Exponentially Clustered Synchronization of a Stochastic Complex Network with Reaction–Diffusion Terms and Time Delays via a Pinning Boundary Control

Abstract

A pinning boundary control strategy that can achieve the exponentially clustered synchronization of a specific class of complex networks is developed. Firstly, the studied model captures the essential features of networks, including spatial dependence, stochastic switching, noise perturbation, and time delays. Secondly, the proposed control algorithm can save the implementation cost and overcome environmental constraint by acting on the boundary of a few nodes. Thirdly, an average state related to the directed topology of the nodes in the same cluster is calculated as the target network. Finally, nonlinear simulations show that the proposed controller can solve the cluster synchronization of a directed coupled reaction–diffusion neural network with Markovian switching, stochastic noise and time delay.

1. Introduction

In the vast landscape of complex network dynamics, the research of synchronization phenomena has garnered significant attention due to its profound implications across various fields, ranging from neuroscience to engineering systems. For example, the coordinated firing of neurons in brain networks enables information processing, and synchronized operation of nodes in the power grid ensures stable energy supply. Over the past few decades, significant progress has been made in understanding synchronization phenomena in complex networks (please refer to [1,2,3,4,5], etc.).

Early works focused primarily on complete synchronization, where all nodes in a network converge to a common state. However, as research evolved, the focus shifted towards more nuanced forms of synchronization, such as cluster synchronization. Cluster synchronization stands out as a particularly intriguing phenomenon, where nodes within a network are grouped into clusters and exhibit synchronized behavior within each cluster while remaining unsynchronized across different clusters. This form of synchronization is particularly relevant in scenarios where networks possess inherent community structures, such as social networks (where users within the same interest group share information synchronously), brain networks (where functionally related neuron clusters coordinate activities), and power grids (where regional sub-grids operate in synchronization while maintaining independent regulation). To date, several analytical and numerical methods have been developed to investigate cluster synchronization in complex networks (see [6,7,8,9,10,11], etc.).

Recently, some researchers have begun to incorporate stochastic factors, reaction–diffusion terms, and time delays into their models of the synchronization or stabilization in complex networks [12,13,14,15,16,17]. These factors, prevalent in real-world systems, introduce complexities that significantly alter the dynamical behavior of networks. Stochastic perturbations or stochastic switching, for instance, can disrupt synchronized states, while reaction–diffusion terms capture the spatial spread of information or substances within networks. Time delays, on the other hand, arise naturally in communication networks and biological systems, where signals or interactions do not propagate instantaneously. These extensions have provided deeper insights into the robustness and resilience of synchronized states in real-world systems. Therefore, the effects of stochastic factors, reaction–diffusion terms or time delays on the cluster synchronization of networks have been, respectively, considered in the literature [18,19,20,21,22,23,24,25,26,27] and so forth.

Despite the advances made in the field, several gaps persist in our understanding of the interplay between stochastic factors, reaction–diffusion terms, time delays and their combined effects on the cluster synchronization in complex networks. For instance, while individual factors have been studied extensively, their combined effects on network dynamics are less well-understood. This is because real-world networks rarely encounter these factors in isolation. For example, a smart grid simultaneously faces stochastic renewable energy inputs, spatial energy diffusion, and communication delays between control centers and distributed nodes. Additionally, existing studies for cluster synchronization often adopt target states that are pre-defined isolated node states [18,19,20,22,25], which lack adaptability to the inherent topological characteristics of clusters.

Furthermore, the control of cluster synchronization in such networks remains a challenging problem. Traditional control strategies, such as pinning control, as a promising alternative that balances control efficiency and implementation cost, have emerged as a dominant approach for synchronizing large-scale complex networks [11,18,19,21,24,28,29,30]. The core idea of pinning control is to impose control actions on only a small subset of pinned nodes, with the synchronized behavior propagating to the remaining unpinned nodes through network interactions. However, most pinning controllers for the reaction–diffusion networks require imposing control on the entire spatial domain (see [21,24]), making such control strategies unfeasible for real-world deployment. For example, in a large-scale water distribution network (a typical reaction–diffusion system), installing sensors and actuators across the entire spatial domain would incur enormous costs and maintenance burdens.

Boundary control approaches, on the other hand, offer promising alternatives by applying the control actions only at the spatial boundaries of the nodes [13,14,15,16,17,22]. More recently, boundary control has been integrated with pinning control for reaction–diffusion networks [31,32,33], where control actions are applied only at the spatial boundaries of pinned nodes. A key practical advantage of this pinning boundary control strategy translates to a significant reduction in the number of required actuators and sensors. For example, in a regional power grid (a reaction–diffusion network with stochastic renewable inputs), the pinning boundary control strategy only requires actuators at the boundary nodes of each sub-grid cluster, rather than every node in the entire grid. This not only lowers hardware procurement and installation costs but also reduces the probability of system failures caused by malfunctions of a large number of actuators. This enhances the robustness and operational reliability of the control system.

By utilizing an adaptive boundary controller and a pinning adaptive boundary controller, respectively, the authors in [31] have solved the synchronization problem of a reaction–diffusion complex network. Through appropriate pinning boundary control strategies, the authors in [32] have achieved the synchronization of coupled reaction–diffusion neural networks, and the authors in [33] have studied the finite–time stabilization of a semi-Markov reaction–diffusion memristive neural network. While these pinning boundary control strategies reduce spatial actuation requirements, they still have limitations: some focus on deterministic networks and do not account for stochastic factors [31,32], and none have systematically integrated the combined effects of stochasticity, reaction–diffusion, and time delays for cluster synchronization.

Motivated by these gaps in the literature and the pressing practical needs of real-world complex networks, the present study aims to investigate the cluster synchronization in stochastic complex networks with reaction–diffusion terms and time delays. Specifically, we focus on developing a pinning boundary control strategy to steer such networks towards exponentially clustered synchronized states.

1.1. Contributions and Novelties

Compared to existing solutions, the present work exhibits three distinct advantages that enhance its theoretical significance and practical value:

(i) Wider applicability of the model: The model to cluster synchronization captures the essential features of many real-world networks, including stochastic noise, Markovian switching, reaction–diffusion, and time delays. This makes the model is more general than the ones in [6,7,8,9,10,11,18,19,20,21,22,23,24,25,26,27] and enables the model to accurately characterize the dynamical behavior of complex networks in practical scenarios (e.g., smart grids, brain neural networks).

(ii) Lower implementation cost of the controller: Compared to full spatial domain pinning controllers [21,24], the proposed pinning boundary control strategy applies control only to partial nodes and their spatial boundaries, significantly reducing the number of required actuators and sensors. Additionally, the switched constant gains in the controllers can take different values in each switching state, allowing the controller to adapt to dynamic changes in network states (e.g., sudden fluctuations in wind power output), which further improves its practical applicability.

(iii) Higher adaptability of the target states: Unlike existing studies that adopt pre-defined isolated node states as synchronization targets [18,19,20,22,25], the proposed target states are cluster-specific average states associated with the directed topology of all nodes within the same cluster. This approach not only simplifies theoretical analysis and numerical computation but also enhances adaptability to the inherent community structures of real-world networks, overcoming the poor adaptability of traditional target state designs.

The remainder of this paper is organized as follows. Section 2 presents the considered models and some necessary preliminaries. To achieve cluster synchronization, a pinning boundary controller with distributed measurement and a pinning boundary controller with spatial sampled-data are designed, respectively, in Section 3 and Section 4. Numerical simulations are presented in Section 5 to validate the theoretical results. Finally, we conclude the paper in Section 6 with a summary of our findings and directions for future research.

