1 Introduction

Since the memristor was found by Chua [1] and was achieved by HP Laboratory [2], memristive neural networks (MNNs) have acquired much attention for their application superiority in different fields, including associative memory [3], pattern recognition [4] and secure communication [5]. Especially, synchronization of MNNs has become an important field of complex networks and dynamic system. Up to now, many results have been obtained for the deterministic MNNs [6,7,8]. Robust analysis approach was investigated in [6], which realized finite-time synchronization (FTS) of the master–slave delayed MNNs. The article achieved exponential synchronization of MNNs with time-varying delay via the linear feedback control [7]. However, due to the error of modeling and the influence of perturbations, practical systems often exist some uncertain factors, such as unknown parameters and stochastic perturbations. These uncertain factors make the system unstable, chaotic, and even oscillating. Thus, it is very practical to investigate uncertain MNNs. Moreover, time delay affects the synchronization and stability of system due to signal transmission is not instantaneous [9], such as remote control system. And since the existence of parallel paths, distributed delays are also common [10,11,12,13]. The author studied adaptive synchronization of MNNs which include mixed delays and stochastic perturbation [14]. In practice, stochastic perturbations may lead to instability and performance degradation of neural networks. In fields such as secure communication, risk control, and bioinformatics, stochastic neural networks are more suitable for practical applications. Therefore, it is necessary to study the memristor-based stochastic neural networks. This article discusses the asymptotical synchronization for memristor-based multi-layers networks with delays under stochastic noise [15]. However, there exist few results that based on MNNs synchronization with unknown parameters, stochastic perturbations and mixed delays. It is essential to give relevant results.

For uncertain systems, effective control methods can make system run more stable. In recent years, several effective control schemes have been found, containing impulsive control [16], fuzzy control [17], SMC [18, 19] and adaptive control [20, 21], which have been proposed to deal with MNNs synchronization. It should be emphasized that SMC has great anti-interference ability and rapid dynamic response property, which has been widely used in uncertain networks synchronization. For example, the SMC strategy was designed for the discrete-time NNs synchronization, and sufficient conditions are given in [19]. When some parameters of system are unknown or unattainable, the SMC is ineffective. In contrast, ASMC provides a feasible scheme. This control scheme has both the online identification function of adaptive control and the anti-interference property of SMC, and its performance in uncertain system is satisfactory [22,23,24,25]. For example, the researchers [22] concentrated on ASM observer for the stability of uncertain nonlinear systems, and estimated the unknown disturbance in finite time. The multiple chaotic systems synchronization with perturbations is discussed in [23], and the adaptive laws and SM controllers were designed to deal with perturbations. However, above literature rarely consider the synchronization time, which cannot guarantee fast convergence of the error system for MNNs.

Since Kanebkov proposed finite-time stability [26] for fast convergence problem, and many related results subsequently appeared [27,28,29]. In [27], the FTS criterion of delayed NNs is obtained by Lyapunov functions and integral formula method. Liu et al. [28] designed two controllers with saturation function to avoid chattering, which realize bipartite FTS of MNNs. Nevertheless, the ST of FTS is relevant to initial states and the FTS is disabled when the initial values are huge or unknown. To overcome this defect, Polyakov [30] put forward fixed-time stability theory. So it’s not surprising that there are many results about fixed-time synchronization (FXTS) [31,32,33,34,35,36]. For instance, an adaptive pinning control scheme was designed to save resources and achieve the self-regulation function, which ensured FXTS for delayed NNs in [33]. Xiao et al. [34] studyed FXTS of multidimension-valued NNs, and applied spreading Cauchy-Schwarz inequality to design the nonlinear controllers. The problem of improved fixed-time stability for delayed fractional-order systems is studied in [35], and the results are extended to study fractional-order NNs with time-delays in FXTS. Therefore, FXTS is more flexible and has broader application prospects compared with FTS. Through above analysis, it is very interesting to study the FXTS of uncertain MNNs with mixed-delays. But, the ST of FXTS only give a maximum upper bound of stable time, which is often much larger than actual stable time. In order to further accelerate convergence speed of the system, it is urgent to explore an approach to adjust ST directly. To be sure, preassigned-time synchronization (PATS) is not relevant to any initial values and arbitrary parameters, which attracts people’s attention in recent years [37,38,39,40]. The preassigned-time stability of the discontinuous system was discussed, and the new theorem was developed by hyperbolic-tangent function [38]. Hu et al. [40] developed fresh conditions of MNNs PATS and a new controller without linear feedback term, the ST is more precise than [33].

To the best of our knowledge, there are no relevant results concerning FXTS and PATS of MNNs. It is challenging to develop concise and precise estimations on ST. Motivated by the above discussions, we throw light on the FXTS and PATS of MNNs. The contributions are organized as follows.

  1. (i)

    The FXTS and PATS framework of uncertain MNNs are constructed. The relationships among unknown weight, mixed delays, stochastic perturbations and synchronization capability are discussed. Compared with [20, 21], which contains only unknown parameters, the model designed is more general in this paper.

  2. (ii)

    An ASMC scheme is proposed to track unknown parameters and solve the mismatched parameters problem, which guarantees the fast convergence of error system.

  3. (iii)

    Some synchronization criteria of FXTS and PATS are derived. Compared to results about finite-time synchronization of NNs in [20], The obtained ST of FXTS overcome the dependence on initial values. And obtained ST of PATS guarantee the arbitrary setting of ST, then the degree of freedom and practicability of system synchronization are greatly increased.

2 Model Formulations and Preliminaries

Consider the drive system as

$$\begin{aligned}{} & {} {d{x_p}\left( t \right) = }{\left[ { - {{\textbf{c}}_p}{x_p}\left( t \right) } \right. + \sum \limits _{q = 1}^n {{{\textbf{a}}_{pq}}\left( {{x_q}\left( t \right) } \right) } {f_q}\left( {{x_q}\left( t \right) } \right) + \sum \limits _{q = 1}^n {{{\textbf{b}}_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) } {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) }\nonumber \\{} & {} { + \sum \limits _{q = 1}^n {{{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{x_q}\left( s \right) } \right) \left. {ds} \right] } } dt + \sum \limits _{q = 1}^n {{\chi _{pq}}(t,{x_q}(t),{x_q}(t - {\tau _q}(t)))d{w_q}\left( t \right) } } \end{aligned}$$
(1)

where \( p,q=1,2,3,...,n \), \( {x_p}\left( t \right) \) is \( {p_{th}} \) neuron state. \( {f_q}\left( \cdot \right) \) and \( {g_q}\left( \cdot \right) \) are bounded activation functions, and satisfy Lipschitz condition with \( {f_q}\left( 0 \right) = {g_q}\left( 0 \right) = 0 \), where \( \left| {{f_q}\left( \cdot \right) } \right| \le {M_1} \), \( \left| {{g_q}\left( \cdot \right) } \right| \le {M_2} \), \({M_1}> 0,{M_2} > 0 \). \( {{\textbf{a}}_{pq}}\left( \cdot \right) ,{{\textbf{b}}_{pq}}\left( \cdot \right) \) and \({{\textbf{r}}_{pq}}\left( \cdot \right) \) are connection weight coefficients, and \({\textbf{c}_p} > 0 \). \( {\tau (t)} \) is discrete delay and holds \(0 \le \tau \left( t \right) \le m. \) \( h\left( t \right) \) is distributed delay and meets \( 0 \le h\left( t \right) \le d \), where \( m> 0,d > 0 \). \({\chi _{pq}}\left( \cdot \right) \in {R^n} \) is noise intensity function. Complete probability space \( (\Psi ,F,P) \) occures brownian motion \( {w_q}\left( t \right) \). Initial value of (1) is \({x_p}\left( s \right) = {\Omega _p}\left( s \right) = {\left( {{\Omega _1}\left( s \right) ,{\Omega _2}\left( s \right) , \ldots ,{\Omega _p}\left( s \right) } \right) ^T} \in C\left( {\left[ { - \ell ,0} \right] ,{R^n}} \right) ,\ell \mathrm{{ = max}}\left\{ {d,m} \right\} ,\) \({{\textbf{a}}_{pq}}\left( {{x_p}\left( t \right) } \right) ,{{\textbf{b}}_{pq}}\left( {{x_p}\left( t \right) } \right) \) and \({{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) \) are given as

$$\begin{aligned}&\displaystyle \begin{array}{*{20}{l}} {{{\textbf{a}}_{pq}}\left( {{x_p}\left( t \right) } \right) = \frac{{{T_{pq}}}}{{{C_p}}} \times sig{n_{pq}},}&{{{\textbf{b}}_{pq}}\left( {{x_p}\left( t \right) } \right) = \frac{{{{T'}_{pq}}}}{{{C_p}}} \times sig{n_{pq}},} \end{array}\\&\displaystyle \begin{array}{*{20}{l}} {{{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) = \frac{{{{T''}_{pq}}}}{{{C_p}}} \times sig{n_{pq}},} \end{array}\end{aligned}$$

