Fixed/Predefined-Time Synchronization for Delayed Memristive Reaction-Diffusion Neural Networks Subject to Stochastic Disturbances

Abstract

This paper investigates the fixed-time (FXT) and predefined-time (PDT) synchronization of memristive neural networks (MNNs) subject to stochastic disturbances, reaction-diffusion terms, and time delays. First, a new PDT stability criterion is established for stochastic nonlinear systems, which permits a priori assignment of the settling time bound regardless of initial conditions, and offers a more concise form than prior results. Second, by leveraging Green’s formula, integral inequality, and stochastic analysis, some sufficient conditions are derived to guarantee FXT and PDT synchronization of introduced stochastic MNNs with reaction-diffusion terms. Finally, numerical simulations are given to validate the effectiveness of the proposed synchronization scheme.

1. Introduction

Over the past thirty years, neural networks have attracted considerable attention due to their powerful information processing capabilities and wide applications in pattern recognition, signal processing, associative memory, and optimization [1,2]. Therefore, the various dynamical behavior of neural networks such as stability, periodicity, and synchronization has been extensively studied [3]. Meanwhile, in 1971, Chua predicted the existence of the memristor as the fourth basic circuit element [4], and its physical realization in 2008 [5] opened new avenues for neuromorphic computing. In fact, memristors can exhibit synapse-like learning and memory properties, making them an ideal tool for constructing MNNs where connection weights vary according to the history of neuronal activity [6,7,8]. MNNs have shown great potential in various applications including pattern recognition [9], signal processing [10], and associative memory [11].

It is well known that time delays are inherent in neural systems due to finite signal transmission speeds along axons, synaptic processing times, and feedback loops. The presence of delays can significantly affect network dynamics, potentially causing oscillations, instability, or bifurcations. The analysis of delay differential equations constitutes a vibrant field of research, with applications spanning from predator–prey systems [12] to neural networks. Therefore, introducing time delays into MNNs models is essential for accurate description of real neural circuits and hardware implementations. Besides time delay, real neural systems are inevitably subject to various stochastic perturbations, such as thermal noise in electronic components, random release of neurotransmitters at synapses, and environmental fluctuations. In fact, stochastic modeling is crucial for robustness analysis, as demonstrated in diverse fields including epidemiological dynamics [13]. For MNNs, stochastic disturbances must be explicitly considered to ensure reliable performance under realistic noisy conditions [14].

In addition, a reaction-diffusion process is a common phenomenon of spatial diffusion observed in the information transmission of biological neurons and the movement of electrons in non-uniform electromagnetic fields [15,16]. Reaction-diffusion neural network models have been proposed to capture this spatial-temporal coupling, where the state evolution depends on both spatial and temporal variables [17,18]. In recent decades, reaction-diffusion effects have been incorporated into MNNs modeling, leading to significant progress in the dynamics analysis of MNNs with reaction-diffusion [19,20,21,22,23]. In [24], by designing a pinning feedback controller, the author investigated the synchronization problem of a stochastic delay memristive reaction-diffusion neural network. In [25], the authors analyzed the global exponential stability problem for stochastic MNNs with reaction-diffusion terms and hybrid delays.

In practical applications, we usually aim to achieve synchronization of the drive–response neural network system in a bounded time [26,27]. In contrast to infinite-time synchronization, finite-time (FNT) synchronization inherently requires a faster convergence rate. However, the settling time of FNT synchronization depends on the initial conditions, which cause a difficulty to estimate or adjust in practice. To address this issue, Polyakov introduced the FXT convergence concept [28], where the settling time is bounded and independent of the system’s initial conditions. Currently, FXT synchronization of various types of neural networks including MNNs have be investigated [29,30,31,32,33].

Despite the considerable advances in FXT control for MNNs, its settling time still depends on system parameters and thus it is challenging to adjust to meet the engineering requirements. To address this issue, PDT stability is introduced where system solutions can reach zero point before the given predefined time and it is independent of any system parameters and initial values. Owing to this advantageous feature, various PDT control schemes have been developed [34,35] and applied in many fields [36]. However, there are few results addressing the FXT and PDT synchronization problems for stochastic MNNs with reaction-diffusion terms and time delays. Motivated by the discussion above, this paper studies the FXT and PDT synchronization of delayed stochastic MNNs with reaction-diffusion terms. The main contributions of it are summarized below: (1) A novel PDT stability criterion is proposed for stochastic nonlinear systems, which has a more concise form. (2) By integrating the Lyapunov function method with the divergence theorem and inequality techniques, several sufficient conditions for ensuring the FXT/PDT synchronization of the addressed system are introduced. (3) Some numerical simulations are given to verify the theoretical results.

Notations: Let $\mathsf{\Omega }=\left\{w={(w_1,w_2,\dots ,w_l)}^T\mid |w_k|<h_k,k=1,2,\dots ,l\right\}$ be a bounded compact set with a smooth boundary $\partial \mathsf{\Omega }$ and $mes\mathsf{\Omega }>0$, where $mes\mathsf{\Omega }$ denotes the Lebesgue measure of $\mathsf{\Omega }$, and $R^n$ the n-dimensional Euclidean space. Some key parameters used in the paper are explained in

Table 1. Key parameters and descriptions.

Symbol	Description
$a_{mn}(·),b_{mn}(·)$	State-dependent memristive connection weight
${\mathring{a}}_{mn}$, ${\breve{a}}_{mn}$	Values of $a_{mn}(·)$ below and above the switching threshold $T_m$
${\mathring{b}}_{mn}$, ${\breve{b}}_{mn}$	Values of $b_{mn}(·)$ below and above the switching threshold $T_m$
${\overline{a}}_{mn}$, ${\underline{a}}_{mn}$	Maximum and minimum absolute values of ${\mathring{a}}_{mn}$ and ${\breve{a}}_{mn}$
${\overline{b}}_{mn}$, ${\underline{b}}_{mn}$	Maximum and minimum absolute values of ${\mathring{b}}_{mn}$ and ${\breve{b}}_{mn}$
$L_{mn}^a,L_{mn}^b$	Variation range of the memristive weight $a_{mn}(·),b_{mn}(·)$
${\gamma }_m$	Key parameter in the fixed/predefined-time stability condition

.

2. Model Description and Preliminaries

We consider the following stochastic delayed memristive reaction-diffusion neural networks model

$$\begin{array}{cc}\hfill d{ħ}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{ħ}_m(t,w)}{\partial w_k^2}}-c_m{ħ}_m(t,w)+\sum _{n=1}^N{a}_{mn}\left({ħ}_m(t,w)\right)f_n\left({ħ}_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^N{b}_{mn}\left({ħ}_m(t,w)\right)g_n\left({ħ}_n(t-\aleph ,w)\right)\right]dt+{\varrho }_m(t,{ħ}_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(1)

where $m,n\in \mathbf{I}\triangleq \{1,2,\dots ,N\}$, with N standing for the total neuron number. ${ħ}_m(t,w)$ represents the state of the m-th neuron at time t and spatial point $w\in \mathsf{\Omega }\subseteq R^l$. $d_{mk}^{*}\ge 0$ is the transmission diffusion coefficient, $c_m>0$ is the self-inhibition coefficient. $\aleph >0$ denotes the constant transmission delay between neurons. The functions $f_n\left({ħ}_n(t,w)\right)$ and $g_n\left({ħ}_n(t-\aleph ,w)\right)$ represent the neuronal activation functions of the n-th neuron at the times t and $t-\aleph$, respectively. Furthermore, ${\varrho }_m(·)$ denotes the noise intensity function, and ${\overline{w}}_m\left(t\right)$ is a scalar standard Brownian motion defined on the complete probability space $\left(\mathsf{\Omega },\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge t_0},\mathcal{P}\right)$. The memristive connection weights $a_{mn}\left({ħ}_m(t,w)\right)$ and $b_{mn}\left({ħ}_m(t,w)\right)$ are defined as follows:

$$a_{mn}\left({ħ}_m(t,w)\right)=\left\{\begin{array}{cc}{\mathring{a}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|\le T_m,\hfill \\ {\breve{a}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|>T_m,\hfill \end{array}\right.$$

$$b_{mn}\left({ħ}_m(t,w)\right)=\left\{\begin{array}{cc}{\mathring{b}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|\le T_m,\hfill \\ {\breve{b}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|>T_m,\hfill \end{array}\right.$$

where $T_m>0$ denotes the switching jump; ${\mathring{a}}_{mn}$, ${\breve{a}}_{mn}$, ${\mathring{b}}_{mn}$, and ${\breve{b}}_{mn}$ for $m,n=1,2,\dots ,N$ are known constants associated with memristances. The boundary conditions and initial values of system (1) are specified as follows:

$$\begin{array}{cc}\hfill {ħ}_m(t,w)& =0,(t,w)\in [-\aleph ,+\infty )\times \partial \mathsf{\Omega },\hfill \\ \hfill {ħ}_m(s,w)& ={\varphi }_m(s,w),(s,w)\in [-\aleph ,0]\times \mathsf{\Omega },\hfill \end{array}$$

(2)

where ${\varphi }_m(s,w)$ for $m=1,2,\dots ,N$ is a continuous and bounded function.

