Adaptive Control for Multi-Agent Systems Governed by Fractional-Order Space-Varying Partial Integro-Differential Equations

Abstract

This paper investigates a class of multi-agent systems (MASs) governed by nonlinear fractional-order space-varying partial integro-differential equations (SVPIDEs), which incorporate both nonlinear state terms and integro terms. Firstly, a distributed adaptive control protocol is developed for leaderless fractional-order SVPIDE-based MASs, aiming to achieve consensus among all agents without a leader. Then, for leader-following fractional-order SVPIDE-based MASs, the protocol is extended to account for communication between the leader and follower agents, ensuring that the followers reach consensus with the leader. Finally, three examples are presented to illustrate the effectiveness of the proposed distributed adaptive control protocols.

1. Introduction

Multi-agent systems (MASs) allow agents to work together autonomously, synchronize their actions, and tackle complex challenges that would be too difficult or impossible for a single agent to manage independently [1,2]. The robustness and expandability of MASs make them indispensable in domains like robotics [3], traffic management [4], distributed computing, smart grid [5], smart energy systems [6], sensor networks [7], e-commerce [8], and remote health monitoring [9].

By now, most literature holds that MASs are usually modeled by ordinary differential equations [10,11,12]. With the rapid scientific and technological development, more agents with flexible materials are being introduced, such as flexible manipulators and flexible tubing [13]. Almost all behaviors depend not only on time but also on spatial factors [14,15,16]. Consequently, MASs, modeled by partial differential equations (PDEs) [17], or partial integro-differential equations (PIDEs) [18], are critical to solve these challenges.

Recently, several significant studies have focused on PDE-based MASs (PDEsMASs) [19]. Ferrari-Trecate et al. introduced functional analysis for the coordination of leaderless and leader–follower PDEsMASs using automatic control [20]. Qi et al. proposed an approach to track a target with a formation for PDEsMASs using a simple distributed control [21]. Meurer and Krstic investigated a distributed-parameter setting and a flatness-based approach for motion planning of PDEsMASs. Man et al. studied heterogeneous nonlinear PDEsMASs by using single-point and double-boundary control for finite-time deployment [22]. First-order linear hyperbolic PDEsMASs were developed by Wang and Huang for consensus via boundary control [23]. Wan et al. explored the output-feedback adaptive control of PDEsMASs with time delays [24]. Li and Liu proposed consensus tracking control of nonlinear PDEsMASs by using Hamilton’s principle to deal with flexible manipulators [25]. Compared to PDEs, PIDEs integrate both differential and integral operators, enabling them to capture phenomena with non-local interactions and memory effects [26]. PIDE-based multi-agent systems (PIDEsMASs) have attracted significant attention. Dai et al. proposed a proportional–spatial differential control method for time-delayed parabolic PIDEsMASs [18]. Gabriel studied cooperative output regulation of general heterodirectional hyperbolic PIDE–ODE-based MASs with disturbances [27]. These illustrated the good properties and results of MASs with spatio-temporal characterization.

Fractional-order systems display dynamics with non-integer orders, enabling more adaptable modeling of intricate behaviors relative to integer-order systems, and offering improved abilities by incorporating memory and historical effects [28]. Yan et al. proposed two boundary control approaches for fractional-order PDEsMASs under two different measurement forms [29], considered PDE-ODE models [30], and further studied observer-based boundary control [31]. Yang et al. proposed impulsive control protocols [32] and Zhao et al. studied boundary control for event-triggered consensus of fractional-order PDEsMASs [33]. These studies successfully addressed the consensus problem for fractional-order PDEsMASs. Space-varying PDEs are characterized by coefficients that differ across different regions in space, which occurs due to the properties of the medium, such as material density, thermal conductivity, or stiffness [34,35,36,37]. Such spatial variations introduce increasing complexity into the behavior and solutions of PDEs. The state of an agent may be influenced not only by time but also by spatial variations and past decisions, and these historical and spatial dependencies can be effectively modeled using integral terms and fractional-order techniques. However, consensus control of MASs governed by nonlinear fractional-order space-varying partial integro-differential equations (SVPIDEs) has not been solved, which is still a challenge.

This paper studies a class of semilinear fractional-order SVPIDE-based MASs, where the integro-term results from the product of two nonlinear functions. Initially, a distributed adaptive controller is developed for leaderless fractional-order SVPIDE-based MASs. For leader-following fractional-order SVPIDE-based MASs, a distributed adaptive controller is designed to further account for communication both among the following agents and between the leader and following agents. Using Lyapunov’s function and fractional-order inequalities, sufficient conditions for consensus of both leaderless and leader-following fractional-order SVPIDE-based MASs are derived.