where \( {T_{pq}},{{T'}_{pq}},{{T''}_{pq}} \) represent resistances of memristors \( {W_{pq}},{{W'}_{pq}},{{W''}_{pq}} \) respectively. \( sig{n_{pq}} = 1 \) when \( p=q \), otherwise, \( sig{n_{pq}} = -1 \). \( {W_{pq}},{{W'}_{pq}},{{W''}_{pq}} \) represent memristors between \( {f_q}\left( {{x_q}\left( t \right) } \right) \) and \( {{x_p}\left( t \right) } \), \( {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) \) and \( {{x_p}\left( t \right) } \), \( \int _{t - h\left( t \right) }^t {{f_q}\left( {{x_q}\left( s \right) } \right) } ds \) and \( {{x_p}\left( t \right) } \) respectively. \( {C_p} \) is capacitor, the \( {x_p}\left( t \right) \) is the \({\mathrm{{C}}_{\mathrm{{p }}}}^\prime s\) voltage. The connection weights are used as

where are all constants. \( {H_q} \) is a positive constant and \( {H _q} \) represents jump threshold of memristor. The set-valued mappings are described as

where \(\overline{co} \) is convex closure. According to the theory of differential inclusion, (1) is transformed into

$$\begin{aligned}{} & {} {d{x_p}\left( t \right) \in }{\left[ { - {{\textbf{c}}_p}{x_p}\left( t \right) } \right. + \sum \limits _{q = 1}^n {{{\textbf{a}}_{pq}}\left( {{x_q}\left( t \right) } \right) } {f_q}\left( {{x_q}\left( t \right) } \right) + \sum \limits _{q = 1}^n {{{\textbf{b}}_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) } {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) }\nonumber \\{} & {} { + \sum \limits _{q = 1}^n {{{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{x_q}\left( s \right) } \right) \left. {ds} \right] } } dt + \sum \limits _{q = 1}^n {{\chi _{pq}}(t,{x_q}(t),{x_q}(t - {\tau _q}(t)))d{w_q}\left( t \right) ,} } \end{aligned}$$
(2)

where \( K\left[ {{{\textbf{a}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] ,K\left[ {{{\textbf{b}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \) and \(K\left[ {{{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \) represent the convex closures of the sets \(\left[ {{{\textbf{a}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \), \(\left[ {{{\textbf{b}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \), \( \left[ {{{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \) With \({a_{pq}}\left( {{x_p}\left( t \right) } \right) \in K\left[ {{{\textbf{a}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \), \({b_{pq}}\left( {{x_p}\left( t \right) } \right) \in K\left[ {{{\textbf{b}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \), \({r_{pq}}\left( {{x_p}\left( t \right) } \right) \in K\left[ {{{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) } \right] \), the (2) is equivalent to

$$\begin{aligned}{} & {} {d{x_p}\left( t \right) = }\left[ { - {c_p}{x_p}\left( t \right) } \right. + \sum \limits _{q = 1}^n {{a_{pq}}\left( {{x_q}\left( t \right) } \right) } {f_q}\left( {{x_q}\left( t \right) } \right) \nonumber \\{} & {} \quad + \sum \limits _{q = 1}^n {{b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) } {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) \nonumber \\{} & {} \quad + \sum \limits _{q = 1}^n {{r_{pq}}\left( {{x_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{x_q}\left( s \right) } \right) \left. {ds} \right] } } dt \nonumber \\{} & {} \quad + \sum \limits _{q = 1}^n {{\chi _{pq}}(t,{x_q}(t),{x_q}(t - {\tau _q}(t)))d{w_q}\left( t \right) .} \end{aligned}$$
(3)

where \( {{\textbf{a}}_{pq}}\left( \cdot \right) ,{{\textbf{b}}_{pq}}\left( \cdot \right) \) and \({{\textbf{r}}_{pq}}\left( \cdot \right) \) are unkown connection weights.

Assumption 1

\( {\chi _{pq}} =:{R_ + } \times R \times R \rightarrow R \) satisfies Lipschitz condition and linear growth condition with \( {\chi _{pq}}\left( {t,0,0} \right) = 0, \) it has

$$\begin{aligned}\begin{array}{*{20}{l}} {{{\left| {{\chi _{pq}}(t,{y_q}(t),{y_q}(t - {\tau _q}(t))) - {\chi _{pq}}(t,{x_q}(t),{x_q}(t - {\tau _q}(t)))} \right| }^2}}\\ { \le {\eta _{pq}}{{\left| {{y_q}(t) - {x_q}(t)} \right| }^2} + {\mu _{pq}}{{\left| {{y_q}(t - {\tau _q}(t)) - {x_q}(t - {\tau _q}(t))} \right| }^2}.} \end{array}\end{aligned}$$

where \( {\eta _{pq}}> 0,{\mu _{pq}} > 0 \). The response system is described as

$$\begin{aligned} d{y_p}\left( t \right)= & {} {\left[ { - {{{\hat{c}}}_p}(t){y_p}\left( t \right) } \right. + \sum \limits _{q = 1}^n {{{{\hat{a}}}_{pq}}} (t){f_q}\left( {{y_q}\left( t \right) } \right) + \sum \limits _{q = 1}^n {{{{\hat{b}}}_{pq}}(t)} {g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) }\nonumber \\{} & {} {} + \sum \limits _{q = 1}^n {{{{\hat{r}}}_{pq}}\left( {{y_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{y_q}\left( s \right) } \right) \left. {ds + {u_p}(t)} \right] } } dt \nonumber \\{} & {} + \sum \limits _{q = 1}^n {{\chi _{pq}}(t,{y_q}(t),{y_q}(t - {\tau _q}(t)))d{w_q}\left( t \right) } \end{aligned}$$
(4)

where \({{u_p}(t)}\) are suitable adaptive control inputs, \( {{{\hat{c}}}_p}(t),{{{\hat{a}}}_{pq}}(t) \) and \( {{{{\hat{b}}}_{pq}}(t)} \) are tunable weights.

Define the synchronization error as \( {e_p}\left( t \right) = {y_p}\left( t \right) - {x_p}\left( t \right) \), and it yields

$$\begin{aligned} \begin{array}{*{20}{l}} {d{e_p}\left( t \right) = \left\{ { - {c_p}{e_p}(t) - \left[ {{{{\hat{c}}}_p}(t) - {c_p}} \right] {y_p}(t) + \sum \limits _{p = 1}^n {{a_{pq}}\left( {{x_q}\left( t \right) } \right) } \left[ {{f_q}\left( {{y_q}\left( t \right) } \right) - {f_q}\left( {{x_q}\left( t \right) } \right) } \right] } \right. }\\ {\begin{array}{*{20}{c}} {}&{}{}&{}{\begin{array}{*{20}{l}} \begin{array}{l} + \sum \limits _{q = 1}^n {\left[ {{{{\hat{a}}}_{pq}}(t) - {a_{pq}}\left( {{x_q}\left( t \right) } \right) } \right] } {f_q}\left( {{y_q}\left( t \right) } \right) + \sum \limits _{q = 1}^n {\left[ {{{{\hat{b}}}_{pq}}(t) - {b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) } \right] {g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) } \\ + \sum \limits _{q = 1}^n {{b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) \left[ {{g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) - {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) } \right] } \end{array}\\ {\left. { + \sum \limits _{q = 1}^n {\left[ {{{{\hat{r}}}_{pq}}\left( {{y_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{y_q}\left( s \right) } \right) ds - \sum \limits _{q = 1}^n {{r_{pq}}\left( {{x_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{x_q}\left( s \right) } \right) ds} } } } \right] + {u_p}(t)} } \right\} dt} \end{array}} \end{array}}\\ {\begin{array}{*{20}{c}} {}&{}{}&{}{}&{}+ \end{array}\sum \limits _{q = 1}^n {{\sigma _{pq}}(t,{e_q}(t),{e_q}(t - {\tau _q}(t)))d{w_q}\left( t \right) } }, \end{array}\nonumber \\ \end{aligned}$$
(5)

where \( {\sigma _{pq}}(t,{e_q}(t),{e_q}(t - {\tau _q}(t))) = {\chi _{pq}}(t,{y_q}(t),{y_q}(t - {\tau _q}(t))) - {\chi _{pq}}(t,{x_q}(t),{x_q}(t - {\tau _q}(t)))\). Consider stochastic nonlinear systems as

$$\begin{aligned} d\theta \left( t \right) = \Psi \left( {t,\theta \left( t \right) } \right) dt + \Upsilon \left( {t,\theta \left( t \right) } \right) dw\left( t \right) ,\end{aligned}$$
(6)

where \(\theta (t)\) denotes system state with \(\theta \left( {{t_0}} \right) = {\theta _0}\), \(\Psi \left( \cdot \right) \) and \(\Upsilon \left( \cdot \right) \) are nonlinear continuous function and noise intensity function respectively.

