Boundary Control-Based Finite-Time Passivity for Fractional Spatiotemporal Directed Networks with Multiple Weights

Abstract

This article mainly delves into finite-time passivity and finite-time synchronization of fractional-order spatiotemporal directed networks with multiple weights via boundary control schemes. Firstly, several concepts of finite-time passivity for fractional spatiotemporal models are proposed by virtue of a fractional-order differential inequality. Next, a boundary controller is presented to realize the passivity, which only relies on the information of the state at the spatial boundary. Subsequently, by constructing the Lyapunov function and leveraging some inequality techniques, some sufficient conditions in the form of linear matrix inequalities are established to ensure the finite-time strict passivity, finite-time input strict passivity, and finite-time output strict passivity. As a consequence of the derived result of finite-time output strict passivity, several criteria are obtained for realizing finite-time synchronization of the addressed fractional spatiotemporal networks. A numerical example is eventually presented to verify the developed control protocol and the theoretical criteria.

1. Introduction

Numerous studies in recent decades have confirmed that a multitude of practical systems in nature and society can be depicted by complex networks, such as transportation networks [1], social networks [2], and mobile communication networks [3]. In recent years, complex networks have garnered extensive attention and have been applied in flexible robotic arms [4], image encryption [5], and combinatorial optimization [6]. Furthermore, in actual life, the dynamic behaviors of complex dynamic networks not only rest on temporal information but also depend on spatial positions, resulting in a type of network called spatiotemporal networks modeled by partial differential equations. For instance, in applications including traffic systems [7], epidemic spreading [8], urban planning [9], and more, the modeling with spatiotemporal networks can more accurately capture and predict the dynamic processes of entities or information flow. Consequently, the study of spatiotemporal networks has practical significance and broad application prospects.

As is widely known, traditional integral calculus serves a vital role in describing engineering systems and biological phenomena. Nonetheless, for more complex practical systems, the traditional integer-order theory is no longer the best choice to accurately depict their dynamic performance. For example, in materials science, the stress–strain relationships of some polymer materials show obvious hysteresis [10], which is difficult to be completely described by integral calculus. In order to solve these kinds of problems, fractional calculus has emerged. Fractional-order systems described by fractional calculus have some unique properties: infinite memory [11], heredity [12], and non-locality [13], which enable them to simulate real models [14,15,16] more accurately and exhibit more complex dynamical behaviors. Therefore, introducing fractional calculus into spatiotemporal networks to construct fractional-order spatiotemporal networks (FSNs) is an important breakthrough, and it is essential to investigate the dynamic behaviors of FSNs.

Passivity, a special instance of dissipativity, originates from circuit theory [17]. It relates the energy storage function of a system with external input and output information to characterize the energy decay of a system in the process of motion evolution. Passivity theory has been effectively utilized in neural networks [18], agent systems [19], and electro-mechanical systems [20], since passivity can guarantee the internal stability of a system. To date, plenty of meaningful results on the passivity of fractional-order systems have been published [21,22,23]. In comparison with bountiful achievements on passivity of fractional-order systems without spatial effect, the results on passivity of FSNs are quite limited due to analytical complexity arose by spatial factors. Moreover, in the study of synchronization within complex networks, passivity has continuously served as a powerful tool. At present, the synchronization issue of passive complex networks has received considerable attention [24,25,26]. In particular, passivity and passivity-based synchronization were explored in [26] for directed reaction–diffusion neural networks through event-triggered pinning control protocol.

It is not difficult to find that the above studies investigated passivity or synchronization of network models in the asymptotic sense. However, considering the limitation of the machine’s operating cycle or control accuracy, asymptotic passivity or synchronization is not desirable in certain practical applications. As an illustration, in intercontinental remote surgeries, doctors must synchronize robotic arm controls precisely within a finite time to ensure surgical safety and success [27]. In light of this, the concept of finite-time (FT) convergence has been presented [28]. Over the recent years, many researchers have carried out in-depth research on the FT passivity or FT synchronization of fractional-order systems and have achieved highly influential results [29,30,31,32], several of which centered around FSNs [30,31,32]. In [30], FT passivity was discussed for fractional BAM neural networks with diffusion efforts by applying Gauss’s formula and Jensen’s inequality. The FT synchronization issue for FSNs was explored in [31] by means of an event-triggered impulsive control strategy. In [32], FT synchronization for complex-valued FSNs was studied via intermittent control. The authors in [31,32] dealt with FT synchronization of FSNs by constructing the Lyapunov function. Unlike the above research method of FT synchronization, there were some achievements that leveraged the FT passivity of complex networks to explore its FT synchronization [33,34]. In [33], FT passivity for complex-valued networks was investigated through an order reduction approach, and FT synchronization was discussed based on the derived FT passivity results. FT passivity and FT passivity-based FT synchronization were studied in [34] for coupled complex-valued memristive neural networks by constructing a complex-value state feedback controller.