2. Problem Formulation

One fractional-order SVPIDE-based MAS with spatio-temporal characteristics and semilinear terms is studied as

$$\left\{\begin{array}{c}{}_{t_0}{}^{c }D_t^{\alpha }{y}_i(\epsilon ,t)={\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial y_i(\epsilon ,t)}{\partial \epsilon }}\right)+{\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial y_i(\epsilon ,t)}{\partial \epsilon }}+A\left(\epsilon \right)y_i(\epsilon ,t)+f\left(y_i(\epsilon ,t)\right)\hfill \\ +B\left(\epsilon \right){\int }_0^{\epsilon }{y}_i(\rho ,t)d\rho +C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho +u_i(\epsilon ,t)\hfill \\ {\displaystyle \frac{\partial y_i(0,t)}{\partial \epsilon }}=0,{\displaystyle \frac{\partial y_i(L,t)}{\partial \epsilon }}=0,\hfill \\ y_i(\epsilon ,0)=y_i^0\left(\epsilon \right),\hfill \end{array}\right.$$

(1)

where $y_i(\epsilon ,t)\in {\mathbb{R}}^n$ stands for the state, $(\epsilon ,t)\in [0,L]\times [0,\infty ]$ represents the spatial and temporal variables, ${}_{t_0}{}^{c }D_t^{\alpha }$ is the Caputo fractional-order derivative, $\alpha \in (0,1)$, $0<L\in \mathbb{R}$, $f(·)$ and $g(·)$ are nonlinear functions, $u_i(\epsilon ,t)$ is the input, ${\Theta }_1\left(\epsilon \right)>0,{\Theta }_2$, $A\left(\epsilon \right),B\left(\epsilon \right),$$C\left(\epsilon \right)\in {\mathbb{R}}^{n\times n}$ denotes known matrix functions, $i\in \{1,2,\cdots ,N\}$, and N represents the total number of agents.

Remark 1.

This paper focuses on the distributed adaptive consensus control of fractional-order SVPIDE-based MASs, considering both leaderless and leader-following scenarios. The proposed methods effectively account for the complex dynamics introduced by fractional-order terms and spatial variations in the system.

Remark 2.

More recently, only a limited number of significant studies have focused on researching fractional-order MASs modeled by PDEs [29,30,38,39]. Based on those results, this paper further researched the fractional-order MASs modeled by PIDEs.

Definition 1

([40]). The Caputo fractional-order derivative of $h(\epsilon ,t)$ concerning time t is expressed as

$$\begin{array}{c}{}_{t_0}{}^{c }D_t^{\alpha }h(\epsilon ,t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_{t_0}^t{\displaystyle \frac{\partial h(\epsilon ,\kappa )}{\partial \kappa }}{\displaystyle \frac{1}{{(t-\kappa )}^{\alpha }}}d\kappa ,\hfill \end{array}$$

(2)

where $0<\alpha <1$.

Definition 2.

The leaderless fractional-order PIDEsMASs (2) reach consensus if

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|\right|y_i(\epsilon ,t)-{\displaystyle \frac{1}{N}}\sum _{j=1}^N{y}_j(\epsilon ,t)\left|\right|=0.\end{array}$$

(3)

Assumption 1.

Suppose for any scalars $s_1$ and $s_2$, there is a positive scalar γ satisfying

$$\begin{array}{c}\hfill \left|\right|f\left(s_1\right)-f\left(s_2\right){\left|\right|}_2\le \gamma \left|\right|s_1-s_2{\left|\right|}_2.\end{array}$$

(4)

Lemma 1

([41]). Given a differential function $p(\epsilon ,t)$, the following holds:

$${}_{t_0}{}^{c }D_t^{\alpha }\left(p^T(\epsilon ,t)p(\epsilon ,t)\right)\le 2p^T{(\epsilon ,t)}_{t_0}^c{D}_t^{\alpha }p(\epsilon ,t).$$

(5)

3. Consensus of Leaderless Fractional-Order SVPIDE-Based MASs

Letting the consensus error of leaderless fractional-order SVPIDE-based MASs be $e_i(\epsilon ,t)\triangleq y_i(\epsilon ,t)-{\displaystyle \frac{1}{N}}{\sum }_{j=1}^N{y}_j(\epsilon ,t),$ a distributed adaptive controller is studied as

$$\begin{array}{cc}\hfill & u_i(\epsilon ,t)=p_i\left(t\right)\sum _{j=1}^N{a}_{ij}(y_j(\epsilon ,t)-y_i(\epsilon ,t)),\hfill \\ \hfill & \dot{p_i}\left(t\right)={\tau }_i\sum _{j=1}^N{l}_{ij}{\int }_0^L{e}_i^T(\epsilon ,t)e_j(\epsilon ,t)d\epsilon ,\hfill \end{array}$$

(6)

where $p_i\left(t\right)$ represents the control gain and ${\tau }_i>0$ is any real number. $\mathcal{A}={\left(a_{ij}\right)}_{N\times N}$ is defined such that $a_{ij}=a_{ji}>0$ when i is connected with j; otherwise, $a_{ij}=0$. Here, $\mathcal{L}={\left(l_{ij}\right)}_{N\times N}$ is a Laplacian matrix, where ${\mathcal{L}}_{ii}={\displaystyle \sum _{j=1}^N}{a}_{ij}$ and ${\mathcal{L}}_{ij}=-a_{ij}$ when $i\ne j$.

Remark 3.

Space-varying PIDEs have coefficients differing across different regions in space, introducing increasing complexity into the behaviors. This paper has managed to build a distributed adaptive control for the consensus of fractional-order SVPIDE-based MASs.