Remark 1

In recent years, the dynamics of memristor-based neural networks have been extensively studied, which have led to many successful applications in a variety of areas, including signal processing, pattern recognition, prediction model, etc. In practice, stochastic perturbations may lead to instability and performance degradation of neural networks. In risk control, oncology, bioinformatics and other related fields, stochastic neural networks are more suitable for practical applications than ordinary neural networks Therefore, it is necessary to study the memristor-based stochastic neural networks. It should be noted that the synchronization problem of stochastic neural networks studied in our paper focuses more on theoretical analysis and does not explore it from a practical application perspective. This will also be one of our important future work. In order to achieve fixed-time stability and preassigned-time stability of error system (5), some definitions and lemmas are taken into account.

Definition 1

[41] Where \( {T_\nu }> 0,{\Gamma _p} > 0 \), for any initial state \( {\theta _0} \) of \( {\theta _0}\left( t \right) \in {R^n} \), there are conditions satisfy: (i) The origin can achieve the globally stochastically finite-time stable. (ii)The expectation of \( {T\left( {{\theta _0},w\left( t \right) } \right) } \) meets

$$\begin{aligned}{T_\nu } = \textrm{E}\left( {T\left( {{\theta _0},w} \right) } \right) \le {\Gamma _p},\end{aligned}$$

where \( {\Gamma _p} \) denotes the maximum value of the ST.

Definition 2

[20] Define the vector function \( \Phi \left( \sigma \right) \) as

$$\begin{aligned}\Phi \left( \sigma \right) = \left\{ \begin{array}{l} {\left\| \sigma \right\| ^{ - 2}}\sigma ,\sigma \ne 0,\\ 0,\begin{array}{*{20}{c}} {}&{}{}&{}{\sigma = 0.} \end{array} \end{array} \right. \end{aligned}$$

Lemma 1

[42] If the Lyapunov function \( V\left( {t,\theta \left( t \right) } \right) \in {C^{2,1}}:{R^ + } \times {R^n} \rightarrow {R_ + } \) satisfies

$$\begin{aligned}{{\mathcal {L}}}V\left( {\theta \left( t \right) } \right) \le - {c_1}{V^{{\phi _1}}}\left( {\theta \left( t \right) } \right) - {c_2}{V^{{\phi _2}}}\left( {\theta \left( t \right) } \right) ,\end{aligned}$$

then the stochastic system (6) realizes fixed-time stability, and the ST is calculated as

$$\begin{aligned}{T_\nu } = \textrm{E}\left( {T\left( {{\theta _0},w} \right) } \right) \le \frac{1}{{{c_1}\left( {1 - {\phi _1}} \right) }} + \frac{1}{{{c_2}\left( {{\phi _2} - 1} \right) }},\end{aligned}$$

where \( {c_1}> 0,{c_2} > 0 \), \(0< {\phi _1} < 1,{\phi _2} > 1\).

Lemma 2

[39]For (6), if Lyapunov function \( V\left( \cdot \right) \) is a continuous strictly monotonically decreased function and satisfies: (i)\( {T_c} \in \left\{ {{b_1},...,{b_n}} \right\} > 0 \) is a parameter that can be set. (ii)For any \( V\left( \cdot \right) > 0 \) and satisfies

$$\begin{aligned}{{\mathcal {L}}}V\left( {\theta \left( t \right) } \right) \le - \frac{v}{{{T_c}}}\left[ {{b_1}{V^{{\varphi _1}}}\left( {\theta \left( t \right) } \right) + {b_2}{V^{{\varphi _2}}}\left( {\theta \left( t \right) } \right) } \right] ,\end{aligned}$$

where \(\varpi = 1 + \frac{{{\varphi _2} - 1}}{{1 - {\varphi _1}}},v = \frac{1}{{1 - {\varphi _1}}}\frac{{{2^{\varpi - 1}}}}{{{b_2}^{\frac{1}{\varpi }}\left( {\varpi - 1} \right) }}{b_1}^{\frac{{1 - \varpi }}{\varpi }},\) \(0< {\varphi _1} < 1,{\varphi _2} > 1\), and \( {{b_1}},{{b_2}} \) are positive constants. Then the system will achieve globally preassigned-time stable.

Lemma 3

[20] For any positive constants \({{\textrm{K}_p}}\) and \(0 < {\upsilon _1} \le 1,{\upsilon _2} > 1\), it gets

$$\begin{aligned}\sum \limits _{p = 1}^n {\mathrm{{K}}_p^{{\upsilon _1}}} \ge {\left( {\sum \limits _{p = 1}^n {{\mathrm{{K}}_p}} } \right) ^{{\upsilon _1}}},\sum \limits _{p = 1}^n {\mathrm{{K}}_p^{{\upsilon _2}}} \ge {n^{1 - {\upsilon _2}}}{\left( {\sum \limits _{p = 1}^n {{\mathrm{{K}}_p}} } \right) ^{{\upsilon _2}}}.\end{aligned}$$

3 Main Results

In this part, the error system achieves FXT stability and PAT stability by designing the ASMC controller. And sufficient criteria of FXTS and PATS are deduced.

3.1 Fixed-Time Synchronization of Stochastic MNNs

Construct the sliding-mode surface (SMS) as

$$\begin{aligned} {s_p}(t) = {e_p}(t) + m\int _0^t {\left( {{{\left| {{e_p}(\tau )} \right| }^{{\theta _{\mathrm{{3}}}}}}\mathrm{{ + }}{{\left| {{e_p}(\tau )} \right| }^{{\theta _4}}}} \right) } sign\left( {{e_p}(\tau )} \right) d\tau ,\end{aligned}$$
(7)

where \(m> 0,0< {\theta _{\mathrm{{3}}}} < 1,{\theta _4} > 1\). Based on the designed sliding mode control scheme, FXTS of MNNs will be ahieved. Then in order to prove fixed-time accessibility of SMS, consider the control input as

$$\begin{aligned} \begin{array}{*{20}{l}} {{u_p}(t) = - {\alpha _p}(t){e_p}(t) - {\beta _p}sign({s_p}(t)){{\left| {{s_p}(t)} \right| }^{{\theta _1}}} - {{\beta '}_p}sign({s_p}(t)){{\left| {{s_p}(t)} \right| }^{{\theta _2}}}}\\ \;\;\;\;\;\;\;{\begin{array}{*{20}{c}} {}&{}{}&{}{ - {l_p}\omega \Phi ({s_p}(t)) - {{\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{k_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] }^{\frac{{{\theta _1} + 1}}{2}}}\Phi ({s_p}(t))} \end{array}}\\ \;\;\;\;\;\;\;{\begin{array}{*{20}{c}} {}&{}{}&{}{ - {{\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{{k'}_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] }^{\frac{{{\theta _2} + 1}}{2}}}\Phi ({s_p}(t)) - {\delta _p}e_p^2(t)\Phi ({s_p}(t))} \end{array}}\\ \end{array} \end{aligned}$$
(8)
$$\begin{aligned}\begin{array}{l} - {\gamma _p}sign({s_p}(t)) - {\varepsilon _p}{e_p}(t) - {\lambda _p}\sum \limits _{q = 1}^n {e_q^2(t - \tau (t))\Phi ({s_p}(t))} \\ - msign({e_p}(t)){\left| {{e_p}(t)} \right| ^{{\theta _3}}} - msign({e_p}(t)){\left| {{e_p}(t)} \right| ^{{\theta _4}}}, \end{array}\end{aligned}$$

where \( {\beta _p},\beta '_p,{l_p},{k_p},{{k'}_p},m \) are positive constants, \( {\alpha _p}(t) \) is the gain of adaptive controller, \( 0< {\theta _1} < 1,{\theta _2} > 1,\) and \(\omega = {\left( {\left\| {{\hat{C}}(t)} \right\| + {\theta _C}} \right) ^{{\theta _1} + 1}} + {\left( {\left\| {{\hat{C}}(t)} \right\| + {\theta _C}} \right) ^{{\theta _2} + 1}}\). \({\hat{C}}(t) = diag\left\{ {{{{\hat{c}}}_1}(t),{{{\hat{c}}}_1}(t),...,{{{\hat{c}}}_1}(t)} \right\} \in {R^{n \times n}}\) is estimated of C. In order to achieve estimation for unknown weights, the following adaptive laws for weights and controller are given

$$\begin{aligned}{} & {} {{{\mathop {{\hat{c}}}\limits ^ \cdot }_p}(t) = {y_p}(t){s_p}(t),{{\mathop {{\hat{a}}}\limits ^ \cdot }_{pq}}(t) = - {f_q}({y_q}(t)){s_p}(t),{{\mathop {{\hat{b}}}\limits ^ \cdot }_{pq}}(t) = - {g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) {s_p}(t),}\nonumber \\ {}{} & {} {{{{\dot{\alpha }} }_p}\left( t \right) = {s_p}(t){e_p}(t) - {\phi _p}sign({\alpha _p}(t)){{\left| {{\alpha _p}(t)} \right| }^{{\theta _1}}} - {{\phi '}_p}sign({\alpha _p}(t)){{\left| {{\alpha _p}(t)} \right| }^{{\theta _2}}},} \end{aligned}$$
(9)

where \( {\phi _p}> 0,\phi '_p > 0. \)

Remark 2

Due to the error of modeling and the influence of perturbations, practical systems often exist some uncertain factors, such as unknown parameters and stochastic perturbations. SMC has strong anti-interference ability and fast dynamic response characteristics, and has been widely used in uncertain network synchronization. Meanwhile, the adaptive strategy is the powerful tool of identifying unkown parameters and can automatically adjust by different updating laws. The drive system weights are unknown in this paper. Therefore, the weight update laws are constructed to identify the unkown weights of drive system, and controller update law promotes system convergence.