Due to the discontinuous nature of the right-hand side in the switching system (1), regular solutions are not defined. Therefore, from the theory of differential inclusions and Filippov solutions, system (1) can be re-expressed as

$$\begin{array}{cc}\hfill d{ħ}_m(t,w)& \in \left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{ħ}_m(t,w)}{\partial w_k^2}}-c_m{ħ}_m(t,w)+\sum _{n=1}^{N}K\left[a_{mn}\left({ħ}_m(t,w)\right)\right]f_n\left({ħ}_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^{N}K\left[b_{mn}\left({ħ}_m(t,w)\right)\right]g_n\left({ħ}_n(t-\aleph ,w)\right)\right]dt+{\varrho }_m(t,{ħ}_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(3)

where

$$K\left[a_{mn}\left({ħ}_m(t,w)\right)\right]=\left\{\begin{array}{cc}{\mathring{a}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|<T_m,\hfill \\ \overline{co}\left\{{\mathring{a}}_{mn},{\breve{a}}_{mn}\right\},\hfill & \left|{ħ}_m(t,w)\right|=T_m,\hfill \\ {\breve{a}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|>T_m,\hfill \end{array}\right.$$

and

$$K\left[b_{mn}\left({ħ}_m(t,w)\right)\right]=\left\{\begin{array}{cc}{\mathring{b}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|<T_m,\hfill \\ \overline{co}\left\{{\mathring{b}}_{mn},{\breve{b}}_{mn}\right\},\hfill & \left|{ħ}_m(t,w)\right|=T_m,\hfill \\ {\breve{b}}_{mn},\hfill & \left|{ħ}_m(t,w)\right|>T_m,\hfill \end{array}\right.$$

where $\overline{\mathrm{co}}$ denotes the convex closure of a set.

For simplicity, let ${\overline{a}}_{mn}=max\{|{\mathring{a}}_{mn}|,|{\breve{a}}_{mn}|\}$, ${\underline{a}}_{mn}=min\{|{\mathring{a}}_{mn}|,|{\breve{a}}_{mn}|\}$, ${\overline{b}}_{mn}=max\{|{\mathring{b}}_{mn}|,|{\breve{b}}_{mn}|\}$, ${\underline{b}}_{mn}=min\{|{\mathring{b}}_{mn}|,|{\breve{b}}_{mn}|\}$, $L_{mn}^a={\overline{a}}_{mn}-{\underline{a}}_{mn}$, $L_{mn}^b={\overline{b}}_{mn}-{\underline{b}}_{mn}$. Then, according to the measurable selection theorem, there exist two functions ${\tilde{a}}_{mn}\left(t\right)\in K\left[a_{mn}\left({ħ}_m(t,w)\right)\right]$ and ${\tilde{b}}_{mn}\left(t\right)\in K\left[b_{mn}\left({ħ}_m(t,w)\right)\right]$ such that

$$\begin{array}{cc}\hfill d{ħ}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{ħ}_m(t,w)}{\partial w_k^2}}-c_m{ħ}_m(t,w)+\sum _{n=1}^N{\tilde{a}}_{mn}\left(t\right)f_n\left({ħ}_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^N{\tilde{b}}_{mn}\left(t\right)g_n\left({ħ}_n(t-\aleph ,w)\right)\right]dt+{\varrho }_m(t,{ħ}_m(t,w))d{\overline{w}}_m\left(t\right).\hfill \end{array}$$

(4)

By considering system (1) as the drive system, we introduce the response system under control inputs as:

$$\begin{array}{cc}\hfill d{z}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{z}_m(t,w)}{\partial w_k^2}}-c_m{z}_m(t,w)+\sum _{n=1}^N{a}_{mn}\left(z_m(t,w)\right)f_n\left(z_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^N{b}_{mn}\left(z_m(t,w)\right)g_n\left(z_n(t-\aleph ,w)\right)+u_m(t,w)\right]dt+{\varrho }_m(t,z_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(5)

where $u_m(t,w)$ is the control input to be designed.

The boundary conditions and initial values for system (6) are given as follows:

$$\begin{array}{cc}\hfill z_m(t,w)& =0,(t,w)\in [-\aleph ,+\infty )\times \partial \mathsf{\Omega },\hfill \\ \hfill z_m(s,w)& ={\phi }_m(s,w),(s,w)\in [-\aleph ,0]\times \mathsf{\Omega },\hfill \end{array}$$

(6)

where the function ${\phi }_m(s,w)$$(m=1,2,\dots ,N)$ is continuous and bounded.

Similarly, there exist measurable functions ${\widehat{a}}_{mn}\left(t\right)\in K\left[a_{mn}\left(z_m(t,w)\right)\right]$ and ${\widehat{b}}_{mn}\left(t\right)\in K\left[b_{mn}\left(z_m(t,w)\right)\right]$ such that

$$\begin{array}{cc}\hfill d{z}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{z}_m(t,w)}{\partial w_k^2}}-c_m{z}_m(t,w)+\sum _{n=1}^N{\widehat{a}}_{mn}\left(t\right)f_n\left(z_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^N{\widehat{b}}_{mn}\left(t\right)g_n\left(z_n(t-\aleph ,w)\right)+u_m(t,w)\right]dt+{\varrho }_m(t,z_m(t,w))d{\overline{w}}_m\left(t\right).\hfill \end{array}$$

(7)

Let $e_m(t,w)=z_m(t,w)-{ħ}_m(t,w)$ be the synchronization error, then the error system can be derived as:

$$\begin{array}{cc}\hfill d{e}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{e}_m(t,w)}{\partial w_k^2}}-c_m{e}_m(t,w)+\sum _{n=1}^N{\widehat{a}}_{mn}\left(t\right)[f_n\left(z_n(t,w)\right)-f_n\left({ħ}_n(t,w)\right)]\right.\hfill \\ & +\sum _{n=1}^N[{\widehat{a}}_{mn}\left(t\right)-{\tilde{a}}_{mn}\left(t\right)]f_n\left({ħ}_n(t,w)\right)+\sum _{n=1}^N[{\widehat{b}}_{mn}\left(t\right)-{\tilde{b}}_{mn}\left(t\right)]g_n\left({ħ}_n(t-\aleph ,w)\right)\hfill \\ & \left.+\sum _{n=1}^N{\widehat{b}}_{mn}\left(t\right)[g_n\left(z_n(t-\aleph ,w)\right)-g_n\left({ħ}_n(t-\aleph ,w)\right)]+u_m(t,w)\right]dt\hfill \\ & +{\tilde{\varrho }}_m(t,e_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(8)

where ${\tilde{\varrho }}_m(t,e_m(t,w))={\varrho }_m(t,z_m(t,w))-{\varrho }_m(t,{ħ}_m(t,w))$. The structural framework of the drive and response systems is illustrated in Figure 1.

Remark 1.

Compared with the models studied in [15,16,17], system (1) exhibits a higher level of generality since it incorporates reaction-diffusion terms, time delays, and stochastic disturbances. When the memristive connection weights are set to constant values, system (1) can be reduced to a common reaction-diffusion neural network models with stochastic disturbances. In fact, system (1) integrates several physically meaningful features: (1) The memristive weights $a_{mn}(·)$ and $b_{mn}(·)$ emulate synaptic plasticity, whose connection strengths can be dynamically adjusted according to the activity state of neurons; (2) The reaction-diffusion terms ${\sum }_{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{ħ}_m(t,w)}{\partial w_k^2}}$ describe the propagation process of spatial signals, such as the conduction process of electrical potential in neural tissue; (3) The stochastic disturbance term ${\varrho }_{m}d{\overline{w}}_m\left(t\right)$ corresponds to various noise sources, such as the random release process of neurotransmitters.

Table 2. Comparisons between existing studies and our work.

	[5]	[18]	[23]	[31]	[37]	This Paper
memristive items	✓	×	✓	✓	×	✓
reaction-diffusion items	×	✓	✓	×	×	✓
stochastic disturbances	✓	×	×	×	✓	✓
FXT stability	✓	✓	×	✓	✓	✓
PDT stability	✓	×	×	×	✓	✓

presents comparisons between existing studies and our work.

