Mittag–Leffler Synchronization of Caputo-Delayed Quaternion BAM Neural Networks via Adaptive and Linear Feedback Control Designs

Abstract

The Mittag–Leffler synchronization (MLS) issue for Caputo-delayed quaternion bidirectional associative memory neural networks (BAM-NNs) is studied in this paper. Firstly, a novel lemma is proved by the Laplace transform and inverse transform. Then, without decomposing a quaternion system into subsystems, the adaptive controller and the linear controller are designed to realize MLS. According to the proposed lemma, constructing two different Lyapunov functionals and applying the fractional Razumikhin theorem and inequality techniques, the sufficient criteria of MLS on fractional delayed quaternion BAM-NNs are derived. Finally, two numerical examples are given to illustrate the validity and practicability.

1. Introduction

In 1987, Kosko first proposed BAM-NNs [1,2] playing a significant role in NNs theory. BAM-NNs are double-layer bidirectional networks, which consist of several groups of typical input and output vectors to form the weight matrix [3]. When an input signal is added to one layer, the other layer can obtain the output. Since the initial mode can act on any layer of the network, information can spread in both directions, and there is no clear input layer or output layer. Therefore, the dynamics of BAM-NNs have attracted extensive attention from many scholars. In [4], according to inequality technique and fractional calculus theory, the sufficient conditions for finite-time stabilization on fractional fuzzy BAM-NNs are derived. In [5], by Homomorphic mapping theorem and Lyapunov function, the Mittag–Leffler stability criteria of impulsive Caputo complex-valued delayed BAM-NNs are obtained. In [6], the quaternion system is decomposed into eight real-valued systems, and Lyapunov Krasovskii functions are constructed, in which MLS criteria for fractional quaternion BAM-NNs are established by using inequality techniques.

Compared with classical calculus, fractional calculus has better genetic and memory properties, and more degrees of freedom in studying electric circuits [7], viscoelastic materials [8], hydrodynamics [9], biological systems [10,11] and so on, which can accurately describe the signal transmission relationship between neurons of NNs [12,13,14,15]. Fractional-order NNs can reveal the dynamical behavior of NNs in [4,5,6,12,13,14,15].

In fact, due to the limitation of information transmission speed and signal processing between neurons, NNs usually result in oscillation, bifurcation, and chaos. The studies of dynamical systems have been reported including discrete delays [16], time-varying delays [17], leakage delays [18,19], distributed delays [20,21,22,23], and so on. For various reasons, time delays are also inevitable in the synchronization. Therefore, the research on dynamical systems with delays is of great significance in the field of theory and applications. However, in the existing references, there are few studies on the Caputo quaternion BAM-NNs with delays.

Synchronization is an interesting and important phenomenon in networks. Due to its practical applications and important theoretical significance in digital cryptography [24], signal processing [25], secure communication [26], and other fields, more and more scholars are interested in synchronization phenomena, such as quasi-uniform synchronization [18,27], projective synchronization [13,28], finite-time synchronization [26,29], MLS [5,6,13,23,30,31], asymptotic synchronization [32], phase synchronization [33], exponential synchronization [34], and dissipative synchronization [35]. Particularly, in the literature of MLS [5,6,13,23,30,31], such factors of the models have not been simultaneously considered including time delay, quaternion, and BAM. In various synchronization schemes, the controllers with sign function are generally used in [36]. Obviously, this class of controllers is inconvenient and undesirable in applications.

Different from the Homomorphic mapping theorem [5], decomposition method [6], linear matrix inequality (LMI) [21], differential inclusion [27,32,37], and comparison principle [38], a new Caputo differential inequality is constructed by Laplace transform; thus, an algebraic inequality resulting in Lemma 5 is derived, which is very convenient to derive MLS of Caputo fractional delayed quaternion BAM-NNs. This paper mainly studies the MLS for fractional delayed quaternion BAM-NNs, according to the new proposed lemma, Lyapunov functional, and inequality techniques, the sufficient criteria of MLS are derived.

The highlights and innovations of this paper are presented as follows:

• The impacts of delay, quaternion and BAM on MLS are simultaneously considered, then the discussed models are more general.
• Without decomposing, the quaternion-value is regarded as a compact whole, which reduces the complexity of calculation and the difficulty of theoretical analysis.
• A new lemma is proved by the Laplace transform, and MLS criteria of fractional delayed quaternion BAM-NNs are obtained by using this new lemma.
• The adaptive feedback and linear feedback controllers are designed, and the corresponding Lyapunov functionals are constructed, which can improve control efficiency and reduce control cost.

2. Preliminaries

In handling fractional-order electrical circuit issues, the Caputo derivative is very efficient in theoretical analysis and numerical simulations, and provides a powerful tool to explore fractional-order circuit systems theory and applications (see [7] and references therein). In this section, the related definitions, lemmas and Caputo-delayed quaternion BAM-NNs are introduced.

Definition 1

([39]). The Riemann-Liouville’s integral for fractional $\gamma >0$ of $v(·)\in \mathbf{C}\left[\left[0,+\infty \right),\mathbb{R}\right]$ is

$$\begin{array}{c}\hfill {}_0{D}_t^{-\gamma }v\left(s\right)=\frac{1}{\mathsf{\Gamma }\left(\gamma \right)}{\int }_0^t{(s-t)}^{\gamma -1}v\left(t\right)dt.\end{array}$$

Definition 2

([39]). The Caputo derivative of fractional-order $\gamma \in (k-1,k)$ for $v(·)\in {\mathbf{C}}^k$$([t_0,+\infty ),\mathbb{R})$ is defined as

$$\begin{array}{c}\hfill {}_{t_0}^C{D}_t^{\gamma }v\left(s\right)=\frac{1}{\mathsf{\Gamma }(k-\gamma )}{\int }_{t_0}^t\frac{v^{\left(k\right)}\left(\tau \right)}{{(s-\tau )}^{\gamma -k+1}}d\tau ,t⩾t_0.\end{array}$$