1.2. Notations

$\mathcal{N}=\{1,2,\cdots \},\mathcal{R}=(-\infty ,+\infty ),{\mathcal{R}}^{+}=(0,+\infty )$, ${\mathcal{R}}_0^{+}=[0,+\infty )$. For each $n\in \mathcal{N},{\mathcal{N}}_n=\{1,\cdots ,n\}$, ${\mathcal{R}}^n$ denotes the n-dimensional real Euclidean space, and ${\mathcal{R}}^{n\times n}$ is the set of all $n\times n$ real matrices. For a real matrix $A,$ $A>0(A<0)$ means that A is positive definite (negative definite), $A^T$ means its transpose, tr(A) is the trace of A, $A^s=(A+A^T)/2$, ${\displaystyle {\parallel A\parallel }_{max}=max\left\{\right|a_{ij}|:i\in {\mathcal{N}}_m,j\in {\mathcal{N}}_n\}}$ for $A={\left(a_{ij}\right)}_{m\times n}$, ${\lambda }_{max}\left(A\right)$ and ${\lambda }_{min}\left(A\right)$ denote the largest and smallest eigenvalues of A if $A^T=A$, respectively. For a vector $x\in {\mathcal{R}}^n$, $x>0$ means that each element of x is positive and, $\parallel x\parallel =\sqrt{x^{T}x}$. ${\mathbf{0}}_n={(0,\cdots ,0)}^T\in {\mathcal{R}}^n$, ${\mathbf{1}}_n={(1,\cdots ,1)}^T\in {\mathcal{R}}^n$, $I_n$ is a n-order identity matrix, diag$(·)$ presents a diagonal matrix, and ⊗ denotes the Kronecker product of two matrices. For any $x_i\in \mathcal{R}$, ${\displaystyle {\left(x_i\right)}_{i\in {\mathcal{N}}_n}={(x_1,x_2,\cdots ,x_n)}^T\in {\mathcal{R}}^n}$. For a positive constant l, ${\mathcal{L}}^2\left([0,l]\right)$ is the Hilbert space of square integrable vector functions $z:[0,l]\to {\mathcal{R}}^n$ with the norm ${\displaystyle {\parallel z\parallel }_2={\left({\int }_0^l{\parallel z\left(x\right)\parallel }^2\mathrm{d}x\right)}^{\frac{1}{2}}}$; ${\displaystyle {\mathcal{H}}^1\left([0,l]\right)=\{z\in {\mathcal{L}}^2\left([0,l]\right)\mid \frac{\mathrm{d}z}{\mathrm{d}x}\in {\mathcal{L}}^2\left([0,l]\right)\}}$ is the Sobolev space. $\mathrm{E}(·)$ is the mathematical expectation operator.

$\{m\left(t\right)\in {\mathcal{N}}_S:t\ge 0\}$ is a right-continuous Markov chain defined on a complete probability space $(\overline{\Omega },\mathfrak{F},P)$ with a natural filtration ${\left\{{\mathfrak{F}}_t\right\}}_{t\ge 0}$. And the transition probability is given by

$$P\{m(t+\delta )=\nu |m\left(t\right)=\mu \}=\left\{\begin{array}{cc}\hfill & {\sigma }_{\mu \nu }\delta +o\left(\delta \right),\mu \ne \nu ,\hfill \\ \hfill & 1+{\sigma }_{\mu \nu }\delta +o\left(\delta \right),\mu =\nu ,\hfill \end{array}\right.$$

in which $\delta >0$ and ${\left({\sigma }_{\mu \nu }\right)}_{S\times S}$ is a generator with the transition rate ${\sigma }_{\mu \nu }\ge 0$ from $\mu$ to $\nu$ if $\mu \ne \nu$, and ${\displaystyle {\sigma }_{\mu \mu }=-{\displaystyle \sum _{\nu =1,\nu \ne \mu }^S}{\sigma }_{\mu \nu }}$.

2. Model Description and Preliminaries

Suppose that the stochastic complex network with reaction–diffusion term and time-varying delays (R-D SCN) considered in this paper has $N\in \mathcal{N}$ nodes, which can be divided into $K\in {\mathcal{N}}_N$ clusters: $C_1=\{1,\cdots ,d_1\},$ $C_2=\{d_1+1,\cdots ,d_1+d_2\},\cdots$, $C_k=\{{\sum }_{i=1}^{k-1}{d}_i+1,\cdots ,{\sum }_{i=1}^k{d}_i\},\cdots$,$C_K=\{{\sum }_{i=1}^{K-1}{d}_i+1,\cdots ,N\}$, where $d_k\in {\mathcal{N}}_N$ is the number of nodes in the cluster $C_k$ for all $k\in {\mathcal{N}}_K$. Obviously, ${\sum }_{k=1}^K{d}_k=N$. In the following contents, k and $k^{\prime }$ are assumed to satisfy $k,k^{\prime }\in {\mathcal{N}}_K$.

In the kth cluster $C_k$ of the network, the dynamic equation of the ith node can be expressed by

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{d}{z}_i(t,x)=[D\left(m\left(t\right)\right)\frac{{\partial }^2{z}_i(t,x)}{\partial x^2}+f^k(t,z_i(t,x),z_i(t-{\tau }_1\left(t\right),x),m\left(t\right))}\hfill \\ \hfill & {\displaystyle +c{\displaystyle \sum _{j=1}^N}{a}_{ij}\Gamma \left(m\left(t\right)\right)(z_j(t,x)-z_i(t,x))]\mathrm{d}t}\hfill \\ \hfill & {\displaystyle +h^k(t,z_i(t,x),z_i(t-{\tau }_2\left(t\right),x),m\left(t\right))\mathrm{d}y\left(t\right),}\hfill \end{array}$$

(1)

in which $a_{ij}$ denotes the coupled strength. Denote

$$A={\left(a_{ij}\right)}_{N\times N}=\left[\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1K}\\ A_{21}& A_{22}& \cdots & A_{2K}\\ ⋮& ⋮& ⋮& ⋮\\ A_{K1}& A_{K2}& \cdots & A_{KK}\end{array}\right]$$

with $A_{k{k}^{\prime }}={\left(a_{ij}\right)}_{d_k\times d_{k^{\prime }}}\in {\mathcal{R}}^{d_k\times d_{k^{\prime }}}$, and introduce the following assumptions about the network topology and the coupled strength.

Assumption A1.

The graph of the network composed of nodes belonging to the same cluster $C_k$ is strongly connected.

Assumption A2.

For any $i,j\in {\mathcal{N}}_N(i\ne j)$, $a_{ij}\ne 0$ if there is a link from node i to node j, otherwise, $a_{ij}=0$; moreover, $a_{ij}\ge 0$ if $i,j\in C_k$, and ${\displaystyle \sum _{\ell \in C_{k^{\prime }}}}{a}_{i\ell }=0$.

Remark 1.

In the existing references on cluster synchronization, the models discussed in [6,7,8,9,10,11] are directed or undirected complex networks; the models in [18,19,20,25,26] are some complex networks with Markovian switching parameters, stochastic noise, or time delays; the models in [21,22,23,24] are some types of reaction–diffusion complex networks. Different from the above works, the network (1) studied in this paper considers not only the directed topology and reaction–diffusion, but also stochastic noise, Markovian switching and time delays. Therefore, the network (1) is more general and representative than the ones in the above literature.

Remark 2.