Assumption 1.

The activation functions $f_m$ and $g_m$ are globally Lipschitz continuous and bounded, i.e., there exist positive constants $L_m^f$, $L_m^g$, $G_m^f$, and $G_m^g$ such that for any $x,y,\theta \in R$ and $m=1,2,\dots ,N$,

$$|f_m\left(x\right)-f_m\left(y\right)|\le L_m^f|x-y|,|g_m\left(x\right)-g_m\left(y\right)|\le L_m^g|x-y|,$$

and

$$|f_m\left(\theta \right)|\le G_m^f,|g_m\left(\theta \right)|\le G_m^g.$$

Assumption 2.

The noise intensity function ${\varrho }_m(t,e_m\left(t\right))$ satisfies the uniformly Lipschitz condition:

$$trace\left[{\tilde{\varrho }}_i^T(t,e_m\left(t\right)){\tilde{\varrho }}_m(t,e_m\left(t\right))\right]\le L_m^{\varrho }{e}_m^2\left(t\right),$$

(9)

where $L_m^{\varrho }$ is a positive constant.

Take the following stochastic nonlinear system into consideration:

$$\begin{array}{cc}\hfill d\mu & =\mathcal{F}\left(\mu \right)dt+\mathcal{G}\left(\mu \right)d\overline{w},\hfill \end{array}$$

where $\mu \left(0\right)={\mu }_0$, $\mu \left(t\right)\in R^n$ is the system state, and $\overline{w}\left(t\right)$ is a scalar Brownian motion defined on the complete probability space $\left(\mathsf{\Omega },\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge t_0},\mathcal{P}\right)$. The functions $\mathcal{F}:R^n\to R^n$ and $\mathcal{G}:R^n\to R^{n\times r}$ are continuous, with $\mathcal{F}\left(0\right)=0$ and $\mathcal{G}\left(0\right)=0$.

Definition 1

([38]). For a function $V\left(\mu \right)$ that is twice continuously differentiable, the infinitesimal generator $\mathcal{L}$ corresponding to system (9) is defined as follows: $\mathcal{L}V=\left({\displaystyle \frac{\partial V}{\partial \mu }}\right)·\mathcal{F}+{\displaystyle \frac{1}{2}}\mathit{Tr}\left\{{\mathcal{G}}^{\top }\left({\displaystyle \frac{{\partial }^{2}V}{\partial {\mu }^2}}\right)\mathcal{G}\right\}.$

Definition 2

([39]). The trivial solution of system (9) is said to be stochastically FNT stable, if the stochastic system (9) admits a solution for any initial value $\mu \left(0\right)\in R^n$, and the following statements hold:

(i) 

FNT attractiveness in probability: For every initial value ${\mu }_0\in {\mathrm{R}}^n$, and any solution $\mu \left(t;{\mu }_0\right)$, the first hitting time of $\mu \left(t;{\mu }_0\right)$, i.e., ${\tau }_{{\mu }_0}=inf\left\{t\ge 0:\mu \left(t;{\mu }_0\right)=0\right\}$, called stochastic ST, is finite almost surely, that is $\mathrm{P}\left({\tau }_{{\mu }_0}<\infty \right)=1$. Furthermore, $\mu \left(t+{\tau }_{{\mu }_0};{\mu }_0\right)=0,\mathrm{a}.\mathrm{s}.,\forall t\ge 0$.

(ii) 

Stability in probability: For any solution $\mu \left(t;{\mu }_0\right)$, and every pair of $\epsilon \in (0,1)$ and $r>0$, there exists a $\delta (\epsilon ,r)>0$ such that $\mathrm{P}\left(\left|\mu \left(t;{\mu }_0\right)\right|<r\right)\ge 1-\epsilon$ whenever $\left|{\mu }_0\right|<\delta$.

Definition 3

([40]). The trivial solution of system (9) is said to be stochastically FXT stable if the following conditions hold:

(i) 

The trivial solution is stochastically FNT stable.

(ii) 

There exists a constant $T_0>0$, which is independent of the initial condition ${\mu }_0$, such that ${sup}_{{\mu }_0\in R^n}E\left[\sigma \left({\mu }_0\right)\right]\le T_0.$

Definition 4

([41]). System (9) is said to be stochastically PDT stable if it is stochastically FXT stable, and any $T_p>0$

$$\underset{{\mu }_0\in {\mathbb{R}}^n\setminus \left\{0\right\}}{sup}\mathcal{E}T\left({\mu }_0\right)\le T_p.$$

Lemma 1

([42]). Let $h\left(\mu \right)$ denote a real-valued continuous function defined on Ω with ${h\left(\mu \right)|}_{\partial \mathsf{\Omega }}=0$. Then,

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{h}^2\left(\mu \right)d\mu & \le h_k^2{\int }_{\mathsf{\Omega }}{\left({\displaystyle \frac{\partial h}{\partial {\mu }_k}}\right)}^{2}d\mu .\hfill \end{array}$$

Lemma 2

([37]). Suppose $V\left(\mu \left(t\right)\right):R^n\to R$ is a C-regular function and satisfies the differential inequality presented as follows:

$$\begin{array}{cc}\hfill \mathcal{L}V\left(\mu \right(t\left)\right)& \le \gamma V\left(\mu \left(t\right)\right)-{\delta }_1{V}^{{\delta }_3}\left(\mu \left(t\right)\right)-{\delta }_2{V}^{{\delta }_4}\left(\mu \left(t\right)\right),\mu \left(t\right)\in R^{\mathrm{n}}\setminus \left\{0\right\},\hfill \end{array}$$

where $\gamma <0$, ${\delta }_1>0$, ${\delta }_2>0$, ${\delta }_3>1$, and $0<{\delta }_4<1$. Then, the zero equilibrium of system (9) achieves FXT stability in probability, with the ST being estimable by

$$\begin{array}{c}\hfill E\left[T\left({\mu }_0,\omega \right)\right]<T_{max}={\displaystyle \frac{1}{\gamma {\lambda }_1(1-{\delta }_3)}}ln\left(1-{\displaystyle \frac{\gamma }{{\delta }_2}}{\left({\displaystyle \frac{{\delta }_2}{{\delta }_1}}\right)}^{{\lambda }_1}\right),\end{array}$$

where ${\lambda }_1={\displaystyle \frac{1-{\delta }_4}{{\delta }_3-{\delta }_4}}$.

Lemma 3

([41]). Given any $T_p>0$, if there exits a positive and radially unbounded Lyapunov function $V:R^n\to R_{+}$${\mathcal{C}}^2$ such that

$$\begin{array}{c}\hfill \mathcal{L}V\left(\mu \right)\le -{\displaystyle \frac{1}{T_p}}H\left(V\left(\mu \right)\right),\mu \in R^n\setminus \left\{0\right\},\end{array}$$

where $H:R_{+}\to R_{+}$ is a function satisfying $\dot{H}(·)\ge 0$, $H\left(s\right)>0$ for all $s>0$, and ${\int }_0^{\infty }{\displaystyle \frac{1}{H\left(s\right)}}ds\le 1$. Then, system (9) is stochastically PDT stabilizable, with ${sup}_{{\mu }_0\in {\mathbb{R}}^n\setminus \left\{0\right\}}ET\left({\mu }_0\right)\le T_p$.

Lemma 4.

For system (9), suppose there exists a positive definite and radially unbounded Lyapunov function $V\left(\mu \right)$. Given any $T_p>0$, a control input $u^{*}\left(\mu \left(t\right)\right)\in \mathcal{U}$ exists such that when applied to system (9), the resulting state satisfies

$$\begin{array}{cc}\hfill \mathcal{L}V\left(\mu \right)& \le {\displaystyle \frac{1}{T_p}}\left(\gamma V\left(\mu \left(t\right)\right)-{\delta }_1{V}^{{\delta }_3}\left(\mu \left(t\right)\right)-{\delta }_2{V}^{{\delta }_4}\left(\mu \left(t\right)\right)\right),\hfill \end{array}$$

where $\gamma <0$, ${\delta }_1>0$, ${\delta }_3>1$, and $0<{\delta }_4<1$, ${\lambda }_1={\displaystyle \frac{1-{\delta }_4}{{\delta }_3-{\delta }_4}}$ and ${\delta }_2={\left[{\displaystyle \frac{{\delta }_1^{{\lambda }_1}}{\gamma }}\left(1-e^{\gamma {\lambda }_1(1-{\delta }_3)}\right)\right]}^{{\displaystyle \frac{1}{{\lambda }_1-1}}}$. Then, system (9) achieves stochastic PDT stability, i.e., ${sup}_{{\mu }_0\in R^n\setminus \left\{0\right\}}\mathcal{E}T\left({\mu }_0\right)\le T_p$ holds true.