Assumption 2

The matrice C is bounded with \( {\theta _C} > 0 \) and satisfies \(\left\| C \right\| \le {\theta _C}.\)

Based on the adaptive control input (8), the fixed-time synchronization of MNNs is discussed.

Theorem 1

The error system (5) will arrive SMS under controller (8), if the following conditions hold

$$\begin{aligned} \left\{ \begin{array}{l} {\varepsilon _p} \ge \sum \limits _{q = 1}^n {\left( {\frac{{a_{pq}^{\max }{\rho _q} + b_{pq}^{\max }{\zeta _q}}}{2}} \right) - \frac{{{c_p}}}{2},} \\ {\gamma _p} \ge \sum \limits _{q = 1}^n {\left( {a_{pq}^{\max }{M_1} + a_{pq}^{\max }{M_2} + 2r_{pq}^{\max }{M_1}m} \right) ,} \\ {\lambda _p} \ge \sum \limits _{q = 1}^n {\left( {\frac{{b_{pq}^{\max }{\zeta _q} + {\mu _{pq}}}}{2}} \right) ,} \\ {\delta _p} \ge \sum \limits _{q = 1}^n {\left( {\frac{{{\eta _{qp}} + a_{qp}^{\max }{\rho _p}}}{2}} \right) - \frac{{{c_p}}}{2},} \end{array} \right. \end{aligned}$$
(10)

and the ST can be calculated as

$$\begin{aligned} {T_{\nu 1}} = E\left( {T\left( {{\theta _0},w} \right) } \right) \le \frac{1}{{{i_1}\left( {1 - {\theta _1}} \right) }} + \frac{1}{{{i_2}\left( {{\theta _2} - 1} \right) }}, \end{aligned}$$
(11)

where \( {i_1} = \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\beta _p},{l_p},{k_p},{\phi _p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}},{i_2} = {n^{\frac{{1 - {\theta _2}}}{2}}}\mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\beta '}_p},{l_p},{{k'}_p},{{\phi '}_p}} \right\} {2^{\frac{{{\theta _2} + 1}}{2}}},0< {\theta _1} < 1,{\theta _2} > 1. \)

Proof

The Lyapunov function is considered as

$$\begin{aligned} V\left( t \right) = {V_1}\left( t \right) + {V_2}\left( t \right) + {V_3}\left( t \right) \end{aligned}$$
(12)

where

$$\begin{aligned} {V_1}\left( t \right)= & {} \frac{1}{2}\sum \limits _{p = 1}^n {s_p^2(t)}, {{V_2}\left( t \right) = \frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} + \frac{1}{2}\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } },\\{} & {} {{V_3}\left( t \right) = \frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2(t)} }. \end{aligned}$$

Based on \( It{\hat{o}} \) formula and Assumption 1, it can deduce that

$$\begin{aligned}{} & {} {{\mathcal {L}}}{V_1}\left( t \right) \nonumber \\= & {} \sum \limits _{p = 1}^n {{s_p}(t)\left\{ { - {c_p}{e_p}(t) - \left[ {{{{\hat{c}}}_p}(t) - {c_p}} \right] {y_p}(t) + \sum \limits _{q = 1}^n {{a_{pq}}\left( {{x_q}\left( t \right) } \right) } \left[ {{f_q}\left( {{y_q}\left( t \right) } \right) - {f_q}\left( {{x_q}\left( t \right) } \right) } \right] } \right. } \nonumber \\{} & {} + \sum \limits _{q = 1}^n {\left[ {{{{\hat{a}}}_{pq}}(t) - {a_{pq}}\left( {{x_q}\left( t \right) } \right) } \right] } {f_q}\left( {{y_q}\left( t \right) } \right) \nonumber \\{} & {} + \sum \limits _{q = 1}^n {\left[ {{{{\hat{b}}}_{pq}}(t) - {b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) } \right] {g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) } \nonumber \\{} & {} + \sum \limits _{j = 1}^n {{b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) \left[ {{g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) - {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) } \right] } \nonumber \\{} & {} + \sum \limits _{q = 1}^n {\left[ {{{{\hat{r}}}_{pq}}\left( {{y_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{y_q}\left( s \right) } \right) ds - \sum \limits _{q = 1}^n {{r_{pq}}\left( {{x_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{x_q}\left( s \right) } \right) ds} } } } \right] } \nonumber \\{} & {} - {\alpha _p}(t){e_p}(t) - {\beta _p}sign({s_p}(t)){\left| {{s_p}(t)} \right| ^{{\theta _1}}} - {{\beta '}_p}sign({s_p}(t)){\left| {{s_p}(t)} \right| ^{{\theta _2}}} - {l_p}\omega \Phi ({s_p}(t))\nonumber \\{} & {} - {\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{k_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _1} + 1}}{2}}}\Phi ({s_p}(t)) \nonumber \\{} & {} - {\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{{k'}_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _2} + 1}}{2}}}\Phi ({s_p}(t))\nonumber \\{} & {} \left. { - {\delta _p}e_p^2(t)\Phi ({s_p}(t)) - {\gamma _p}sign({s_p}(t)) - {\varepsilon _p}{e_p}(t) - {\lambda _p}\sum \limits _{q = 1}^n {e_q^2(t - \tau (t))\Phi ({s_p}(t))} } \right\} \nonumber \\{} & {} + \sum \limits _{q = 1}^n {\frac{{{\mu _{pq}}}}{2}} e_q^2(t - \tau t)) + \sum \limits _{q = 1}^n {\frac{{{\eta _{qp}}}}{2}} e_p^2(t)\end{aligned}$$
(13)
$$\begin{aligned} {{{\mathcal {L}}}{V_2}\left( t \right) }= & {} \sum \limits _{p = 1}^n {\left[ {{{{\hat{c}}}_p}(t) - {c_p}} \right] {y_p}(t){s_p}(t)} \nonumber \\{} & {} + \sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {\left[ { - {{{\hat{a}}}_{pq}}(t){f_q}\left( {{y_q}\left( t \right) } \right) {s_p}(t) - {{{\hat{b}}}_{pq}}(t){g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) {s_p}(t)} \right] } }\end{aligned}$$
(14)
$$\begin{aligned} {{\mathcal {L}}}{V_3}\left( t \right)= & {} \sum \limits _{p = 1}^n {{\alpha _p}\left( t \right) {{{\dot{\alpha }} }_p}\left( t \right) } \nonumber \\{} & {} \le - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\phi _p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] ^{\frac{{{\theta _1} + 1}}{2}}} \nonumber \\{} & {} - \mathop {\min }\limits _{1 \le i \le n} \left\{ {{{\phi '}_p}} \right\} {n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] ^{\frac{{{\theta _2} + 1}}{2}}}\nonumber \\{} & {} + \sum \limits _{p = 1}^n {{\alpha _p}\left( t \right) {s_p}(t){e_p}(t)}. \end{aligned}$$
(15)