By applying the control protocol (6), the behavior of $e_i(\epsilon ,t)$ can be obtained

$$\left\{\begin{array}{c}{}_{t_0}{}^{c }D_t^{\alpha }{e}_i(\epsilon ,t)={\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}\right)+{\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}+A\left(\epsilon \right)e_i(\epsilon ,t)\hfill \\ +f\left(y_i(\epsilon ,t)\right)-{\displaystyle \frac{1}{N}}\sum _{j=1}^{N}f\left(y_j(\epsilon ,t)\right)+B\left(\epsilon \right){\int }_0^{\epsilon }{e}_i(\rho ,t)d\rho +C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho \hfill \\ -{\displaystyle \frac{1}{N}}\sum _{j=1}^{N}C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_j(\rho ,t)\right)d\rho +u_i(\epsilon ,t),\hfill \\ {\displaystyle \frac{\partial e(0,t)}{\partial \epsilon }}={\displaystyle \frac{\partial e(L,t)}{\partial \epsilon }}=0,\hfill \end{array}\right.$$

(7)

where $e\stackrel{\Delta }{=}{[e_1^T,\cdots ,e_N^T]}^T.$

Theorem 1.

Under Assumption 1, the leaderless fractional-order SVPIDE-based MASs (1) can reach consensus by using the distributed adaptive controller (6).

Proof. 

Let the Lyapunov function be

$$\begin{array}{cc}\hfill V_1\left(t\right)=& 0.5\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)e_i(\epsilon ,t)d\epsilon +\sum _{i=1}^N{\displaystyle \frac{{(p_i\left(t\right)-\beta )}^2}{2{\tau }_i}}.\hfill \end{array}$$

(8)

Fractional-order differentiating $V_1\left(t\right)$ yields

$$\begin{array}{cc}{}_{t_0}{}^{c }D_t^{\alpha }{V}_1\left(t\right)\le \hfill & \sum _{i=1}^N{\int }_0^L{e}_i^T{(\epsilon ,t)}_{t_0}^c{D}_t^{\alpha }{e}_i(\epsilon ,t)d\epsilon +\sum _{i=1}^N{\displaystyle \frac{p_i\left(t\right)-\beta }{{\tau }_i}}\dot{p_i}\left(t\right)\hfill \\ \hfill =& \sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t){\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}\right)d\epsilon +\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t){\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}d\epsilon \hfill \\ & +\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)A\left(\epsilon \right)e_i(\epsilon ,t)d\epsilon +\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)f\left(y_i(\epsilon ,t)\right)d\epsilon \hfill \\ & -{\displaystyle \frac{1}{N}}\sum _{i=1}^N\sum _{j=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)f\left(y_j(\epsilon ,t)\right)d\epsilon \hfill \\ & +\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)B\left(\epsilon \right){\int }_0^{\epsilon }{e}_i(\rho ,t)d\rho d\epsilon \hfill \\ & +\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho d\epsilon \hfill \\ & -{\displaystyle \frac{1}{N}}\sum _{i=1}^N\sum _{j=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_j(\rho ,t)\right)d\rho d\epsilon \hfill \\ & +\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)u_i(\epsilon ,t)d\epsilon \hfill \\ & +\sum _{i=1}^N{\int }_0^L\sum _{i=1}^N(p_i\left(t\right)-\beta )\sum _{j=1}^N{l}_{ij}{e}_i^T(\epsilon ,t)e_j(\epsilon ,t)d\epsilon .\hfill \end{array}$$

(9)

Using integration by parts, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^L{e}_i^T(\epsilon ,t){\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}\right)d\epsilon \hfill \\ \hfill =& e_i^T(\epsilon ,t){\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}{|}_{\epsilon =0}^{\epsilon =L}-{\int }_0^L{\displaystyle \frac{\partial \left(e_i^T(\epsilon ,t)\right)}{\partial \epsilon }}{\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}d\epsilon \hfill \\ \hfill =& -{\int }_0^L{\displaystyle \frac{\partial e_i^T(\epsilon ,t)}{\partial \epsilon }}{\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}d\epsilon \hfill \end{array}$$

(10)

Applying the triangle inequality [42],

$$\begin{array}{c}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t){\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}d\epsilon \hfill \\ \le 0.5inf{\left({\Theta }_1\left(\epsilon \right)\right)}^{-1}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t){\Theta }_2\left(\epsilon \right){\Theta }_2^T\left(\epsilon \right)e_i(\epsilon ,t)d\epsilon \hfill \\ +0.5inf\left({\Theta }_1\left(\epsilon \right)\right)\sum _{i=1}^N{\int }_0^L{\displaystyle \frac{\partial e_i^T(\epsilon ,t)}{\partial \epsilon }}{\displaystyle \frac{\partial e_i(\epsilon ,t)}{\partial \epsilon }}d\epsilon .\hfill \end{array}$$

(11)

where $inf(·)$ denotes the infimum of the corresponding function.

