Synchronization of Networked Reaction-Diffusion Nonlinear Systems via Hybrid Control

Abstract

This paper addresses the asymptotic synchronization problem for networked nonlinear reaction–diffusion systems under a novel hybrid control strategy. The hybrid controller consists of two components: impulsive control and continuous feedback control. By combining the comparison principle of impulsive systems with the introduction of a time-varying function and a power exponent to flexibly adjust the system, sufficient conditions for synchronization of networked nonlinear reaction–diffusion systems are derived, ensuring that the error dynamics between the network nodes converge to zero. Numerical simulations of a representative example are presented to demonstrate the practical validity and effectiveness of the proposed theoretical control scheme, confirming that the hybrid controller successfully achieves synchronization.

1. Introduction

This section first provides a background introduction to the investigated system and control approach, analyzing the key challenges in the study to derive the motivation of this work, and discussing the limitations of existing research. Subsequently, the main contributions of this paper are summarized. Finally, the organizational structure and notation used throughout the paper are presented.

1.1. Motivation and Related Work

Nonlinear reaction–diffusion systems are widely observed in nature and engineering, such as signal transmission and synchronous firing among neurons in neuroscience [1], species diffusion and population competition in ecology [2], and pathogen propagation over spatial networks in epidemiology [3]. The core feature of such systems lies in the coupling between local nonlinear reactions and spatial diffusion, which gives rise to complex spatiotemporal dynamical behaviors.

Traditional reaction–diffusion models are often formulated in continuous spatial domains (e.g., partial differential equations). However, in practical scenarios, systems are frequently composed of discrete nodes (such as neurons or cities) connected by edges (such as synaptic links or transportation networks), forming complex networks. Networked reaction–diffusion systems integrate discrete network topologies with continuous diffusion processes, where adjacency matrices characterize the interactions among nodes. This modeling framework provides a closer representation of real-world systems, such as information diffusion in social networks [4] and cascading failures in power systems [5].

In addition to theoretical research on dynamical behavior, the development of reaction–diffusion systems has been greatly advanced by numerical and analytical methods for partial differential equations. For nonlinear PDEs, various effective solution frameworks have been proposed, such as the Double Sumudu Transform Iterative Method for nonlinear coupled Sine-Gordon equations [6], and comparative studies analyzing the performance of explicit finite-difference schemes and physics-informed neural networks (PINNs) for Burgers’ equations [7]. For more complex nonlinear evolution equations, approaches such as PINNs and Fourier spectral methods have been successfully applied to generalized Korteweg-de Vries equations [8].

For linear diffusion-type systems, finite-difference schemes remain essential numerical tools, as demonstrated by diffusion-equation-based modeling of gas-sensitive thin-film semiconductors [9]. Collectively, these linear and nonlinear PDE methodologies enrich the theoretical and computational foundations of reaction–diffusion systems and provide essential technical support for understanding their spatiotemporal evolution and developing corresponding control frameworks.

Synchronization serves as the foundation for the coordinated operation of complex systems, with applications ranging from information encoding and transmission in neural networks [10] to cooperative task execution in multi-agent systems [11]. Nevertheless, achieving synchronization in nonlinear reaction–diffusion systems is particularly challenging due to high dimensionality (node states incorporating spatial distributions), time-varying topologies (dynamically changing network connections), and external disturbances (such as noise and delays).

Current control methods can generally be divided into two major categories: continuous control [12] and discrete control [13]. Feedback control [14], one of the most widely used continuous control strategies, relies on the deviation between the system’s real-time output and the desired target. By continuously or periodically adjusting the control input, the system state is gradually steered toward the ideal trajectory. Owing to its robustness, feedback control effectively handles internal parameter variations and external disturbances, and has been extensively applied in industrial processes such as temperature and pressure regulation in chemical production. On the other hand, impulsive control [15], as a discrete strategy, applies instantaneous control inputs at specific moments, allowing for fast and efficient adjustments of system states. This makes it particularly suitable for scenarios requiring rapid transient responses under limited control resources. For example, in specific power electronic systems, pulse-width modulation (PWM) leverages impulsive control to achieve efficient energy conversion and precise distribution.

However, as system complexity increases and control objectives become more diverse, the limitations of relying on a single traditional control strategy become evident. Although continuous feedback control can provide real-time and stable regulation, it requires continuous sampling and transmission of control signals. This leads to significant communication overhead, high control energy consumption, and increased implementation difficulty in large-scale networks or high-dimensional systems. Moreover, under strong nonlinearities or sudden disturbances, continuous feedback tends to respond with a delay, making it difficult to rapidly suppress error propagation within a short time. Pure impulsive control, which applies instantaneous regulation only at discrete moments, effectively reduces energy consumption and communication load. However, between two impulsive instants, the system evolves freely without active regulation. This lack of continuous suppression of nonlinear coupling and diffusion disturbances may cause error amplification or oscillations during the inter-impulse intervals.

In existing studies on synchronization control of networked reaction–diffusion systems, most results rely on a single type of controller. Typically, either continuous feedback control or pure impulsive control is employed to achieve synchronization, while a systematic analytical framework that integrates both mechanisms is still lacking. For example, in [16], impulsive controllers were used to realize impulsive synchronization of two identical reaction–diffusion neural networks with discrete and unbounded distributed delays, where relatively conservative criteria were established and the influence of diffusion terms was taken into account. However, continuous feedback was not incorporated into the controller design, and therefore the synchronization behavior during the continuous-time evolution of the systems was not optimized. Although some existing research has applied hybrid controllers—combining continuous control and impulsive control—to networked systems, these works have not considered the influence of reaction–diffusion terms. For instance, Ref. [17] used hybrid control to study cooperative behaviors of networked systems under continuously connected, impulsively connected, or jointly connected topologies, and provided sufficient conditions for synchronization under both fixed and switching structures. Nevertheless, for PDE-type networked systems with spatiotemporal dynamics, such as reaction–diffusion networks, the advantages of hybrid control have not yet been systematically explored.

To overcome these limitations and harness the advantages of both methods, this study proposes a hybrid control strategy that integrates impulsive control with feedback control. By combining the instantaneous adjustment capability of impulsive control with the continuous corrective mechanism of feedback control, the hybrid approach [18] aims to achieve more precise, efficient, and robust control of complex systems. Specifically, impulsive control provides strong regulation by injecting control signals at predetermined moments. In contrast, feedback control continuously monitors the system state and fine-tunes the control input in real time, thereby ensuring that the system remains stably aligned with the desired trajectory.

In previous studies, considerable attention has been devoted to networked reaction–diffusion systems. For example, Ref. [19] investigated the optimal mixed control of networked reaction–diffusion systems, where reaction control and diffusion control were combined to regulate chemical reaction models toward the desired state. Ref. [20] addressed the control problem of networked systems based on the original Turing reaction–diffusion model and derived the minimal control placement conditions for diffusion systems. Ref. [21] examined the conditions for the emergence of Turing instabilities in reaction–diffusion systems defined on complex networks and modified according to the Cattaneo formulation. These works mainly focus on the intrinsic control properties of traditional models, without providing an in-depth exploration of the cooperative control associated with network topologies and information transmission. Moreover, they generally overlook uncertainties such as parameter perturbations arising during system evolution. In contrast, the present study investigates asymptotic synchronization of networked reaction–diffusion systems via hybrid control. The proposed model explicitly incorporates network topology, offering a more comprehensive framework. Furthermore, the hybrid controller, which integrates continuous control with impulsive discrete control, enables more effective regulation of cooperative information transmission and exhibits dynamic adaptability.