According to (12)-(15), it has

$$\begin{aligned} {{\mathcal {L}}}V\left( t \right)= & {} \sum \limits _{p = 1}^n {{s_p}(t)\left\{ { - {c_p}{e_p}(t) + \sum \limits _{q = 1}^n {{a_{pq}}\left( {{x_q}\left( t \right) } \right) } \left[ {{f_q}\left( {{y_q}\left( t \right) } \right) - {f_q}\left( {{x_q}\left( t \right) } \right) } \right] } \right. } \nonumber \\{} & {} - \sum \limits _{q = 1}^n {{a_{pq}}\left( {{x_q}\left( t \right) } \right) } {f_q}\left( {{y_q}\left( t \right) } \right) - \sum \limits _{q = 1}^n {{b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) {g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) } \nonumber \\{} & {} + \sum \limits _{q = 1}^n {{b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) \left[ {{g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) - {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) } \right] } \nonumber \\{} & {} + \sum \limits _{q = 1}^n {\left[ {{{{\hat{r}}}_{pq}}\left( {{y_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{y_q}\left( s \right) } \right) ds - \sum \limits _{q = 1}^n {{r_{pq}}\left( {{x_p}\left( t \right) } \right) \int _{t - h\left( t \right) }^t {{f_q}\left( {{x_q}\left( s \right) } \right) ds} } } } \right] } \nonumber \\{} & {} - {\beta _p}sign({s_p}(t)){\left| {{s_p}(t)} \right| ^{{\theta _1}}} - {{\beta '}_p}sign({s_p}(t)){\left| {{s_p}(t)} \right| ^{{\theta _2}}} - {l_p}\omega \Phi ({s_p}(t)) \nonumber \\{} & {} - {\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{k_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _1} + 1}}{2}}}\Phi ({s_p}(t)) - {\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{{k'}_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _2} + 1}}{2}}}\Phi ({s_p}(t)) \nonumber \\{} & {} -\left. { {\delta _p}e_p^2(t)\Phi ({s_p}(t)) - {\gamma _p}sign({s_p}(t)) - {\varepsilon _p}{e_p}(t) - {\lambda _p}\sum \limits _{q = 1}^n {e_q^2(t - \tau (t))\Phi ({s_p}(t))} } \right\} \nonumber \\{} & {} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\phi _p}} \right\} {\left[ {\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] ^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\phi '}_p}} \right\} {n^{\frac{{1 - {\theta _2}}}{2}}}{\left[ {\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] ^{\frac{{{\theta _2} + 1}}{2}}} \nonumber \\{} & {} + \sum \limits _{q = 1}^n {\frac{{{\mu _{pq}}}}{2}} e_q^2(t - \tau (t)) + \sum \limits _{q = 1}^n {\frac{{{\eta _{qp}}}}{2}} e_p^2(t). \end{aligned}$$
(16)

Using Young’s inequality, one obtains

$$\begin{aligned} \begin{array}{*{20}{l}} {\sum \limits _{p = 1}^n {{s_p}(t)\sum \limits _{q = 1}^n {{a_{pq}}\left( {{x_q}\left( t \right) } \right) } \left[ {{f_q}\left( {{y_q}\left( t \right) } \right) - {f_q}\left( {{x_q}\left( t \right) } \right) } \right] } }\\ { \le \sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {a_{pq}^{\max }} } {\rho _q}\left| {{s_p}(t)} \right| \left| {{e_q}(t)} \right| }\\ { \le \sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {\left[ {\frac{{a_{pq}^{\max }{\rho _q}}}{2}s_p^2(t) + \frac{{a_{qp}^{\max }{\rho _p}}}{2}e_p^2(t)} \right] } } } \end{array} \end{aligned}$$
(17)

and

$$\begin{aligned} \begin{array}{*{20}{l}} {\sum \limits _{p = 1}^n {{s_p}(t)\sum \limits _{q = 1}^n {{b_{pq}}\left( {{x_q}\left( {t - \tau (t)} \right) } \right) \left[ {{g_q}\left( {{y_q}\left( {t - \tau \left( t \right) } \right) } \right) - {g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) } \right] } } }\\ { \le \sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {b_{pq}^{\max }{\zeta _q}\left| {{s_p}(t)} \right| \left| {{e_q}(t - \tau (t)} \right| } } }\\ { \le \sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {\frac{{b_{pq}^{\max }{\zeta _q}}}{2}\left( {{{\left| {{s_p}(t)} \right| }^2} + {{\left| {{e_q}(t - \tau (t)} \right| }^2}} \right) } },} \end{array} \end{aligned}$$
(18)

where \({\rho _q}\) and \({\zeta _q}\) are positive Lipschitz constants. With Assumption 2, it yields

$$\begin{aligned} \left\| {{\hat{C}}(t) - C} \right\| \le \left\| {{\hat{C}}(t)} \right\| + \left\| C \right\| \le \left\| {{\hat{C}}(t)} \right\| + {\theta _C}.\end{aligned}$$
(19)

Therefore, it obtains

$$\begin{aligned}{} & {} - \sum \limits _{p = 1}^n {{s_p}(t){l_p}\omega \Phi ({s_p}(t))} \nonumber \\{} & {} \quad { \le - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} \left[ {{{\left\| {{\hat{C}}(t) - C} \right\| }^{{\theta _1} + 1}} + {{\left\| {{\hat{C}}(t) - C} \right\| }^{{\theta _2} + 1}}} \right] }\nonumber \\{} & {} \quad \le - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] }^{\frac{{{\theta _1} + 1}}{2}}} \nonumber \\{} & {} \quad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} {n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] }^{\frac{{{\theta _2} + 1}}{2}}} \end{aligned}$$
(20)

Substituting (17)-(20) into (16), it gains

$$\begin{aligned} \begin{array}{*{20}{l}} {{{\mathcal {L}}}V\left( t \right) }&{}{ \le \sum \limits _{p = 1}^n {\left\{ {\left[ { - \frac{{{c_p}}}{2} + \sum \limits _{q = 1}^n {\left( {\frac{{a_{pq}^{\max }{\rho _q} + b_{pq}^{\max }{\zeta _q}}}{2}} \right) } - {\varepsilon _p}} \right] s_p^2(t)} \right. } }\\ {}&{}{\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} { + \left[ {\sum \limits _{q = 1}^n {\left( {a_{pq}^{\max }{M_1} + a_{pq}^{\max }{M_2} + 2r_{pq}^{\max }{M_1}m} \right) - {\gamma _p}} } \right] \left| {{s_p}(t)} \right| }\\ { + \left[ {\sum \limits _{q = 1}^n {\left( {\frac{{b_{pq}^{\max }{\zeta _j} + {\mu _{pq}}}}{2}} \right) - {\lambda _p}} } \right] e_q^2(t - \tau (t))\left. { + \left[ {\sum \limits _{q = 1}^n {\frac{{{\eta _{qp}} + a_{qi}^{\max }{\rho _p}}}{2} - {\delta _p}} } \right] e_p^2(t)} \right\} } \end{array}}\\ \begin{array}{l} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{k_p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _1} + 1}}{2}}}\\ - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{k'}_p}} \right\} {n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _2} + 1}}{2}}} \end{array}\\ { \;- \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\beta _p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {s_p^2\left( t \right) } } \right] }^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\beta '}_p}} \right\} {n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {s_i^2\left( t \right) } } \right] }^{\frac{{{\theta _2} + 1}}{2}}}} \end{array}}\\ {}&{}{\begin{array}{*{20}{l}} { \;- \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] }^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} {n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] }^{\frac{{{\theta _2} + 1}}{2}}}}\\ { \;- \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\phi _p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] }^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\phi '}_p}} \right\} {n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] }^{\frac{{{\theta _2} + 1}}{2}}}.} \end{array}} \end{array}\nonumber \\ \end{aligned}$$
(21)

Further calculation under Theorem 1, the inequality (21) gets

$$\begin{aligned} {{\mathcal {L}}}V\left( t \right) \le - {i_1}{\left( {V\left( t \right) } \right) ^{\frac{{{\theta _1} + 1}}{2}}} - {i_2}{\left( {V\left( t \right) } \right) ^{\frac{{{\theta _2} + 1}}{2}}}.\end{aligned}$$
(22)

where \( {i_1} = \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\beta _p},{l_p},{k_p},{\phi _p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}},{i_2} = {n^{\frac{{1 - {\theta _2}}}{2}}}\mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\beta '}_p},{l_p},{{k'}_p},{{\phi '}_p}} \right\} {2^{\frac{{{\theta _2} + 1}}{2}}},0< {\theta _1} < 1,{\theta _2} > 1. \) The ST is calculated as (11). Next, it is proved that the error on SMS converges to 0 in fixed-time. \(\square \)

Theorem 2

The error system (5) can converge to 0 on the SMS (7). And the \( {T_{\nu 2}} \) is

$$\begin{aligned} {T_{\nu 2}} = E\left( {T\left( {{\theta _0},w} \right) } \right) \le \frac{1}{{{m_1}\left( {1 - {\theta _3}} \right) }} + \frac{1}{{{m_2}\left( {{\theta _4} - 1} \right) }}, \end{aligned}$$
(23)

where \( {m_1}> 0,{m_2}> 0,0< {\theta _3} < 1,{\theta _4} > 1. \) By using the conditions of \( {s_p}(t) = 0,{{\dot{s}}_p}(t) = 0 \), and it follows that

$$\begin{aligned} {{\dot{e}}_{p}}(t) = - m\left( {{{\left| {{e_p}(\tau )} \right| }^{{\theta _{\mathrm{{3}}}}}}\mathrm{{ + }}{{\left| {{e_p}(\tau )} \right| }^{{\theta _4}}}} \right) sign\left( {{e_p}(\tau )} \right) . \end{aligned}$$
(24)