In [10,19], a strong connectivity requirement is imposed on all nodes within the network. By contrast, Assumption 1 in this paper implies that strong connectivity is only required for nodes belonging to the same cluster.

From Assumption 2, ${\displaystyle \sum _{j=1}^N}{a}_{ij}={\displaystyle \sum _{j\in C_k}}{a}_{ij}+{\displaystyle \sum _{k^{\prime }=1,k^{\prime }\ne k}^K}{\displaystyle \sum _{j\in C_{k^{\prime }}}}{a}_{ij}=0,$ and the directed network (1) can be rewritten to

$$\begin{array}{cc}\hfill \mathrm{d}{z}_i(t,x)& {\displaystyle =[D\left(m\left(t\right)\right)\frac{{\partial }^2{z}_i(t,x)}{\partial x^2}+f^k(t,z_i(t,x),z_i(t-{\tau }_1\left(t\right),x),m\left(t\right))}\hfill \\ & {\displaystyle +c{\displaystyle \sum _{j=1}^N}{a}_{ij}\Gamma \left(m\left(t\right)\right)z_j(t,x)]\mathrm{d}t}\hfill \\ & {\displaystyle +h^k(t,z_i(t,x),z_i(t-{\tau }_2\left(t\right),x),m\left(t\right))\mathrm{d}y\left(t\right).}\hfill \end{array}$$

(2)

In the models (1) and (2), $t\in [0,+\infty )$ is the time variable with initial-time $0\in {\mathcal{R}}_0^{+}$, $x\in [0,l]$ is the space variable, and ${\tau }_1\left(t\right)$ and ${\tau }_2\left(t\right)$ are time-varying delays; $z_i(·,x)={\left(z_i^r(·,x)\right)}_{r=1}^n\in {\mathcal{H}}^1\left([0,l]\right)$ denotes the state vector; n-order diagonal matrix $D(·)>0$ represents the transmission diffusion coefficient matrix; $f^k(·,·,·,·)\in {\mathcal{R}}^n$ is a continuous function vector described the inherent dynamic behavior of the nodes in the kth cluster; $c\in {\mathcal{R}}^{+}$ denotes the overall coupling strength; $\Gamma (·)\in {\mathcal{R}}^{n\times n}$ is the inner coupled matrix and $\Gamma (·)>0$; $y\left(t\right)\in {\mathcal{R}}^{\tilde{n}}$ is a Wiener process with $\mathrm{E}y\left(t\right)={\mathbf{0}}_{\tilde{n}}$. $h^k(·,·,·,·)=\left(h_{rj}^k\right)\in {\mathcal{R}}^{n\times \tilde{n}}$ represents the noise intensity function matrix where $h_{rj}^k=h_{rj}^k(t,z_i^r(t,x),z_i^r(t-\tau \left(t\right),x),m\left(t\right))$. The Wiener process $y\left(t\right)$ is assumed to be independent on the Markov chain $\{m\left(t\right)\in {\mathcal{N}}_S:t\ge 0\}$.

The following assumptions about the time-varying delays, inherent dynamic function vector $f^k$ and noise intensity function matrix $h^k={\left(h_{rj}^k\right)}_{n\times \tilde{n}}$ are given.

Assumption A3.

There exists constant $\overline{\tau }$ such that $0\le {\tau }_1\left(t\right),{\tau }_2\left(t\right)\le \overline{\tau }.$

Remark 3.

The time-varying delays in this paper only need to be bounded, are not required to be differentiable as the ones in [17,33,34] and so on.

Assumption A4.

For any ${\mu }_1,{\mu }_2,{\nu }_1,{\nu }_2\in {\mathcal{R}}^n,\rho \in {\mathcal{N}}_S$ and n-order matrix $B>0$, there exist constants $L_1^k>0,{\widehat{L}}^k,{\check{L}}^k$ such that

$$\begin{array}{cc}\hfill & {\displaystyle {[f^k(t,{\mu }_1,{\nu }_1,\rho )-f^k(t,{\mu }_2,{\nu }_2,\rho )]}^{T}B[f^k(t,{\mu }_1,{\nu }_1,\rho )-f^k(t,{\mu }_2,{\nu }_2,\rho )]}\hfill \\ \hfill & {\displaystyle \le L_1^k[{({\mu }_1-{\mu }_2)}^{T}B({\mu }_1-{\mu }_2)+{({\nu }_1-{\nu }_2)}^{T}B({\nu }_1-{\nu }_2)],}\hfill \end{array}$$

$${\widehat{L}}^k({\mu }_1-{\mu }_2+{\nu }_1-{\nu }_2)\le [h_{rj}^k(t,{\mu }_1,{\nu }_1,\rho )-h_{rj}^k(t,{\mu }_2,{\nu }_2,\rho )]\le {\check{L}}^k({\mu }_1-{\mu }_2+{\nu }_1-{\nu }_2),$$

in which $r\in {\mathcal{N}}_n,$$j\in {\mathcal{N}}_{\tilde{n}}$.

Equip network (2) with the initial condition

$${\displaystyle z_i(s,x)={\phi }_i(s,x),(s,x)\in [-\overline{\tau },0]\times [0,l]\mathrm{and}m\left(0\right)={\rho }_0\in {\mathcal{N}}_S,}$$

(3)

and the Neumann boundary condition

$${\displaystyle \frac{\partial z_i(t,x)}{\partial x}{|}_{x=0}={\mathbf{0}}_n,\frac{\partial z_i(t,x)}{\partial x}{|}_{x=l}=u_i(t,m\left(t\right)),t\in [-\overline{\tau },+\infty ),}$$

(4)

where $u_i(t,m\left(t\right))$ is a boundary controller which will be designed later, ${\phi }_i(s,x)\in {\mathcal{C}}_n$ and ${\mathcal{C}}_n=\mathcal{C}([-\overline{\tau },0]\times [0,l],{\mathcal{R}}^n)$ is a Banach space composed of all continuous functions with the norm ${\displaystyle \parallel {\phi }_i{\parallel }_{\overline{\tau }}=sup\left\{\parallel {\phi }_i{(s,·)\parallel }_2:s\in [-\overline{\tau },0]\right\}.}$

To the coupled strength matrix in (2), A is not required to be symmetric, that is, the coupling topology of node (2) is directed. For the directed network, the following lemma will be used.

Lemma 1

([28]). Let $A={\left(a_{ij}\right)}_{d\times d}$ be a weighted adjacency matrix of a directed network which is strongly connected, and ${\sum }_{j=1}^d{a}_{ij}=0$ with $a_{ij}\ge 0(i\ne j)$, then there exists a vector ${\displaystyle \psi ={\left({\psi }_i\right)}_{i=1}^d>0}$ such that

(i)

$A^T\psi ={\mathbf{0}}_d$;

(ii)

$\Psi A+{\left(\Psi A\right)}^T={\left({\widehat{a}}_{ij}\right)}_{d\times d}$ is symmetric and ${\sum }_{j=1}^d{\widehat{a}}_{ij}={\sum }_{j=1}^d{\widehat{a}}_{ji}=0,i\in {\mathcal{N}}_d,$ where $\Psi =\mathrm{diag}({\psi }_1,{\psi }_2,\cdots ,{\psi }_d)$.

