\mathcal{E}}_2\left(t\right)=\sqrt{{\sum }_{\ell =6}^{11}\left|\right|x_{\ell }\left(t\right)-{\pi }_2\left(t\right){\left|\right|}^2}$, which stand for the synchronized errors of the groups ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ in system (40), respectively. Figure 3 shows the time evolution of ${\mathcal{E}}_1\left(t\right)$ and ${\mathcal{E}}_2\left(t\right)$ for system (40) without control, which indicates the heterogeneous FODN (40) can not reach cluster synchronization spontaneously. Figure 4 and Figure 5 depict, respectively, the time evolution of the state vectors $x_{\ell }\left(t\right)$ and the adaptive control gains $d_{\ell }\left(t\right)$ for system (40) under the controller (3) and (4), where $\ell =1,2,\dots ,11$. Figure 6 portrays the time evolution of ${\mathcal{E}}_1\left(t\right)$ and ${\mathcal{E}}_2\left(t\right)$ for system (40) under the controller (3) and (4). These figures clearly illustrate cluster synchronization is reached within ${\mathcal{T}}_1=2.1636$, and the feasibility of our theoretical results is further demonstrated.

Figure 3. Time evolution of ${\mathcal{E}}_1\left(t\right)$ and ${\mathcal{E}}_2\left(t\right)$ for system (40) without control.

Figure 4. Time evolution of the state vectors $x_{\ell }\left(t\right)$ for system (40) under the controller (3) and (4), where $\ell =1,2,\dots ,11$.

Figure 5. Time evolution of the adaptive control gains $d_{\ell }\left(t\right)$ for system (40) under the controller (3) and (4), where $\ell =1,2,\dots ,11$.

Figure 6. Time evolution of ${\mathcal{E}}_1\left(t\right)$ and ${\mathcal{E}}_2\left(t\right)$ for system (40) under the controller (3) and (4).

In order to reveal the influences of the order of the derivative q and the parameters $\mathsf{\Theta }$, ${\alpha }^{*}$ on the convergence time of cluster synchronization, we first keep other parameters unchanged and vary q from 0.6 to 1 and plot the relation curve between the estimated CSST ${\mathcal{T}}_1$ and q in Figure 7, where the blue curve depicts the values of ${\mathcal{T}}_1$ with different q. From Figure 7, we can observe that ${\mathcal{T}}_1$ decreases with increasing q, suggesting the larger the q is, the smaller the CSST is, namely that the shorter the convergence time to cluster synchronization is. Next, keeping other parameters fixed, let $\mathsf{\Theta }$ vary within a proper range, and the corresponding ${\mathcal{T}}_1$ is determined. As shown graphically in Figure 8, a bigger $\mathsf{\Theta }$ yields a smaller ${\mathcal{T}}_1$, which results in reaching FnT cluster synchronization sooner. Finally, we let the parameter ${\alpha }^{*}$ vary from 1 to 25 and remain other parameters unchanged. By calculating the corresponding ${\mathcal{T}}_1$, from Figure 9, we can see that a smaller ${\mathcal{T}}_1$ could be obtained with a bigger ${\alpha }^{*}$, manifesting the convergence time to cluster synchronization becomes shorter as the parameters ${\alpha }_{\ell }$ ($\ell =1,2,\dots ,11$) increase. That is to say, the FnT cluster synchronization can be achieved more quickly with the increasing of ${\alpha }_{\ell }$ ($\ell =1,2,\dots ,11$).

Figure 7. The relation curve between the estimated CSST ${\mathcal{T}}_1$ and the order of the derivative q.

Figure 8. The relation curve between estimated CSST ${\mathcal{T}}_1$ and the parameter $\mathsf{\Theta }$.

Figure 9. The relation curve between estimated CSST ${\mathcal{T}}_1$ and the parameter ${\alpha }^{*}$.

5. Conclusions

In this paper, in order to better characterize the feature of real-world DNs, a heterogeneous FODN with community structure and co-competition interactions is first proposed. Then, an adaptive controller is designed and applied to steer such a heterogeneous FODN towards reaching cluster synchronization over a FnT interval. Based on the Lyapunov method and reduction to absurdity, some generic FnT cluster synchronization criteria are derived and the CSSTs are determined theoretically. In addition, the effects of the order of the derivative and the control parameters on the CSST are also analyzed in detail. Finally, numerical simulations are given to validate the theoretical results.

It is noted that, for realizing FnT cluster synchronization, the proposed adaptive controller should be added to all network nodes. For large-scale DNs, however, this strategy is not easy to implement. Hence, it is meaningful to discuss the problem of FnT cluster synchronization for FODNs with community structure and co-competition interactions by applying the designed adaptive controller to partial network nodes, which will be our future research.