Proof. 

Let $H\left(V\left(\mu \right)\right)=-\gamma V\left(\mu \left(t\right)\right)+{\delta }_1{V}^{{\delta }_3}\left(\mu \left(t\right)\right)+{\delta }_2{V}^{{\delta }_4}\left(\mu \left(t\right)\right)$, obviously $H\left(V\right(\mu \left)\right)>0$ and $\dot{H}\left(V\left(\mu \left(t\right)\right)\right)>0$, moreover,

$$\begin{array}{cc}\hfill {\int }_0^{+\infty }{\displaystyle \frac{dV\left(\mu \right)}{H\left(V\right(\mu \left)\right)}}& ={\int }_0^{+\infty }{\displaystyle \frac{dV\left(\mu \right)}{-\gamma V\left(\mu \left(t\right)\right)+{\delta }_1{V}^{{\delta }_3}\left(\mu \left(t\right)\right)+{\delta }_2{V}^{{\delta }_4}\left(\mu \left(t\right)\right)}}\hfill \\ & \le {\displaystyle \frac{1}{\gamma {\lambda }_1(1-{\delta }_3)}}ln\left(1-{\displaystyle \frac{\gamma }{{\delta }_2}}{\left({\displaystyle \frac{{\delta }_2}{{\delta }_1}}\right)}^{{\lambda }_1}\right)\hfill \\ & =1.\hfill \end{array}$$

Therefore, from Lemma 3, the conclusion of Lemma 4 is satisfied. □

Remark 2.

Compared with the

$$\mathcal{L}V\left(\mu \right)\le -{\displaystyle \frac{a-a{b}_1}{\alpha \left(a-a{b}_1-1\right)\left(b_2-1\right)}}V{\left(\mu \right)}^{b_2}-{\displaystyle \frac{a}{\alpha }}V{\left(\mu \right)}^{b_1},$$

in Corollary 2.3 of Reference [41], the requirement of

$$\mathcal{L}V\left(\mu \right)\le {\displaystyle \frac{1}{T_p}}\left(rV\left(\mu \left(t\right)\right)-{\delta }_1{V}^{{\delta }_3}\left(\mu \left(t\right)\right)-{\delta }_2{V}^{{\delta }_4}\left(\mu \left(t\right)\right)\right),$$

in this paper is more general.

Remark 3.

Lemma 4 establishes a Lyapunov-based criterion for stochastic PDT stability. When the integral condition ${\int }_0^{+\infty }{\displaystyle \frac{dV\left(\mu \right)}{H\left(V\right(\mu \left)\right)}}\le 1$ holds, the system (9) achieves stochastic PDT stability. If the condition is relaxed to ${\int }_0^{+\infty }{\displaystyle \frac{dV\left(w\right)}{H\left(V\right(\mu \left)\right)}}\le \infty$, then system (9) can be in stochastic FXT stable.

3. Main Results

3.1. Fixed-Time Synchronization

In this subsection, we derive some FXT synchronization criteria between drive–response systems (1) and (5). To achieve this goal, we design the controller in the response system (5) as follows:

$$u_m(t,w)=\left\{\begin{array}{c}-{\lambda }_m\mathrm{sign}\left(e_m(t,w)\right)-{\displaystyle \sum _{n=1}^N}{\sigma }_m\mathrm{sign}\left(e_m(t,w)\right)|e_n(t-\aleph ,w)|\hfill \\ -v_m{\widehat{e}}^{\alpha -1}\left(t\right)e_m(t,w)-p_m{\widehat{e}}^{\beta -1}\left(t\right)e_m(t,w),\hfill & e_m(t,w)\ne 0,\hfill \\ 0,\hfill & e_m(t,w)=0,\hfill \end{array}\right.$$

(10)

where $\widehat{e}\left(t\right)={\left({\int }_{\mathsf{\Omega }}{\displaystyle \sum _{m=1}^N}{e}_m^2(t,w)dw\right)}^{{\displaystyle \frac{1}{2}}}$, $\alpha >1$, $0<\beta <1$, ${\lambda }_m$, ${\sigma }_m$, $v_m$ and $p_m$ are positive constants.

Remark 4.

In this paper, the sign function terms are introduced into the controller design. These terms play a crucial role in establishing the synchronization of the proposed drive–response system within the FXT/PDT framework. Although the inclusion of sign functions may introduce minor chattering in the control signal, such chattering remains within acceptable limits. In practical applications, to further attenuate chattering, the sign functions can be substituted with saturation functions, which can be defined in the following forms:

$$Sat\left(\mu \right)=\left\{\begin{array}{cc}\mathrm{sign}\left(\mu \right),\hfill & |\mu |\ge \theta ,\hfill \\ {\left({\displaystyle \frac{\mu }{\theta }}\right)}^{{\displaystyle \frac{\alpha }{\beta }}},\hfill & |\mu |<\theta ,\hfill \end{array}\right.$$

where $\mu \in \mathbb{R}$, $0<\alpha <\beta$ are odd numbers, $\theta >0$. Investigating FXT/PDT synchronization of stochastic MNNs via such smooth saturation-based control schemes constitutes an important direction for future research, and it will be one of our future research directions.

For convenience, denote

$$\begin{array}{cc}\hfill {\gamma }_m& =L_m^{\varrho }-2c_m-\sum _{k=1}^l{\displaystyle \frac{2d_{mk}^{*}}{h_k^2}}+\sum _{n=1}^N({\overline{a}}_{mn}{L}_n^f+{\overline{a}}_{nm}{L}_m^f).\hfill \end{array}$$

(11)

Theorem 1.

Under the Assumptions 1 and 2, if control gains ${\gamma }_m$, ${\lambda }_m$ and ${\sigma }_m$ of controller (10) satisfy the following inequalities:

$$\begin{array}{cc}\hfill {\lambda }_m& \ge L_{mn}^a{G}_n^f+L_{mn}^b{G}_n^g,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\sigma }_m& \ge {\overline{b}}_{mn}{\rho }_n,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \underset{1\le m\le N}{max}& \left\{{\gamma }_m\right\}<0,\hfill \end{array}$$

(14)

then the drive–response systems (1) and (5) can realize FXT synchronization in probability, with ST estimation $T_{max}$.

Proof. 

Select the Lyapunov function given below:

$$\begin{array}{cc}\hfill V\left(t\right)& ={\int }_{\mathsf{\Omega }}\sum _{m=1}^N{e}_m^2(t,w)dw.\hfill \end{array}$$

(15)

Calculating the $\mathcal{L}V\left(t\right)$ along the solution of system (8), we get

$$\begin{array}{cc}\hfill \mathcal{L}V\left(t\right)& ={\int }_{\mathsf{\Omega }}\left[2\sum _{m=1}^N{e}_m(t,w)\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{e}_m(t,w)}{\partial w_k^2}}-\sum _{m=1}^{N}2c_m{e}_m^2(t,w)\right.\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w){\widehat{a}}_{mn}\left(t\right)\left[f_n\left(z_n(t,w)\right)-f_n\left({ħ}_n(t,w)\right)\right]\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w)({\widehat{a}}_{mn}\left(t\right)-{\tilde{a}}_{mn}\left(t\right))f_n\left({ħ}_n(t,w)\right)\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w)({\widehat{b}}_{mn}\left(t\right)-{\tilde{b}}_{mn}\left(t\right))g_n\left({ħ}_n(t-\aleph ,w)\right)\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w){\widehat{b}}_{mn}\left(t\right)\left[g_n\left(z_n(t-\aleph ,w)\right)-g_n\left({ħ}_n(t-\aleph ,w)\right)\right]\hfill \\ & -\sum _{m=1}^{N}2{\lambda }_m\left|e_m(t,w)\right|-\sum _{m=1}^N\sum _{n=1}^{N}2{\sigma }_m\left|e_m(t,w)\right|\left|e_n(t-\aleph ,w)\right|\hfill \\ & -\sum _{m=1}^{N}2e_m(t,w)v_m{\widehat{e}}^{\alpha -1}\left(t\right)e_m(t,w)-\sum _{m=1}^{N}2e_m(t,w)p_m{\widehat{e}}^{\beta -1}\left(t\right)e_m(t,w)\hfill \\ & \left.+trace\left[{\tilde{\varrho }}_m^T(t,e(t,w)){\tilde{\varrho }}_m(t,e(t,w))\right]\phantom{\sum _{m=1}^N\sum _{k=1}^l}\right]dw.\hfill \end{array}$$

(16)