1.2. Contributions

This paper investigates the impulsive synchronization of networked reaction–diffusion systems under hybrid control. It presents a practical framework based on the comparison principle, which is applicable to a wide range of models. The main contributions of this work are as follows:

• We present a novel and practical controller for networked reaction–diffusion systems. The proposed hybrid controller combines continuous feedback control with discrete impulsive control, enabling continuous monitoring of the system states while performing instantaneous adjustments at specific moments to accelerate system synchronization. Furthermore, the impulsive instant interval in this paper is time-varying, which enables the stable conditions more flexible.
• By integrating the Lyapunov function with the comparison principle, a comparison system is constructed to transform the analysis of complex networked systems into the study of simpler functional models, thereby simplifying the derivation process of the system dynamics.
• We propose less conservative conditions for the impulsive synchronization of networked reaction–diffusion systems via a dynamically adaptive approach. In the constructed comparison system, an impulsive function is introduced to dynamically scale the system size, facilitating flexible expansion and contraction. Moreover, a power exponent is incorporated into the Lyapunov-based system state to flexibly adjust the growth rate of the state variables.

1.3. Organization and Notation

The organization of the paper is as follows. Section 2 mainly introduces the model studied in this paper, as well as several existing theoretical results that serve as the foundation for the subsequent proofs. The main results, corresponding proofs, and corollaries are presented in Section 3. A numerical example is provided in Section 4 to demonstrate the feasibility and practicality of the proposed conditions. Finally, Section 5 summarizes the main findings of this work and outlines potential directions for future research.

Notations: The notation employed throughout this paper all conform to the standard, where ${\mathbb{R}}^{+}$ denotes the collection of non-negative real values as well as ${\mathbb{R}}^n$ is the Euclidean space of dimension n. Let $I_n$ means identity matrix, ${\lambda }_{max}(·)$ and ${\lambda }_{min}(·)$ correspond to the maximum and the minimum eigenvalue of a symmetric matrix, respectively. The notation $C^{1,1}({\mathbb{R}}^n\times {\mathbb{R}}^{+};{\mathbb{R}}^{+})$ describes the collection of all non-negative function mappings $V(x,t)$ on ${\mathbb{R}}^n\times {\mathbb{R}}^{+}$ that that possess continuous first-order differentiability in both x and t. $\Omega =\{s=(s_1,s_2,\cdots ,s_q):\left|s_k\right|<{\gamma }_k,{\gamma }_k\in {\mathbb{R}}^{+},k=1,2,\cdots ,q\}$ is an open bounded region in ${\mathbb{R}}^q$ with smooth boundary $\partial \Omega$.

2. Preliminaries and Model Formulation

In this section, the system model and controller design are introduced. In addition, the definitions, lemmas, and assumptions relevant to the main results are presented, serving as the theoretical foundation for the analysis in the next section.

2.1. Model Description

To capture the spatiotemporal evolution inherent in many real-world phenomena—particularly those involving spatial diffusion and nonlinear local interactions—we extend the conventional network model by introducing nonlinear reaction–diffusion dynamics. In this networked reaction–diffusion nonlinear system, each node represents an individual subsystem, while the diffusion component characterizes the spatial coupling among neighboring nodes.

The governing equation of the master subsystem (or reference network) is formulated as

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial x_i(s,t)}{\partial t}}& =\sum _{k=1}^q{\displaystyle \frac{\partial }{\partial s_k}}\left[D_k{\displaystyle \frac{\partial x_i(s,t)}{\partial s_k}}\right]+B_i{x}_i(s,t)+{\tilde{f}}_i\left(x_i(s,t)\right)\hfill \\ \hfill & +\sum _{j=1}^N{a}_{ij}{\tilde{H}}_{ij}(x_i(s,t),x_j(s,t)),\hfill \\ \hfill x_i(s,t_0)& =x_0^{\left(i\right)},i\in \mathcal{V},\hfill \end{array}\right.$$

(1)

subject to the boundary condition $x_i(s,t)=0,(s,t)\in \partial \Omega \times [-\infty ,+\infty )$, where $s=(s_1,s_2,\cdots ,s_q)\in \Omega \subseteq {\mathbb{R}}^q$ is the space coordinate. The state variable $x_i(s,t)\in {\mathbb{R}}^n$ describes the dynamic behavior of node i at location s, ${\displaystyle \sum _{k=1}^q}\partial \left[D_k\partial x_i(s,t)/\phantom{\partial x_i(s,t)\partial s_k}\partial s_k\right]/\phantom{\partial \left[D_k\partial x_i(s,t)/\phantom{\partial x_i(s,t)\partial s_k}\partial s_k\right]\partial s_k}\partial s_k$ represents the nonlinear diffusion term and $D_k=diag\{d_1,d_2,\cdots ,d_n\}$ is the diffusion coefficient matrix along the kth spatial direction. $B_i$ denotes the internal feedback matrix, ${\tilde{f}}_i\in {\mathbb{R}}^n$ is a continuous nonlinear function, ${\tilde{H}}_{ij}$ characterizes the coupling between nodes i and j with coupling strength $a_{ij}$. The vector $x_0^{\left(i\right)}\in {\mathbb{R}}^n$ specifies the initial state of the node i.

Correspondingly, the slave subsystem (or controlled network) can be

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial y_i(s,t)}{\partial t}}& =\sum _{k=1}^q{\displaystyle \frac{\partial }{\partial s_k}}\left[D_k{\displaystyle \frac{\partial y_i(s,t)}{\partial s_k}}\right]+B_i{y}_i(s,t)+{\tilde{f}}_i\left(y_i(s,t)\right)\hfill \\ \hfill & +\sum _{j=1}^N{a}_{ij}{\tilde{H}}_{ij}(y_i(s,t),y_j(s,t))+u_i(s,t),\hfill \\ \hfill y_i(s,t_0)& =y_0^{\left(i\right)},i\in \mathcal{V},\hfill \end{array}\right.$$

(2)

where $y_i(s,t)\in {\mathbb{R}}^n$ denotes the state of the corresponding node in the slave network and the descriptions of the remaining parameters are the same as above, besides $u_i(s,t)$ represents the control input to be designed later.

Defining the state deviation between the two subsystems as ${\eta }_i(s,t)\triangleq y_i(s,t)-x_i(s,t)$, and subsequently the error system is expressed as

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial t}}& =\sum _{k=1}^q{\displaystyle \frac{\partial }{\partial s_k}}\left[D_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}\right]+B_i{\eta }_i(s,t)+f_i\left({\eta }_i(s,t)\right)\hfill \\ \hfill & +\sum _{j=1}^N{a}_{ij}{H}_{ij}({\eta }_i(s,t),{\eta }_j(s,t))+u_i(s,t),\hfill \\ \hfill {\eta }_i(s,t_0)& ={\eta }_0^{\left(i\right)},i\in \mathcal{V},\hfill \end{array}\right.$$

(3)

where

$$f_i\left({\eta }_i(s,t)\right)\triangleq {\tilde{f}}_i\left(y_i(s,t)\right)-{\tilde{f}}_i\left(x_i(s,t)\right),$$

$$\begin{array}{cc}\hfill H_{ij}({\eta }_i(s,t),{\eta }_j(s,t))\triangleq & {\tilde{H}}_{ij}(y_i(s,t),y_j(s,t))-{\tilde{H}}_{ij}(x_i(s,t),x_j(s,t)).\hfill \end{array}$$

2.2. Controller Design

In the control of networked nonlinear systems, using a single control mechanism, either continuous or discrete, is often insufficient to achieve both stability and efficiency. Pure continuous control requires uninterrupted monitoring and signal transmission, leading to high communication and energy costs in large-scale or spatially distributed systems. Moreover, it lacks flexibility in responding to abrupt disturbances. On the other hand, pure discrete (impulsive) control applies corrections only at isolated time instants, which may cause large deviations between impulses and insufficient regulation of the system’s transient behavior.