Note that the aforementioned results mainly concentrated on single-weighted network models. Practically speaking, multi-weighted networks are more reasonable to depict several complex practical problems than single-weighted networks [35]. Take transportation networks as an example. Cities are connected to each other through diverse modes of transportation, such as railways, highways, aviation, waterways, etc. Each mode is given a certain weight according to its importance or efficiency, thereby justifying the use of networks with multi-weights to describe transportation networks, within which varying coupling terms reflect differing influential factors. Hence, the investigation of multi-weighted dynamic networks is significant from both theoretical and practical perspectives, and numerous researchers have explored the dynamics of fractional-order networks with multiple weights, including passivity and synchronization [36,37,38,39]. Specially, the authors in [36,37] respectively discussed asymptotic passivity or synchronization of FSNs with multiple weights. In [36], the issue of passivity was discussed for multi-weighted FSNs with directed topology by decomposing inner coupling matrices. The authors in [37] explored the passivity and synchronization for FSNs with multi-adaptive couplings by virtue of the Lyapunov function method. For FT passivity or synchronization that is more robust and convergent faster, less research has been reported on multi-weighted fractional networks [40,41], even fewer on FT passivity-based FT synchronization of multi-weighted networks [42]. In [40], FT passivity was concerned for fractional multi-weighted networks under a state-feedback controller. FT passivity and FT synchronization were investigated in [41] for fractional-order multi-weighted complex networks via adaptive control strategy. The authors in [42] discussed the FT passivity and FT passivity-based FT synchronization for fractional-order multi-weighted coupled neural networks through state feedback control schemes.

In the present investigation of dynamic control of FSNs, internal control was predominantly used, as adopted in [30,36,37], which requires deploying numerous actuators and sensors throughout the entire spatial domain. Unlike internal control, boundary control is a quite effective strategy that only needs to place sensors and actuators on certain specified spatial boundaries, greatly reducing the difficulty of implementation and saving the control cost. To give an example, when heating a metal bar, heating only one end can make the whole bar evenly heated. Boundary control, initially proposed in [43], has been drawing increasing attention in the following decades. The authors in [44] explored the passivity and robust passivity of FSNs under boundary control. In [45], the issue of asymptotical synchronization was investigated for reaction–diffusion systems through boundary control strategy. It is noteworthy that the boundary controllers designed in [44,45] rely on the whole spatial information about error states. To improve this kind of control design, the authors in [46] developed a new boundary controller that merely depends on the boundary state information to study the synchronization of FSNs. This type of boundary control reduces the need to monitor the internal state of the system and lowers the complexity of data processing.

To sum up, even though passivity of fractional-order systems has been widely explored [21,22,23,29,30,36,37,40,41,42], the spatial diffusion phenomenon, network topology, and node connection patterns were mostly neglected in these network models, and the convergence time is mainly infinite [21,22,23,36,37]. Furthermore, in the study of passivity for FSNs, most control strategies were implemented on the spatial domain [30,36,37], which are uneconomic and cubersome. On the other hand, FT synchronization based on FT passivity for multi-weighted FSNs is also a challenging and interesting problem. Motivated by the aforementioned discussions, the FT passivity and FT passivity-based FT synchronization of multi-weighted FSNs with directed topology are investigated in this paper by designing boundary control schemes. The primary innovations of this paper are outlined below.

(1) A type of directed FSNs with multiple weights is proposed in this paper, which can better describe the spatiotemporal dependence characteristics and memory of individuals compared with the temporal networks independent of space [38,40,41,42] and integer-order networks [26,32,45].

(2) Several definitions of FT passivity are given by extending the definitions of passivity for integer-order systems and combining a fractional differential inequality with FT convergence, which are different from the previous FT passivity definitions of fractional models given in [40,42].

(3) A type of boundary controller is developed to investigate the FT passivity of a fractional spatiotemporal system, which is only related to the state information on the boundary. Compared with the internal control schemes [36,37,39] and the boundary control designs dependent on the entire state information on the boundary given in [44,45], the boundary control protocol presented in this article significantly lowers the difficulty of implementation and reduces the control cost.

(4) Several criteria of FT passivity for multi-weighted FSNs are established by leveraging the Lyapunov technique and the LMI method, which improve the passivity results in the asymptotic sense [21,22,23,24,25,26] in terms of robustness and convergence accuracy and rate. Additionally, the FT synchronization is ensured on the basis of FT output strict passivity, the relation between FT passivity and FT synchronization is revealed, and the processing method differs from the traditional Lyapunov function-based method in synchronization analysis in [31,32].