By applying the divergence theorem on the basis of boundary conditions, we get

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}\sum _{m=1}^{N}2e_m(t,w)\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{e}_m(t,w)}{\partial w_k^2}}dw& ={\int }_{\mathsf{\Omega }}\sum _{m=1}^{N}2e_m(t,w)\left[\nabla ·{\left(d_{mk}^{*}{\displaystyle \frac{\partial e_m(t,w)}{\partial w_k}}\right)}_{k=1}^l\right]dw\hfill \\ \hfill & ={\int }_{\partial \mathsf{\Omega }}2\sum _{m=1}^N\left[{\left(d_{mk}^{*}{e}_m(t,w){\displaystyle \frac{\partial e_m(t,w)}{\partial w_k}}\right)}_{k=1}^l·\mathbf{N}\right]dw\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}2\sum _{m=1}^N\sum _{k=1}^l{d}_{mk}^{*}{\left({\displaystyle \frac{\partial e_m(t,w)}{\partial w_k}}\right)}^{2}dw\hfill \\ \hfill & =-{\int }_{\mathsf{\Omega }}2\sum _{m=1}^N\sum _{k=1}^l{d}_{mk}^{*}{\left({\displaystyle \frac{\partial e_m(t,w)}{\partial w_k}}\right)}^{2}dw,\hfill \end{array}$$

(17)

where “·” represents the inner product, $\mathbf{N}$ denotes the outward-pointing unit normal field corresponding to the boundary $\partial \mathsf{\Omega }$, $\nabla =\left({\displaystyle \frac{\partial }{\partial w_1}},{\displaystyle \frac{\partial }{\partial w_2}},\dots ,{\displaystyle \frac{\partial }{\partial w_l}}\right)$ stands for the gradient operator, and ${\left({\displaystyle \frac{\partial e_m(t,w)}{\partial w_k}}\right)}_{k=1}^l$ is defined as the transposed vector ${\left({\displaystyle \frac{\partial e_m(t,w)}{\partial w_1}},{\displaystyle \frac{\partial e_m(t,w)}{\partial w_2}},\dots ,{\displaystyle \frac{\partial e_m(t,w)}{\partial w_l}}\right)}^{\mathrm{T}}$.

From Lemma 1, we get

$$\begin{array}{cc}\hfill -{\int }_{\mathsf{\Omega }}2\sum _{m=1}^N\sum _{k=1}^l{d}_{mk}^{*}{\left({\displaystyle \frac{\partial e_m(t,w)}{\partial w_k}}\right)}^{2}dw& \le -{\int }_{\mathsf{\Omega }}\sum _{m=1}^N\sum _{k=1}^l{\displaystyle \frac{2d_{mk}^{*}}{h_k^2}}{e}_m^2(t,w)dw.\hfill \end{array}$$

(18)

Using the Assumption 1 and inequality $2{\alpha }_1{\alpha }_2\le {\alpha }_1^2+{\alpha }_2^2$ for $w_1>0,w_2>0$, one has

$$\begin{array}{cc}\hfill \sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w){\widehat{a}}_{mn}\left(t\right)(f_n\left(z_n(t,w)\right)-f_n\left({ħ}_n(t,w)\right))& \le \sum _{m=1}^N\sum _{n=1}^{N}2\left|e_m(t,w)\right|{\overline{a}}_{mn}{L}_n^f\left|e_n(t,w)\right|\hfill \\ & \le \sum _{m=1}^N\sum _{n=1}^N({\overline{a}}_{mn}{L}_n^f+{\overline{a}}_{nm}{L}_m^f)e_m^2(t,w),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w)({\widehat{a}}_{mn}\left(t\right)-{\tilde{a}}_{mn}\left(t\right))f_n\left({ħ}_n(t,w)\right)& \le \sum _{m=1}^N\sum _{n=1}^{N}2\left|e_m(t,w)\right|L_{mn}^a{G}_n^f,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w)({\widehat{b}}_{mn}\left(t\right)-{\tilde{b}}_{mn}\left(t\right))g_n\left({ħ}_n(t-\aleph ,w)\right)& \le \sum _{m=1}^N\sum _{n=1}^{N}2\left|e_m(t,w)\right|L_{mn}^b{G}_n^g,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w){\widehat{b}}_{mn}\left(t\right)[g_n\left(z_n(t-\aleph ,w)\right)-g_n\left({ħ}_n(t-\aleph ,w)\right)]& \le \sum _{m=1}^N\sum _{n=1}^{N}2\left|e_m(t,w)\right|\hfill \\ & \times {\overline{b}}_{mn}{L}_n^g\left|e_n(t-\aleph ,w)\right|.\hfill \end{array}$$

(22)

Based on Assumption 2, we derive

$$\begin{array}{cc}\hfill trace\left[{\tilde{\varrho }}_m^T\left(t,e_m(t,w)\right){\tilde{\varrho }}_m\left(t,e_m(t,w)\right)\right]& \le \sum _{m=1}^N{L}_m^{\varrho }{e}_m^2(t,w).\hfill \end{array}$$

(23)

Furthermore, we have

$$\begin{array}{cc}\hfill -{\int }_{\mathsf{\Omega }}\sum _{m=1}^{N}2e_m(t,w)v_m{\widehat{e}}^{\alpha -1}\left(t\right)e_m(t,w)dw& =-2{\widehat{e}}^{\alpha -1}\left(t\right){\int }_{\mathsf{\Omega }}\sum _{m=1}^N{v}_m{e}_m^2(t,w)dw\hfill \\ & \le -2\underset{1\le m\le N}{min}\left\{v_m\right\}{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right).\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill -{\int }_{\mathsf{\Omega }}\sum _{m=1}^{N}2e_m(t,w)p_m{\widehat{e}}^{\beta -1}\left(t\right)e_m(t,w)dw& =-2{\widehat{e}}^{\beta -1}\left(t\right){\int }_{\mathsf{\Omega }}\sum _{m=1}^N{p}_m{e}_m^2(t,w)dw\hfill \\ & \le -2\underset{1\le m\le N}{min}\left\{p_m\right\}{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right).\hfill \end{array}$$

(25)

Substituting (17)–(25) into (16), it gains

$$\begin{array}{cc}\hfill \mathcal{L}V\left(t\right)& \le {\int }_{\mathsf{\Omega }}\sum _{m=1}^N\left[L_m^{\varrho }-\sum _{k=1}^l{\displaystyle \frac{2d_{mk}^{*}}{h_k^2}}-2c_m+\sum _{n=1}^N({\overline{a}}_{mn}{L}_n^f+{\overline{a}}_{nm}{L}_m^f)\right]e_m^2(t,w)dw\hfill \\ & +{\int }_{\mathsf{\Omega }}\sum _{m=1}^N\sum _{n=1}^N(2L_{mn}^a{G}_n^f+2L_{mn}^b{G}_n^g-2{\lambda }_m)\left|e_m(t,w)\right|dw\hfill \\ & +{\int }_{\mathsf{\Omega }}\sum _{m=1}^N\sum _{m=1}^N(2{\overline{b}}_{mn}{L}_n^g-2{\sigma }_m)\left|e_m(t,w)\right|\left|e_n(t-\aleph ,w)\right|dw\hfill \\ & -2\underset{1\le m\le N}{min}\left\{v_m\right\}{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2\underset{1\le m\le N}{min}\left\{p_m\right\}{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right)\hfill \\ & \le {\gamma }_{m}V\left(t\right)-2\underset{1\le m\le N}{min}\left\{v_m\right\}{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2\underset{1\le m\le N}{min}\left\{p_m\right\}{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right)\hfill \\ & \le \gamma V\left(t\right)-2v{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2p{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right),\hfill \end{array}$$

(26)

where $\gamma =\underset{1\le m\le N}{max}\left\{{\gamma }_m\right\}$, $v=\underset{1\le m\le N}{min}\left\{v_m\right\}>0$ and $p=\underset{1\le m\le N}{min}\left\{p_m\right\}>0$.

Let ${\delta }_1=2v$, ${\delta }_2=2p$, ${\delta }_3={\displaystyle \frac{\alpha +1}{2}}$, and ${\delta }_4={\displaystyle \frac{\beta +1}{2}}$. By inequality (17), we can infer that $\gamma <0$. Therefore, based on Lemma 2, the drive–response systems (1) and (5) can realize FXT synchronization in probability, and the ST is $T_{max}$, as defined in Lemma 2. The proof is completed. □

Remark 5.

Notably, the parameter $L_m^{\varrho }$ imposed on the constraint in Assumption 2 is incorporated as a positive term of ${\gamma }_m$. This indicates that under the conditions of Assumption 2, the effect of noise is suppressed via selecting the feasible range of the parameter ${\gamma }_m$.