To overcome these limitations, this study adopts a hybrid control strategy that integrates continuous feedback control with impulsive control. The continuous feedback component provides smooth and real-time stabilization, ensuring that the system state evolves toward a state of synchronization. The impulsive control component performs rapid state corrections at specific instants, effectively suppressing accumulated errors and reducing communication burden. By combining both mechanisms, the hybrid controller achieves a desirable balance between control performance, resource efficiency, and robustness, making it well-suited for practical networked systems with nonlinear and spatially distributed dynamics.

The hybrid controller, combining feedback control and pulse control, is described as

$$u_i(s,t)=\sum _{k=1}^{\infty }\delta (t-t_k)h_i{\eta }_i(s,t)+{\kappa }_i{\eta }_i(s,t),i\in \mathcal{V},k\in \mathbb{Z},$$

(4)

where ${\kappa }_i$ represents the strength of the continuous feedback control, $h_i\in (-2,0)$ represents the impulse gain that regulates the strength of impulsive control, $\delta (·)$ denotes the Dirac delta function to apply the impulsive control at pulse moments. The impulse instant sequence ${\left\{t_k\right\}}_{k=1}^{\infty }$ satisfies $t_{k-1}<t_k$, $\underset{k\to \infty }{lim}{t}_k=+\infty$ and for a given constant $\epsilon$, we have $t_{2k+1}-t_{2k}\le \epsilon (t_{2k}-t_{2k-1})$. Denote ${\Delta }_1\triangleq \underset{j}{sup}\left\{t_{2j+1}-t_{2j}\right\}\le \infty ,{\Delta }_2\triangleq \underset{j}{sup}\left\{t_{2j}-t_{2j-1}\right\}\le \infty$, where ${\Delta }_1$, ${\Delta }_2$ are positive.

Then, the state deviation system (3) under controller (4) is rewritten as

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial t}}& =\sum _{k=1}^q{\displaystyle \frac{\partial }{\partial s_k}}\left[D_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}\right]+B_i{\eta }_i(s,t)+{\kappa }_i{\eta }_i(s,t)\hfill \\ \hfill & +f_i\left({\eta }_i(s,t)\right)+\sum _{j=1}^N{a}_{ij}{H}_{ij}({\eta }_i(s,t),{\eta }_j(s,t)),t\ne t_k\hfill \\ \hfill {\eta }_i(s,{t_k}^{+})& -{\eta }_i(s,{t_k}^{-})=h_i{\eta }_i(s,t_k),t=t_k\hfill \\ \hfill {\eta }_i(s,t_0)& ={{\eta }_0}^{\left(i\right)}\hfill \end{array}\right.$$

(5)

where ${\eta }_i(s,t_k)={\eta }_i(s,t_k^{-})$ and ${\eta }_i(s,t_k^{+})$ exists.

2.3. Preliminary Results and Assumptions

Definition 1. 

The two networked reaction–diffusion nonlinear systems will achieve asymptotic synchronization if the following condition is satisfied

$$\underset{t\to \infty }{lim}∥{\eta }_i\left(t\right)∥=\underset{t\to \infty }{lim}∥y_i\left(t\right)-x_i\left(t\right)∥=0,i\in \mathcal{V},\forall t\ge 0,$$

(6)

where $∥·∥$ stands for the Euclidean norm in ${\mathbb{R}}^n$.

Definition 2  

([22]). Consider a function $\Psi :{\mathbb{R}}^{+}\times {\mathbb{R}}^n\to {\mathbb{R}}^{+}$ which satisfying the following conditions:

(1) 

on each set $(t_{k-1},t_k]\times {\mathbb{R}}^n$ the function Ψ maintain continuous and when $t=t_k^{+}$, for each $x\in {\mathbb{R}}^n$:

$$\underset{(t,y)\to ({t_k}^{+},x)}{lim}\Psi (t,y)=\Psi ({t_k}^{+},x)$$

exists;

(2) 

in $x\in {\mathbb{R}}^n$, Ψ is locally Lipschitz continuous and satisfies $\Psi (t,0)\equiv 0$ for all $t\ge t_0$,

then we can say $\Psi \in {\ominus }_0$.

Remark 1.  

Definition 2 specifies the requirements for a Lyapunov-type function $\Psi (t,x)$ that must remain valid across both continuous intervals and impulsive instants. This property is essential when dealing with systems exhibiting discontinuities or non-smooth evolution, as it guarantees that the Lyapunov framework can still be applied to assess stability under mixed continuous–discrete dynamics.

Since the system under study operates under impulsive control, it naturally exhibits discontinuous dynamics, rendering traditional derivatives inapplicable. Hence, a generalized differentiation concept—specifically the right and upper Dini derivatives—is adopted to characterize the variation rate of the Lyapunov function along system trajectories.

Definition 3  

([22]). If $\Psi \in {\ominus }_0$, for each $(t,x)\in (t_{k-1},t_k]\times {\mathbb{R}}^n$, define the Dini’s derivative as

$$D^{+}\Psi (t,x\left(t\right))=\underset{\Delta \to 0^{+}}{lim}sup{\displaystyle \frac{1}{\Delta }}[\Psi (t+\Delta ,x(t+\Delta ))-\Psi (t,x\left(t\right))].$$

Remark 2.  

To achieve the stability of system (5), this paper primarily employs a comparison-based analytical framework. The central concept is to construct an appropriate auxiliary comparison system whose stability properties can be used to infer those of the original impulsive network. This technique effectively converts the complex, high-dimensional coupled model into a lower-dimensional and analytically manageable system, making it possible to derive synchronization criteria through simpler stability analysis.

Existing research results can be utilized to guide the construction of the comparison system and to derive the corresponding stability conditions.

Definition 4  

([23]). The classical impulsive control system can be described by

$$\left\{\begin{array}{c}\dot{\omega }\left(t\right)=W(t,\omega \left(t\right)),t\ne t_k\hfill \\ \Delta \omega \left(t\right)\stackrel{\Delta }{=}\omega \left(t^{+}\right)-\omega \left(t^{-}\right)=I_k\left(\omega \right),t=t_k,k\in \mathbb{Z}.\hfill \end{array}\right.$$

(7)

Let $\Psi \in {\ominus }_0$ and assume that

$$\left\{\begin{array}{c}{D}^{+}\Psi (t,\omega )\le \varphi (t,\Psi (t,\omega )),t\ne t_k\hfill \\ \Psi (t,\omega +I_k\left(\omega \right))\le h_k(\Psi (t,\omega )),t=t_k,\hfill \end{array}\right.$$

where ψ represents a continuous function satisfying ${\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to \mathbb{R}$ and ${\Psi }_k$ is defined as a non-decreasing function whose mapping is $:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$. Then we can get the following system

$$\left\{\begin{array}{c}\dot{\varpi }=\varphi (t,\varpi ),t\ne t_k\hfill \\ \varpi \left({t_k}^{+}\right)=h_k\left(\varpi \left(t_k\right)\right)\hfill \\ \varpi \left({t_0}^{+}\right)={\varpi }_0\ge 0\hfill \end{array}\right.$$

(8)

to be the comparison system of (7).