Lemma 1

([40]). The Laplace transform of fractional derivative is

$$\begin{array}{c}\hfill L\left\{{}_0{D}_t^{\gamma }f\left(s\right)\right\}=t^{\gamma }F\left(t\right)-\sum _{\xi =0}^{j-1}{t}^{\gamma -\xi -1}{f}^{\left(\xi \right)}\left(0\right),\gamma \in (j-1,j).\end{array}$$

Lemma 2

([41]). The Laplace transform of the Mittag–Leffler function is

$$\begin{array}{c}\hfill L\left\{s^{u\xi +v-1}{E}_{u,v}^{\left(\xi \right)}(\pm a{s}^u)\right\}=\frac{\xi !t^{u-v}}{{(t^u\mp a)}^{\xi +1}}.\end{array}$$

Lemma 3

([38]). If $v(·)\in \mathbb{Q}$ is a continuous analytic function, then

$$\begin{array}{c}\hfill {}_{t_0}^C{D}_t^{\gamma }v\left(\varsigma \right)\overline{v\left(\varsigma \right)}⩽v\left(\varsigma \right){}_{t_0}^C{D}_t^q\overline{v\left(\varsigma \right)}+\overline{v\left(\varsigma \right)}{}_{t_0}^C{D}_t^{q}v\left(\varsigma \right),\end{array}$$

where $t⩾t_0,0<\gamma <1,$$\mathbb{Q}$ denotes the quaternion field.

Lemma 4

([38]). For any $\alpha ,\beta \in \mathbb{Q}$, if there exists $\eta >0$, then the inequality holds:

$$\begin{array}{c}\hfill \alpha \overline{\beta }+\overline{\alpha }\beta ⩽\eta \alpha \overline{\alpha }+\frac{1}{\eta }\beta \overline{\beta }.\end{array}$$

Lemma 5.

Suppose that $H(·)$ and $U(·)$ are nonnegative continuous functions. If

$${}_{t_0}^C{D}_t^{\gamma }\left[H\left(t\right)+U\left(t\right)\right]⩽-\sigma H\left(t\right),0<\gamma <1,\sigma >0,$$

(1)

then

$$\begin{array}{c}\hfill H\left(t\right)⩽\left[H\left(t_0\right)+U\left(t_0\right)\right]E_{\gamma }\left[-\sigma {(t-t_0)}^{\gamma }\right].\end{array}$$

Proof. 

According to inequality (1), there exists the function $\omega \left(t\right)⩾0$, such that

$$\begin{array}{c}\hfill {}_{t_0}^C{D}_t^{\gamma }\left[H\left(t\right)+U\left(t\right)\right]+\omega \left(t\right)=-\sigma H\left(t\right).\end{array}$$

From Lemma 1, we have

$$\begin{array}{c}\hfill s^{\gamma }\left[H\left(s\right)+U\left(s\right)\right]-s^{\gamma -1}\left[H\left(t_0\right)+U\left(t_0\right)\right]+\omega \left(s\right)=-\sigma H\left(s\right).\end{array}$$

Then, we obtain

$$\begin{array}{c}\hfill H\left(s\right)=\frac{s^{\gamma -1}\left[H\left(t_0\right)+U\left(t_0\right)\right]-\omega \left(s\right)-s^{\gamma }U\left(s\right)}{s^{\gamma }+\sigma }.\end{array}$$

It is known from Lemma 2 that

$$\begin{array}{c}\hfill H\left(t\right)⩽\left[H\left(t_0\right)+U\left(t_0\right)\right]E_{\gamma }\left[-\sigma {\left(t-t_0\right)}^{\gamma }\right].\end{array}$$

Then, the proof of Lemma 5 is completed. □

Consider the following fractional delayed quaternion BAM-NNs:

$$\begin{array}{cc}\hfill & {}_{t_0}^C{D}_t^{\gamma }{p}_{\tau i}\left(t\right)=-c_i{p}_{\tau i}\left(t\right)+\sum _{\hslash =1}^n{a}_{i\hslash }{z}_{\hslash }\left(q_{v\hslash }\left(t\right)\right)+\sum _{\hslash =1}^n{d}_{i\hslash }{z}_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)+U_i\left(t\right),\hfill \\ \hfill & {}_{t_0}^C{D}_t^{\gamma }{q}_{v\hslash }\left(t\right)=-c_{\hslash }{q}_{v\hslash }\left(t\right)+\sum _{i=1}^m{\tilde{a}}_{\hslash i}{\ell }_i\left(p_{\tau i}\left(t\right)\right)+\sum _{i=1}^m{\tilde{d}}_{\hslash i}{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)+J_{\hslash }\left(t\right),\hfill \end{array}$$

(2)

where $\hslash =1,2,\cdots n$, $0<\gamma <1$, $i=1,2,\cdots m$; $c_i$ and $c_{\hslash }$ are positive constants, $p_{\tau i}(·)=\left(p_{\tau 1}(·),p_{\tau 2}(·),\cdots \right.,\left.p_{\tau m}(·)\right)\in {\mathbb{Q}}^m$ and $q_{v\hslash }(·)=(q_{v1}(·),q_{v2}(·),\cdots ,q_{vn}(·))\in {\mathbb{Q}}^n$ refer to the state variables of layer-i and layer-ℏ, respectively; $a_{i\hslash },d_{i\hslash },{\tilde{a}}_{\hslash i},{\tilde{d}}_{\hslash i}$ are connection weights, ${\varrho }_{\tau }$ and ${\varrho }_v$ are time delays, and $\varrho =max\left\{{\varrho }_{\tau },{\varrho }_v\right\}$; $U_i\left(t\right)$ and $J_{\hslash }\left(t\right)$ represent external input bias, system (2) is the master system; $z_{\hslash }\left(q_{v\hslash }\left(t\right)\right),{\ell }_i\left(p_{\tau i}\left(t\right)\right),z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right),{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)$ denote the activation functions satisfying

$$\begin{array}{c}\hfill ∥z\left(t_1\right)-z\left(t_2\right)∥⩽M∥t_1-t_2∥,∥l\left(t_1\right)-l\left(t_2\right)∥⩽N∥t_1-t_2∥,\end{array}$$

where $M>0,N>0$ are Lipschitz constants.