Since ${\sum }_{i=1}^N{e}_i(\epsilon ,t)=0$, one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)\sum _{j=1}^{N}f\left(y_j(\epsilon ,t)\right)d\epsilon \hfill \\ \hfill =& \sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)f({\displaystyle \frac{1}{N}}\sum _{j=1}^N{y}_j(\epsilon ,t))d\epsilon \hfill \\ \hfill =& 0.\hfill \end{array}$$

(12)

According to Assumption 1 and (12), one has

$$\begin{array}{cc}\hfill & \sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)f\left(y_i(\epsilon ,t)\right)d\epsilon -{\displaystyle \frac{1}{N}}\sum _{i=1}^N\sum _{j=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)f\left(y_j(\epsilon ,t)\right)d\epsilon \hfill \\ \hfill =& \sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)(f\left(y_i(\epsilon ,t)\right)-{\displaystyle \frac{1}{N}}\sum _{j=1}^{N}f\left(y_j(\epsilon ,t)\right))d\epsilon \hfill \\ \hfill =& \sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)(f\left(y_i(\epsilon ,t)\right)-f({\displaystyle \frac{1}{N}}\sum _{j=1}^N{y}_j(\epsilon ,t)))d\epsilon \hfill \\ \hfill \le & {\gamma }_1\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)e_i(\epsilon ,t)d\epsilon .\hfill \end{array}$$

(13)

Using Wirtinger’s inequality [43] and the triangle inequality, one has

$$\begin{array}{c}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)B\left(\epsilon \right){\int }_0^{\epsilon }{e}_i(\rho ,t)d\rho d\epsilon \hfill \\ \le 0.5\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)B\left(\epsilon \right)B^T\left(\epsilon \right)e_i(\epsilon ,t)d\epsilon +0.5\sum _{i=1}^N{\int }_0^L{\int }_0^{\epsilon }{e}_i^T(\rho ,t)d\rho {\int }_0^{\epsilon }{e}_i(\rho ,t)d\rho d\epsilon \hfill \\ \le 0.5\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)B\left(\epsilon \right)B^T\left(\epsilon \right)e_i(\epsilon ,t)d\epsilon +2{\pi }^{-2}{L}^2\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)e_i(\epsilon ,t)d\epsilon .\hfill \end{array}$$

(14)

Using Assumption 1 and the triangle inequality, one has

$$\begin{array}{c}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho d\epsilon \hfill \\ -{\displaystyle \frac{1}{N}}\sum _{i=1}^N\sum _{j=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_j(\rho ,t)\right)d\rho d\epsilon \hfill \\ \le {\displaystyle \frac{1}{2}}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)C\left(\epsilon \right)C^T\left(\epsilon \right)e_i(\epsilon ,t)d\epsilon +{\displaystyle \frac{1}{2}}\sum _{i=1}^N{\int }_0^L{\int }_0^{\epsilon }G\left(e_i^T(\rho ,t)\right)d\rho {\int }_0^{\epsilon }G\left(e_i(\rho ,t)\right)d\rho d\epsilon \hfill \\ \le {\displaystyle \frac{1}{2}}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)C\left(\epsilon \right)C^T\left(\epsilon \right)e_i(\epsilon ,t)d\epsilon +2{\pi }^{-2}{L}^2\sum _{i=1}^N{\int }_0^{L}G\left(e_i^T(\epsilon ,t)\right)G\left(e_i(\epsilon ,t)\right)d\epsilon \hfill \\ \le {\displaystyle \frac{1}{2}}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)C\left(\epsilon \right)C^T\left(\epsilon \right)e_i(\epsilon ,t)d\epsilon +2{\pi }^{-2}{L}^2{\gamma }_2^2\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)e_i(\epsilon ,t)d\epsilon ,\hfill \end{array}$$

(15)

and $G\left(e\right)\stackrel{\Delta }{=}{[G^T\left(e_1\right),\cdots ,G^T\left(e_N\right)]}^T$ and $G\left(e_i\right)\stackrel{\Delta }{=}g\left(y_i\right)-{\displaystyle \frac{1}{N}}{\sum }_{j=1}^{N}g\left(y_j\right)$.

Using the property of Laplace matrices,

$$\begin{array}{c}\sum _{i=1}^N{\int }_0^L{e}_i^T(\epsilon ,t)u_i(\epsilon ,t)d\epsilon \hfill \\ +\sum _{i=1}^N{\int }_0^L(p_i\left(t\right)-\beta )\sum _{j=1}^N{l}_{ij}{e}_i^T(\epsilon ,t)e_j(\epsilon ,t)d\epsilon \hfill \\ =-\beta \sum _{i=1}^N{\int }_0^L\sum _{j=1}^N{l}_{ij}{e}_i^T(\epsilon ,t)e_j(\epsilon ,t)d\epsilon \hfill \\ \le -\beta {\lambda }_2\left(\mathcal{L}\right){\int }_0^L{e}^T(\epsilon ,t)e(\epsilon ,t)d\epsilon .\hfill \end{array}$$

(16)

where ${\lambda }_2\left(\mathcal{L}\right)$ denotes the smallest non-zero eigenvalue of the Laplacian matrix.