From Lemma 1 and Assumption 1, there exists ${\psi }^k={\left({\psi }_i\right)}_{i\in C_k}>0$ such that ${\left({\psi }^k\right)}^T{A}_{kk}={\mathbf{0}}_{d_k}^T$. Let ${\xi }_i={\psi }_i{\left({\displaystyle \sum _{j\in C_k}}{\psi }_j\right)}^{-1},w^k(t,x)={\left(w_r^k(t,x)\right)}_{r=1}^n={\displaystyle \sum _{i\in C_k}}{\xi }_i{z}_i(t,x),$ then $w_r^k(t,x)={\displaystyle \sum _{i\in C_k}}{\xi }_i{z}_i^r(t,x)$, ${\displaystyle \sum _{i\in C_k}}{\xi }_i=1$ with $0<{\xi }_i<1$, and ${\displaystyle \sum _{i\in C_k}}{\xi }_i{a}_{ij}=0$ for all $j\in C_k$. And the target network used for synchronization can be obtained as

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{d}{w}^k(t,x)}\hfill \\ \hfill & {\displaystyle =\left\{D\left(m\left(t\right)\right)\frac{{\partial }^2{w}^k(t,x)}{\partial x^2}+{\displaystyle \sum _{i\in C_k}}{\xi }_i{f}^k(t,z_i(t,x),z_i(t-\tau \left(t\right),x),m\left(t\right))\right\}\mathrm{d}t}\hfill \\ \hfill & {\displaystyle +{\displaystyle \sum _{i\in C_k}}{\xi }_i{h}^k(t,z_i(t,x),z_i(t-{\tau }_2\left(t\right),x),m\left(t\right))\mathrm{d}y\left(t\right).}\hfill \end{array}$$

(5)

Remark 4.

The target state for cluster synchronization in this study is the average state $w^k(t,x)$ of nodes within the same cluster. This target state can be derived directly from the existing nodes in the network, requiring no additional pre-specified information. By contrast, the cluster synchronization target states in [18,19,20,22,25] must be separately pre-specified as isolated node states.

For any $i\in C_k$, let $e_i^k(t,x)=z_i(t,x)-w^k(t,x)$, then ${\displaystyle {\displaystyle \sum _{i\in C_k}}{\xi }_i{e}_i^k(t,x)={\mathbf{0}}_n}$. Denote

$$\begin{array}{cc}\hfill & {\displaystyle F_i^k(t,e_i^k(t,x),e_i^k(t-{\tau }_1\left(t\right),x),m\left(t\right))}\hfill \\ \hfill & {\displaystyle =f^k(t,z_i(t,x),z_i(t-{\tau }_1\left(t\right),x),m\left(t\right))+c{\displaystyle \sum _{j=1}^N}{a}_{ij}\Gamma \left(m\left(t\right)\right)z_j(t,x)}\hfill \\ \hfill & {\displaystyle -{\displaystyle \sum _{l\in C_k}}{\xi }_l{f}^k(t,z_l(t,x),z_l(t-{\tau }_1\left(t\right),x),m\left(t\right)),}\hfill \\ \hfill & {\displaystyle H_i^k(t,e_i^k(t,x),e_i^k(t-{\tau }_2\left(t\right),x),m\left(t\right))}\hfill \\ \hfill & {\displaystyle =h^k(t,z_i(t,x),z_i(t-{\tau }_2\left(t\right),x),m\left(t\right))-{\displaystyle \sum _{l\in C_k}}{\xi }_l{h}^k(t,z_l(t,x),z_l(t-{\tau }_2\left(t\right),x),m\left(t\right)),}\hfill \end{array}$$

and $H_i^k(t,e_i^k(t,x),e_i^k(t-{\tau }_2\left(t\right),x),m\left(t\right))$ is abbreviated as $H_i^k(t,x,m\left(t\right))$, the following error system can be derived:

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{d}{e}_i^k(t,x)}\hfill \\ \hfill & {\displaystyle =\left[D\left(m\left(t\right)\right)\frac{{\partial }^2{e}_i^k(t,x)}{\partial x^2}+F_i^k(t,e_i^k(t,x),e_i^k(t-{\tau }_1\left(t\right),x),m\left(t\right))\right]\mathrm{d}t}\hfill \\ \hfill & {\displaystyle +H_i^k(t,e_i^k(t,x),e_i^k(t-{\tau }_2\left(t\right),x),m\left(t\right))\mathrm{d}y\left(t\right)}\hfill \end{array}$$

(6)

with

$${\displaystyle e_i^k(s,x)={\phi }_i(s,x)-{\displaystyle \sum _{l\in C_k}}{\xi }_l{\phi }_l(s,x),(s,x)\in [-\overline{\tau },0]\times [0,l]\mathrm{and}m\left(0\right)={\rho }_0\in {\mathcal{N}}_S,}$$

(7)

$${\displaystyle \frac{\partial e_i^k(t,x)}{\partial x}{|}_{x=0}={\mathbf{0}}_n,\frac{\partial e_i^k(t,x)}{\partial x}{|}_{x=l}=u_i(t,m\left(t\right))-{\displaystyle \sum _{l\in C_k}}{\xi }_l{u}_l(t,m\left(t\right)),t\in [-\overline{\tau },+\infty ).}$$

(8)

Subsequently, one basic definition, several crucial lemmas together with two propositions are offered, all of which are indispensable for exporting our main results.

Definition 1.

The directed R-D SCN (2) with conditions (3) and (4) is said to achieve exponentially clustered synchronization (ECS) in mean square, if there exist constants $\lambda >0$ and $M\ge 1$ such that

$$\mathrm{E}\parallel e_i^k{(t,·)\parallel }_2\le M\mathrm{E}{\parallel \phi \parallel }_{\overline{\tau }}exp(-\lambda t)$$

for all $i\in C_k$, in which $\phi ={\left({\phi }_i(s,x)-{\displaystyle \sum _{l\in C_k}}{\xi }_l{\phi }_l(s,x)\right)}_{i\in {\mathcal{N}}_N}$.

Furthermore, let ${\displaystyle e(t,x)={\left(e_i^k(t,x)\right)}_{i=1}^N}$, ${\displaystyle F={(D\frac{{\partial }^2{e}_i^k}{\partial x^2}+F_i^k)}_{i=1}^N}$ and

$$H=\mathrm{diag}\left(H_1^1,\cdots ,H_{d_1}^1,H_{d_1+1}^2,\cdots ,H_N^K\right),$$

then $e(t,x)={\left(e^k(t,x)\right)}_{k=1}^K$ with $e^k(t,x)={\left(e_i^k(t,x)\right)}_{i\in C_k}$, and the system (6) can be rewritten as

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{d}e(t,x)}\hfill \\ \hfill & {\displaystyle =F(t,e(t,x),e(t-{\tau }_1\left(t\right),x),m\left(t\right))\mathrm{d}t+H(t,e(t,x),e(t-{\tau }_2\left(t\right),x),m\left(t\right))({\mathbf{1}}_N\otimes \mathrm{d}y\left(t\right)).}\hfill \end{array}$$

(9)

Lemma 2

([29]). For any $u\in {\mathcal{R}}^m$, $v\in {\mathcal{R}}^n$ and $A=\left(a_{ij}\right)\in {\mathcal{R}}^{m\times n}$,

$$u^{T}Av\le {\displaystyle \frac{1}{2}}{max\{m,n\}\parallel A\parallel }_{max}{(\parallel u\parallel }^2+{\parallel v\parallel }^2).$$

Lemma 3

([30]). Suppose that ${\displaystyle A=\left(a_{ij}\right)\in {\mathcal{R}}^{n\times n}}$ is a irreducible matrix with ${\displaystyle a_{ii}=-{\displaystyle \sum _{j=1,j\ne i}^n}{a}_{ij}}$ and ${\displaystyle a_{ij}=a_{ji}\ge 0(i\ne j)}$, ${\displaystyle Q=diag(q_1,q_2,\cdots ,q_l,0,\cdots ,0)}$ with $q_i>0,1\le l\le n$, then ${\lambda }_{max}(A-Q)<0$.