Lemma 1  

([24]). Suppose that the following three conditions hold:

(i) 

$\Psi :{\mathbb{R}}^{+}\times {\mathbb{R}}^n\to {\mathbb{R}}^{+}$ and $\Psi \in {\ominus }_0$, then there exists an integer number $\beta \ge 1$, such that:

$$\begin{array}{c}\hfill R\left(t\right)D^{+}\Psi (t,{\omega }^{\beta })+D^{+}R\left(t\right)\Psi (t,{\omega }^{\beta })\le \varphi (t,R\left(t\right)\Psi (t,{\omega }^{\beta })),t\ne t_k\end{array}$$

where ${\omega }^{\beta }={({{\omega }_1}^{\beta },{{\omega }_2}^{\beta },\cdots ,{{\omega }_n}^{\beta })}^T$, g is a continuous function in $(t_{k-1},t_k]\times {\mathbb{R}}^n$ for each $\omega \in {\mathbb{R}}^n$, also

$$R\left(t\right)\ge m>0,\underset{t\to {t_k}^{-}}{lim}R\left(t\right)=R\left(t_k\right),\underset{t\to {t_k}^{+}}{lim}R\left(t\right)\mathit{exists},$$

and

$$D^{+}R\left(t\right)=\underset{\Delta \to 0^{+}}{lim}sup{\displaystyle \frac{1}{\Delta }}[R(t+\Delta )-R\left(t\right)];$$

(ii) 

$R\left({t_k}^{+}\right)\Psi ({t_k}^{+},{(\omega +I_k\left(\omega \right))}^{\beta })\le h_k(R\left(t_k\right)\Psi (t_k,{\omega }^{\beta }))$;

(iii) 

on ${\mathbb{R}}^{+}\times {\mathbb{R}}^n$, $\Psi (t,0)=0$ as well as $\psi \left(∥{\omega }^{\beta }∥\right)\le \Psi (t,{\omega }^{\beta })$ always holds, where $\psi (·)\in \mathcal{K}$ (a class where ψ is strictly increasing, $\psi \in C[{\mathbb{R}}^{+},{\mathbb{R}}^{+}]$, and $\psi \left(0\right)=0$).

Therefore, from the globally asymptotically stable solution of system (8), it can be derived that the original impulsive system (7) also exhibits global asymptotic stability.

Lemma 2  

([24]). Define $\varphi (t,\varpi )=\dot{\phi }\left(t\right)\varpi$, where $\phi \in C^1[{\mathbb{R}}^{+},{\mathbb{R}}^{+}]$, $h_k\left(\varpi \right)={\zeta }_k\varpi$ with ${\zeta }_k\ge 0$, $k\in \mathbb{Z}$. If the requirements in Lemma 1 as well as the following additional constraints are satisfied, the origin system (7) can get globally asymptotically stability.

(i) 

$\phi \left(t\right)$ is a non-decreasing function, $\underset{t\to t_k-}{lim}\phi \left(t\right)=\phi \left(t_k\right)$, and $\underset{t\to {t_k}^{+}}{lim}\phi \left(t\right)=\phi \left({t_k}^{+}\right)$ exists, for all $k\in \mathbb{Z}$;

(ii) 

${sup}_k\{{\zeta }_{k}exp(\phi \left(t_{k+1}\right)-\phi \left({t_k}^{+}\right))\}={\epsilon }_0<\infty$;

(iii) 

there exists $r>1$ satisfying $\phi \left(t_{2k+3}\right)+\phi \left(t_{2k+2}\right)+ln\left(r{\zeta }_{2k+2}{\zeta }_{2k+1}\right)\le \phi \left({t_{2k+2}}^{+}\right)+\phi \left({t_{2k+1}}^{+}\right)$, in which ${\zeta }_{2k+2}{\zeta }_{2k+1}\ne 0$, $k\in \mathbb{Z}$, or there exists $r>1$ satisfying $\phi \left(t_{k+1}\right)+ln\left(r{\zeta }_k\right)\le \phi \left({t_k}^{+}\right)$, $k\in \mathbb{Z}$;

(iv) 

on ${\mathbb{R}}^{+}\times {\mathbb{R}}^n$, $\psi \left(∥\omega ∥\right)\le \Psi (t,\omega )$ as well as $\Psi (t,0)=0$, where $\psi (·)\in \mathcal{K}$.

Remark 3.  

Lemma 1 offers a constructive approach for determining the stability of the original system from the derived comparison dynamics. The introduced parameter β plays a key role in regulating the convergence rate of the system states.

Lemma 2 introduces a specific formulation of the comparison system. When the original system satisfies this structure, its stability can be inferred in a straightforwardly manner. Moreover, Lemma 2 highlights that the gain applied at impulsive instants influences the overall stability behavior of the system.

Lemma 3  

([25]). Let $l\left(s\right)$ be a real-valued function which satisfying $l\left(s\right)\in C_1\left(S\right)$ and ${\left.l\left(s\right)\right|}_{\partial S}=0$, where S is a cube $\left|s_k\right|<\left|{\gamma }_k\right|,k=1,2,\cdots ,q$. Then

$${\int }_S{l}^2\left(s\right)ds\le {{\gamma }_k}^2{\int }_S{\left|{\displaystyle \frac{\partial l\left(s\right)}{\partial s_k}}\right|}^{2}ds$$

Assumption 1.  

For every $x,y\in {\mathbb{R}}^n$, there exists a positive constant ρ, such that

$${\left[y-x\right]}^T\left[f(t,y)-f(t,x)\right]\le \rho {\left[y-x\right]}^T\left[y-x\right],\forall t\in {\mathbb{R}}^n.$$

Assumption 2.  

For all ${\eta }_i,{\eta }_j\in {\mathbb{R}}^n$, there exist positive constants ${\tilde{\Pi }}_{ij}$, ${\widehat{\Pi }}_{ij}$, so that

$$H_{ij}({\eta }_i,{\eta }_j)\le {\tilde{\Pi }}_{ij}{\eta }_i+{\widehat{\Pi }}_{ij}{\eta }_j,$$

where $H_{ij}$ is a function which is Lipschitz continuous in η.

3. Main Results

This section establishes a set of sufficient criteria ensuring the asymptotic stability of system (5). The overall approach proceeds as follows: an auxiliary function is first formulated by combining the Lyapunov function with a time-dependent scaling term $L\left(t\right)$. Next, the evolution of this function is examined by deriving its upper and lower bounds within both continuous intervals and impulsive instants. Using the preceding definitions and lemmas, a suitable comparison system corresponding to system (5) is then developed. Finally, by incorporating the system parameters into the inequality constraints provided in Lemma 2, the sufficient conditions guaranteeing the asymptotic stability of the original system (5) are obtained.

Theorem 1.  