The rest of this paper is arranged as follows: Several essential preliminaries are provided in Section 2. In Section 3, the FT passivity and FT passivity-based FT synchronization of directed FSNs with multiple weights are separately investigated, and the corresponding criteria are given. A numerical example is shown to validate the theoretical analysis in Section 4. Eventually, the conclusion is reached.

Notations: $\mathcal{I}=\{1,2,\cdots ,n\}$, $\mathbb{R}=(-\infty ,+\infty )$, ${\mathbb{R}}^{+}=[0,+\infty )$, ${\mathbb{R}}^n$ indicates a space consisting of all n-dimensional real vectors. For a vector $x={(x_1,x_2,\dots ,x_n)}^T\in {\mathbb{R}}^n$, $\mathrm{sign}\left(x\right)={(\mathrm{sign}\left(x_1\right),\mathrm{sign}\left(x_2\right),\dots ,\mathrm{sign}\left(x_n\right))}^T$, ${\parallel x\parallel }_1={\sum }_{i=1}^n\left|x_i\right|$. ${\mathbb{R}}^{n\times n}$ stands for a set which is composed of all $n\times n$ real matrices. ${\mathbb{R}}^{n\times n}\ni A>0$$({\mathbb{R}}^{n\times n}\ni A<0)$ means that A is a positive (negative) definite matrix. For a real matrix A, $A^T$ denotes its transpose. $\mathrm{diag}\{·\}$ represents a diagonal matrix. $0_n$ is the n-dimensional full 0 column vector, $I_n$ is the n-dimensional identity matrix. ⊗ denotes the Kronecker product. Notation ∗ represents the term of symmetric block matrix. For a vector function $\epsilon (\mathfrak{s},t)={({\epsilon }_1(\mathfrak{s},t),{\epsilon }_2(\mathfrak{s},t),\dots ,{\epsilon }_n(\mathfrak{s},t))}^T\in {\mathbb{R}}^n$ with $(\mathfrak{s},t)\in [0,l]\times [t_0,+\infty )$, the ${\mathcal{L}}_2$-norm is defined by

$${\parallel \epsilon (\mathfrak{s},t)\parallel }_{{\mathcal{L}}_2}={\left({\int }_0^l\sum _{i=1}^n{\epsilon }_i^2(\mathfrak{s},t)d\mathfrak{s}\right)}^{{\displaystyle \frac{1}{2}}}.$$

2. Preliminaries and Network Model

In this section, fundamental knowledge about fractional calculus and the considered network model in this article is provided.

2.1. Preliminaries

Definition 1

([47]). The Caputo fractional derivative on the time t for the binary function $\nu :[0,l]\times [t_0,+\infty )\to \mathbb{R}$ is defined as

$$\stackrel{C}{t_0}{D}_t^{\kappa }\nu (\mathfrak{s},t)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\kappa )}}{\int }_{t_0}^t{\displaystyle \frac{\partial \nu (\mathfrak{s},\varsigma )}{\partial \varsigma }}{\displaystyle \frac{d\varsigma }{{(t-\varsigma )}^{\kappa }}},$$

where $0<\kappa <1$ and $\mathrm{\Gamma }(·)$ represents the Gamma function. Particularly, if $\nu (\mathfrak{s},t)$ is independent on $\mathfrak{s}$, the Caputo fractional derivative for $\nu \left(t\right)$ is defined by

$$\stackrel{C}{t_0}{D}_t^{\kappa }\nu \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\kappa )}}{\int }_{t_0}^t{\displaystyle \frac{{\nu }^{\prime }\left(\varsigma \right)}{{(t-\varsigma )}^{\kappa }}}d\varsigma .$$

Lemma 1

([48]). For $\kappa \in (0,1)$ and the binary function vector $\nu :[0,l]\times [t_0,+\infty )\to {\mathbb{R}}^n$, one has

$$\stackrel{C}{t_0}{D}_t^{\kappa }\left({\nu }^T(\mathfrak{s},t)\nu (\mathfrak{s},t)\right)\le 2{\nu }^T(\mathfrak{s},t)\left(\stackrel{C}{t_0}{D}_t^{\kappa }\nu (\mathfrak{s},t)\right).$$

Lemma 2

([49] (Wirtinger’s inequality)). For any continuous vector function $\psi \left(\theta \right)$ defined on $[0,l]$ with $\psi \left(0\right)=0_m$ or $\psi \left(l\right)=0_m$, and for any real matrix $Q\ge 0$, the following inequality holds:

$${\int }_0^l{\psi }^T\left(\theta \right)Q\psi \left(\theta \right)d\theta \le {\displaystyle \frac{4l^2}{{\pi }^2}}{\int }_0^l{\left({\displaystyle \frac{d\psi \left(\theta \right)}{d\theta }}\right)}^{T}Q\left({\displaystyle \frac{d\psi \left(\theta \right)}{d\theta }}\right)d\theta .$$