Lemma 4

([34]). Let $\widehat{D}\in {\mathcal{R}}^{n\times n}$ satisfied $\widehat{D}\ge 0$, $z\left(x\right)\in {\mathcal{H}}^1\left([0,l]\right)$ be a vector function with $z\left(0\right)={\mathbf{0}}_n$ or $z\left(l\right)={\mathbf{0}}_n$, then

$${\int }_0^l{z}^T\left(x\right)\widehat{D}z\left(x\right)\mathrm{d}x\le {\displaystyle \frac{4l^2}{{\pi }^2}}{\int }_0^l{\left({\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}\right)}^T\widehat{D}{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}\mathrm{d}x.$$

Lemma 5

([35] Halanay Inequality). For any constants $t_0\ge 0$ and $\tau >0$, let $V\left(t\right)$ is a nonnegative continuous function on $[t_0-\tau ,+\infty )$. If there exist constants ${\alpha }_1$ and ${\alpha }_2$ such that ${\alpha }_1>{\alpha }_2>0$ and

$${\displaystyle \frac{\mathrm{d}V\left(t\right)}{\mathrm{d}t}}\le -{\alpha }_{1}V\left(t\right)+{\alpha }_{2}sup\left\{V\left(s\right):s\in [t-\tau ,t]\right\}$$

for all $t\ge t_0$, then

$$V\left(t\right)\le sup\left\{V\left(s\right):s\in [t_0-\tau ,t_0]\right\}exp(-\gamma (t-t_0)),$$

in which γ satisfies $\gamma -{\alpha }_1+{\alpha }_{2}exp\left(\gamma {\alpha }_1\right)=0$.

Proposition 1.

Under Assumption 4, the following inequality holds for each $\rho \in {\mathcal{N}}_S$ and n-order diagonal matrix $B>0$:

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{tr}\left[{\left(H_i^k(t,x,\rho )\right)}^{T}B{H}_i^k(t,x,\rho )\right]}\hfill \\ \hfill & {\displaystyle \le 2\tilde{n}{L}_2^k\left[{\left(e_i^k(t,x)\right)}^{T}B{e}_i^k(t,x)+e_i^k{(t-{\tau }_2\left(t\right),x)}^{T}B{e}_i^k(t-{\tau }_2\left(t\right),x)\right],}\hfill \end{array}$$

where ${\displaystyle L_2^k={\left(max\left\{\right|{\widehat{L}}^k|,|{\check{L}}^k\left|\right\}\right)}^2.}$

Proof. 

See Appendix A. □

Proposition 2.

Denote $e^k(t,x)={\left(e_i^k(t,x)\right)}_{i\in C_k}$, ${\xi }^k=diag({\xi }_i:i\in C_k)$ and $\widehat{d}=max\{d_k:k\in {\mathcal{N}}_K\}$. For any n-order matrix $B>0$,

$$\begin{array}{cc}\hfill & {\displaystyle {\displaystyle \sum _{k=1}^K}{\displaystyle \sum _{i\in C_k}}{\displaystyle \sum _{j=1}^N}{a}_{ij}{\xi }_i{\left(e_i^k(t,x)\right)}^{T}B{e}_j(t,x)}\hfill \\ \hfill & {\displaystyle \le {\displaystyle \sum _{k=1}^K}{\left(e^k(t,x)\right)}^T\left[{\left({\xi }^k{A}_{kk}\right)}^s\otimes B+p(\widehat{d}{\xi }^k+I_{d_k})\otimes I_n\right]e^k(t,x)}\hfill \end{array}$$

with $p={n\parallel A\parallel }_{max}{\parallel B\parallel }_{max}(K-1)/2$.

Proof. 

See Appendix B. □

3. Distributed Measurement

To achieve the ECS of the directed R-D SCN (2), a switched pinning boundary controller with distributed measurement is designed as

$${\displaystyle u_i(t,m\left(t\right))=-q_i^k\left(m\left(t\right)\right){\int }_0^l{e}_i^k(t,x)\mathrm{d}x,t\in [-\overline{\tau },+\infty ),i\in C_k,m\left(t\right)\in {\mathcal{N}}_S.}$$

(10)

In the controller (10), $q_i^k\left(m\left(t\right)\right)$ is a constant control gain and satisfies $q_i^k\left(m\left(t\right)\right)>0$ when node i is pinned, $q_i^k\left(m\left(t\right)\right)=0$ when node i is not pinned.

For any $\rho \in {\mathcal{N}}_S$, let $B^k\left(\rho \right)>0$ is n-order diagonal matrix, denote

$$Q^k\left(\rho \right)=\mathrm{diag}(q_i^k\left(\rho \right):i\in C_k),p_1={n\parallel A\parallel }_{max}{\parallel B^k\left(\rho \right)\Gamma \left(\rho \right)\parallel }_{max}(K-1)/2,$$

$${\mathcal{Q}}^k\left(\rho \right)=c{\left({\xi }^k{A}_{kk}\right)}^s\otimes \left(B^k\left(\rho \right)\Gamma \left(\rho \right)\right)-\left({\xi }^k{Q}^k\left(\rho \right)\right)\otimes \left(B^k\left(\rho \right)D\left(\rho \right)\right),$$

$$\begin{array}{cc}\hfill & {\displaystyle {\alpha }_1^k\left(\rho \right)=(1-\theta ){\lambda }_{max}\left({\mathcal{Q}}^k\left(\rho \right)\right){\lambda }_{max}^{-1}({\xi }^k\otimes B^k\left(\rho \right))+\frac{1}{2}(1+L_1^k)+2\tilde{n}{L}_2^k}\hfill \\ \hfill & {\displaystyle +{\lambda }_{min}^{-1}\left(B^k\left(\rho \right)\right)[c{p}_1(\widehat{d}+{\lambda }_{min}^{-1}\left({\xi }^k\right))+{\displaystyle \sum _{\nu =1}^S}{\sigma }_{\rho \nu }{\lambda }_{max}\left(B^k\left(\nu \right)\right)],}\hfill \end{array}$$

$${\beta }^k\left(\rho \right)=max\{{\displaystyle \frac{1}{2}}{L}_1^k,2\tilde{n}{L}_2^k\},$$

$${\alpha }_1\left(\rho \right)=-max\{{\alpha }_1^k\left(\rho \right):k\in {\mathcal{N}}_K\},\beta \left(\rho \right)=max\{{\beta }^k\left(\rho \right):k\in {\mathcal{N}}_K\},$$

where $\theta$ is a constant. It should be noted that ${\lambda }_{max}\left({\mathcal{Q}}^k\left(\rho \right)\right)<0$ thanks to Assumptions 1 and 2 and Lemma 3.

Theorem 1.

Under Assumptions 1–4 and the controller (10), the ECS in mean square of the directed R-D SCN (2) can be achieved if for any $k\in {\mathcal{N}}_K$ and $\rho \in {\mathcal{N}}_S$, there exist control gain matrix $Q^k\left(\rho \right)$, free-weighting matrix $B^k\left(\rho \right)$ and regulated constant $0<\theta <1$ such that

$${\displaystyle \theta {\mathcal{Q}}^k\left(\rho \right)+\frac{l^2}{{\pi }^2}\left({\xi }^k{\left(Q^k\left(\rho \right)\right)}^2\right)\otimes \left(B^k\left(\rho \right)D\left(\rho \right)\right)<0,}$$

(11)

$${\displaystyle {\alpha }_1\left(\rho \right)-2\beta \left(\rho \right)>0.}$$

(12)

Proof. 