Remark 6.

It is easy to conclude from Theorem 1 that the upper bound of ST is closely associated with the controller parameters α, β, ${\lambda }_m$, ${\sigma }_m$, $v_m$, and $p_m$. Thus, by properly tuning these controller parameters, we can achieve the optimal convergence performance.

When the memristor-based connection weights become constants, i.e., $a_{mn}(·)=a_{mn}$ and $b_{mn}(·)=b_{mn}$ and the delayed feedback connection weights by $b_{mn}$, systems (1) and (5) are transformed into the subsequent form.

$$\begin{array}{cc}\hfill d{ħ}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{ħ}_m(t,w)}{\partial w_k^2}}-c_m{ħ}_m(t,w)+\sum _{n=1}^N{a}_{mn}{f}_n\left({ħ}_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^N{b}_{mn}{g}_n\left({ħ}_n(t-\aleph ,w)\right)\right]dt+{\varrho }_m(t,{ħ}_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill d{z}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{z}_m(t,w)}{\partial w_k^2}}-c_m{z}_m(t,w)+\sum _{n=1}^N{a}_{mn}{f}_n\left(z_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^N{b}_{mn}{g}_n\left(z_n(t-\aleph ,w)\right)+u_m(t,w)\right]dt+{\varrho }_m(t,z_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(28)

consequently, we can deduce the error systems as

$$\begin{array}{cc}\hfill d{e}_m(t,w)& =\left[\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{e}_m(t,w)}{\partial w_k^2}}-c_m{e}_m(t,w)+\sum _{n=1}^N{a}_{mn}[f_n\left(z_n(t,w)\right)-f_n\left({ħ}_n(t,w)\right)]\right.\hfill \\ & \left.+\sum _{n=1}^N{b}_{mn}[g_n\left(z_n(t-\aleph ,w)\right)-g_n\left({ħ}_n(t-\aleph ,w)\right)]+u_m(t,w)\right]dt\hfill \\ & +{\varrho }_m(t,e_m(t,w))d{\overline{w}}_m\left(t\right).\hfill \end{array}$$

(29)

Correspondingly, the controller is designed as follows:

$$u_m(t,w)=\left\{\begin{array}{c}-{\displaystyle \sum _{n=1}^N}{\sigma }_m\mathrm{sign}\left(e_m(t,w)\right)|e_n(t-\aleph ,w)|\hfill \\ -v_m{\widehat{e}}^{\alpha -1}\left(t\right)e_m(t,w)-p_m{\widehat{e}}^{\beta -1}\left(t\right)e_m(t,w),\hfill & e_m(t,w)\ne 0,\hfill \\ 0,\hfill & e_m(t,w)=0,\hfill \end{array}\right.$$

(30)

where $\widehat{e}\left(t\right)={\left({\int }_{\mathsf{\Omega }}{\displaystyle \sum _{m=1}^N}{e}_m^2(t,w)dx\right)}^{{\displaystyle \frac{1}{2}}},$$\alpha >1$, $0<\beta <1$, ${\sigma }_m$, $v_m$ and $p_m$ are positive constants.

Corollary 1.

Suppose that the Assumptions 1 and 2 are satisfied and control gains ${\gamma }_m$ and ${\sigma }_m$ of the controller (30) satisfy inequality (13) and (14). Then, under controller (30), the drive–response systems (27) and (28) can achieve FXT synchronization in probability with ST estimation $T_{max}$.

Proof. 

Take into account the Lyapunov function (15). Then, we can calculate the $\mathcal{L}V\left(t\right)$ along the solution of system (29) as

$$\begin{array}{cc}\hfill \mathcal{L}V\left(t\right)& \le {\int }_{\mathsf{\Omega }}\sum _{m=1}^N\left[L_m^{\varrho }-\sum _{k=1}^l{\displaystyle \frac{2d_{mk}^{*}}{h_k^2}}-2c_m+\sum _{n=1}^N(a_{mn}{L}_n^f+a_{nm}{L}_m^f)\right]e_m^2(t,w)dw\hfill \\ & +{\int }_{\mathsf{\Omega }}\sum _{m=1}^N\sum _{n=1}^N(2b_{mn}{L}_n^g-2{\sigma }_m)\left|e_m(t,w)\right|\left|e_n(t-\aleph ,w)\right|dw\hfill \\ & -2\underset{1\le m\le N}{min}\left\{v_m\right\}{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2\underset{1\le m\le N}{min}\left\{p_m\right\}{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right)\hfill \\ & \le {\gamma }_{m}V\left(t\right)-2\underset{1\le m\le N}{min}\left\{v_m\right\}{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2\underset{1\le m\le N}{min}\left\{p_m\right\}{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right)\hfill \\ & \le \gamma V\left(t\right)-2v{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2p{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right).\hfill \end{array}$$

(31)

The remaining steps are identical to those in Theorem 1 and thus are omitted herein. □

3.2. Predefined-Time Synchronization

This section provides analysis of the PDT synchronization problem for the drive–response systems (1) and (5).

First, we introduce the following PDT controller

$$u_m(t,w)=\left\{\begin{array}{c}-{\lambda }_m\mathrm{sign}\left(e_m(t,w)\right)-{\displaystyle \sum _{n=1}^N}{\sigma }_m\mathrm{sign}\left(e_m(t,w)\right)|e_n(t-\aleph ,w)|\hfill \\ -{\displaystyle \frac{1}{T_p}}{v}_m{\widehat{e}}^{\alpha -1}\left(t\right)e_m(t,w)-{\displaystyle \frac{1}{T_p}}{p}_m{\widehat{e}}^{\beta -1}\left(t\right)e_m(t,w),\hfill & e_m(t,w)\ne 0,\hfill \\ 0,\hfill & e_m(t,w)=0,\hfill \end{array}\right.$$

(32)

where $\widehat{e}\left(t\right)={\left({\int }_{\mathsf{\Omega }}{\displaystyle \sum _{m=1}^N}{e}_m^2(t,w)dw\right)}^{{\displaystyle \frac{1}{2}}}$, $p_m={\displaystyle \frac{1}{2}}{\left[{\displaystyle \frac{{\delta }_1^{{\lambda }_1}}{\gamma }}\left(1-e^{\gamma {\lambda }_1(1-{\delta }_3)}\right)\right]}^{{\displaystyle \frac{1}{{\lambda }_1-1}}}$, $0<\alpha <1$, $\beta >1$, ${\lambda }_m$, ${\sigma }_m$, and $v_m$ are positive constants, and $T_p>0$ is any given time.

Remark 7.

The power terms and parameter $p_m$ in the controller are derived based on the predefined-time stability Lemma 4. Specifically, the power structures with $0<\alpha <1$ and $\beta >1$, together with the expression of $p_m={\displaystyle \frac{1}{2}}{\left[{\delta }_1^{{\lambda }_1}/\gamma \left(1-e^{\gamma {\lambda }_1\left(1-{\delta }_3\right)}\right)\right]}^{1/\left({\lambda }_1-1\right)}$, jointly guarantee that the system achieves synchronization convergence within the PDT.

Denote

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\gamma }_m}{T_p}}& =L_m^{\varrho }-2c_m-\sum _{k=1}^l{\displaystyle \frac{2d_{mk}^{*}}{h_k^2}}+\sum _{n=1}^N({\overline{a}}_{mn}{L}_n^f+{\overline{a}}_{nm}{L}_m^f).\hfill \end{array}$$

(33)

Theorem 2.

If Assumptions 1 and 2 are valid and the control gains ${\gamma }_m$, ${\lambda }_m$, and ${\sigma }_m$ meet the requirements specified in conditions (12)–(14), then the drive–response systems (1) and (5) will attain PDT synchronization in a probabilistic sense under controller (32).

Proof. 