Lemma 3

([50]). Let $V\left(\epsilon \right):{\mathbb{R}}^m\to {\mathbb{R}}^{+}$ with $\epsilon :[t_0,+\infty )\to {\mathbb{R}}^m$ be continuous and differentiable, if there has a positive constant ℓ such that

$$\stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon \left(t\right)\right)\le -\mathcal{l},\epsilon \left(t\right)\in {\mathbb{R}}^m\setminus \left\{0_m\right\},$$

then $V\left(\epsilon \right(t\left)\right)=0$ for $t\ge T\left(\epsilon \left(t_0\right)\right)$, where $\kappa \in (0,1)$ and

$$T\left(\epsilon \left(t_0\right)\right)=t_0+{\left({\displaystyle \frac{V\left(\epsilon \left(t_0\right)\right)\mathrm{\Gamma }(\kappa +1)}{\mathcal{l}}}\right)}^{{\displaystyle \frac{1}{\kappa }}}.$$

Lemma 4

([51]). Suppose that function $\nu :[0,l]\times [t_0,+\infty )\to \mathbb{R}$ is integrable on $[0,l]$ and is differentiable with respect to t. Denote

$$\mathrm{\Lambda }\left(t\right)={\int }_0^l\nu (\mathfrak{s},t)d\mathfrak{s},$$

then

$$\stackrel{C}{t_0}{D}_t^{\kappa }\mathrm{\Lambda }\left(t\right)={\int }_0^l\stackrel{C}{t_0}{D}_t^{\kappa }\nu (\mathfrak{s},t)d\mathfrak{s}.$$

Lemma 5

([52]). Assuming that matrix ${\mathbb{R}}^{n\times n}\ni B={\left(b_{ij}\right)}_{n\times n}$ is irreducible and satisfies ${\sum }_{j=1}^n{b}_{ij}=0$ with $b_{ij}\ge 0(i\ne j)$, then there exists a positive vector $\alpha ={({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)}^T\in {\mathbb{R}}^n$ such that

(1) $B^T\alpha =0_n.$

(2) $\widehat{B}={\left({\widehat{b}}_{ij}\right)}_{n\times n}=\widehat{\alpha }B+B^T\widehat{\alpha }$ is symmetric and ${\sum }_{j=1}^n{\widehat{b}}_{ij}={\sum }_{j=1}^n{\widehat{b}}_{ji}=0$, where ${\mathbb{R}}^{n\times n}\ni \widehat{\alpha }=\mathrm{diag}\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\}$, $i\in \mathcal{I}.$

2.2. Network Model

A class of multi-weighted FSNs considered in this paper is described as

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }{z}_i(\mathfrak{s},t)=& R{\displaystyle \frac{{\partial }^2{z}_i(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}}+A{z}_i(\mathfrak{s},t)+Ef\left(z_i(\mathfrak{s},t)\right)\hfill \\ \hfill & +\sum _{s=1}^{\sigma }\sum _{j=1}^n{c}^s{G}_{ij}{\mathrm{\Gamma }}^s{z}_j(\mathfrak{s},t)+H{v}_i(\mathfrak{s},t),i\in \mathcal{I},\hfill \end{array}$$

(1)

where $\kappa \in (0,1),\mathfrak{s}\in [0,l]$ is the space variable and $t\in [t_0,+\infty )$ is the time variable, $z_i(\mathfrak{s},t)\in {\mathbb{R}}^m$ represents the state vector of the ith vertex, $v_i(\mathfrak{s},t)\in {\mathbb{R}}^m$ is the external input, $0<R=\mathrm{diag}\{r_1,r_2,\cdots ,r_m\}\in {\mathbb{R}}^{m\times m}$ is a diffusion coefficient matrix, $A=\mathrm{diag}\{a_1,a_2,\dots ,a_m\}\in {\mathbb{R}}^{m\times m}$, $E\in {\mathbb{R}}^{m\times m}$ and $H\in {\mathbb{R}}^{m\times m}$ are constant matrices, $f\in {\mathbb{R}}^m$ refers to a continuous nonlinear vector function, $0<c^s\in \mathbb{R}$ and $0<{\mathrm{\Gamma }}^s\in {\mathbb{R}}^{m\times m}$, respectively, denote the coupling strength and inner coupling matrix of the sth coupling form, ${\mathbb{R}}^{n\times n}\ni G={\left(G_{ij}\right)}_{n\times n}$ represents the outer coupling matrix, which is expressed as follows: if there is a link from vertex j to vertex i, then $G_{ij}>0$, otherwise, $G_{ij}=0(i\ne j)$, the diagonal elements are defined by