The system (5) will get asymptotic stability if there exists a $r>1$ and a function $R\left(t\right)$ that is non-increasing and differentiable at $t=t_k$ as well as satisfying the conditions of Lemma 1, such that:

(T1) 

for any $k\in \mathbb{Z}$, there exists a constant $\sigma >0$, such that ${\int }_{\Omega }\left[{{\eta }_i}^T(s,{t_k}^{+}){\eta }_i(s,{t_k}^{+})\right]ds\le \sigma {\int }_{\Omega }\left[{{\eta }_i}^T(s,t_k){\eta }_i(s,t_k)\right]ds$, where $\sigma \ge {max}_{i\in \mathcal{V}}{(1+{\eta }_i)}^2,i\in \mathcal{V}$;

(T2) 

$$\underset{k}{sup}\left\{{\sigma }^{\beta }exp\left[2\mu \beta (t_{k+1}-{t_k}^{+})+ln{\displaystyle \frac{R\left(t_{k+1}\right)}{R\left({t_k}^{+}\right)}}\right]\right\}<\infty$$

where $\mu =-\gamma +{\lambda }_{max}\left(B_i\right)+{max}_{i\in \mathcal{V}}\left({\kappa }_i\right)+\rho +\lambda$, $\gamma ={\lambda }_{min}\left\{{\displaystyle \sum _{k=1}^q}{\displaystyle \frac{D_k}{{{\gamma }_k}^2}}\right\}$, $\lambda =max\left\{{\displaystyle \sum _{j=1}^N}{a}_{ij}({\tilde{\Pi }}_{ij}+{\displaystyle \frac{\nu }{2}}{\widehat{\Pi }}_{ij})+{\displaystyle \sum _{k=1}^N}{a}_{ki}{\displaystyle \frac{1}{2\nu }}{\widehat{\Pi }}_{ki}\right\}$;

(T3) 

$$\begin{array}{c}\hfill 2\mu \beta (1+\epsilon ){\Delta }_2+ln\left[{\displaystyle \frac{R\left(t_{2k+3}\right)}{R\left(t_{2k+2}^{+}\right)}}{\displaystyle \frac{R\left(t_{2k+2}\right)}{R\left(t_{2k+1}^{+}\right)}}\right]\le -ln\left(r{\sigma }^{2\beta }\right),\end{array}$$

or

$$\begin{array}{c}\hfill 2\mu \beta max({\Delta }_1,{\Delta }_2)+ln\left[{\displaystyle \frac{R\left(t_{k+1}\right)}{R\left(t_k^{+}\right)}}\right]\le -ln\left(r{\sigma }^{\beta }\right);\end{array}$$

(T4) 

$$2\mu \beta +{\displaystyle \frac{\dot{R}\left(t\right)}{R\left(t\right)}}\ge 0.$$

Proof. 

The Lyapunov function we chosen is defined as

$$V(\eta ,t)=\sum _{i=1}^N{\int }_{\Omega }[{{\eta }_i}^T(s,t){\eta }_i(s,t)]ds.$$

For $t\in [t_{k-1},t_k)$, define the function $M\left(t\right)=R\left(t\right)V({\eta }^{\beta },t)$, $R\left(t\right)$ is differentiable at $t\ne t_k$, then differentiating $M\left(t\right)$ along system (5), we can get

$$\begin{array}{cc}\hfill \dot{M}\left(t\right)& =R\left(t\right)D^{+}V({\eta }^{\beta },t)+D^{+}R\left(t\right)V({\eta }^{\beta },t)\hfill \\ & =R\left(t\right)\{2\beta \sum _{i=1}^N{\int }_{\Omega }[{{\eta }_i}^T(s,t){\eta }_i(s,t){]}^{\beta -1}{{\eta }_i}^T(s,t){\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial t}}ds\}+D^{+}R\left(t\right)V({\eta }^{\beta },t)\hfill \\ & =2\beta R\left(t\right)\sum _{i=1}^N{\int }_{\Omega }[{{\eta }_i}^T(s,t){\eta }_i(s,t){]}^{\beta -1}{{\eta }_i}^T(s,t)·[\sum _{k=1}^q{\displaystyle \frac{\partial }{\partial s_k}}\left[D_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}\right]+B_i{\eta }_i(s,t)\hfill \\ & +{\kappa }_i{\eta }_i(s,t)+f_i\left({\eta }_i(s,t)\right)+\sum _{j=1}^N{a}_{ij}{H}_{ij}({\eta }_i(s,t),{\eta }_j(s,t))]ds+D^{+}R\left(t\right)V({\eta }^{\beta },t).\hfill \end{array}$$

By the Neumann boundary condition and Green’s formula, we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {{\eta }_i}^T(s,t)\sum _{k=1}^q{\displaystyle \frac{\partial }{\partial s_k}}\left[D_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}\right]ds\hfill \\ & =-\sum _{k=1}^q{\int }_{\Omega }{\displaystyle \frac{\partial {{\eta }_i}^T(s,t)}{\partial s_k}}·\left[D_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}\right]ds+{\int }_{\partial \Omega }{{\eta }_i}^T(s,t)\left[D_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial t}}\right]n_{k}dS\hfill \\ & =-\sum _{k=1}^q{\int }_{\Omega }{\displaystyle \frac{\partial {{\eta }_i}^T(s,t)}{\partial s_k}}·\left[D_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}\right]ds,\hfill \end{array}$$

where $n_k$ is the normal vector satisfying $n_k\notin \partial \Omega$.

Since $D_k$ is a symmetric positive definite matrix, we can define $D_k={Z_k}^T{Z}_k$ with Z is a real matrix. Then

$$\begin{array}{cc}& \sum _{k=1}^q{\int }_{\Omega }{\displaystyle \frac{\partial {{\eta }_i}^T(s,t)}{\partial s_k}}{D}_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}ds=\sum _{k=1}^q{\int }_{\Omega }{\displaystyle \frac{\partial {\left[Z{\eta }_i(s,t)\right]}^T}{\partial s_k}}{\displaystyle \frac{\partial \left[Z{\eta }_i(s,t)\right]}{\partial s_k}}ds.\hfill \end{array}$$

Let $z_i(s,t)=Z{\eta }_i(s,t)$. Through Lemma 3, we have

$$\begin{array}{cc}\hfill \sum _{k=1}^q{\int }_{\Omega }{\displaystyle \frac{\partial {z_i}^T(s,t)}{\partial s_k}}{\displaystyle \frac{\partial z_i(s,t)}{\partial s_k}}ds& \ge \sum _{k=1}^q{\displaystyle \frac{1}{{{\gamma }_k}^2}}{\int }_{\Omega }{z_i}^T(s,t)z_i(s,t)ds\hfill \\ & =\sum _{k=1}^q{\displaystyle \frac{1}{{{\gamma }_k}^2}}{\int }_{\Omega }{{\eta }_i}^T(s,t)D_k{\eta }_i(s,t)ds.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}& -\sum _{k=1}^q{\int }_{\Omega }{\displaystyle \frac{\partial {{\eta }_i}^T(s,t)}{\partial s_k}}{D}_k{\displaystyle \frac{\partial {\eta }_i(s,t)}{\partial s_k}}ds\le -{\int }_{\Omega }{{\eta }_i}^T(s,t)\left[\sum _{k=1}^q{\displaystyle \frac{D_k}{{{\gamma }_k}^2}}\right]{\eta }_i(s,t)ds.\hfill \end{array}$$

According to the property of the matrix $B_i$, we can get

$$\begin{array}{cc}& {\int }_{\Omega }{{\eta }_i}^T(s,t)(B_i+k_i){\eta }_i(s,t)ds\le {\int }_{\Omega }[{\lambda }_{max}\left(B_i\right)+\underset{i\in \mathcal{V}}{max}\left({\kappa }_i\right)]{{\eta }_i}^T(s,t){\eta }_i(s,t)ds.\hfill \end{array}$$