The initial conditions to system (2) are chosen as $p_{\tau i}\left(s\right)={\phi }_{\tau i}^Q\left(s\right)$, $q_{v\hslash }\left(s\right)={\varphi }_{v\hslash }^Q\left(s\right)$, where $s\in \left[-\varrho ,0\right)$.

The slave system of system (2) is

$$\begin{array}{cc}\hfill & {}_{t_0}^C{D}_t^{\gamma }{\overline{p}}_{\tau i}\left(t\right)=-c_i{\overline{p}}_{\tau i}\left(t\right)+\sum _{\hslash =1}^n{a}_{i\hslash }{z}_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)+\sum _{\hslash =1}^n{d}_{i\hslash }{z}_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)+U_i\left(t\right)+I_i\left(t\right),\hfill \\ \hfill & {}_{t_0}^C{D}_t^{\gamma }{\overline{q}}_{v\hslash }\left(t\right)=-c_{\hslash }{\overline{q}}_{v\hslash }\left(t\right)+\sum _{i=1}^m{\tilde{a}}_{\hslash i}{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)+\sum _{i=1}^m{\tilde{d}}_{\hslash i}{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)+J_{\hslash }\left(t\right)+I_{\hslash }\left(t\right),\hfill \end{array}$$

(3)

where $I_i(·)$ and $I_{\hslash }(·)$ are external controllers; ${\overline{p}}_{\tau i}(·)\in {\mathbb{Q}}^m$ and ${\overline{q}}_{v\hslash }(·)\in {\mathbb{Q}}^n$ are the state variables.

Let $w_{\tau i}(·)={\overline{p}}_{\tau i}(·)-p_{\tau i}(·)$, $w_{v\hslash }(·)={\overline{q}}_{v\hslash }(·)-q_{v\hslash }(·)$, then the error systems are described as

$$\begin{array}{cc}\hfill {}_{t_0}^C{D}_t^{\gamma }{w}_{\tau i}\left(t\right)=& -c_i{w}_{\tau i}\left(t\right)+\sum _{\hslash =1}^n{a}_{i\hslash }\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]+\sum _{\hslash =1}^n{d}_{i\hslash }\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]+I_i\left(t\right),\hfill \\ \hfill {}_{t_0}^C{D}_t^{\gamma }{w}_{v\hslash }\left(t\right)=& -c_{\hslash }{w}_{v\hslash }\left(t\right)+\sum _{i=1}^m{\tilde{a}}_{\hslash i}\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]+\sum _{i=1}^m{\tilde{d}}_{\hslash i}\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]+I_{\hslash }\left(t\right).\hfill \end{array}$$

(4)

Definition 3

([42]). Under the initial states $w_{\tau }\left(0\right)$ and $w_v\left(0\right)$ to system (4), if there exist two constants ${\delta }_1$ and ${\delta }_2$, such that

$$\begin{array}{c}\hfill ∥w_{\tau }\left(t\right)∥+∥w_v\left(t\right)∥⩽{\left\{\left[\phi \left(w_{\tau }\left(0\right)\right)+\varphi \left(w_v\left(0\right)\right)\right]E_{\gamma }(-{\delta }_1{t}^{\gamma })\right\}}^{{\delta }_2},\end{array}$$

where $t⩾0$, $\phi \left(w_{\tau }\left(0\right)\right)⩾0$, $\varphi \left(w_v\left(0\right)\right)⩾0$, $\phi \left(0\right)=0$ and $\varphi \left(0\right)=0$, then systems (2) and (3) achieve MLS.

Remark 1.

In the literature on MLS, the impacts of time delay, quaternion and BAM on MLS have not been simultaneously considered. For example, MLS issue for quaternion BAM-NNs without delay was discussed in [6,43]. MLS problems of fractional NNs were discussed in [5,30,36], but the number fields are concerned with real-value or complex-value fields. In contrast to the existing results, system (2) is more general.

3. Main Results

In this section, the MLS criteria for systems (2) and (3) are proposed by using the proposed Lemma 5, Lyapunov functional approach, fractional Razumikhin theorem and inequality technique.

The following adaptive feedback controllers is designed

$$\begin{array}{c}\hfill I_i\left(t\right)=-{\eta }_i\left(t\right)w_{\tau i}\left(t\right),I_{\hslash }\left(t\right)=-{\kappa }_{\hslash }\left(t\right)w_{v\hslash }\left(t\right),\end{array}$$

(5)

where ${}_{t_0}^C{D}_t^{\gamma }{\eta }_i\left(t\right)={\rho }_i{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}$, ${}_{t_0}^C{D}_t^{\gamma }{\kappa }_{\hslash }\left(t\right)={\pi }_{\hslash }{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}$, ${\rho }_i>0,$${\pi }_{\hslash }>0$ are constants, ${\eta }_i\left(t_0\right)=0,{\kappa }_{\hslash }\left(t_0\right)=0.$ Figure 1 represents the control framework of adaptive feedback controller (5).

Theorem 1.