Proof

Lyapunov functional is provided as

$$\begin{aligned} V\left( t \right) = \frac{1}{2}\sum \limits _{p = 1}^n {e_p^2(t)}. \end{aligned}$$
(25)

The derivation of \( {V\left( t \right) } \) is

$$\begin{aligned} \begin{array}{*{20}{l}} {{{\mathcal {L}}}V\left( t \right) = \sum \limits _{p = 1}^n {{e_p}\left( t \right) {{\dot{e}}_p}\left( t \right) } }\\ \;\;\;\;\;\;\;{\begin{array}{*{20}{c}} {}&{}{}&{} = \end{array}\sum \limits _{p = 1}^n {{e_p}\left( t \right) \left( { - m{{\left| {{e_p}\left( t \right) } \right| }^{{\theta _3}}}sign\left( {{e_p}\left( t \right) } \right) - m{{\left| {{e_p}\left( t \right) } \right| }^{{\theta _4}}}sign\left( {{e_p}\left( t \right) } \right) } \right) } }\\ \;\;\;\;\;\;\;{\begin{array}{*{20}{c}} {}&{}{}&{}{ \le - m\sum \limits _{p = 1}^n {{{\left| {{e_p}\left( t \right) } \right| }^{{\theta _3} + 1}}} - m\sum \limits _{p = 1}^n {{{\left| {{e_p}\left( t \right) } \right| }^{{\theta _4} + 1}}} } \end{array}}\\ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {}&{}{}&{} \;\;\;\;\;\;\le \end{array} - m{2^{\frac{{{\theta _3} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {e_p^2\left( t \right) } } \right] }^{\frac{{{\theta _3} + 1}}{2}}} - m{2^{\frac{{{\theta _4} + 1}}{2}}}{n^{\frac{{1 - {\theta _4}}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {e_p^2\left( t \right) } } \right] }^{\frac{{{\theta _4} + 1}}{2}}}}\\ {\begin{array}{*{20}{c}} {}&{}{}&{}{ \;\;\;\;\;\;= - {m_1}{{\left( {V\left( t \right) } \right) }^{\frac{{{\theta _3} + 1}}{2}}} - {m_2}{{\left( {V\left( t \right) } \right) }^{\frac{{{\theta _4} + 1}}{2}}}.} \end{array}} \end{array}} \end{array} \end{aligned}$$
(26)

where \( {m_1} = m{2^{\frac{{{\theta _3} + 1}}{2}}},{m_2} = m{2^{\frac{{{\theta _4} + 1}}{2}}}{n^{\frac{{1 - {\theta _4}}}{2}}} \). Through the above proof that the error on SMS converges to 0 in fixed-time under adaptive controller,and the ST is shown in (23). \(\square \)

Remark 3

From (7), a first-order SMS is constructed, that is, the integral SMS, which the degree of freedom of sliding variable s is 1. The time evolution of the system state is driven to slide along the SMS and stay on the SMS. In accordance with Theorem 1, the error system slides along the SMS in \( {T_{\nu 1}} \); based on Theorem 2, the error system evolves in a neighborhood around the SMS in \( {T_{\nu 2}} \). Consequently, the total time is \( T = {T_{\nu 1}} + {T_{\nu 2}} \). And it can be known from (11) and (23) that the ST of FXTS is correlated with the controller parameters \({\beta _p},{{\beta '}_p},{l_p},{k_p},{{k'}_p},{\phi _p},{{\phi '}_p}\). When chattering occures, \({\mathop {{\textrm{sign}}}} ( \cdot )\) in the controller (8) can be replaced by \(\frac{{{e_p}\left( t \right) }}{{\left| {{e_p}\left( t \right) } \right| + 0.01}}\).

Remark 4

For [21], the asymptotic synchronization of MNNs is investigated, which ST of asymptotic synchronization may be infinitely long. Compared with FTS in [27, 28], the FXTS can make the system errors tend to 0 faster, which has more practical application value. The ST of FXTS compared to FTS can be adjusted by controller parameters regardless of the system initial values.

Remark 5

Different from [20], which NNs only considered time-delayed and unknown parameters, this paper considers unknown parameters, mixed delays and stochastic perturbations, so the obtained results are more comprehensive and general. In other words, [20] can be regarded as a special case of this paper.

This part discusses the synchronization of MNNs in fixed-time, and the ST can be regulated by parameters. However, the ST that can be set arbitrarily by users is worth studying in the current. Next part will study PATS of MNNs.

3.2 Preassigned-Time Synchronization of Sochastic MNNs

This part presents a novel SMS that guarantee the system achieving PATS.

$$\begin{aligned} {s_p}(t) = {e_p}(t) + \frac{{{v_2}}}{{{T_{c2}}}}m\int _0^t {\left( {{{\left| {{e_p}(\tau )} \right| }^{{\theta _{\mathrm{{3}}}}}}\mathrm{{ + }}{{\left| {{e_i}(\tau )} \right| }^{{\theta _4}}}} \right) } sign\left( {{e_p}(\tau )} \right) d\tau , \end{aligned}$$
(27)

where \(m> 0,0< {\theta _{\mathrm{{3}}}} < 1,{\theta _4} > 1\). To prove the preassigned-time accessibility of SMS, the adaptive controller is designed as

$$\begin{aligned} \begin{array}{*{20}{l}} {{u_p}(t) = - {\alpha _p}(t){e_p}(t) - \frac{{{v_1}}}{{{T_{c1}}}}{\beta _p}sign({s_p}(t)){{\left| {{s_p}(t)} \right| }^{{\theta _1}}} - \frac{{{v_1}}}{{{T_{c1}}}}{{\beta '}_p}sign({s_p}(t)){{\left| {{s_p}(t)} \right| }^{{\theta _2}}}}\\ {\;\;\;\;\;\begin{array}{*{20}{c}} {}&{}{}&{}{ - \frac{{{v_1}}}{{{T_{c1}}}}{l_p}\omega \Phi ({s_p}(t)) - \frac{{{v_1}}}{{{T_{c1}}}}{{\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{k_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] }^{\frac{{{\theta _1} + 1}}{2}}}\Phi ({s_p}(t))} \end{array}}\\ {\;\;\;\;\;\begin{array}{*{20}{c}} {}&{}{}&{}{ - \frac{{{v_1}}}{{{T_{c1}}}}{{\left[ {\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {{{k'}_p}({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] }^{\frac{{{\theta _2} + 1}}{2}}}\Phi ({s_p}(t)) - {\delta _p}e_p^2(t)\Phi ({s_p}(t))} \end{array}}\\ {\;\;\;\;\;\begin{array}{*{20}{c}} {}&{}{}&{}{ - {\gamma _p}sign({s_p}(t)) - {\varepsilon _p}{e_p}(t) - {\lambda _p}\sum \limits _{q = 1}^n {e_q^2(t - \tau (t))\Phi ({s_p}(t))} } \end{array}}\\ {\;\;\;\;\;\begin{array}{*{20}{c}} {}&{}{}&{}{ - \frac{{{v_2}}}{{{T_{c2}}}}msign({e_p}(t)){{\left| {{e_p}(t)} \right| }^{{\theta _3}}} - \frac{{{v_2}}}{{{T_{c2}}}}msign({e_p}(t)){{\left| {{e_p}(t)} \right| }^{{\theta _4}}},} \end{array}} \end{array} \end{aligned}$$
(28)

the controller parameters are mentioned as (8). The adaptive laws \(\mathop {{{{\hat{a}}}_{pq}}}\limits ^ \cdot (t),\mathop {{{{\hat{b}}}_{pq}}}\limits ^ \cdot (t)\) and \(\mathop {{\hat{c}}{}_p}\limits ^ \cdot \left( t \right) \) are consistent with (9), and

$$\begin{aligned} {{{{\dot{\alpha }} }_p}\left( t \right) = {s_p}(t)e_p^2(t) - \frac{{{v_1}}}{{{T_{c1}}}}{\phi _p}sign({\alpha _p}(t)){{\left| {{\alpha _p}(t)} \right| }^{{\theta _1}}} - \frac{{{v_1}}}{{{T_{c1}}}}{{\phi '}_p}sign({\alpha _p}(t)){{\left| {{\alpha _p}(t)} \right| }^{{\theta _2}}},} \end{aligned}$$
(29)

where \( {\phi _p}> 0,{\phi '_p} > 0. \)

Based on the controller (28), the following results are derived.