According to Assumption 1, we have

$$\begin{array}{c}\hfill \sum _{i=1}^N{\int }_{\Omega }{{\eta }_i}^T(s,t)f_i\left({\eta }_i(s,t)\right)ds\le \rho \sum _{i=1}^N{\int }_{\Omega }{{\eta }_i}^T(s,t){\eta }_i(s,t)ds.\end{array}$$

Moreover, from Assumption 2 and Young’s inequality, we can derive

$$\begin{array}{cc}& \sum _{i=1}^N{\int }_{\Omega }{{\eta }_i}^T(s,t)\sum _{j=1}^N{a}_{ij}{H}_{ij}({\eta }_i(s,t),{\eta }_j(s,t))ds\hfill \\ & \le \sum _{i=1}^N{\int }_{\Omega }\left[\sum _{j=1}^N{a}_{ij}{\tilde{\Pi }}_{ij}{{\eta }_i}^T(s,t){\eta }_i(s,t)+\sum _{j=1}^N{a}_{ij}{\widehat{\Pi }}_{ij}{{\eta }_i}^T(s,t){\eta }_j(s,t)\right]ds\hfill \\ & \le {\int }_{\Omega }\sum _{i=1}^N\left[\sum _{j=1}^N{a}_{ij}{\tilde{\Pi }}_{ij}{{\eta }_i}^T(s,t){\eta }_i(s,t)+\sum _{j=1}^N{a}_{ij}{\widehat{\Pi }}_{ij}({\displaystyle \frac{\nu }{2}}{{\eta }_i}^T(s,t){\eta }_i(s,t)+{\displaystyle \frac{1}{2\nu }}{{\eta }_j}^T(s,t){\eta }_j(s,t))\right]ds\hfill \\ & ={\int }_{\Omega }\left[\sum _{i=1}^N\sum _{j=1}^N{a}_{ij}({\tilde{\Pi }}_{ij}+{\displaystyle \frac{\nu }{2}}{\widehat{\Pi }}_{ij}){{\eta }_i}^T(s,t){\eta }_i(s,t)\right]+\left[\sum _{i=1}^N\sum _{j=1}^N{a}_{ij}{\displaystyle \frac{1}{2\nu }}{\widehat{\Pi }}_{ij}{{\eta }_j}^T(s,t){\eta }_j(s,t)\right]ds\hfill \end{array}$$

$$\begin{array}{cc}& ={\int }_{\Omega }\left[\sum _{i=1}^N\sum _{j=1}^N{a}_{ij}({\tilde{\Pi }}_{ij}+{\displaystyle \frac{\nu }{2}}{\widehat{\Pi }}_{ij})\right]{{\eta }_i}^T(s,t){\eta }_i(s,t)+\left[\sum _{k=1}^N\sum _{i=1}^N{a}_{ki}{\displaystyle \frac{1}{2\nu }}{\widehat{\Pi }}_{ki}\right]{{\eta }_i}^T(s,t){\eta }_i(s,t)ds\hfill \\ & \le \lambda \sum _{i=1}^N{\int }_{\Omega }{{\eta }_i}^T(s,t){\eta }_i(s,t)ds,\hfill \end{array}$$

where $\lambda =max\{{\displaystyle \sum _{j=1}^N}{a}_{ij}({\tilde{\Pi }}_{ij}+{\displaystyle \frac{\nu }{2}}{\widehat{\Pi }}_{ij})+{\displaystyle \sum _{k=1}^N}{a}_{ki}{\displaystyle \frac{1}{2\nu }}{\widehat{\Pi }}_{ki}\}$.

Let $\gamma ={\lambda }_{min}\left\{{\displaystyle \sum _{k=1}^q}{\displaystyle \frac{D_k}{{{\gamma }_k}^2}}\right\}$, then we can get

$$\begin{array}{cc}& \dot{M}\left(t\right)\hfill \\ & \le 2\beta R\left(t\right)\sum _{i=1}^N{\int }_{\Omega }[{{\eta }_i}^T(s,t){\eta }_i(s,t){]}^{\beta -1}\left[{\lambda }_{max}\left(B_i\right)+\underset{i\in \mathcal{V}}{max}\left({\kappa }_i\right)-\gamma +\rho +\lambda \right]{{\eta }_i}^T(s,t){\eta }_i(s,t)ds\hfill \\ & +D^{+}R\left(t\right)V({\eta }^{\beta },t)\hfill \\ & \le 2\beta R\left(t\right)\mu \sum _{i=1}^N{\int }_{\Omega }{\left[{{\eta }_i}^T(s,t){\eta }_i(s,t)\right]}^{\beta }ds+D^{+}RV({\eta }^{\beta },t)\hfill \\ & \le \left[2\mu \beta +{\displaystyle \frac{\dot{R}\left(t\right)}{R\left(t\right)}}\right]M\left(t\right)\hfill \end{array}$$

where $\mu =-\gamma +{\lambda }_{max}\left(B_i\right)+{max}_{i\in \mathcal{V}}\left({\kappa }_i\right)+\rho +\lambda$.

When $t=t_k$, from the system (5), we can get that ${\eta }_i(s,{t_k}^{+})={\eta }_i(s,{t_k}^{-})+h_i{\eta }_i(s,t_k)=(1+h_i){\eta }_i(s,t_k)$. Then, we have

$$\begin{array}{cc}\hfill V_i({\eta }_i(s,{t_k}^{+}),{t_k}^{+})& ={\int }_{\Omega }{{\eta }_i}^T(s,{t_k}^{+}){\eta }_i(s,{t_k}^{+})ds\hfill \\ & ={\int }_{\Omega }{(1+h_i)}^2{\eta }_i^T(s,t_k){\eta }_i(s,t_k)ds\le \sigma V_i({\eta }_i\left(s,t_k\right),t_k)\hfill \end{array}$$

Then, the condition (T1) in Theorem 1 is satisfied. We can use the above inequality and $R\left(t\right)$ is non-increasing to get that

$$\begin{array}{cc}\hfill M\left({t_k}^{+}\right)& =R\left({t_k}^{+}\right)V({\eta }^{\beta },{t_k}^{+})=R\left({t_k}^{+}\right)\sum _{i=1}^N{\int }_{\Omega }[{{\eta }_i(s,{t_k}^{+})}^T{\eta }_i(s,{t_k}^{+}){]}^{\beta }ds\hfill \\ & \le {\sigma }^{\beta }R\left(t_k\right)V({\eta }^{\beta },t_k)={\sigma }^{\beta }M\left(t_k\right)\hfill \end{array}$$

Then the comparison system of system (5) is

$$\left\{\begin{array}{c}\dot{\varpi }\left(t\right)=\left[2\mu \beta +{\displaystyle \frac{\dot{R}\left(t\right)}{R\left(t\right)}}\right]\varpi \left(t\right),t\ne t_k\hfill \\ \varpi \left({t_k}^{+}\right)={\sigma }^{\beta }\varpi \left(t_k\right),t=t_k\hfill \\ \varpi \left({t_0}^{+}\right)={\varpi }_0\ge 0.\hfill \end{array}\right.$$

(9)

Let $\dot{\phi }\left(t\right)=2\mu \beta +{\displaystyle \frac{\dot{R}\left(t\right)}{R\left(t\right)}}$, ${\zeta }_k={\sigma }^{\beta }$, it follows from Lemma 2 that

$$\begin{array}{cc}& \underset{k}{sup}\left\{{\sigma }^{\beta }exp\left[2\mu \beta (t_{k+1}-{t_k}^{+})+ln{\displaystyle \frac{R\left(t_{k+1}\right)}{R\left({t_k}^{+}\right)}}\right]\right\}\hfill \\ & ={\sigma }^{\beta }exp\left[2\mu \beta max({\Delta }_1,{\Delta }_2)\right],\hfill \end{array}$$

which means that (T2) holds.