Substituting (10)–(16) into (9) yields

$$\begin{array}{cc}{}_{t_0}{}^{c }D_t^{\alpha }{V}_1\left(t\right)\le \hfill & {\int }_0^L{\displaystyle \frac{{\partial }^{T}e(\epsilon ,t)}{\partial \epsilon }}(-I_N\otimes {\Theta }_1\left(\epsilon \right)+0.5inf\left({\Theta }_1\left(\epsilon \right)I\right)){\displaystyle \frac{\partial e(\epsilon ,t)}{\partial \epsilon }}d\epsilon \hfill \\ & +{\int }_0^L{e}^T(\epsilon ,t)(I_N\otimes (0.5A\left(\epsilon \right)+0.5A^T\left(\epsilon \right)+0.5B\left(\epsilon \right)B^T\left(\epsilon \right)+0.5C\left(\epsilon \right)C^T\left(\epsilon \right)\hfill \\ & +0.5inf{\left({\Theta }_1\left(\epsilon \right)\right)}^{-1}{\Theta }_2\left(\epsilon \right){\Theta }_2^T\left(\epsilon \right))+({\gamma }_1+2{\pi }^{-2}{L}^2+2{\pi }^{-2}{L}^2{\gamma }_2^2)I)e(\epsilon ,t)d\epsilon \hfill \\ & -\beta {\lambda }_2\left(\mathcal{L}\right){\int }_0^L{e}^T(\epsilon ,t)e(\epsilon ,t)d\epsilon .\hfill \end{array}$$

(17)

Choosing $\beta >{\displaystyle \frac{{\lambda }_{max}(0.5I_N\otimes (A+A^T+B{B}^T+C{C}^T+inf{\left({\Theta }_1\left(\epsilon \right)\right)}^{-1}{\Theta }_2{\Theta }_2^T)+({\gamma }_1+2{\pi }^{-2}{L}^2+2{\pi }^{-2}{L}^2{\gamma }_2^2)I)}{{\lambda }_2\left(\mathcal{L}\right)}},$ it can be derived from (17) that ${}_{t_0}{}^{c }D_t^{\alpha }{V}_1\left(t\right)<0$, which implies $\left|\right|e(\epsilon ,t)\left|\right|\to 0$ as $t\to \infty$, and makes SVPIDE-based MASs reach consensus. □

Remark 4.

Unlike traditional methods [10,11,12], which primarily consider time-based dynamics, the proposed method accounts for both historical (past states) and spatial (space-varying) dependencies in the agents’ behavior. This is achieved through the use of fractional-order calculus and integral terms, which capture the memory and spatial variations that are often present in real-world systems. This makes the model more versatile and accurate for systems with complex temporal and spatial dynamics.

4. Consensus of the Leader-Following Fractional-Order SVPIDE-Based MASs

The dynamic for the i-th follower of the leader-following fractional-order SVPIDE-based MASs is expressed as

$$\left\{\begin{array}{c}{}_{t_0}{}^{c }D_t^{\alpha }{y}_i(\epsilon ,t)={\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial y_i(\epsilon ,t)}{\partial \epsilon }}\right)+{\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial y_i(\epsilon ,t)}{\partial \epsilon }}+A\left(\epsilon \right)y_i(\epsilon ,t)+f\left(y_i(\epsilon ,t)\right),\hfill \\ +B\left(\epsilon \right){\int }_0^{\epsilon }{y}_i(\rho ,t)d\rho +C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho +{\tilde{u}}_i(\epsilon ,t),\hfill \\ {\displaystyle \frac{\partial y_i(0,t)}{\partial \epsilon }}=0,{\displaystyle \frac{\partial y_i(L,t)}{\partial \epsilon }}=0,\hfill \\ y_i(\epsilon ,t)=y_i^0\left(\epsilon \right).\hfill \end{array}\right.$$

(18)

The dynamic of the leader is expressed as

$$\left\{\begin{array}{c}{}_{t_0}{}^{c }D_t^{\alpha }{y}_0(\epsilon ,t)={\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial y_0(\epsilon ,t)}{\partial \epsilon }}\right)+{\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial y_0(\epsilon ,t)}{\partial \epsilon }}+A\left(\epsilon \right)y_0(\epsilon ,t)+f\left(y_0(\epsilon ,t)\right)\hfill \\ +B\left(\epsilon \right){\int }_0^{\epsilon }{y}_0(\rho ,t)d\rho +C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_0(\rho ,t)\right)d\rho ,\hfill \\ {\displaystyle \frac{\partial y_0(0,t)}{\partial \epsilon }}=0,{\displaystyle \frac{\partial y_0(L,t)}{\partial \epsilon }}=0,\hfill \\ y_0(\epsilon ,t)=y_0^0\left(\epsilon \right).\hfill \end{array}\right.$$

(19)

The error of the leader-following fractional-order SVPIDE-based MASs is described as $\tilde{e}(\epsilon ,t)\triangleq y_i(\epsilon ,t)-y_0(\epsilon ,t),$ and a distributed adaptive controller is defined as

$$\left\{\begin{array}{c}{\tilde{u}}_i(\epsilon ,t)=p_i\left(t\right)[\sum _{j=1}^N{a}_{ij}(y_j(\epsilon ,t)-y_i(\epsilon ,t))+{\overline{a}}_i(y_0(\epsilon ,t)-y_i(\epsilon ,t))],\hfill \\ \dot{p_i}\left(t\right)={\tau }_i\sum _{j=1}^N{h}_{ij}{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t){\tilde{e}}_j(\epsilon ,t)d\epsilon ,\hfill \end{array}\right.$$

(20)

where if $y_i$ can obtain state information from $y_0$, then ${\overline{a}}_i>0$; otherwise, ${\overline{a}}_i=0$. $\mathcal{H}={\left(h_{ij}\right)}_{N\times N}$ is a positive definite symmetric matrix, in which $h_{ii}={\displaystyle \sum _{j=1}^N}{a}_{ij}+{\overline{a}}_i$ and $h_{ij}=-a_{ij}$ when $i\ne j$.