For $1⩽i⩽m$ and $1⩽\hslash ⩽n$, if there exist constants $\xi >1$, ${\eta }_i\ast >0$ and ${\kappa }_{\hslash }\ast >0$, such that $\lambda >\varpi \xi ,$ then systems (2) and (3) realize MLS under controller (5), where $\lambda =min\left\{{\lambda }_i,{\lambda }_{\hslash }\right\},$ and

$$\begin{array}{cc}\hfill {\lambda }_i=& c_i+\overline{c_i}+2{\eta }_i\ast -N^2-\sum _{\hslash =1}^n\left[a_{i\hslash }\overline{a_{i\hslash }}+d_{i\hslash }\overline{d_{i\hslash }}\right]>0,\hfill \\ \hfill {\lambda }_{\hslash }=& c_{\hslash }+\overline{c_{\hslash }}+2{\kappa }_{\hslash }\ast -M^2-\sum _{i=1}^m\left[{\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}+{\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}\right]>0,\hfill \\ \hfill \varpi =& max\left\{m{M}^2,n{N}^2\right\}.\hfill \end{array}$$

Proof. 

Selecting the Lyapunov function $V\left(t\right)=V_1\left(t\right)+V_2\left(t\right),$ where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & V_1\left(t\right)=\sum _{i=1}^m{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+\sum _{\hslash =1}^n{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)},\hfill \\ \hfill & V_2\left(t\right)=\sum _{i=1}^m\frac{1}{{\rho }_i}{[{\eta }_i\left(t\right)-{\eta }_i\ast ]}^2+\sum _{\hslash =1}^n\frac{1}{{\pi }_{\hslash }}{[{\kappa }_{\hslash }\left(t\right)-{\kappa }_{\hslash }\ast ]}^2.\hfill \end{array}\end{array}$$

From Lemma 3, we obtain

$$\begin{array}{ll}{}_{t_0}^C{D}_t^{\gamma }V\left(t\right)⩽& {\displaystyle \sum _{i=1}^m}\left[w_{\tau i}\left(t\right){}_{t_0}^C{D}_t^{\gamma }\overline{w_{\tau i}\left(t\right)}+\overline{w_{\tau i}\left(t\right)}{}_{t_0}^C{D}_t^{\gamma }{w}_{\tau i}\left(t\right)\right]+\sum _{\hslash =1}^n\left[w_{v\hslash }\left(t\right){}_{t_0}^C{D}_t^{\gamma }\overline{w_{v\hslash }\left(t\right)}+\overline{w_{v\hslash }\left(t\right)}{}_{t_0}^C{D}_t^{\gamma }{w}_{v\hslash }\left(t\right)\right]\\ & +\sum _{i=1}^m\frac{2}{{\rho }_i}({\eta }_i\left(t\right)-{\eta }_i\ast ){}_{t_0}^C{D}_t^{\gamma }{\eta }_i\left(t\right)++\sum _{\hslash =1}^n\frac{2}{{\pi }_{\hslash }}({\kappa }_{\hslash }\left(t\right)-{\kappa }_{\hslash }\ast ){}_{t_0}^C{D}_t^{\gamma }{\kappa }_{\hslash }\left(t\right)\\ & ⩽-\sum _{i=1}^m\left(c_i+\overline{c_i}+2{\eta }_i\ast \right)w_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+\sum _{i=1}^m\left\{\sum _{\hslash =1}^n\overline{a_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]}\right.\\ & +\sum _{\hslash =1}^n\overline{d_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]}+\sum _{\hslash =1}^n{a}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]\\ & \left.+\sum _{\hslash =1}^n{d}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]\right\}\\ & -\sum _{\hslash =1}^n\left(c_{\hslash }+\overline{c_{\hslash }}+2{\kappa }_{\hslash }\ast \right)w_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}+\sum _{\hslash =1}^n\left\{\sum _{i=1}^m\overline{{\tilde{a}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]}\right.\\ & +\sum _{i=1}^m\overline{{\tilde{d}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]}+\sum _{i=1}^m{\tilde{a}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]\\ & \left.+\sum _{i=1}^m{\tilde{d}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]\right\}.\end{array}$$

An application of Lemma 4 yields that

$$\begin{array}{cc}\hfill & \sum _{\hslash =1}^n\overline{a_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]}+\sum _{\hslash =1}^n{a}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]\hfill \\ \hfill ⩽& \sum _{\hslash =1}^n\left(a_{i\hslash }\overline{a_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}\right.+\left.\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]\right)\hfill \\ \hfill ⩽& \sum _{\hslash =1}^n\left[a_{i\hslash }\overline{a_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+M^2{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}\right].\hfill \end{array}$$

(6)

Similarly,

$$\begin{array}{cc}\hfill & \sum _{i=1}^m\overline{{\tilde{a}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]}+\sum _{i=1}^m{\tilde{a}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]\hfill \\ \hfill ⩽& \sum _{i=1}^m\left[{\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}+N^2{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}\right],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \sum _{\hslash =1}^n\overline{d_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]}+\sum _{\hslash =1}^n{d}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]\hfill \\ \hfill ⩽& \sum _{\hslash =1}^n\left[d_{i\hslash }\overline{d_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+M^2{w}_{v\hslash }(t-{\varrho }_v)\overline{w_{v\hslash }(t-{\varrho }_v)}\right],\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \sum _{i=1}^m\overline{{\tilde{d}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]}+\sum _{i=1}^m{\tilde{d}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]\hfill \\ \hfill ⩽& \sum _{i=1}^m{\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}+N^2{w}_{\tau i}(t-{\varrho }_{\tau })\overline{w_{\tau i}(t-{\varrho }_{\tau })}.\hfill \end{array}$$

(9)