Moreover,

$$\begin{array}{cc}& {\int }_{t_{2k+2}}^{t_{2k+3}}\dot{\phi }\left(t\right)dt+{\int }_{t_{2k+1}}^{t_{2k+2}}\dot{\phi }\left(t\right)dt+ln\left(r{\sigma }^{2\beta }\right)\le 0\hfill \end{array}$$

and

$${\int }_{{t_k}^{+}}^{t_{k+1}}\dot{\phi }\left(t\right)dt+ln\left(r{\sigma }^{\beta }\right)\le 0,r>1\mathrm{hold},$$

thus we have

$$\begin{array}{cc}& 2\mu \beta \left(t_{2k+3}-t_{2k+2}\right)+2\mu \beta \left(t_{2k+2}-t_{2k+1}\right)+ln\left[{\displaystyle \frac{R\left(t_{2k+3}\right)}{R\left(t_{2k+2}^{+}\right)}}\right]+ln\left[{\displaystyle \frac{R\left(t_{2k+2}\right)}{R\left(t_{2k+1}^{+}\right)}}\right]+ln\left(r{\sigma }^{2\beta }\right)\hfill \\ & \le 2\mu \beta \left({\Delta }_1+{\Delta }_2\right)+ln\left[{\displaystyle \frac{R\left(t_{2k+3}\right)}{R\left(t_{2k+2}^{+}\right)}}\right]+ln\left[{\displaystyle \frac{R\left(t_{2k+2}\right)}{R\left(t_{2k+1}^{+}\right)}}\right]+ln\left(r{\sigma }^{2\beta }\right)\hfill \\ & \le 2\mu \beta (1+\epsilon ){\Delta }_2+ln\left[{\displaystyle \frac{R\left(t_{2k+3}\right)}{R\left(t_{2k+2}^{+}\right)}}{\displaystyle \frac{R\left(t_{2k+2}\right)}{R\left(t_{2k+1}^{+}\right)}}\right]+ln\left(r{\sigma }^{2\beta }\right)\le 0\hfill \end{array}$$

and

$$\begin{array}{cc}& 2\mu \beta \left(t_{k+1}-t_k\right)+ln\left[{\displaystyle \frac{R\left(t_{k+1}\right)}{R\left(t_k^{+}\right)}}\right]+ln\left(r{\sigma }^{\beta }\right)\hfill \\ & \le 2\mu \beta max({\Delta }_1,{\Delta }_2)+ln\left[{\displaystyle \frac{R\left(t_{k+1}\right)}{R\left(t_k^{+}\right)}}\right]+ln\left(r{\sigma }^{\beta }\right)\le 0\hfill \end{array}$$

with

$$2\mu \beta +{\displaystyle \frac{\dot{R}\left(t\right)}{R\left(t\right)}}\ge 0,$$

then the origin system (5) can achieve asymptotic stability. The condition (T2), (T3), (T4) in Theorem 1 are all satisfied. The proof is completed. □

Remark 4.  

While both studies adopt networked reaction–diffusion systems as the underlying framework, the models in [26,27] aim to minimize a cost functional through reaction–diffusion or simplicial complex control. In contrast, our work establishes sufficient conditions for asymptotic synchronization via hybrid impulsive and continuous feedback control. In addition, unlike the control-dependent Laplacian decomposition and simplex-weight modulation in [27], the coupling structure in this paper allows for nonlinear inter-node interaction functions, which makes the coupling mechanism more general and capable of capturing nonlinear spatial dependencies.

Remark 5.  

If $R\left({t_{2k+2}}^{+}\right)=R\left(t_{2k+2}\right)$, from the condition (T3) in Theorem 1, we can get that $V(\eta ,t)$ only needs to be non-increasing on the odd-numbered time subsequence, rather than over the entire switching sequence, leading to our result being less conservative.

If $\beta =1$ in Theorem 1, the theorem in [28] can be obtained. Furthermore, if $N=1$ (i.e., the system is not a network), $\beta =1$ and $R\left(t\right)\equiv 1$ in Theorem 1, we can get the theorem in [29] by Theorem 1. In the proposed hybrid control scheme, the impulsive control instants are time-varying, which differs from the fixed-step impulsive control used in [26]. This allows for more flexible and adaptive synchronization in practical systems with varying temporal demands.

Remark 6.  

The comparison principle adopted in this paper enables the transformation of the analysis of complex reaction–diffusion networks into that of a simpler comparison system, thereby allowing the derivation of sufficient conditions for asymptotic stability. The introduction of the function $R\left(t\right)$ increases the flexibility of the stability conditions. Many existing studies on impulsive systems employing the comparison principle [30,31] can be considered as special cases with $R\left(t\right)=1$ when $R\left(t\right)$ is not introduced.

If the linear feedback term $B_i{\eta }_i$ is removed, the proposed system can be reduced to a similar framework analyzed in [21]. Compared to the reaction–diffusion systems with finite propagation studied in [21], which introduce inertial times to model relativistic effects, our model does not assume finite propagation speeds but instead incorporates spatial diffusion with impulsive jumps. This simplifies the dynamics while still capturing essential spatiotemporal coupling. Furthermore, the introduction of the scaling function $R\left(t\right)$ and the power exponent β in our Lyapunov-based comparison system provides a more flexible and less conservative stability condition than the classical Turing instability analysis used in their work.

4. Numerical Examples

In this section, two numerical examples with different N values are provided to validate the feasibility and universality of our results.

Example 1.  

In this case, a one-dimensional networked reaction–diffusion nonlinear system consisting of $N=4$ nodes is considered. The spatial domain is selected as $\Omega =[-2,2]$ with homogeneous Neumann boundary conditions and $q=1$. The diffusion coefficient is set as $D_1=0.1$, and the local linear term as $B_i=-I$, so ${\lambda }_{max}\left(B_i\right)=-1$. The coupling topology between nodes is defined by the adjacency matrix:

$$A={\left(a_{ij}\right)}_{N\times N}=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 0\end{array}\right]$$

The local reaction dynamics of each node are described by the nonlinear function ${\tilde{f}}_i\left(x\right)=\alpha tanh\left(x\right)$, where the nonlinearity coefficient is chosen as $\alpha =2$. What’s more, the coupling function is set as ${\tilde{H}}_{ij}\left(x\right)=sin(x_j-x_i)$. The continuous feedback and impulsive control gains are chosen as ${\kappa }_i=2$ and $h_i=-0.6$, respectively.