By applying the controller (20), the behavior of ${\tilde{e}}_i(\epsilon ,t)$ yields

$$\left\{\begin{array}{c}{}_{t_0}{}^{c }D_t^{\alpha }{\tilde{e}}_i(\epsilon ,t)={\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial {\tilde{e}}_i(\epsilon ,t)}{\partial \epsilon }}\right)+{\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial {\tilde{e}}_i(\epsilon ,t)}{\partial \epsilon }}+A\left(\epsilon \right)e_i(\epsilon ,t)+f\left(y_i(\epsilon ,t)\right)\hfill \\ -f\left(y_0(\epsilon ,t)\right)+B\left(\epsilon \right){\int }_0^{\epsilon }{\tilde{e}}_i(\rho ,t)d\rho +C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho \hfill \\ -C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_0(\rho ,t)\right)d\rho +{\tilde{u}}_i(\epsilon ,t),\hfill \\ {\displaystyle \frac{\partial {\tilde{e}}_i(0,t)}{\partial \epsilon }}={\displaystyle \frac{\partial {\tilde{e}}_i(L,t)}{\partial \epsilon }}=0.\hfill \end{array}\right.$$

(21)

Theorem 2.

Under Assumption 1, the following fractional-order SVPIDE-based MASs (18) can reach consensus with the leader (19) by using the distributed adaptive controller (20).

Proof. 

Let the Lyapunov function be

$$\begin{array}{cc}\hfill V_2\left(t\right)=& {\displaystyle \frac{1}{2}}\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t){\tilde{e}}_i(\epsilon ,t)d\epsilon +\sum _{i=1}^N{\displaystyle \frac{{(p_i\left(t\right)-\beta )}^2}{2{\tau }_i}}.\hfill \end{array}$$

(22)

Taking the derivative of $V_2\left(t\right)$ yields

$$\begin{array}{cc}\hfill {}_{t_0}{}^{c }D_t^{\alpha }{V}_2\left(t\right)\le & {\int }_0^L{\tilde{e}}_i^T{(\epsilon ,t)}_{t_0}^c{D}_t^{\alpha }{\tilde{e}}_i(\epsilon ,t)d\epsilon +\sum _{i=1}^N{\displaystyle \frac{p_i\left(t\right)-\beta }{{\tau }_i}}\dot{p_i}\left(t\right)\hfill \\ \hfill =& \sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t){\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial {\tilde{e}}_i(\epsilon ,t)}{\partial \epsilon }}\right)d\epsilon +\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t){\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial {\tilde{e}}_i(\epsilon ,t)}{\partial \epsilon }}d\epsilon \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t)A\left(\epsilon \right){\tilde{e}}_i(\epsilon ,t)d\epsilon +\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t)f\left(y_i(\epsilon ,t)\right)d\epsilon \hfill \\ \hfill & -\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t)f\left(y_0(\epsilon ,t)\right)d\epsilon +\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t)B\left(\epsilon \right){\int }_0^{\epsilon }{\tilde{e}}_i(\rho ,t)d\rho d\epsilon \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t)C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho d\epsilon \hfill \\ \hfill & -\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t)C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_0(\rho ,t)\right)d\rho d\epsilon \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t){\tilde{u}}_i(\epsilon ,t)d\epsilon \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^L(p_i\left(t\right)-\beta )\sum _{j=1}^N{l}_{ij}{\tilde{e}}_i^T(\epsilon ,t){\tilde{e}}_j(\epsilon ,t)d\epsilon .\hfill \end{array}$$

(23)

Using the property of Laplace matrices,

$$\begin{array}{c}\sum _{i=1}^N{\int }_0^L{\tilde{e}}_i^T(\epsilon ,t){\tilde{u}}_i(\epsilon ,t)d\epsilon +\sum _{i=1}^N{\int }_0^L\sum _{i=1}^N(p_i\left(t\right)-\beta )\sum _{j=1}^N{l}_{ij}{\tilde{e}}_i^T(\epsilon ,t){\tilde{e}}_j(\epsilon ,t)d\epsilon \hfill \\ =-\beta {\int }_0^L{\tilde{e}}^T(\epsilon ,t)(\mathcal{H}\otimes I_n)\tilde{e}(\epsilon ,t)d\epsilon \hfill \\ \le -\beta {\lambda }_{min}\left(\mathcal{H}\right){\int }_0^L{\tilde{e}}^T(\epsilon ,t)\tilde{e}(\epsilon ,t)d\epsilon ,\hfill \end{array}$$

(24)

where $\tilde{e}\stackrel{\Delta }{=}[{\tilde{e}}_1^T,{\tilde{e}}_2^T,\cdots ,{\tilde{e}}_N^T].$