Select the Lyapunov function presented below:

$$\begin{array}{cc}\hfill V\left(t\right)& ={\int }_{\mathsf{\Omega }}\sum _{m=1}^N{e}_m^2(t,w)dw.\hfill \end{array}$$

(34)

By calculating $\mathcal{L}V\left(t\right)$, we get

$$\begin{array}{cc}\hfill \mathcal{L}V\left(t\right)& ={\int }_{\mathsf{\Omega }}\left[2\sum _{m=1}^N{e}_m(t,w)\sum _{k=1}^l{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{e}_m(t,w)}{\partial w_k^2}}-\sum _{m=1}^{N}2c_m{e}_m^2(t,w)\right.\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w){\widehat{a}}_{mn}\left(t\right)[f_n\left(z_n(t,w)\right)-f_n\left({ħ}_n(t,w)\right)]\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w)({\widehat{a}}_{mn}\left(t\right)-{\tilde{a}}_{mn}\left(t\right))f_n\left({ħ}_n(t,w)\right)\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w)({\widehat{b}}_{mn}\left(t\right)-{\tilde{b}}_{mn}\left(t\right))g_n\left({ħ}_n(t-\aleph ,w)\right)\hfill \\ & +\sum _{m=1}^N\sum _{n=1}^{N}2e_m(t,w){\widehat{b}}_{mn}\left(t\right)[g_n\left(z_n(t-\aleph ,w)\right)-g_n\left({ħ}_n(t-\aleph ,w)\right)]\hfill \\ & -\sum _{m=1}^{N}2{\lambda }_m\left|e_m(t,w)\right|-\sum _{m=1}^N\sum _{n=1}^{N}2{\sigma }_m\left|e_m(t,w)\right|\left|e_n(t-\aleph ,w)\right|\hfill \\ & -\sum _{m=1}^{N}2{\displaystyle \frac{1}{T_p}}{e}_m(t,w)v_m{\widehat{e}}^{\alpha -1}\left(t\right)e_m(t,w)-\sum _{m=1}^{N}2{\displaystyle \frac{1}{T_p}}{e}_m(t,w)p_m{\widehat{e}}^{\beta -1}\left(t\right)e_m(t,w)\hfill \\ & \left.+trace\left[{\varrho }_m^T\left(t,e_m(t,w)\right){\varrho }_m\left(t,e_m(t,w)\right)\right]\phantom{\sum _{m=1}^N}\right]dw.\hfill \end{array}$$

(35)

Similar to the proof of Theorem 1, we get

$$\begin{array}{cc}\hfill -{\displaystyle \frac{1}{T_p}}{\int }_{\mathsf{\Omega }}\sum _{m=1}^{N}2e_m(t,w)v_m{\widehat{e}}^{\alpha -1}\left(t\right)e_m(t,w)dw& =-{\displaystyle \frac{1}{T_p}}2{\widehat{e}}^{\alpha -1}\left(t\right){\int }_{\mathsf{\Omega }}\sum _{m=1}^N{v}_m{e}_m^2(t,w)dw\hfill \\ & \le -2{\displaystyle \frac{1}{T_p}}\underset{1\le m\le N}{min}\left\{v_m\right\}{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill -{\displaystyle \frac{1}{T_p}}{\int }_{\mathsf{\Omega }}\sum _{m=1}^{N}2e_m(t,w)p_m{\widehat{e}}^{\beta -1}\left(t\right)e_m(t,w)dw& =-2{\displaystyle \frac{1}{T_p}}{\widehat{e}}^{\beta -1}\left(t\right){\int }_{\mathsf{\Omega }}\sum _{m=1}^N{p}_m{e}_m^2(t,w)dw\hfill \\ & \le -2{\displaystyle \frac{1}{T_p}}\underset{1\le m\le N}{min}\left\{p_m\right\}{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right).\hfill \end{array}$$

(37)

Substituting (36) and (37) into (35), yields

$$\begin{array}{cc}\hfill \mathcal{L}V\left(t\right)& \le {\int }_{\mathsf{\Omega }}\sum _{m=1}^N\left[L_m^{\varrho }-\sum _{k=1}^l{\displaystyle \frac{2d_{mk}^{*}}{h_k^2}}-2c_m+\sum _{n=1}^N({\overline{a}}_{mn}{L}_n^f+{\overline{a}}_{nm}{L}_m^f)\right]e_m^2(t,w)dw\hfill \\ & +{\int }_{\mathsf{\Omega }}\sum _{m=1}^N\sum _{n=1}^N(2L_{mn}^a{G}_n^f+2L_{mn}^b{G}_n^g-2{\lambda }_m)\left|e_m(t,w)\right|dw\hfill \\ & +{\int }_{\mathsf{\Omega }}\sum _{m=1}^N\sum _{m=1}^N(2{\overline{b}}_{mn}{L}_n^g-2{\sigma }_m)\left|e_m(t,w)\right|\left|e_n(t-\aleph ,w)\right|dw\hfill \\ & -2{\displaystyle \frac{1}{T_p}}\underset{1\le m\le N}{min}\left\{v_m\right\}{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2{\displaystyle \frac{1}{T_p}}\underset{1\le m\le N}{min}\left\{p_m\right\}{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right)\hfill \\ & \le {\displaystyle \frac{\gamma }{T_p}}V\left(t\right)-2{\displaystyle \frac{1}{T_p}}v{V}^{{\displaystyle \frac{\alpha +1}{2}}}\left(t\right)-2{\displaystyle \frac{1}{T_p}}p{V}^{{\displaystyle \frac{\beta +1}{2}}}\left(t\right)\hfill \\ & \le {\displaystyle \frac{1}{T_p}}\left[\gamma V\left(t\right)-{\delta }_1{V}^{{\delta }_3}\left(t\right)-{\delta }_2{V}^{{\delta }_4}\left(t\right)\right],\hfill \end{array}$$

(38)

where ${\delta }_1=2v$, ${\delta }_2=2p={\left[{\displaystyle \frac{{\delta }_1^{{\lambda }_1}}{\gamma }}\left(1-e^{\gamma {\lambda }_1(1-{\delta }_3)}\right)\right]}^{{\displaystyle \frac{1}{{\lambda }_1-1}}}$, ${\lambda }_1={\displaystyle \frac{1-{\delta }_4}{{\delta }_3-{\delta }_4}}$, ${\delta }_3={\displaystyle \frac{\alpha +1}{2}}$, ${\delta }_4={\displaystyle \frac{\beta +1}{2}}$, $\gamma =\underset{1\le m\le N}{max}\left\{{\gamma }_m\right\}$, $v=\underset{1\le m\le N}{min}\left\{v_m\right\}$ and $p=\underset{1\le m\le N}{min}\left\{p_m\right\}$.

Since ${sup}_{w_0\in {\mathcal{R}}^n\setminus \left\{0\right\}}\mathcal{E}T\left(w_0\right)\le T_p$ can be insured from Lemma 4, thus the drive–response systems (1) and (5) achieve PDT synchronization in probability. The proof is completed. □

4. Numerical Examples and Simulations

Example 1.

For $N=3$, consider the following reaction-diffusion MNNs with stochastic disturbances

$$\begin{array}{cc}\hfill d{ħ}_m(t,w)& =\left[\sum _{k=1}^3{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{ħ}_m(t,w)}{\partial w_k^2}}-c_m{ħ}_m(t,w)+\sum _{n=1}^3{a}_{mn}\left({ħ}_m(t,w)\right)f_n\left({ħ}_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^3{b}_{mn}\left({ħ}_m(t,w)\right)g_n\left({ħ}_n(t-\aleph ,w)\right)\right]dt+{\varrho }_m(t,{ħ}_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill d{z}_m(t,w)& =\left[\sum _{k=1}^3{d}_{mk}^{*}{\displaystyle \frac{{\partial }^2{z}_m(t,w)}{\partial w_k^2}}-c_m{z}_m(t,w)+\sum _{n=1}^3{a}_{mn}\left(z_m(t,w)\right)f_n\left(z_n(t,w)\right)\right.\hfill \\ & \left.+\sum _{n=1}^3{b}_{mn}\left(z_m(t,w)\right)g_n\left(z_n(t-\aleph ,w)\right)+u_m(t,w)\right]dt+{\varrho }_m(t,z_m(t,w))d{\overline{w}}_m\left(t\right),\hfill \end{array}$$