$$G_{ii}=-\sum _{\stackrel{j=1}{j\ne i}}^n{G}_{ij},i\in \mathcal{I}.$$

The initial value and boundary value related to the multi-weighted FSN (1) are provided as

$$\left\{\begin{array}{c}{z}_i(\mathfrak{s},t_0)={\mathrm{\Theta }}_i\left(\mathfrak{s}\right),\mathfrak{s}\in [0,l],\hfill \\ {\displaystyle \frac{\partial z_i(\mathfrak{s},t)}{\partial \mathfrak{s}}{|}_{\mathfrak{s}=0}=0,\frac{\partial z_i(\mathfrak{s},t)}{\partial \mathfrak{s}}{|}_{\mathfrak{s}=l}=u_i\left(t\right),i\in \mathcal{I},}\hfill \end{array}\right.$$

where ${\mathrm{\Theta }}_i\left(\mathfrak{s}\right)$ is bounded and continuous, $u_i\left(t\right)$ is a boundary control input to be designed.

Assumption 1.

The topological structure of the network (1) is strongly connected.

Assumption 2.

$f(·)$ satisfies the following inequality:

$$\parallel f\left({\gamma }_1\right)-f\left({\gamma }_2\right)\parallel \le \omega \parallel {\gamma }_1-{\gamma }_2\parallel$$

for any ${\gamma }_1,{\gamma }_2\in {\mathbb{R}}^m$, in which $0<\omega \in \mathbb{R}$.

From Assumption 1 and Lemma 5, there exists a positive vector $\alpha ={({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)}^T$

$\in {\mathbb{R}}^n$ such that $G^T\alpha =0_n$ and ${\sum }_{j=1}^n{\widehat{G}}_{ij}={\sum }_{j=1}^n{\widehat{G}}_{ji}=0$, where $\widehat{G}={\left({\widehat{G}}_{ij}\right)}_{n\times n}=\widehat{\alpha }G+G^T\widehat{\alpha }$, $\widehat{\alpha }=\mathrm{diag}\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\}$. Let ${\mathrm{\xi }}_i={\displaystyle \frac{{\alpha }_i}{{\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n}}(i=1,2,\dots ,n)$, then ${\sum }_{i=1}^n{\mathrm{\xi }}_i{G}_{ij}=0$ and ${\sum }_{i=1}^n{\mathrm{\xi }}_i=1$.

Define $\widehat{z}(\mathfrak{s},t)={\displaystyle \sum _{\rho =1}^n}{\mathrm{\xi }}_{\rho }{z}_{\rho }(\mathfrak{s},t)$, one has

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }\widehat{z}(\mathfrak{s},t)=& R{\displaystyle \frac{{\partial }^2\widehat{z}(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}}+A\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{z}_{\rho }(\mathfrak{s},t)+E\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }f\left(z_{\rho }(\mathfrak{s},t)\right)\hfill \\ \hfill & +\sum _{s=1}^{\sigma }\sum _{j=1}^n{c}^s\left(\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{G}_{\rho j}\right){\mathrm{\Gamma }}^s{z}_j(\mathfrak{s},t)+H\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{v}_{\rho }(\mathfrak{s},t)\hfill \\ \hfill =& R{\displaystyle \frac{{\partial }^2\widehat{z}(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}}+A\widehat{z}(\mathfrak{s},t)+E\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }f\left(z_{\rho }(\mathfrak{s},t)\right)+H\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{v}_{\rho }(\mathfrak{s},t).\hfill \end{array}$$

(2)

Let ${\mathbb{R}}^m\ni {\epsilon }_i(\mathfrak{s},t)=z_i(\mathfrak{s},t)-\widehat{z}(\mathfrak{s},t)$, then

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }{\epsilon }_i(\mathfrak{s},t)=& R{\displaystyle \frac{{\partial }^2{\epsilon }_i(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}}+A{\epsilon }_i(\mathfrak{s},t)+Ef\left(z_i(\mathfrak{s},t)\right)\hfill \\ \hfill & -E\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }f\left(z_{\rho }(\mathfrak{s},t)\right)+\sum _{s=1}^{\sigma }\sum _{j=1}^n{c}^s{G}_{ij}{\mathrm{\Gamma }}^s{\epsilon }_j(\mathfrak{s},t)\hfill \\ \hfill & +H{v}_i(\mathfrak{s},t)-H\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{v}_{\rho }(\mathfrak{s},t),i\in \mathcal{I}.\hfill \end{array}$$