Theorem 3

If the inequalitis (10) hold, the PATS of MNNs can be achieved at preassigned-time \( {T_{c1}}\), and \({v_1}\) satisfy

$$\begin{aligned} {v_1} = \frac{1}{{1 - {\theta _1}}}\frac{{{2^{\varpi - 1}}}}{{{i_4}^{\frac{1}{\varpi }}\left( {\varpi - 1} \right) }}{i_3}^{\frac{{1 - \varpi }}{\varpi }} \end{aligned}$$
(30)

where \(\varpi = 1 + \frac{{{\theta _2} - 1}}{{1 - {\theta _1}}}\), \( {i_3} = \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\beta _p},{l_p},{k_p},{\phi _p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}},{i_4} = {n^{\frac{{1 - {\theta _2}}}{2}}}\mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\beta '}_p},{l_p},{{k'}_p},{{\phi '}_p}} \right\} {2^{\frac{{{\theta _2} + 1}}{2}}},0< {\theta _1} < 1,{\theta _2} > 1. \)

Proof

Consider appropriate Lyapunov function as (12), it yields

$$\begin{aligned}{} & {} {{{\mathcal {L}}}V\left( t \right) } \le \sum \limits _{p = 1}^n \left\{ {\left[ { - \frac{{{c_p}}}{2} + \sum \limits _{q = 1}^n {\left( {\frac{{a_{pq}^{\max }{\rho _q} + b_{pq}^{\max }{\zeta _q}}}{2}} \right) } - {\varepsilon _p}} \right] s_p^2(t)} \right. \nonumber \\{} & {} \quad { + \left[ {\sum \limits _{q = 1}^n {\left( {a_{pq}^{\max }{M_1} + a_{pq}^{\max }{M_2} + 2r_{pq}^{\max }{M_1}m} \right) - {\gamma _p}} } \right] \left| {{s_p}(t)} \right| }\nonumber \\{} & {} \quad + \left[ {\sum \limits _{q = 1}^n {\left( {\frac{{b_{pq}^{\max }{\zeta _q} + {\mu _{pq}}}}{2}} \right) - {\lambda _p}} } \right] e_q^2(t - \tau (t))\left. { + \left[ {\sum \limits _{q = 1}^n {\frac{{{\eta _{qp}} + a_{qp}^{\max }{\rho _p}}}{2} - {\delta _p}} } \right] e_p^2(t)} \right\} \nonumber \\{} & {} \quad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{k_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _1} + 1}}{2}}}\nonumber \\{} & {} \quad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{k'}_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _2} + 1}}{2}}} \nonumber \\{} & {} \quad { - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\beta _p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {s_p^2\left( t \right) } } \right] }^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\beta '}_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {s_i^2\left( t \right) } } \right] }^{\frac{{{\theta _2} + 1}}{2}}}} \nonumber \\ \end{aligned}$$
$$\begin{aligned}{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] ^{\frac{{{\theta _1} + 1}}{2}}}\nonumber \\{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] ^{\frac{{{\theta _2} + 1}}{2}}} \nonumber \\{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\phi _p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] ^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\phi '}_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] ^{\frac{{{\theta _2} + 1}}{2}}}\nonumber \\{} & {} \quad \le - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{k_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _1} + 1}}{2}}}\nonumber \\{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{k'}_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\sum \limits _{q = 1}^n {({\hat{a}}_{pq}^2(t) + {\hat{b}}_{pq}^2(t))} } } \right] ^{\frac{{{\theta _2} + 1}}{2}}}\nonumber \\{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\beta _p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {s_p^2\left( t \right) } } \right] ^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\beta '}_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {s_p^2\left( t \right) } } \right] ^{\frac{{{\theta _2} + 1}}{2}}}\nonumber \\{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] ^{\frac{{{\theta _1} + 1}}{2}}}\nonumber \\{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{l_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {{{\left( {{{{\hat{c}}}_p}(t) - {c_p}} \right) }^2}} } \right] ^{\frac{{{\theta _2} + 1}}{2}}} \nonumber \\{} & {} \qquad - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\phi _p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{2^{\frac{{{\theta _1} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _i^2\left( t \right) } } \right] ^{\frac{{{\theta _1} + 1}}{2}}} - \mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\phi '}_p}} \right\} \frac{{{v_1}}}{{{T_{c1}}}}{n^{\frac{{1 - {\theta _2}}}{2}}}{2^{\frac{{{\theta _2} + 1}}{2}}}{\left[ {\frac{1}{2}\sum \limits _{p = 1}^n {\alpha _p^2\left( t \right) } } \right] ^{\frac{{{\theta _2} + 1}}{2}}}\nonumber \\{} & {} \quad \le - \frac{{{v_1}}}{{{T_{c1}}}}\left[ {{i_3}{{\left( {V\left( t \right) } \right) }^{\frac{{{\theta _1} + 1}}{2}}} + {i_4}{{\left( {V\left( t \right) } \right) }^{\frac{{{\theta _2} + 1}}{2}}}} \right] . \end{aligned}$$
(31)

where \( {i_3} = \mathop {\min }\limits _{1 \le p \le n} \left\{ {{\beta _p},{l_p},{k_p},{\phi _p}} \right\} {2^{\frac{{{\theta _1} + 1}}{2}}},{i_4} = {n^{\frac{{1 - {\theta _2}}}{2}}}\mathop {\min }\limits _{1 \le p \le n} \left\{ {{{\beta '}_p},{l_p},{{k'}_p},{{\phi '}_p}} \right\} {2^{\frac{{{\theta _2} + 1}}{2}}},0< {\theta _1} < 1,{\theta _2} > 1. \) The above derivation satisfies the condition of lemma 2, and the error reaches SMS at preassigned-time \( {T_{c1}}\) under Theorem 3. Next the following is to prove that the error on SMS converges to 0 at preassigned-time \( {T_{c2}}\).

Theorem 4

The system (5) converges to 0 on the SMS (26) at preassigned-time \( {T_{c2}}\), and \({v_2}\) satisfy

$$\begin{aligned} {v_2} = \frac{1}{{1 - {\theta _3}}}\frac{{{2^{\varpi - 1}}}}{{{m_4}^{\frac{1}{\varpi }}\left( {\varpi - 1} \right) }}{m_3}^{\frac{{1 - \varpi }}{\varpi }},\end{aligned}$$
(32)

where \(\varpi = 1 + \frac{{{\theta _4} - 1}}{{1 - {\theta _3}}}\), \( 0< {\theta _3} < 1,{\theta _4} > 1. \) According to the condition \( {s_p}(t) = 0,{{\dot{s}}_p}(t) = 0 \), we get

$$\begin{aligned} {{\dot{e}}_{p'}}(t) = - \frac{{{v_2}}}{{{T_{c2}}}}{{\dot{e}}_{p}}(t). \end{aligned}$$
(33)

where \({{\dot{e}}_{p}}(t)\) is described as formula (24).

Proof

The following Lyapunov functional is developed

$$\begin{aligned} V\left( t \right) = \frac{1}{2}\sum \limits _{p = 1}^n {e_{p'}^2(t)}. \end{aligned}$$
(34)

The derivation of \( {V\left( t \right) } \) is calculated in the same way as (26), the following formula is calculated.

$$\begin{aligned} {{\mathcal {L}}}V\left( {\theta \left( t \right) } \right) \le - \frac{{{v_2}}}{{{T_{c2}}}}\left[ {{m_3}{V^{\frac{{{\theta _3} + 1}}{2}}}\left( t \right) + {m_4}{V^{\frac{{{\theta _4} + 1}}{2}}}\left( t \right) } \right] .\end{aligned}$$
(35)

where \( {m_3} = m{2^{\frac{{{\theta _3} + 1}}{2}}},{m_4} = m{2^{\frac{{{\theta _4} + 1}}{2}}}{n^{\frac{{1 - {\theta _4}}}{2}}} \). The above proof is derived that the error on SMS converges to 0 at preassigned-time \( {T_{c2}}\) under adaptive controller. \(\square \)

Remark 6

From [27] and [28], the ST of FTS connects with initial values. In addition, the error system cannot converge in finite time, when the condition of large initial values. In cases where the initial values are difficult to obtain, then FXTS can be used. However, compared with the ST that can not guarantee the accuracy in FXTS, the ST of PATS can be set directly. In this literature, when PATS of MNNs is realized, the error system slides along the SMS within preassigned-time \( {T_{c1}}\) and the error system evolves in a neighborhood around the SMS within preassigned-time \( {T_{c2}}\). Compared with FTS and FXTS, \( {T_{c1}}\) and \( {T_{c2}}\) is independent of initial values and controller parameters, which can be set directly. As a result, it is not difficult to conclude the total time \( T = {T_{c1}} + {T_{c2}} \). PATS of MNNs is more practical compared with FXTS from the above.