Therefore, from Lemma 3 and Assumptions 1, and 2, we can get $\theta =2$, ${\tilde{\Pi }}_{ij}={\widehat{\Pi }}_{ij}=1$, ${\gamma }_1=2$. Then, we can calculate that $\gamma ={\displaystyle \frac{1}{40}}$, $\lambda =0.8$, so that $\mu =-\gamma +{\lambda }_{max}\left(B_i\right)+{max}_{i\in \mathcal{V}}\left({\kappa }_i\right)+\theta +\lambda =3.775$, $\sigma \ge {(1+h_i)}^2=0.16$. Choose $r=1.1$, $\beta =1$, $\epsilon =0.5$, and

$$R\left(t\right)=\left\{\begin{array}{c}{e}^{-k}+1,t=t_k\\ e^{-t+t_k-k}+1,t\in (t_{k-1},t_k).\end{array}\right.$$

Therefore, according to the condition(T3)in Theorem 1, we can derive that $t_{2k}-t_{2k-1}\le {\Delta }_2\le -ln\left(r{\sigma }^2\right)/2\mu \beta (1+\epsilon )\approx 0.315$, ${\Delta }_1\le \epsilon {\Delta }_2=0.158$. Therefore, all the parameters satisfy Theorem 1.

For the master subsystem, the initial spatial distribution of each node is given by

$$x_i(s,0)=0.5sin\left({\displaystyle \frac{\pi is}{2}}\right)$$

while for the slave subsystem, the initial spatial distribution is

$$y_i(s,0)=4cos\left({\displaystyle \frac{\pi is}{4}}\right)$$

Figure 1 consists of four subplots corresponding to the three-dimensional spatial state evolution of the master system, slave system, and their deviation system for four network nodes during the simulation. The color scale represents the magnitude of each node’s state value. It can be observed that the master and slave systems exhibit large state differences at the initial time; however, under the designed control scheme, their spatial waveforms gradually converge and become nearly identical as time progresses. Meanwhile, the error system tends toward zero, as reflected by the color transition to neutral tones, demonstrating that synchronization between the two systems has been successfully achieved. These results demonstrate that the proposed control strategy achieves satisfactory synchronization performance under the combined action of continuous feedback and impulsive control. The smooth spatiotemporal evolution further reflects the regulating effect of the reaction–diffusion term on spatial balance.

Figure 2 presents the spatial cross-sectional distributions of the master and slave systems at the end of the simulation period. The blue solid lines denote the master system, while the red dashed lines denote the slave system. As shown, the profiles of each node almost completely overlap, further confirming that the two systems have achieved complete spatial synchronization under the designed control strategy.

Figure 3 plots the time evolution of the $L_2$-norm of the synchronization error, defined as

$${∥\eta ∥}_{L_2}=\sqrt{\sum _{i=1}^N{\int }_{\Omega }{\left|y_i(s,t)-x_i(s,t)\right|}^{2}ds}$$

The error norm decreases monotonically and approaches zero after approximately $t\approx 1\mathrm{s}$, confirming the global asymptotic stability of the synchronization manifold. This behavior clearly demonstrates the efficiency and robustness of the proposed impulsive synchronization strategy.

Example 2.  

In this case, to simplify result presentation and computational complexity, a one-dimensional networked reaction–diffusion system with $q=1$ remains selected as the simulation subject, with the number of nodes set to $N=41$. The non-zero elements in the corresponding coupling weight matrix are listed as follows:

• $a_{1,3}=a_{3,14}=a_{14,25}=a_{25,36}=a_{37,26}=a_{26,15}=a_{15,4}=a_{4,1}=a_{1,5}=a_{5,16}=a_{16,27}=a_{27,38}=a_{1,6}=a_{6,17}=a_{17,28}=a_{28,39}=a_{33,22}=a_{22,21}=a_{21,10}=a_{10,1}=2$;
• $a_{2,4}=a_{4,6}=a_{6,8}=a_{8,10}=a_{10,2}=a_{3,11}=a_{11,9}=a_{9,7}=a_{7,5}=a_{5,3}=a_{21,13}=a_{13,15}=a_{15,17}=a_{17,19}=a_{19,21}=a_{14,12}=a_{12,20}=a_{20,18}=a_{18,16}=a_{16,14}=3$;
• $a_{1,11}=a_{11,12}=a_{12,23}=a_{23,34}=a_{1,8}=a_{8,19}=a_{19,30}=a_{30,41}=a_{1,9}=a_{9,20}=a_{20,31}=a_{31,32}=a_{40,29}=a_{29,18}=a_{18,7}=a_{7,1}=a_{1,2}=a_{2,13}=a_{13,24}=a_{24,35}=1$;
• $a_{22,24}=a_{24,26}=a_{26,28}=a_{28,30}=a_{30,22}=a_{25,23}=a_{23,31}=a_{31,29}=a_{29,27}=a_{27,25}=a_{36,34}=a_{34,32}=a_{32,40}=a_{40,38}=a_{38,36}=a_{35,37}=a_{37,39}=a_{39,41}=a_{41,33}=a_{33,35}=1.5$.

In this case, the coupling function is replaced with ${\tilde{H}}_{ij}\left(x\right)={\displaystyle \frac{1}{5}}sin(x_j-x_i)$ and the impulsive control gain is chosen as $h_i=-0.7$. Therefore, ${\tilde{\Pi }}_{ij}={\widehat{\Pi }}_{ij}=0.2$, correspondingly. The other parameters and function selections remain unchanged, such as $\Omega =[-2,2]$, $D_1=0.1$, ${\tilde{f}}_i\left(x\right)=\alpha tanh\left(x\right)=2tanh\left(x\right)$, ${\kappa }_i=2$. Then, we can get $\theta =2$, ${\gamma }_1=2$, $\gamma ={\displaystyle \frac{1}{40}}$, $\lambda =3.5$, so that $\mu =-\gamma +{\lambda }_{max}\left(B_i\right)+{max}_{i\in \mathcal{V}}\left({\kappa }_i\right)+\theta +\lambda =6.475$, $\sigma \ge {(1+h_i)}^2=0.09$. Choose $r=1.1$, $\beta =1$, $\epsilon =0.5$, and $R\left(t\right)=1$, we can derive that $t_{2k}-t_{2k-1}\le {\Delta }_2\le -ln\left(r{\sigma }^2\right)/2\mu \beta (1+\epsilon )\approx 0.0.243$, ${\Delta }_1\le \epsilon {\Delta }_2\approx 0.122$. The initial spatial distribution of each node also remain unchanged.

Figure 4 and Figure 5 present the results of this example. Figure 4 displays the three-dimensional spatiotemporal evolution waveforms of the master system, slave system, and error system for four selected nodes. Figure 5 shows the time evolution of the synchronization error $L_2$-norm integrated across all nodes. These figures demonstrate that despite significantly different initial states across nodes, the synchronization error rapidly converges to zero under the hybrid control scheme proposed in this paper, achieving master-slave synchronization and verifying the effectiveness of our results. Compared with Example 1 which involved fewer nodes, this example employs a larger number of nodes with more complex coupling relationships, yet the proposed method remains applicable, confirming the universality of the research findings.

5. Conclusions

This paper achieves asymptotic synchronization of a networked reaction–diffusion nonlinear system by employing the comparison principle and a hybrid impulsive feedback control strategy, and several sufficient conditions with both flexibility and reduced conservativeness are derived. The lemmas and proof techniques used in this work are applicable to a class of higher-order systems amenable to the comparison principle, thus providing a practical framework for simplifying stability analysis. The simulation results demonstrate that under the proposed hybrid control scheme, the error system converges rapidly, indicating that the synchronization is achieved efficiently and validating the theoretical findings of this paper.

Although the proposed controller has been improved, the selected model has not yet fully incorporated practical factors, such as time-delay effects, which will be the focus of future research.