Substituting (6)–(9) into (6), we have

$$\begin{array}{cc}\hfill {}_{t_0}^C{D}_t^{\gamma }V\left(t\right)⩽& -\sum _{i=1}^m\left[c_i+\overline{c_i}+2{\eta }_i\ast -N^2-\sum _{\hslash =1}^n\left(a_{i\hslash }\overline{a_{i\hslash }}+d_{i\hslash }\overline{d_{i\hslash }}\right)\right]w_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}\hfill \\ \hfill & -\sum _{\hslash =1}^n\left[c_{\hslash }+\overline{c_{\hslash }}+2{\kappa }_{\hslash }\ast -M^2-\sum _{i=1}^m\left({\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}+{\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}\right)\right]w_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}\hfill \\ \hfill & +m\sum _{\hslash =1}^n{M}^2{w}_{v\hslash }\left((t-{\varrho }_v)\right)\overline{w_{v\hslash }\left((t-{\varrho }_v)\right)}+n\sum _{i=1}^m{N}^2{w}_{\tau i}(t-{\varrho }_{\tau })\overline{w_{\tau i}(t-{\varrho }_{\tau })}.\hfill \end{array}$$

By fractional Razumikhin theorem [44], we have

$$\begin{array}{cc}\hfill {}_{t_0}^C{D}_t^{\gamma }V\left(t\right)⩽& -\sum _{i=1}^m{\lambda }_i{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}-\sum _{\hslash =1}^n{\lambda }_{\hslash }{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}\hfill \\ \hfill & +\varpi \left[\sum _{\hslash =1}^n{w}_{v\hslash }\left((t-\varrho )\right)\overline{w_{v\hslash }\left((t-\varrho )\right)}+\sum _{i=1}^m{w}_{\tau i}(t-\varrho )\overline{w_{\tau i}(t-\varrho )}\right]\hfill \\ \hfill ⩽& -\lambda V_1\left(t\right)+\varpi V_1(t-\varrho )\hfill \\ \hfill ⩽& -\left(\lambda -\varpi \xi \right)V_1\left(t\right).\hfill \end{array}$$

From Lemma 5, we can know that

$$\begin{array}{c}\hfill V_1\left(t\right)⩽\left[\sum _{i=1}^m{w}_{\tau i}\left(t_0\right)\overline{w_{\tau i}\left(t_0\right)}+\sum _{\hslash =1}^n{w}_{v\hslash }\left(t_0\right)\overline{w_{v\hslash }\left(t_0\right)}\right]E_{\gamma }(-(\lambda -\varpi \xi ){(t-t_0)}^{\gamma }).\end{array}$$

Moreover, the following inequality also holds:

$$\begin{array}{c}\hfill ∥w_{\tau i}\left(t\right)∥+∥w_{v\hslash }\left(t\right)∥⩽{\left\{2\left[{∥w_{\tau i}\left(t_0\right)∥}^2+{∥w_{v\hslash }\left(t_0\right)∥}^2\right]E_{\gamma }(-(\lambda -\varpi \xi ){(t-t_0)}^{\gamma })\right\}}^{\frac{1}{2}}.\end{array}$$

Then, systems (2) and (3) can achieve MLS. □

Remark 2.

In the existing literature, many results have been obtained by using adaptive controllers in [38,42,44]. For example, Li et al. [38] discussed the global synchronization of fractional quaternion NNs with leakage and discrete delay under adaptive controller. In [42], the adaptive projective synchronization of Caputo quaternion delayed NNs was studied. However, MLS of Caputo quaternion BAM-NNs with time delay under adaptive control has not been studied.

Next, the linear feedback controllers is proposed

$$\begin{array}{c}\hfill I_i\left(t\right)=-{\eta }_i{w}_{\tau i}\left(t\right),I_{\hslash }\left(t\right)=-{\kappa }_{\hslash }{w}_{v\hslash }\left(t\right),\end{array}$$

(10)

where ${\eta }_i\in \mathbb{Q},{\kappa }_{\hslash }\in \mathbb{Q}$ are control gains. Figure 2 represents the control framework of linear feedback controller (10).

Theorem 2.

For $1⩽i⩽m$ and $1⩽\hslash ⩽n$, if there exist ${\xi }^{\prime }>1$, such that $\chi >{\varpi }^{\prime }{\xi }^{\prime },$ then systems (2) and (3) achieve MLS under controller (10), where $\chi =min\left\{{\chi }_i,{\chi }_{\hslash }\right\},$${\chi }_i>0,$${\chi }_{\hslash }>0,$ and

$$\begin{array}{cc}\hfill {\chi }_i=& c_i+\overline{c_i}+{\eta }_i+\overline{{\eta }_i}-\sum _{\hslash =1}^n\left[{\left(a_{i\hslash }\overline{a_{i\hslash }}\right)}^{\frac{1}{2}}+{\left(d_{i\hslash }\overline{d_{i\hslash }}\right)}^{\frac{1}{2}}+N^2{\left({\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}\right)}^{\frac{1}{2}}\right],\hfill \\ \hfill {\chi }_{\hslash }=& c_{\hslash }+\overline{c_{\hslash }}+{\kappa }_{\hslash }+\overline{{\kappa }_{\hslash }}-\sum _{i=1}^m\left[{\left({\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}\right)}^{\frac{1}{2}}+{\left({\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}\right)}^{\frac{1}{2}}+M^2{\left(a_{i\hslash }\overline{a_{i\hslash }}\right)}^{\frac{1}{2}}\right],\hfill \\ \hfill {\varpi }^{\prime }=& max\left\{M^2{\left(d_{i\hslash }\overline{d_{i\hslash }}\right)}^{\frac{1}{2}},N^2{\left({\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}\right)}^{\frac{1}{2}}\right\}.\hfill \end{array}$$

Proof. 