See Appendix C. □

Remark 5.

By using pinning boundary control strategies, the authors in [31] have addressed the synchronization problem of reaction–diffusion complex networks, those in [32] have achieved the synchronization of coupled reaction–diffusion neural networks, and those in [33] have investigated the finite–time stabilization of a semi-Markov reaction–diffusion memristive neural network. However, the models in [31,32] fail to account for the effects of stochastic noise, Markovian switching, and time delays; furthermore, none of these studies have explored the problem of cluster synchronization.

When all nodes in the network (2) belong to the same cluster, i.e., $K=1$, $k=1,C_k={\mathcal{N}}_N,$ then the cluster synchronization of (2) can degenerate into the complete synchronization, and the following result can be obtained.

Corollary 1.

Under $K=1$, Assumptions 1–4 and the controller (10), the exponentially complete synchronization in mean square of the directed R-D SCN (2) can be achieved if for any $\rho \in {\mathcal{N}}_S$, there exist control gain matrix $Q\left(\rho \right)$, n-order diagonal matrix $B\left(\rho \right)>0$ and a regulated constant $0<\theta <1$ such that

$${\displaystyle \theta \mathcal{Q}\left(\rho \right)+\frac{l^2}{{\pi }^2}\left(\xi {\left(Q\left(\rho \right)\right)}^2\right)\otimes \left(B\left(\rho \right)D\left(\rho \right)\right)<0,}$$

$${\displaystyle {\tilde{\alpha }}_1\left(\rho \right)-2\tilde{\beta }\left(\rho \right)>0,}$$

where

$$Q\left(\rho \right)=diag(q_i^1\left(\rho \right):i\in {\mathcal{N}}_N),\xi =diag({\xi }_i:i\in {\mathcal{N}}_N),\tilde{\beta }\left(\rho \right)=max\{{\displaystyle \frac{1}{2}}{L}_1^1,2\tilde{n}{L}_2^1\},$$

$$\mathcal{Q}\left(\rho \right)=c{\left(\xi A\right)}^s\otimes \left(B\left(\rho \right)\Gamma \left(\rho \right)\right)-\left(\xi Q\left(\rho \right)\right)\otimes \left(B\left(\rho \right)D\left(\rho \right)\right),$$

$$\begin{array}{cc}\hfill & {\displaystyle {\tilde{\alpha }}_1\left(\rho \right)=(1-\theta ){\lambda }_{max}\left(\mathcal{Q}\left(\rho \right)\right){\lambda }_{max}^{-1}(\xi \otimes B\left(\rho \right))+\frac{1}{2}(1+L_1^1)+2\tilde{n}{L}_2^1}\hfill \\ \hfill & {\displaystyle +{\lambda }_{min}^{-1}\left(B\left(\rho \right)\right){\displaystyle \sum _{\nu =1}^S}{\sigma }_{\rho \nu }{\lambda }_{max}\left(B\left(\nu \right)\right).}\hfill \end{array}$$

4. Spatial Sampled-Data Measurement

In order to achieve synchronization as efficient as possible, the interval $[0,l]$ is divided into R uniform sampling intervals by inserting $R-1$ points: $0=x_0<x_1<\cdots <x_R=l,$ where R is a finite positive integer. Obviously, $x_{r+1}-x_r=l/R$ for all $r\in {\mathcal{N}}_{R-1}$. Denote $h=l/R$.

Let ${\overline{x}}_r$ is a sampling point arbitrarily chosen on the interval $[x_r,x_{r+1}]$, then a switched pinning boundary controller with spatial sampled-data is designed as

$${\displaystyle u_i(t,m\left(t\right))=-q_i^k\left(m\left(t\right)\right){\displaystyle \sum _{r=0}^{R-1}}{e}_i^k(t,{\overline{x}}_r),t\in [-\overline{\tau },+\infty ),i\in C_k,m\left(t\right)\in {\mathcal{N}}_S,}$$

(13)

where $q_i^k\left(m\left(t\right)\right)$ is consistent with its definition in the controller (10).

Denote

$${\tilde{\mathcal{Q}}}^k\left(\rho \right)=c{\left({\xi }^k{A}_{kk}\right)}^s\otimes \left(B^k\left(\rho \right)\Gamma \left(\rho \right)\right)-{\displaystyle \frac{1}{2h}}\left({\xi }^k{Q}^k\left(\rho \right)\right)\otimes \left(B^k\left(\rho \right)D\left(\rho \right)\right),$$

$$\begin{array}{cc}\hfill & {\displaystyle {\alpha }_2^k\left(\rho \right)={\lambda }_{max}\left({\tilde{\mathcal{Q}}}^k\left(\rho \right)\right){\lambda }_{max}^{-1}({\xi }^k\otimes B^k\left(\rho \right))+\frac{1}{2}(1+L_1^k)+2\tilde{n}{L}_2^k}\hfill \\ \hfill & {\displaystyle +{\lambda }_{min}^{-1}\left(B^k\left(\rho \right)\right)[c{p}_1(\widehat{d}+{\lambda }_{min}^{-1}\left({\xi }^k\right))+{\displaystyle \sum _{\nu =1}^S}{\sigma }_{\rho \nu }{\lambda }_{max}\left(B^k\left(\nu \right)\right)],}\hfill \end{array}$$

$${\alpha }_2\left(\rho \right)=-max\{{\alpha }_2^k\left(\rho \right):k\in {\mathcal{N}}_K\},$$

then ${\lambda }_{max}\left({\tilde{\mathcal{Q}}}^k\left(\rho \right)\right)<0$ due to Lemma 3.

Theorem 2.

Under Assumptions 1–4 and the controller (13), the ECS in mean square of the directed R-D SCN (2) can be achieved if for any $k\in {\mathcal{N}}_K$ and $\rho \in {\mathcal{N}}_S$, there exist control gain matrix $Q^k\left(\rho \right)$, free-weighting matrix $B^k\left(\rho \right)$ such that

$${\displaystyle 2(l^2+h^2)q_i^k\left(\rho \right)-{\pi }^{2}h\le 0,i\in C_k,}$$

(14)

$${\displaystyle {\alpha }_2\left(\rho \right)-2\beta \left(\rho \right)>0.}$$

(15)

Proof. 

See Appendix D. □

Let $R=2,{\overline{x}}_0=x_0=0,{\overline{x}}_1=x_2=l$, then $h=l/2$, and the controller (13) can reduced to

$${\displaystyle u_i(t,m\left(t\right))=-q_i^k\left(m\left(t\right)\right)[e_i^k(t,0)+e_i^k(t,l)],t\in [-\overline{\tau },+\infty ),i\in C_k,m\left(t\right)\in {\mathcal{N}}_S.}$$

(16)

If $R=1,{\overline{x}}_0=x_1=l$, then $h=l$, and

$${\displaystyle u_i(t,m\left(t\right))=-q_i^k\left(m\left(t\right)\right)e_i^k(t,l),t\in [-\overline{\tau },+\infty ),i\in C_k,m\left(t\right)\in {\mathcal{N}}_S.}$$

(17)

The following results can be derived from Theorem 3.

Corollary 2.