Remark 7

Some authors have studied the synchronization by using the linear matrix inequality method, where the computational complexity of the problem increases with the increase of the number of decision variables [44]. When multiple types of delays exist at the same time, the state of the system is seriously delayed, which makes it more challenging to achieve synchronization of drive and response systems. Therefore, based on Lyapunov function and inequality technology, the FXTS and PATS problems are studied by designing adaptive state feedback controller. The discrete time delay and distributed time delay compensation terms are introduced into the controller to overcome the inffuence of mixed time delays, and further realize the synchronization of the system. The conditions obtained are easy to verify.

Fig. 1
figure 1

\( {x_1}\left( t \right) \) and \( {y_1}\left( t \right) \) with controller

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Fig. 2
figure 2

\( {x_2}\left( t \right) \) and \( {y_2}\left( t \right) \) with controller

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Fig. 3
figure 3

The trajectories of FXTS errors of MNNs with controller

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Fig. 4
figure 4

The trajectories of FXTS SMS

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Fig. 5
figure 5

The trajectories of PATS errors of MNNs with controller

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4 Numerical Examples

For drive system and response system, we simplify connection weights as

$$\begin{aligned}\begin{array}{l} {{\textbf{a}}_{pq}}\left( {{x_p}\left( t \right) } \right) = \left\{ \begin{array}{l} {A_1}\left( {pq} \right) ,\left| {{x_p}\left( t \right) } \right| \le {H_q},\\ {A_2}\left( {pq} \right) ,\left| {{x_p}\left( t \right) } \right|> {H_q}, \end{array} \right. \\ \\ {{\textbf{b}}_{pq}}\left( {{x_p}\left( t \right) } \right) = \left\{ \begin{array}{l} {B_1}\left( {pq} \right) ,\left| {{x_p}\left( t \right) } \right| \le {H_q},\\ {B_2}\left( {pq} \right) ,\left| {{x_p}\left( t \right) } \right|> {H_q}, \end{array} \right. \\ \\ {{\textbf{r}}_{pq}}\left( {{x_p}\left( t \right) } \right) = \left\{ \begin{array}{l} {R_1}\left( {pq} \right) ,\left| {{x_p}\left( t \right) } \right| \le {H_q},\\ {R_2}\left( {pq} \right) ,\left| {{x_p}\left( t \right) } \right| > {H_q}, \end{array} \right. \end{array}\end{aligned}$$

where

$$\begin{aligned}{A_1}\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {1.2}&{}{ - 0.2}\\ { - 0.8}&{}{0.8} \end{array}} \right] ,{A_2}\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} 1&{}{ - 0.3}\\ { - 0.4}&{}{0.8} \end{array}} \right] ,\\{B_1}\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} { - 1.2}&{}{0.3}\\ {0.7}&{}{ - 2.2} \end{array}} \right] ,{B_2}\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} { - 1.2}&{}{0.5}\\ {0.4}&{}{ - 2.2} \end{array}} \right] ,\\{R_1}\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} { - 0.5}&{}{0.7}\\ {0.3}&{}{ - 1.8} \end{array}} \right] ,{R_2}\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} { - 0.4}&{}{0.8}\\ {0.5}&{}{ - 1.7} \end{array}} \right] .\end{aligned}$$
Fig. 6
figure 6

The trajectories of PATS SMS

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Fig. 7
figure 7

The trajectories of adaptive laws

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Fig. 8
figure 8

The trajectories of adaptive laws

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Fig. 9
figure 9

The trajectories of adaptive laws

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Fig. 10
figure 10

The trajectories of adaptive laws of the controller

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The switching threshold is \( {H _q} = 1.\) The delays are \( {\tau _1}\left( t \right) = {\tau _2}\left( t \right) = \sin \left( t \right) \) and \({h_1}\left( t \right) = {h_2}\left( t \right) = \frac{{{e^t}}}{{{e^t} + 1}}. \)

The activation functions are

$$\begin{aligned}{f_q}\left( {{x_q}\left( t \right) } \right) = \tanh \left( {{x_q}\left( t \right) } \right) ,{g_q}\left( {{x_q}\left( {t - \tau \left( t \right) } \right) } \right) = \frac{{\left| {{x_q}\left( {t - \tau \left( t \right) } \right) + 1} \right| - \left| {{x_q}\left( {t - \tau \left( t \right) } \right) - 1} \right| }}{2}.\end{aligned}$$

The stochastic perturbation is \({\chi _{11}}(t,{x_1}(t),{x_1}(t - {\tau _1}(t))) = {\chi _{21}}(t,{x_1}(t),{x_1}(t - {\tau _1}(t))) = \sin \left( {{x_1}(t)} \right) + 0.5{x_1}\left( {t - {\tau _1}(t)} \right) ,{\chi _{12}}(t,{x_2}(t),{x_2}(t - {\tau _2}(t))) = {\chi _{22}}(t,{x_2}(t),{x_2}(t - {\tau _2}(t))) = 0.5\sin \left( {{x_2}(t)} \right) + 0.7{x_2}\left( {t - {\tau _2}(t)} \right) . \) The initial values are \( {x_0} = \left[ { 6,- 8} \right] ,{y_0} = \left[ { - 10,13} \right] . \)

To achieve stochastic MNNs FXTS, the parameters are choosed as \( {\rho _1} = {\rho _2} = 1,{\zeta _1} = {\zeta _2} = 1.5,{c_1}\mathrm{{ = }}{c_2}\mathrm{{ = 1}}, {M_1} = {M_2} = 1,m = 1,{\mu _{11}} = {\mu _{12}} = {\mu _{21}} = {\mu _{22}} = 1,{\eta _{11}} = {\eta _{12}} = {\eta _{21}} = {\eta _{22}} = 1. \) And controller parameters are set as \( {\varepsilon _1} \ge 1.55,{\varepsilon _2} \ge 2.52,{\gamma _1} \ge 6,{\gamma _2} \ge 9.4,{\lambda _1} \ge 21.35,{\lambda _2} \ge 22.175,{\delta _1} \ge 15.45,{\delta _2} \ge 15.1. \) Thus, the controller parameters \( {u_p}\left( t \right) \) can be set as \( {\varepsilon _1} = 1.55,{\varepsilon _2} = 2.52,{\gamma _1} = 6,{\gamma _2} = 9.4,{\lambda _1} = 21.35,{\lambda _2} = 22.175,{\delta _1} = 15.45,{\delta _2} = 15.1.\) Furthermore, other parameters can be choosen as \( {\beta _1} = {\beta _2} = 1,{{\beta '}_1} = {{\beta '}_2} = 1.5,{l_1} = {l_2} = 0.15,{k_1} = {k_2} = 1,{{k'}_1} = {{k'}_2} = 1.5,q = 6,{\phi _1} = {\phi _2} = 3,{{\phi '}_1} = {{\phi '}_2} = 3,{\theta _1} = 0.25,{\theta _2} = 1.2,{\theta _3} = 0.5,{\theta _4} = 2.\) Based on above values, the upper bound of ST is about 1.4s. The state curves of \(x_1(t)\) and \(y_1(t)\), \(x_2(t)\) and \(y_2(t)\) finally reach consensus from Figs. 1 and 2. And Fig. 3 indicates the errors converge to 0. From Fig. 4, the SMS tends to be stable. Thus, the theoretical result is effective. However, \({T_{\nu 1}} + {T_{\nu 2}}=25.5s\) is obtained by calculating from (11) and (22), the system synchronization takes at least 25.5s in practice, which means the practical ST is much larger than the theoretical ST. Therefore, in order to realize the fast convergence of the system, it is necessary to investigate PATS of MNNS.

Based on same parameters as FXTS, we set \({T_{c1}} = 0.7s,{T_{c2}} = 0.7s\). As shown in Fig. 5, the errors converge to 0 at preassigned time 1.4s. The ST in PATS is the set value, which is not affected by any parameters. The SMS is shown in Fig. 6. It is not hard to see from Fig. 7 to Fig. 10 that MNNs can implement PATS with adaptive laws of the controller. Constructing weight update laws to identify unknown weights of drive systems. It indicates that the proposed ASMC scheme is effective through above description.

5 Conclusion

The FXTS and PATS of stochastic MNNs is considered in this paper. The influence of mixed delays, stochastic perturbations and unknown parameters are taken into account of MNNs. The adaptive controller and SMS are designed to deal with the uncertainties. Some sufficient conditions and ST of FXTS and PATS are deduced by calculation. The simulation example indicates the applicability through the developed scheme. Event-triggered control (ETC) is an economical control method, which can decrease unnecessary traffic over the network and reduce computational cost. Meanwhile, the fractional-order MNNs (FMNNs) are also derived much attention in synchronization and stability [43]. In addition, in order to achieve the security of information transmission, image encryption has become an important application. Therefore, the synchronization of FMNNs via ETC and its application in image encryption will be our main research direction in the future.