Selecting the Lyapunov function

$$\begin{array}{c}\hfill \overline{V}\left(t\right)=\sum _{i=1}^m{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+\sum _{\hslash =1}^n{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}.\end{array}$$

From Lemma 3, we have

$$\begin{array}{cc}\hfill {}_{t_0}^C{D}_t^{\gamma }\overline{V}\left(t\right)⩽& \sum _{i=1}^m\left[w_{\tau i}\left(t\right){}_{t_0}^C{D}_t^{\gamma }\overline{w_{\tau i}\left(t\right)}+\overline{w_{\tau i}\left(t\right)}{}_{t_0}^C{D}_t^{\gamma }{w}_{\tau i}\left(t\right)\right]+\sum _{\hslash =1}^n\left[w_{v\hslash }\left(t\right){}_{t_0}^C{D}_t^{\gamma }\overline{w_{v\hslash }\left(t\right)}+\overline{w_{v\hslash }\left(t\right)}{}_{t_0}^C{D}_t^{\gamma }{w}_{v\hslash }\left(t\right)\right]\hfill \\ \hfill ⩽& -\sum _{i=1}^m\left(c_i+\overline{c_i}+{\eta }_i+\overline{{\eta }_i}\right)w_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+\sum _{i=1}^m\left\{\sum _{\hslash =1}^n\overline{a_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]}\right.\hfill \\ \hfill & +\sum _{\hslash =1}^n\overline{d_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]}+\sum _{\hslash =1}^n{a}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]\hfill \\ \hfill & \left.+\sum _{\hslash =1}^n{d}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]\right\}\hfill \\ \hfill & -\sum _{\hslash =1}^n\left(c_{\hslash }+\overline{c_{\hslash }}+{\kappa }_{\hslash }+\overline{{\kappa }_{\hslash }}\right)w_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}+\sum _{\hslash =1}^n\left\{\sum _{i=1}^m\overline{{\tilde{a}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]}\right.\hfill \\ \hfill & +\sum _{i=1}^m\overline{{\tilde{d}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]}+\sum _{i=1}^m{\tilde{a}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]\hfill \\ \hfill & \left.+\sum _{i=1}^m{\tilde{d}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]\right\}.\hfill \end{array}$$

(11)

Applications of Lipschitz conditions to activation functions obtain that

$$\begin{array}{cc}\hfill & \sum _{\hslash =1}^n\overline{a_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]}+\sum _{\hslash =1}^n{a}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right]\hfill \\ \hfill ⩽& 2\sum _{\hslash =1}^n{\left(a_{i\hslash }\overline{a_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}\right)}^{{}^{\frac{1}{2}}}{\left[\left(z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right)\overline{\left(z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right)}\right]}^{\frac{1}{2}}\hfill \\ \hfill ⩽& 2\sum _{\hslash =1}^n{\left(a_{i\hslash }\overline{a_{i\hslash }}\right)}^{\frac{1}{2}}\left[\frac{1}{2}\left(w_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}\right)+\frac{1}{2}\left(z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right)\overline{\left(z_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)-z_{\hslash }\left(q_{v\hslash }\left(t\right)\right)\right)}\right]\hfill \\ \hfill ⩽& \sum _{\hslash =1}^n{\left[a_{i\hslash }\overline{a_{i\hslash }}\right]}^{\frac{1}{2}}\left[w_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+M^2{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}\right].\hfill \end{array}$$

(12)

Similarly,

$$\begin{array}{cc}\hfill & \sum _{i=1}^m\overline{{\tilde{a}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]}+\sum _{i=1}^m{\tilde{a}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)-{\ell }_i\left(p_{\tau i}\left(t\right)\right)\right]\hfill \\ \hfill ⩽& \sum _{i=1}^m{\left({\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}\right)}^{\frac{1}{2}}\left[w_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}+N^2{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}\right],\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \sum _{\hslash =1}^n\overline{d_{i\hslash }}{w}_{\tau i}\left(t\right)\overline{\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]}+\sum _{\hslash =1}^n{d}_{i\hslash }\overline{w_{\tau i}\left(t\right)}\left[z_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)-z_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)\right]\hfill \\ \hfill ⩽& \sum _{\hslash =1}^n{\left(d_{i\hslash }\overline{d_{i\hslash }}\right)}^{\frac{1}{2}}\left[w_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}+M^2{w}_{v\hslash }[t-{\varrho }_v)\overline{w_{v\hslash }(t-{\varrho }_v)}\right],\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \sum _{\hslash =1}^n\overline{{\tilde{d}}_{\hslash i}}{w}_{v\hslash }\left(t\right)\overline{\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]}+\sum _{\hslash =1}^n{\tilde{d}}_{\hslash i}\overline{w_{v\hslash }\left(t\right)}\left[{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)-{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)\right]\hfill \\ \hfill ⩽& \sum _{\hslash =1}^n{\left({\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}\right)}^{\frac{1}{2}}\left[w_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}+N^2{w}_{\tau i}(t-{\varrho }_{\tau })\overline{w_{\tau i}(t-{\varrho }_{\tau })}\right].\hfill \end{array}$$

(15)

Substituting (12)–(15) into (11), we obtain

$$\begin{array}{cc}\hfill {}_{t_0}^C{D}_t^{\gamma }\overline{V}\left(t\right)⩽& -\sum _{i=1}^m\left[c_i+\overline{c_i}+{\eta }_i+\overline{{\eta }_i}-\sum _{\hslash =1}^n\left({\left(a_{i\hslash }\overline{a_{i\hslash }}\right)}^{\frac{1}{2}}+{\left(d_{i\hslash }\overline{d_{i\hslash }}\right)}^{\frac{1}{2}}+N^2{\left({\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}\right)}^{\frac{1}{2}}\right)\right]w_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}\hfill \\ \hfill & -\sum _{\hslash =1}^n\left[c_{\hslash }+\overline{c_{\hslash }}+{\kappa }_{\hslash }+\overline{{\kappa }_{\hslash }}-\sum _{i=1}^m\left({\left({\tilde{a}}_{\hslash i}\overline{{\tilde{a}}_{\hslash i}}\right)}^{\frac{1}{2}}+{\left({\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}\right)}^{\frac{1}{2}}+M^2{\left(a_{i\hslash }\overline{a_{i\hslash }}\right)}^{\frac{1}{2}}\right)\right]w_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}\hfill \\ \hfill & +\sum _{i=1}^m\sum _{\hslash =1}^n{M}^2{\left(d_{i\hslash }\overline{d_{i\hslash }}\right)}^{\frac{1}{2}}{w}_{v\hslash }(t-{\varrho }_v)\overline{w_{v\hslash }(t-{\varrho }_v)}\hfill \\ \hfill & +\sum _{\hslash =1}^n\sum _{i=1}^m{N}^2{\left({\tilde{d}}_{\hslash i}\overline{{\tilde{d}}_{\hslash i}}\right)}^{\frac{1}{2}}{w}_{\tau i}(t-{\varrho }_{\tau })\overline{w_{\tau i}(t-{\varrho }_{\tau })}.\hfill \end{array}$$