Similar to the proof of Theorem 1, one has

$$\begin{array}{cc}\hfill {}_{t_0}{}^{c }D_t^{\alpha }{V}_2\left(t\right)\le & {\int }_0^L{\displaystyle \frac{{\partial }^T\tilde{e}(\epsilon ,t)}{\partial \epsilon }}(-I_N\otimes {\Theta }_1\left(\epsilon \right)+0.5inf\left({\Theta }_1\left(\epsilon \right)\right)I){\displaystyle \frac{\partial \tilde{e}(\epsilon ,t)}{\partial \epsilon }}d\epsilon \hfill \\ & +{\int }_0^L{\tilde{e}}^T(\epsilon ,t)(I_N\otimes (0.5A\left(\epsilon \right)+0.5A^T\left(\epsilon \right)+0.5B\left(\epsilon \right)B^T\left(\epsilon \right)+0.5C\left(\epsilon \right)C^T\left(\epsilon \right)\hfill \\ & +0.5inf{\left({\Theta }_1\left(\epsilon \right)\right)}^{-1}{\Theta }_2\left(\epsilon \right){\Theta }_2^T\left(\epsilon \right))+({\gamma }_1+2{\pi }^{-2}{L}^2+2{\pi }^{-2}{L}^2{\gamma }_2^2)I)\tilde{e}(\epsilon ,t)d\epsilon \hfill \\ & -\beta {\lambda }_{min}\left(\mathcal{H}\right){\int }_0^L{\tilde{e}}^T(\epsilon ,t)\tilde{e}(\epsilon ,t)d\epsilon .\hfill \end{array}$$

(25)

Choosing $\beta >{\displaystyle \frac{{\lambda }_{max}(P+({\gamma }_1+2{\pi }^{-2}{L}^2+2{\pi }^{-2}{L}^2{\gamma }_2^2)I)}{{\lambda }_{min}\left(\mathcal{H}\right)}}$, where $P=0.5I_N\otimes (A\left(\epsilon \right)+A^T\left(\epsilon \right)+B\left(\epsilon \right)B^T\left(\epsilon \right)+C\left(\epsilon \right)C^T\left(\epsilon \right)+0.5inf{\left({\Theta }_1\left(\epsilon \right)\right)}^{-1}{\Theta }_2\left(\epsilon \right){\Theta }_2^T\left(\epsilon \right))$ it can be derived from (25) that ${}_{t_0}{}^{c }D_t^{\alpha }{V}_1\left(t\right)<0$, which implies $\left|\right|\tilde{e}(\epsilon ,t)\left|\right|\to 0$ as $t\to \infty$, and makes the SVPIDE-based MASs reach consensus. □

Remark 5.

The states of the agents are not only influenced by time, but may also be related to space-varying factors and past states [29,30,31,32,33]. However, previous studies have not considered this aspect. Therefore, this paper investigates investigate fractional-order MASs based on SVPIDEs.

Remark 6.

Numerous studies on adaptive control for fractional-order MASs modeled by ODEs have achieved significant findings [44,45,46,47,48], while this paper addressed the adaptive control for PIDE-based MASs.

5. Numerical Simulation

Example 1.

To demonstrate the validity of Theorem 1, one nonlinear leaderless SVPIDE-based MAS with agents, random initial conditions and parameters is considered as follows:

$$\begin{array}{cc}\hfill & {\Theta }_1\left(\epsilon \right)=\left[\begin{array}{cc}1.9+0.5sin\left(\epsilon \right)& 0\\ 0& 2.5\end{array}\right],{\Theta }_2\left(\epsilon \right)=\left[\begin{array}{cc}1.2& 0.6\\ 0.8+0.5cos\left(\epsilon \right)& 2.7\end{array}\right],\hfill \\ \hfill & A\left(\epsilon \right)=\left[\begin{array}{cc}3.35& 1.72+0.8tanh\left(\epsilon \right)\\ -1.16& 3.25\end{array}\right],B\left(\epsilon \right)=\left[\begin{array}{cc}2.66& -1.23\\ 1.58& 2.85+2.5sin\left(2\epsilon \right)\end{array}\right],\hfill \\ \hfill & C\left(\epsilon \right)=\left[\begin{array}{cc}3.35+0.5sin\left(\epsilon \right)& 1.72\\ -1.16+0.5cos\left(3\epsilon \right)& 3.25\end{array}\right],\mathcal{A}=\left[\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right],\hfill \\ \hfill & f\left(y_i(\epsilon ,t)\right)=g\left(y_i(\epsilon ,t)\right)=tanh\left(y_i(\epsilon ,t)\right),L=1,\alpha =0.95.\hfill \end{array}$$

(26)

Owing to the property of tanh, ${\gamma }_1$ and ${\gamma }_2=1$ are obtained. Figure 1 shows that fractional-order SVPIDE-based MASs cannot reach consensus without the control. Figure 2 shows that consensus errors reach consensus with the use of the control and ${\tau }_i=0.5$. Figure 3 shows the control input of leaderless fractional-order SVPIDE-based MASs. Figure 4 illustrates the adaptive control gain of fractional-order SVPIDE-based MASs.

Figure 1. $e(\epsilon ,t)$ of fractional-order SVPIDE-based MASs without adaptive control.

Figure 2. $e(\epsilon ,t)$ of the leaderless fractional-order SVPIDE-based MASs with adaptive control.

Figure 3. The adaptive control input of the leaderless fractional-order SVPIDE-based MASs.

Figure 4. The adaptive control gain $p_i\left(t\right)$ of the leaderless fractional-order SVPIDE-based MASs.

Example 2.