Under Assumptions 1–4 and the controller (16), the ECS in mean square of the directed R-D SCN (2) can be achieved if for any $k\in {\mathcal{N}}_K$ and $\rho \in {\mathcal{N}}_S$, there exist control gain matrix $Q^k\left(\rho \right)$, free-weighting matrix $B^k\left(\rho \right)$ such that (15) and

$${\displaystyle 5l{q}_i^k\left(\rho \right)-{\pi }^2\le 0,i\in C_k.}$$

Corollary 3.

Under Assumptions 1–4 and the controller (17), the ECS in mean square of the directed R-D SCN (2) can be achieved if for any $k\in {\mathcal{N}}_K$ and $\rho \in {\mathcal{N}}_S$, there exist control gain matrix $Q^k\left(\rho \right)$, free-weighting matrix $B^k\left(\rho \right)$ such that (15) and

$${\displaystyle 4l{q}_i^k\left(\rho \right)-{\pi }^2\le 0,i\in C_k.}$$

When the network (2) is not under control, i.e., $q_i^k\left(m\left(t\right)\right)\equiv 0$ for all $i\in C_k$, then the boundary condition (4) can degenerate into

$${\displaystyle \frac{\partial z_i(t,x)}{\partial x}{|}_{x=0}={\mathbf{0}}_n,\frac{\partial z_i(t,x)}{\partial x}{|}_{x=l}={\mathbf{0}}_n,t\in [-\overline{\tau },+\infty ),}$$

(18)

and the following results can be given.

Theorem 3.

Under Assumptions 1–4 and (18), the ECS in mean square of the directed R-D SCN (2) can be achieved if for any $k\in {\mathcal{N}}_K$ and $\rho \in {\mathcal{N}}_S$, there exist control free-weighting matrix $B^k\left(\rho \right)$ such that

$${\displaystyle {\alpha }_3\left(\rho \right)-2\beta \left(\rho \right)>0,}$$

(19)

in which

$${\mathcal{D}}^k\left(\rho \right)=c{\left({\xi }^k{A}_{kk}\right)}^s\otimes \left(B^k\left(\rho \right)\Gamma \left(\rho \right)\right)-{\displaystyle \frac{{\pi }^2}{4l^2}}{\xi }^k\otimes \left(B^k\left(\rho \right)D\left(\rho \right)\right),$$

$$\begin{array}{cc}\hfill & {\displaystyle {\alpha }_3^k\left(\rho \right)={\lambda }_{max}\left({\mathcal{D}}^k\left(\rho \right)\right){\lambda }_{max}^{-1}({\xi }^k\otimes B^k\left(\rho \right))+\frac{1}{2}(1+L_1^k)+2\tilde{n}{L}_2^k}\hfill \\ \hfill & {\displaystyle +{\lambda }_{min}^{-1}\left(B^k\left(\rho \right)\right)[c{p}_1(\widehat{d}+{\lambda }_{min}^{-1}\left({\xi }^k\right))+{\displaystyle \sum _{\nu =1}^S}{\sigma }_{\rho \nu }{\lambda }_{max}\left(B^k\left(\nu \right)\right)],}\hfill \end{array}$$

$${\alpha }_3\left(\rho \right)=-max\{{\alpha }_3^k\left(\rho \right):k\in {\mathcal{N}}_K\}.$$

Proof. 

See Appendix E. □

Corollary 4.

Under $K=1$, Assumptions 1–4 and (18), the exponentially complete synchronization in mean square of the directed R-D SCN (2) can be achieved if for any $\rho \in {\mathcal{N}}_S$, there exist n-order diagonal matrix $B\left(\rho \right)>0$ such that

$${\displaystyle {\tilde{\alpha }}_3\left(\rho \right)-2\tilde{\beta }\left(\rho \right)>0,}$$

where

$$\mathcal{D}\left(\rho \right)=c{\left(\xi A\right)}^s\otimes \left(B\left(\rho \right)\Gamma \left(\rho \right)\right)-{\displaystyle \frac{{\pi }^2}{4l^2}}\xi \otimes \left(B\left(\rho \right)D\left(\rho \right)\right),$$

$$\begin{array}{cc}\hfill & {\displaystyle {\tilde{\alpha }}_3\left(\rho \right)={\lambda }_{max}\left(\mathcal{D}\left(\rho \right)\right){\lambda }_{max}^{-1}(\xi \otimes B\left(\rho \right))+\frac{1}{2}(1+L_1^1)+2\tilde{n}{L}_2^1}\hfill \\ \hfill & {\displaystyle +{\lambda }_{min}^{-1}\left(B\left(\rho \right)\right){\displaystyle \sum _{\nu =1}^S}{\sigma }_{\rho \nu }{\lambda }_{max}\left(B\left(\nu \right)\right).}\hfill \end{array}$$

5. Numerical Simulations

In this section, the controller developed in Section 4 is tested for the following directed coupled R-D neural network with time delay, stochastic perturbation and Markovian switching:

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{d}{z}_i(t,x)}\hfill \\ \hfill & {\displaystyle =[D\left(m\left(t\right)\right)\frac{{\partial }^2{z}_i(t,x)}{\partial x}-B_1\left(m\left(t\right)\right)z_i(t,x)+B_2\left(m\left(t\right)\right)f^k\left(z_i(t,x)\right)}\hfill \\ \hfill & {\displaystyle +B_3\left(m\left(t\right)\right)f^k\left(z_i(t-\tau \left(t\right),x)\right)+J\left(m\left(t\right)\right)}\hfill \\ \hfill & {\displaystyle +c{\displaystyle \sum _{j=1}^9}{a}_{ij}\Gamma \left(m\left(t\right)\right)z_j(t,x)]\mathrm{d}t+h^k(z_i(t-\tau \left(t\right),x),m\left(t\right))\mathrm{d}y\left(t\right),i\in C_k.}\hfill \end{array}$$

(20)

In the network (20), $t\ge 0,x\in [0,0.5]$, $i\in C_k$, $k\in \{1,2,3\}$, $C_1=\{1,2,3\}$, $C_2=\{4,5,6\}$, $C_3=\{7,8,9\}$, $z_i(t,x)={(z_{i1}(t,x),z_{i2}(t,x))}^T$ and

$$\begin{array}{cc}\hfill & {\displaystyle {\left(a_{ij}\right)}_{9\times 9}=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]}\hfill \\ \hfill & {\displaystyle =\left[\begin{array}{ccccccccc}-0.5& 0.25& 0.25& 0.5& 0& -0.5& -0.5& 0.5& 0\\ 0.5& -0.5& 0& 0& 0.5& -0.5& 0& 0& 0\\ 0.25& 0& -0.25& -0.5& 0& 0.5& 0& 0& 0\\ -0.5& 0.5& 0& -0.8& 0.5& 0.3& 0& 0.5& -0.5\\ 0& 0& 0& 0.25& -0.75& 0.5& 0& 0& 0\\ 0& -0.5& 0.5& 0.5& 0.3& -0.8& -0.5& 0.5& 0\\ 0& 0& 0& 0& 0& 0& -0.5& 0.5& 0\\ 0& 0& -0.5& 0& 0.5& 0& 0.25& -0.5& 0.25\\ 0& 0& 0& 0& 0& 0& 0& 0.5& -0.5\end{array}\right]}\hfill \end{array}$$

with third–order submatrix $A_{ij}$.

This means Assumption 2 is true, and the model contains nine nodes divided into three clusters and each node has two neurons. The coupled topology of the nine nodes is visualized by Figure 1, which shows that the graph of the network (20) is directed and strongly connected, and Assumption 1 is tenable.