According to fractional Razumikhin theorem in [44], we can obtain

$$\begin{array}{cc}\hfill {}_{t_0}^C{D}_t^{\gamma }\overline{V}\left(t\right)⩽& -\sum _{i=1}^m{\chi }_i{w}_{\tau i}\left(t\right)\overline{w_{\tau i}\left(t\right)}-\sum _{\hslash =1}^n{\chi }_{\hslash }{w}_{v\hslash }\left(t\right)\overline{w_{v\hslash }\left(t\right)}\hfill \\ \hfill & +{\varpi }^{\prime }\left[\sum _{\hslash =1}^n{w}_{v\hslash }\left((t-\varrho )\right)\overline{w_{v\hslash }\left((t-\varrho )\right)}+\sum _{i=1}^m{w}_{\tau i}(t-\varrho )\overline{w_{\tau i}(t-\varrho )}\right]\hfill \\ \hfill ⩽& -\chi \overline{V}\left(t\right)+{\varpi }^{\prime }\overline{V}(t-\varrho )\hfill \\ \hfill ⩽& -\left(\chi -{\varpi }^{\prime }{\xi }^{\prime }\right)\overline{V}\left(t\right).\hfill \end{array}$$

From Lemma 5, we have

$$\begin{array}{c}\hfill \overline{V}\left(t\right)⩽\left[\sum _{i=1}^m{w}_{\tau i}\left(t_0\right)\overline{w_{\tau i}\left(t_0\right)}+\sum _{\hslash =1}^n{w}_{v\hslash }\left(t_0\right)\overline{w_{v\hslash }\left(t_0\right)}\right]E_{\gamma }\left[-(\chi -{\varpi }^{\prime }{\xi }^{\prime }){(t-t_0)}^{\gamma }\right],\end{array}$$

and

$$\begin{array}{c}\hfill ∥w_{\tau i}\left(t\right)∥+∥w_{v\hslash }\left(t\right)∥⩽{\left\{2\left[{∥w_{\tau i}\left(t_0\right)∥}^2+{∥w_{v\hslash }\left(t_0\right)∥}^2\right]E_{\gamma }\left[-(\chi -{\varpi }^{\prime }{\xi }^{\prime }){(t-t_0)}^{\gamma }\right]\right\}}^{\frac{1}{2}}.\end{array}$$

Therefore, systems (2) and (3) can achieve MLS. □

Remark 3.

In [6], the fractional quaternion-valued BAM-NN was decomposed into eight real-valued subsystems. In this paper, without decomposing system (2) into four real-valued subsystems or two complex-valued subsystems, system (2) is regarded as a compact whole, and this approach can reduce the complexity of calculation and the difficulty of theoretical analysis.

Remark 4.

Different from the Homomorphic mapping theorem [5], linear matrix inequality (LMI) [21], differential inclusion [27,32,37] and comparison principle [38], a new Caputo differential inequality is constructed by Laplace transform, Theorems 1 and 2 are derived to realize MLS between systems (2) and (3) based on newly proposed Lemma 5, the Lyapunov functional, fractional Razumikhin theorem and inequality techniques.

4. Examples

Two numerical examples are given to illustrate the validity and practicability of Theorems 1 and 2.

Example 1.

Consider 2-dimensional Caputo-delayed quaternion BAM-NNs:

$$\begin{array}{cc}\hfill & {}_{t_0}^C{D}_t^{\gamma }{p}_{\tau i}\left(t\right)=-c_i{p}_{\tau i}\left(t\right)+\sum _{\hslash =1}^2{a}_{i\hslash }{z}_{\hslash }\left(q_{v\hslash }\left(t\right)\right)+\sum _{\hslash =1}^2{d}_{i\hslash }{z}_{\hslash }\left(q_{v\hslash }(t-{\varrho }_v)\right)+U_i\left(t\right),\hfill \\ \hfill & {}_{t_0}^C{D}_t^{\gamma }{q}_{v\hslash }\left(t\right)=-c_{\hslash }{q}_{v\hslash }\left(t\right)+\sum _{i=1}^2{\tilde{a}}_{\hslash i}{\ell }_i\left(p_{\tau i}\left(t\right)\right)+\sum _{i=1}^2{\tilde{d}}_{\hslash i}{\ell }_i\left(p_{\tau i}(t-{\varrho }_{\tau })\right)+J_{\hslash }\left(t\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {}_{t_0}^C{D}_t^{\gamma }{\overline{p}}_{\tau i}\left(t\right)=-c_i{\overline{p}}_{\tau i}\left(t\right)+\sum _{\hslash =1}^2{a}_{i\hslash }{z}_{\hslash }\left({\overline{q}}_{v\hslash }\left(t\right)\right)+\sum _{\hslash =1}^2{d}_{i\hslash }{z}_{\hslash }\left({\overline{q}}_{v\hslash }(t-{\varrho }_v)\right)+U_i\left(t\right)+I_i\left(t\right),\hfill \\ \hfill & {}_{t_0}^C{D}_t^{\gamma }{\overline{q}}_{v\hslash }\left(t\right)=-c_{\hslash }{\overline{q}}_{v\hslash }\left(t\right)+\sum _{i=1}^2{\tilde{a}}_{\hslash i}{\ell }_i\left({\overline{p}}_{\tau i}\left(t\right)\right)+\sum _{i=1}^2{\tilde{d}}_{\hslash i}{\ell }_i\left({\overline{p}}_{\tau i}(t-{\varrho }_{\tau })\right)+J_{\hslash }\left(t\right)+I_{\hslash }\left(t\right),\hfill \end{array}$$