To demonstrate the validity of Theorem 2, one nonlinear leader-following SVPIDE-based MAS with four following agents and one leader agent is described, where $\alpha =0.6$ and other parameters are the same as those in Example 1.

Figure 5 shows that consensus errors reach consensus with the use of the control and ${\tau }_i=0.5$. Figure 6 shows the control input of leaderless fractional-order SVPIDE-based MASs. Figure 7 illustrates the adaptive control gain of fractional-order SVPIDE-based MASs.

Figure 5. $e(\epsilon ,t)$ of the leader-following fractional-order SVPIDE-based MASs with adaptive control.

Figure 6. The adaptive control input of the leader-following fractional-order SVPIDE-based MASs.

Figure 7. The adaptive control gain $k_i\left(t\right)$ of the leader-following fractional-order SVPIDE-based MASs.

Example 3.

Reaction–Diffusion Neural Networks (RDNNs) are a class of neural networks with spatial and temporal dynamics, commonly used for simulating and processing complex dynamic systems, especially in areas like secure communication [49] and image processing [50,51]. MASs governed by fractional-order PIDEs can be used for synchronization in RDNNs. Consider the following RDNNs:

$$\left\{\begin{array}{c}{}_{t_0}{}^{c }D_t^{\alpha }{y}_i(\epsilon ,t)={\displaystyle \frac{\partial }{\partial \epsilon }}\left({\Theta }_1\left(\epsilon \right){\displaystyle \frac{\partial y_i(\epsilon ,t)}{\partial \epsilon }}\right)+{\Theta }_2\left(\epsilon \right){\displaystyle \frac{\partial y_i(\epsilon ,t)}{\partial \epsilon }}+A\left(\epsilon \right)y_i(\epsilon ,t)+f\left(y_i(\epsilon ,t)\right)\hfill \\ +B\left(\epsilon \right){\int }_0^{\epsilon }{y}_i(\rho ,t)d\rho +C\left(\epsilon \right){\int }_0^{\epsilon }g\left(y_i(\rho ,t)\right)d\rho +u_i(\epsilon ,t)\hfill \\ {\displaystyle \frac{\partial y_i(0,t)}{\partial \epsilon }}=0,{\displaystyle \frac{\partial y_i(L,t)}{\partial \epsilon }}=0,\hfill \\ y_i(\epsilon ,0)=y_i^0\left(\epsilon \right),\hfill \end{array}\right.$$

(27)

where $i=1,2,3,4$, $y_i^0\left(\epsilon \right)$ is random, $\alpha =0.95$, $L=1$, $f\left(y_i(\epsilon ,t)\right)=g\left(y_i(\epsilon ,t)\right)=tanh\left(y_i(\epsilon ,t)\right)$, and ${\Theta }_1\left(\epsilon \right),{\Theta }_2\left(\epsilon \right),A\left(\epsilon \right),B\left(\epsilon \right),C\left(\epsilon \right)and\mathcal{A}$ are chosen as

$$\begin{array}{cc}\hfill & {\Theta }_1\left(\epsilon \right)=\left[\begin{array}{cc}0.3+0.5sin\left(\epsilon \right)& 0\\ 0& 0.5\end{array}\right],{\Theta }_2\left(\epsilon \right)=\left[\begin{array}{cc}0.4& 0.2+0.2cos\left(\epsilon \right)\\ 0.1& 0.7\end{array}\right],\hfill \\ \hfill & A\left(\epsilon \right)=\left[\begin{array}{cc}0.675& 0.211\\ -0.232+0.1sin\left(\epsilon \right)& 0.662\end{array}\right],B\left(\epsilon \right)=\left[\begin{array}{cc}0.843& -0.323+0.15sin\left(2\epsilon \right)\\ 0.411& 0.762\end{array}\right],\hfill \\ \hfill & C\left(\epsilon \right)=\left[\begin{array}{cc}0.532& 0.113\\ -0.095& 0.692+0.23tanh\left(\epsilon \right)\end{array}\right],\mathcal{A}=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 1\\ 0& 1& 0& 0\\ 1& 1& 0& 0\end{array}\right]\hfill \end{array}$$

(28)

By Theorem 1, RDNNs under controller (6) are internally asymptotically stable. The evolutions of the states and inputs of four nodes are depicted in Figure 8 and Figure 9, respectively. Figure 10 shows the trajectory of the adaptive gain $p_i\left(t\right)$.

Figure 8. $e(\epsilon ,t)$ of the RDNNs (27) with adaptive control.

Figure 9. The adaptive control input of the RDNNs (27).

Figure 10. The adaptive control gain $p_i\left(t\right)$ of the RDNNs (27).

6. Conclusions

This paper studied one nonlinear fractional-order SVPIDE-based MAS, containing both nonlinear state terms and nonlinear integro-terms. A distributed adaptive control protocol was developed for leaderless fractional-order SVPIDE-based MASs, utilizing both states and differential components. For leader-following fractional-order SVPIDE-based MASs, another distributed adaptive control protocol was further refined to account for communication both among the following agents and between the leader and following agents. Sufficient conditions for consensus in both leaderless and leader-following fractional-order SVPIDE-based MASs were derived. Finally, an example demonstrated the effectiveness of the proposed adaptive controller. In future, bipartite consensus, group consensus, and finite-time consensus of sensor networks with spatio-temporal characteristics will be studied.