(3)

Apparently, system (3) satisfies the following initial condition:

$$\begin{array}{c}\hfill {\epsilon }_i(\mathfrak{s},t_0)={\mathrm{\Theta }}_i\left(\mathfrak{s}\right)-\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{\mathrm{\Theta }}_{\rho }\left(\mathfrak{s}\right),\mathfrak{s}\in [0,l],\end{array}$$

and the following Neumann boundary condition:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\epsilon }_i(\mathfrak{s},t)}{\partial \mathfrak{s}}}{|}_{\mathfrak{s}=0}=0,{\displaystyle \frac{\partial {\epsilon }_i(\mathfrak{s},t)}{\partial \mathfrak{s}}}{|}_{\mathfrak{s}=l}=u_i\left(t\right)-\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{u}_{\rho }\left(t\right),i\in \mathcal{I}.\end{array}$$

(4)

In order to investigate the FT passivity of system (3), the output vector $y_i(\mathfrak{s},t)\in {\mathbb{R}}^m$ is defined as

$$y_i(\mathfrak{s},t)=M{\epsilon }_i(\mathfrak{s},t)+N{v}_i(\mathfrak{s},t),$$

where M, $N\in {\mathbb{R}}^{m\times m}$ are known real matrices, $i\in \mathcal{I}.$

For brevity, we denote

$$y(\mathfrak{s},t)={(y_1^T(\mathfrak{s},t),y_2^T(\mathfrak{s},t),\dots ,y_n^T(\mathfrak{s},t))}^T\in {\mathbb{R}}^{nm},$$

$$v(\mathfrak{s},t)={(v_1^T(\mathfrak{s},t),v_2^T(\mathfrak{s},t),\dots ,v_n^T(\mathfrak{s},t))}^T\in {\mathbb{R}}^{nm},$$

$$\epsilon (\mathfrak{s},t)={({\epsilon }_1^T(\mathfrak{s},t),{\epsilon }_2^T(\mathfrak{s},t),\dots ,{\epsilon }_n^T(\mathfrak{s},t))}^T\in {\mathbb{R}}^{nm}.$$

Definition 2.

System (3) is said to be strictly passive if there exists a positive semidefinite function $V\left(\epsilon \right)$ and a positive definite function $\phi \left(\epsilon \right)$ such that

$${\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}\ge \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)+\mathcal{l}\phi \left(\epsilon (\mathfrak{s},t)\right),$$

where $\kappa \in (0,1)$ and $0<\mathcal{l}\in \mathbb{R}$. Furthermore, system (3) is said to be FT strictly passive if

$$\phi \left(\epsilon \right)=\left\{\begin{array}{cc}1,\hfill & \epsilon \ne 0_{nm},\hfill \\ 0,\hfill & \epsilon =0_{nm}.\hfill \end{array}\right.$$

(5)

Definition 3.

System (3) is said to be FT input strictly passive if there exists a positive semidefinite function $V\left(\epsilon \right)$ such that

$${\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_1{\int }_0^l{v}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}\ge \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)+\mathcal{l}\phi \left(\epsilon (\mathfrak{s},t)\right),$$

where $\kappa \in (0,1)$, $0<{\varrho }_1,\mathcal{l}\in \mathbb{R}$ and φ is defined in (5).

Definition 4.

System (3) is said to be FT output strictly passive if there exists a positive semidefinite function $V\left(\epsilon \right)$ such that

$${\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_2{\int }_0^l{y}^T(\mathfrak{s},t)y(\mathfrak{s},t)d\mathfrak{s}\ge \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)+\mathcal{l}\phi \left(\epsilon (\mathfrak{s},t)\right),$$

where $\kappa \in (0,1)$, $0<{\varrho }_2,\mathcal{l}\in \mathbb{R}$ and φ is defined in (5).

Remark 1.

Note that the left-hand side of the inequality in Definition 2 is dependent of the time t but independent of the space variable $\mathfrak{s}$, which implies that the right-hand side is also independent of $\mathfrak{s}$. To meet this, $V\left(\epsilon \right(\mathfrak{s},t\left)\right)$ is constructed as ${\sum }_{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}$ in this paper, and φ just relies on the value of the function $\epsilon (\mathfrak{s},t)$ at the boundary l about the variable $\mathfrak{s}$.

Remark 2.