(17)

where $\gamma =0.98$, ${\varrho }_{\tau }=0.5$, ${\varrho }_v=1$, $p\left(t\right),q\left(t\right)\in \mathbb{Q}$, $c_i=4$, $c_{\hslash }=5$, $U=0,J=0$, $M=N=\frac{\sqrt{2}}{2}$, $\xi =1.25$, $a_{11}={\tilde{a}}_{11}=-0.3-0.2i-0.75j-0.2k$, $a_{12}={\tilde{a}}_{21}=-0.2-0.15i-0.2j-0.5k$, $a_{21}={\tilde{a}}_{12}=-0.2-0.3i-0.75j+0.5k$, $a_{22}={\tilde{a}}_{22}=-0.1-0.15i-0.1j+0.5k$, $d_{11}={\tilde{d}}_{11}=-0.3+0.2i+0.15j+k$, $d_{12}={\tilde{d}}_{21}=-0.25+0.1i+0.2j+0.1k$, $d_{21}={\tilde{d}}_{12}=-0.2+0.5i+0.15j+0.25k$, $d_{22}={\tilde{d}}_{22}=-0.1+i+j+0.5k$. Selecting the initial conditions ${\phi }_1\left(s\right)=-0.15+0.2i+0.2j+0.2k,{\phi }_2\left(s\right)=-0.2+0.2i-0.2j+0.2k$, ${\varphi }_1\left(s\right)=0.15+0.3i+0.1j+0.3k,{\varphi }_2\left(s\right)=-0.1+0.3i+0.1j+0.3k$, $s\in (-1,0)$. By calculation, it can be seen that ${\lambda }_1=1.27,{\lambda }_2=3.27,\varpi =1$, $\lambda =1.27>1.25=\varpi \xi$. Then, the conditions of Theorem 1 are satisfied.

Four sub-figures in Figure 3 show the trajectories of state variables of systems (16) and (17) are not synchronized without control when $\gamma =0.98$. Four sub-figures in Figure 4 depict the trajectories of the state variables of systems (16) and (17) under the adaptive controller (5) when $\gamma =0.98$. It can be seen from Figure 4 that the motion trends of the state variables of the master–slave systems (16) and (17) gradually approach and reach synchronization. Figure 5 presents error norm $∥w\left(t\right)∥$ under the adaptive controller (5) when $\gamma =0.98$. Therefore, under the adaptive controller (5), Figure 4 and Figure 5 indicate that systems (16) and (17) can achieve MLS.

Example 2.

Consider 2-dimensional systems (16) and (17). Choose the order $\gamma =0.47$, where ${\varrho }_{\tau }=1$, ${\varrho }_v=0.5$, $\xi =1.2$, $a_{11}={\tilde{a}}_{11}=-0.3-0.2i-0.25j-0.2k$, $a_{12}={\tilde{a}}_{21}=-0.2-0.15i-0.2j-0.5k$, $a_{21}={\tilde{a}}_{12}=-0.2-0.3i-0.175j+0.5k$, $a_{22}={\tilde{a}}_{22}=-0.1-0.15i-0.1j+0.5k$, $d_{11}={\tilde{d}}_{11}=-0.3+0.2i+0.15j+0.1k$, $d_{12}={\tilde{d}}_{21}=-0.25+0.1i+0.2j+0.1k$, $d_{21}={\tilde{d}}_{12}=-0.2+0.5i+0.15j+0.25k$, $d_{22}={\tilde{d}}_{22}=-0.1+0.1i+0.1j+0.5k$. Selecting the initial conditions ${\phi }_1\left(s\right)=-0.5+0.2i+0.2j+0.2k,{\phi }_2\left(s\right)=-0.4+0.4i-0.4j+0.4k$, ${\varphi }_1\left(s\right)=0.5+0.3i+0.2j+0.3k,{\varphi }_2\left(s\right)=-0.3+0.3i+0.1j+0.3k$, $s\in (-1,0)$. The remaining parameters are the same as Example 1. Through calculation, it can be seen that ${\chi }_1=1.25,{\chi }_2=3.25,\varpi =1$, $\chi =1.25>1.2={\varpi }^{\prime }{\xi }^{\prime }$. Thus, the conditions of Theorem 2 are satisfied under controller (11).

Four sub-figures in Figure 6 describe the trajectories of the state variables of systems (16) and (17) gradually approach and reach synchronization under controller (10) when $\gamma =0.47$. Figure 7 characterizes error norm $∥w\left(t\right)∥$ under controller when $\gamma =0.47$. Therefore, Figure 6 and Figure 7 reveal that systems (16) and (17) can realize MLS.

Remark 5.

Inspired by the prediction correction method of dealing with fractional-order delayed differential equations in [45], Examples 1 and 2 characterize the numerical simulation results based on improved prediction correction method. Figure 4, Figure 5, Figure 6 and Figure 7 further confirm the consistency between numerical examples and Theorems 1 and 2.

5. Conclusions

This paper has focused on investigating the MLS issue of fractional delayed quaternion BAM-NNs (2) and (3). Without decomposing system (2) into four real-valued subsystems or two complex-valued subsystems, we have constructed a Caputo differential inequality (1). According to the newly proposed Lemma 5, we have chosen the appropriate Lyapunov functionals, designed the adaptive feedback controller (5) and linear feedback controller (10). By using the inequality techniques and fractional Razumikhin theorem, the concise MLS Theorems 1 and 2 for error system (4) have been established. The validity of the results have been verified by numerical simulations.