Consider the Markov chain $\left\{m\right(t)\in \{1,2\}:t\ge 0\}$ with $\delta =0.5$, ${\sigma }_{11}=-0.6$, ${\sigma }_{12}=0.6$, ${\sigma }_{21}=0.4$ and ${\sigma }_{22}=-0.4$. And the Markov chain $m\left(t\right)$ and the one-dimensional Wiener process $y\left(t\right)$ used in the network (20) can be displayed by Figure 2 and Figure 3.

Let $D\left(1\right)=\mathrm{diag}(0.3,0.285)$, $D\left(2\right)=\mathrm{diag}(0.285,0.3)$, $\Gamma \left(1\right)=\mathrm{diag}(0.06,0.045)$, $\Gamma \left(2\right)=\mathrm{diag}(0.1,0.15)$, $J\left(1\right)={(0.1,-0.1)}^T$, $J\left(2\right)={(-0.1,0.1)}^T$, $\tau \left(t\right)=0.5|sin(t\left)\right|$, $c=0.5$, $B_1\left(1\right)=\mathrm{diag}(0.012,0.02)$, $B_1\left(2\right)=\mathrm{diag}(0.02,0.012)$,

$$B_2\left(1\right)=\left[\begin{array}{cc}0.7& 0.07\\ 0.56& 1.26\end{array}\right],B_2\left(2\right)=\left[\begin{array}{cc}0.54& -0.9\\ 0.42& 0.6\end{array}\right],$$

$$B_3\left(1\right)=\left[\begin{array}{cc}-0.6& -0.06\\ -0.48& 1.08\end{array}\right],B_3\left(2\right)=\left[\begin{array}{cc}0.63& 1.05\\ 0.49& -0.7\end{array}\right],$$

$$f^1\left(z_i(·,x)\right)=0.2tanh\left(z_i(·,x)\right),h^1(z_i(·,x),\rho )=0.6H\left(\rho \right)z_i(t,x),i\in C_1,$$

$$f^2\left(z_i(t,x)\right)=0.3arctan\left(z_i(t,x)\right),h^2(z_i(t,x),\rho )=0.5H\left(\rho \right)sin\left(z_i(t,x)\right),i\in C_2,$$

$$f^3\left(z_i(t,x)\right)=0.1arctan\left(z_i(t,x)\right),h^3(z_i(t,x),\rho )=0.4H\left(\rho \right)cos\left(z_i(t,x)\right),i\in C_3,\rho =1,2$$

with $H\left(1\right)=\mathrm{diag}(0.15,0.16)$, $H\left(2\right)=\mathrm{diag}(0.17,0.16)$. Obviously, Assumptions 3 and 4 are all satisfied.

For any $k=1,2,3$, let $w^k(t,x)={\sum }_{i=3k-2}^{3k}{\xi }_i^k{z}_i(t,x),e_i^k(t,x)=z_i(t,x)-w^k(t,x)$, in which ${\xi }^1=\mathrm{diag}(0.4,0.2,0.4)$, ${\xi }^2=\mathrm{diag}(0.3180,0.3463,0.3357),{\xi }^3=\mathrm{diag}(0.25,0.5,0.25)$ can be calculated to satisfy $(1,1,1){\xi }^k{A}_{kk}=(0,0,0)$. Equip the network (20) with the initial functions

$${\phi }_1(s,x)={\left(-1.5cos\left(0.4\pi x\right),2.5cos\left(0.4\pi x\right)\right)}^T,$$

$${\phi }_2(s,x)={\left(-2.5sin\left(0.4\pi x\right)+1.5,0.5sin\left(0.4\pi x\right)-1\right)}^T,$$

$${\phi }_3(s,x)={\left(-0.5x^2,0.5x^2-2.5\right)}^T,{\phi }_4(s,x)={(-7.5,7.5)}^T,{\phi }_5(s,x)={(10,-10)}^T,$$

$${\phi }_6(s,x)={(-3.5,3.5)}^T,{\phi }_7(s,x)={(7x,-7x)}^T,{\phi }_8(s,x)={(-8,8)}^T,$$

$${\phi }_9(s,x)={\left(9sin\left(0.4\pi x\right),-9sin\left(0.4\pi x\right)\right)}^T$$

for all $(s,x)\in [-0.5,0]\times (0,0.5)$ and $m\left(0\right)=1$. And for all $t\in [-0.5,+\infty )$, equip the network (20) with the Neumann boundary condition

$${\displaystyle \frac{\partial z_i(t,x)}{\partial x}{|}_{x=0}={\mathbf{0}}_n,\frac{\partial z_i(t,x)}{\partial x}{|}_{x=0.5}=u_i(t,m\left(t\right)),}$$

in which $u_i(t,m\left(t\right))$ is the switched pinning boundary controller (13) with spatial sampled-data, and $h=0.05,R=10,x_0=0$ and ${\overline{x}}_r=x_r$. That is

$${\displaystyle u_i(t,m\left(t\right))=-q_i^k\left(m\left(t\right)\right){\displaystyle \sum _{r=0}^9}{e}_i^k(t,x_r),t\in [-0.5,+\infty ),i\in C_k,m\left(t\right)\in {\mathcal{N}}_S.}$$

(21)

Choose $q_i^k\left(\rho \right)=0$ for all $\rho =1,2,i\in C_k$, $k=1,2,3$ in the controller (21), the dynamical behaviors of $e_i^k(t,x)$ and $z_i(t,0)$ are presented in Figure 4 and Figure 5, which indicate that the state vectors $z_i(t,x)$ of the network (20) can not synchronize to the average target states $w^k(t,x)$ if there is no control. Choose $q_2^1\left(\rho \right)=q_4^2\left(\rho \right)=q_6^2\left(\rho \right)=q_8^3\left(\rho \right)=0$, $q_1^1\left(1\right)=q_3^1\left(1\right)=q_5^2\left(1\right)=q_7^3\left(1\right)=q_9^3\left(1\right)=0.97$, $q_1^1\left(2\right)=q_3^1\left(2\right)=q_5^2\left(2\right)=q_7^3\left(2\right)=q_9^3\left(2\right)=0.975$ and $B^k\left(\rho \right)=I_2$, which means that only notes labeled 1, 3, 5, 7 and 9 are controlled, then under the controller (21), the conditions (14) and (15) are all satisfied, and the ECS in mean square of the network (20) can be achieved, which is illustrated by Figure 6 and Figure 7. It can be observed that the synchronization time required for the clusters $C_1$ and $C_3$ is significantly less than that for the cluster $C_2$, due to the fact that in $C_1$ and $C_3$, there are controlled nodes numbered 1, 3 and 7, 9, whereas in $C_2$, there is only one controlled node numbered 5. This also demonstrates that, while nodes with minimal pinning control can indeed facilitate network synchronization, they also prolong the time required to achieve that synchronization.

6. Conclusions

This paper has discussed the exponentially clustered synchronization problem of a directed complex network with reaction–diffusion term, time-delays, stochastic noise and Markovian switching. On the one hand, two pinning boundary control protocols, one with distributed measurement and the other with spatial sampled-data, have been developed, in which the control gain is dependent on the Markovian switching and takes different values in each switching state. On the other hand, an average state of the nodes in the same cluster is taken as the target state of the cluster synchronization.

In addition to the controllers designed in this paper, other control strategies may also be utilized to solve the cluster synchronization problem of the stochastic complex networks with reaction–diffusion term and time-delays. For example, the adaptive boundary control in [17,19], the guaranteed cost boundary control in [22], the adaptive pinning control in [24,29], and so on. This is what we will focus on in the future.