(40)

where $m=1,2,3$. The memristor-based connection weights are given as

$$a_{11}\left({ħ}_1(t,w)\right)=\left\{\begin{array}{cc}2.52,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 2.75,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{12}\left({ħ}_1(t,w)\right)=\left\{\begin{array}{cc}-2.27,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -2.42,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{13}\left({ħ}_1(t,w)\right)=\left\{\begin{array}{cc}-1.59,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -1.75,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{21}\left({ħ}_2(t,w)\right)=\left\{\begin{array}{cc}1.51,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 1.67,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{22}\left({ħ}_2(t,w)\right)=\left\{\begin{array}{cc}2.28,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 2.44,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{23}\left({ħ}_2(t,w)\right)=\left\{\begin{array}{cc}-2.41,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -2.55,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{31}\left({ħ}_3(t,w)\right)=\left\{\begin{array}{cc}0.18,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 0.38,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{32}\left({ħ}_3(t,w)\right)=\left\{\begin{array}{cc}1.15,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 1.32,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$a_{33}\left({ħ}_3(t,w)\right)=\left\{\begin{array}{cc}-1.90,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -1.18,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{11}\left({ħ}_1(t,w)\right)=\left\{\begin{array}{cc}2.52,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 2.56,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{12}\left({ħ}_1(t,w)\right)=\left\{\begin{array}{cc}-2.23,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -2.43,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{13}\left({ħ}_1(t,w)\right)=\left\{\begin{array}{cc}-1.57,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -1.74,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{21}\left({ħ}_2(t,w)\right)=\left\{\begin{array}{cc}1.54,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 1.65,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{22}\left({ħ}_2(t,w)\right)=\left\{\begin{array}{cc}2.22,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 2.43,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{23}\left({ħ}_2(t,w)\right)=\left\{\begin{array}{cc}-2.32,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -2.56,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{31}\left({ħ}_3(t,w)\right)=\left\{\begin{array}{cc}0.22,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 0.42,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{32}\left({ħ}_3(t,w)\right)=\left\{\begin{array}{cc}1.26,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ 1.32,\hfill & \left|{ħ}_1(t,w)\right|>0.3,\hfill \end{array}\right.$$

$$b_{33}\left({ħ}_3(t,w)\right)=\left\{\begin{array}{cc}-1.27,\hfill & \left|{ħ}_1(t,w)\right|\le 0.3,\hfill \\ -1.12,\hfill & \left|{ħ}_1(t,w)\right|>0.3.\hfill \end{array}\right.$$

It is obvious that ${\overline{a}}_{11}=2.75$, ${\underline{a}}_{11}=2.52$, ${\overline{a}}_{12}=2.42$, ${\underline{a}}_{12}=2.27$, ${\overline{a}}_{13}=1.75$, ${\underline{a}}_{13}=1.59$, ${\overline{a}}_{21}=1.67$, ${\underline{a}}_{21}=1.51$, ${\overline{a}}_{22}=2.44$, ${\underline{a}}_{22}=2.28$, ${\overline{a}}_{23}=2.55$, ${\underline{a}}_{23}=2.41$, ${\overline{a}}_{31}=0.38$, ${\underline{a}}_{31}=0.18$, ${\overline{a}}_{32}=1.32$, ${\underline{a}}_{32}=1.15$, ${\overline{a}}_{33}=1.90$, ${\underline{a}}_{33}=1.18$, ${\overline{b}}_{11}=2.56$, ${\underline{b}}_{11}=2.52$, ${\overline{b}}_{12}=2.43$, ${\underline{b}}_{12}=2.23$, ${\overline{b}}_{13}=1.74$, ${\underline{b}}_{13}=1.57$, ${\overline{b}}_{21}=1.65$, ${\underline{b}}_{21}=1.54$, ${\overline{b}}_{22}=2.43$, ${\underline{b}}_{22}=2.22$, ${\overline{b}}_{23}=2.56$, ${\underline{b}}_{23}=2.32$, ${\overline{b}}_{31}=0.42$, ${\underline{b}}_{31}=0.22$, ${\overline{b}}_{32}=1.32$, ${\underline{b}}_{32}=1.26$, ${\overline{b}}_{33}=1.27$ and ${\underline{b}}_{33}=1.12$. By choosing $f_n\left(w\right)=tanh\left(w\right)$ and $g_n\left(w\right)=tanh\left(w\right)$, $d_{mk}=0.005$, $h_k=10$, $c_m=diag(1.817,1.817,1.817)$, ${\varrho }_m(t,w)=6w$, $\aleph =1$, then MATLAB 2020a simulations of system (39) corresponding to the initial conditions ${ħ}_1(t,w)=0.3cos\left(0.5\pi w\right)+0.3$, ${ħ}_2(t,w)=0.5cos\left(0.5\pi w\right)+0.5$ and ${ħ}_3(t,w)=0.7cos\left(0.5\pi w\right)+0.7$ are given in Figure 2, Figure 3 and Figure 4, which verifies that system (40) possesses a chaotic attractor.

We choose the Brownian motion $\overline{w}\left(t\right)$ such that $\Delta \overline{w}=\overline{w}(t+\Delta \Pi )-\overline{w}\left(t\right)=\gimel \sqrt{▵\Pi }$, where $▵\Pi =0.01$ is step size and ℷ obeys a normal distribution N(0,1.28); the simulation of $\overline{w}\left(t\right)$ is shown in Figure 5.

Obviously, we can check that $L_m^f=L_n^f=L_n^g=0.1$, $G_n^f=G_n^g=1$, $m,n=1,2,3$. By selecting the control parameters as ${\lambda }_m=2.5$, ${\sigma }_m=1.8$, $v_m=4$, $\alpha =1.4$, $\beta =0.2$, and $p_m=7.494$, a simple calculation shows that ${\delta }_1=8$, ${\delta }_2=14.988$, ${\delta }_3=1.2$, ${\delta }_4=0.6$, ${\gamma }_1=-0.621$, ${\gamma }_2=-0.401$, ${\gamma }_3=-0.866$. Since $\gamma =max\left\{{\gamma }_1,{\gamma }_2,{\gamma }_3\right\}=-0.401$, it is easy to verify that (12) and (14) hold true. Thus, based on Theorem 1, the drive–response systems (39) and (40) can realize FXT synchronization under controller (10) within ST $T_{max}=1.240$. Figure 6, Figure 7 and Figure 8 describe the time evolutions of drive–response systems (39) and (40) under controller (10).

Next, we investigate the PDT synchronization between the drive–response systems (40) and (41) via controller (32). By choosing PDT $T_p=0.5$, and setting $\alpha =1.4$, $\beta =0.2$, $v_i=2$, we can compute that ${\lambda }_1={\displaystyle \frac{2}{3}}$, ${\displaystyle \frac{1}{{\lambda }_1-1}}=-3$, ${\delta }_1=4$, ${\delta }_2=7.493$, $p_m=3.747$. Therefore, from Theorem 2, we obtain that drive–response systems (39) and (40) achieve PDT synchronization in probability. Figure 9, Figure 10 and Figure 11 show the time evolutions synchronization errors between drive–response systems (39) and (40) under controller (32) with $T_p=0.5$.

Remark 8.

Under comparable parameter settings, we compare the upper bound of ST $T_{max}=1.240$ obtained from Theorem 1 in this paper with the results from [33,37]. The upper bounds of ST in [33,37] were $T_{max}=3.7154$ and $T_{max}=1.596$, respectively. In comparison, the estimation of ST in this paper is significantly more precise.

Furthermore, to validate the chaotic characteristics of our proposed stochastic delayed memristive reaction-diffusion neural networks, we compute the Lyapunov exponents under parameter settings: $d=0.02$, $L_m^{\varrho }=0.15$, $\aleph =1$ and different initial conditions. The results are summarized in

Table 3. Lyapunov exponents of the proposed stochastic delayed memristive reaction-diffusion neural networks (43) under different initial conditions.

Initial Conditions	Lyapunov Exponents	System Behavior
$(0.1,0.2,0.3)$	3.0406, 0.0121, −2.3412	Chaotic
$(0.5,0.5,0.5)$	3.0487, 0.0121, −2.3528	Chaotic
$(1.0,1.0,1.0)$	3.0573, 0.0121, −2.3379	Chaotic
$(0.2,0.4,0.6)$	3.0547, 0.0121, −2.3456	Chaotic

.

5. Conclusions

This paper investigated the FXT and PDT synchronization of a class of delayed MNNs with stochastic disturbances and reaction-diffusion terms. First, a new stochastic PDT stability lemma is introduced based on previous FXT stability results. Then, via the design of innovative controllers and the utilization of analysis techniques, some new FXT and PDT synchronization criteria are derived to insure the FXT and PDT synchronization of delayed MNNs with stochastic disturbances and reaction-diffusion terms. Lastly, the affectivity of the derived results are validated through numerical simulations.

It is worth noting that the spatial norm $\widehat{e}\left(t\right)$ introduced in controllers (10) and (32) requires access to the synchronization error information over the entire spatial domain of the network. In contrast, distributed or decentralized controllers only rely on local error information or the errors of adjacent neurons within a finite spatial radius. So, how to insure the FXT and PDT synchronization of reaction diffusion MNNs with stochastic perturbations is an interesting research topic and it will be one of our future research directions. In addition, the FXT and PDT synchronization framework developed in this paper for MNNs may also provide useful insights for other neural network architectures. In particular, Kolmogorov–Arnold Networks (KANs) [43], which have recently demonstrated effectiveness in highly precise temporal synchronization tasks such as audio-visual speech generation. Extending the current theoretical framework to KANs under stochastic perturbations is a meaningful research direction, and will also be one of our future research directions.