In recent periods, the FT passivity of fractional-order systems has been successfully generalized from integer-order systems and has been deeply explored [29,30,40,42] since its wide applications in FT synchronization. Nonetheless, in most existing outcomes concerning the FT passivity of fractional-order systems [40,42], the definitions of FT passivity, FT input strict passivity, and FT output strict passivity are established grounded in the following fractional-order differential inequality:

$$\stackrel{C}{t_0}{D}_t^{\kappa }V\left(t\right)\le -\lambda V^{\beta }\left(t\right),V\left(t\right)\in {\mathbb{R}}^{+}\setminus \left\{0\right\},$$

where $\lambda >0$ and $0<\beta <\kappa <1$. Note that the above inequality is commonly proved based on the following equality:

$$\stackrel{C}{t_0}{D}_t^{\kappa }{f}^{\mu }\left(t\right)={\displaystyle \frac{\mathrm{\Gamma }(1+\mu )}{\mathrm{\Gamma }(1+\mu -\kappa )}}{f}^{\mu -\kappa }{\left(t\right)}_{t_0}^C{D}_t^{\kappa }f\left(t\right),$$

where $f:[t_0,+\infty )\to \mathbb{R}$, $\mu \in \mathbb{R}$ and $0<\kappa <1$. However, it is not correct. For example, let $f\left(t\right)=t$, $\mu =2$, $\kappa =0.7$, $t_0=0$, $t=1$. On the one hand,

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }{f}^{\mu }\left(t\right)& ={}_0{}^{C}D_1^{0.7}{t}^2\hfill \\ \hfill & {\displaystyle =\frac{1}{\mathrm{\Gamma }(1-0.7)}{\int }_0^1\frac{2\varsigma }{{(1-\varsigma )}^{0.7}}d\varsigma }\hfill \\ \hfill & \approx 0.7142,\hfill \end{array}$$

on the other hand,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{\Gamma }(1+\mu )}{\mathrm{\Gamma }(1+\mu -\kappa )}}{f}^{\mu -\kappa }{\left(t\right)}_{t_0}^C{D}_t^{\kappa }f\left(t\right)& ={\displaystyle \frac{\mathrm{\Gamma }(1+2)}{\mathrm{\Gamma }(1+2-0.7)}}{t}^{1.3}{}_0{}^{C}D_1^{0.7}t\hfill \\ \hfill & {\displaystyle =\frac{\mathrm{\Gamma }\left(3\right)}{\mathrm{\Gamma }\left(2.3\right)}\frac{1}{\mathrm{\Gamma }\left(0.3\right)}{\int }_0^1\frac{1}{{(1-\varsigma )}^{0.7}}d\varsigma }\hfill \\ \hfill & \approx 1.9101.\hfill \end{array}$$

Obviously, the left and right sides of the equality are not equal. Therefore, based on the FT convergence fractional inequality given in Lemma 3, several new definitions of FT passivity are given in this paper.

Remark 3.

The definitions of passivity, input strict passivity, and output strict passivity for integer-order systems were given in [53]. In this article, these definitions are generalized to fractional-order systems, and Definitions 2–4 are introduced. Note that $V\left(\epsilon \right)$ is referred to as a storage function, which signifies the energy stored in the system. In addition, ${\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}$, ${\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_1{\int }_0^l{v}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}$ and ${\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_2{\int }_0^l{y}^T(\mathfrak{s},t)y(\mathfrak{s},t)d\mathfrak{s}$ are the energy supplies injected externally into the system. It can be observed for passive systems that the energy stored inside the system is always no more than the energy externally provided to the system.

3. Main Results

In this section, the FT passivity and FT synchronization of multi-weighted FSNs with directed topology are investigated. By employing the fractional-order differential inequality and certain important lemmas, the FT strict passivity, FT input strict passivity, and FT output strict passivity criteria for FSNs with multiple weights are derived. Additionally, the FT output strict passivity is used to deal with the FT synchronization of multi-weighted FSNs, and several FT synchronization criteria are given.

3.1. FT Passivity

To obtain the FT passivity for system (3) with the Neumann boundary condition (4), the boundary controller is designed as follows:

$$u_i\left(t\right)=-K{\epsilon }_i(l,t)-{\displaystyle \frac{\mathcal{l}}{2\tilde{\mathrm{\xi }}}}{R}^{-1}\mathrm{\Psi }\left({\epsilon }_i(l,t)\right),i\in \mathcal{I},$$

(6)

where $0<K\in {\mathbb{R}}^{m\times m}$ is a control gain matrix to be designed, $\mathcal{l}>0$, $\tilde{\mathrm{\xi }}=\underset{1\le i\le n}{min}\left\{{\mathrm{\xi }}_i\right\}$ and

$$\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\mathrm{sign}\left({\epsilon }_i(l,t)\right)}{\parallel {\epsilon }_i{(l,t)\parallel }_1},}\hfill & {\epsilon }_i(l,t)\ne 0_m,\hfill \\ 0_m,\hfill & {\epsilon }_i(l,t)=0_m.\hfill \end{array}\right.$$

Remark 4.

Boundary control, an effective control strategy, has attracted much attention [45,46,47] since its distinct advantage that the controller works only at the boundary. Note that the boundary controller designed as $-K{\int }_0^l\epsilon (\mathfrak{s},t)d\mathfrak{s}$ in [44,45] depends on the entire state information of $\epsilon (\mathfrak{s},t)$, which requires sensors to be distributed the whole spatial position $\mathfrak{s}\in [0,l]$. Despite the capability of current sensing and measurement technologies to measure the entire state information of system (3), it is still impractical to install actuators at whole spatial domain. Different from this boundary controller, the proposed controller (6) is only relevant to the state at the space boundary, and only a sensor and an actuator need to be placed at the spatial boundary position $\mathfrak{s}=l$, which reduces the control cost and has the work maneuverability.

Theorem 1.

Based on Assumptions 1 and 2, system (3) under the boundary controller (6) is FT strictly passive if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\mathrm{\Phi }=\left(\begin{array}{ccc}{\mathrm{\Phi }}_2& {\displaystyle \frac{2}{l}\mathrm{\xi }\otimes \left(RK\right)}& {\displaystyle \mathrm{\xi }\otimes H-I_n\otimes \frac{M}{2}}\\ \ast & {\mathrm{\Phi }}_3& 0\\ \ast & \ast & {\displaystyle -I_n\otimes \frac{N+N^T}{2}}\end{array}\right)\le 0,$$

(7)

where ${\displaystyle {\mathrm{\Phi }}_2=\mathrm{\xi }\otimes {\mathrm{\Phi }}_1+\sum _{s=1}^{\sigma }{c}^s(\mathrm{\xi }G+G^T\mathrm{\xi })\otimes {\mathrm{\Gamma }}^s}$, ${\displaystyle {\mathrm{\Phi }}_1=-\frac{1}{l}(RK+K^{T}R)+A+A^T+E{E}^T+{\omega }^2{I}_m}$, ${\displaystyle {\mathrm{\Phi }}_3=\mathrm{\xi }\otimes \left(-\frac{1}{l}(RK+K^{T}R)-\frac{{\pi }^2}{2l^2}R\right)}$.

Proof. 

Constructing the following Lyapunov function,

$${\displaystyle V\left(\epsilon (\mathfrak{s},t)\right)=\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}.}$$

In view of Lemmas 1 and 4, for $\epsilon (\mathfrak{s},t)\in {\mathbb{R}}^{nm}\setminus \left\{0_{nm}\right\},$

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)\le & {\displaystyle 2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)[R\frac{{\partial }^2{\epsilon }_i(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}+A{\epsilon }_i(\mathfrak{s},t)+E\left(f\left(z_i(\mathfrak{s},t)\right)-f\left(\widehat{z}(\mathfrak{s},t)\right)\right)}\hfill \\ \hfill & +Ef\left(\widehat{z}(\mathfrak{s},t)\right)-E\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }f\left(z_{\rho }(\mathfrak{s},t)\right)+\sum _{s=1}^{\sigma }\sum _{j=1}^n{c}^s{G}_{ij}{\mathrm{\Gamma }}^s{\epsilon }_j(\mathfrak{s},t)\hfill \\ \hfill & +H{v}_i(\mathfrak{s},t)-H\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{v}_{\rho }(\mathfrak{s},t)]d\mathfrak{s}.\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill \sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i(\mathfrak{s},t)& =\sum _{i=1}^n{\mathrm{\xi }}_i\left(z_i(\mathfrak{s},t)-\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{z}_{\rho }(\mathfrak{s},t)\right)\hfill \\ \hfill & =\sum _{i=1}^n{\mathrm{\xi }}_i{z}_i(\mathfrak{s},t)-\sum _{\rho =1}^n\left(\sum _{i=1}^n{\mathrm{\xi }}_i\right){\mathrm{\xi }}_{\rho }{z}_{\rho }(\mathfrak{s},t)\hfill \\ \hfill & =\sum _{i=1}^n{\mathrm{\xi }}_i{z}_i(\mathfrak{s},t)-\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{z}_{\rho }(\mathfrak{s},t)\hfill \\ \hfill & =0,(\mathfrak{s},t)\in [0,l]\times [t_0,+\infty ),\hfill \end{array}$$

one has

$$\begin{array}{c}\hfill \sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(\mathfrak{s},t)\left(Ef\left(\widehat{z}(\mathfrak{s},t)\right)-E\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }f\left(z_{\rho }(\mathfrak{s},t)\right)\right)=0,\end{array}$$

(8)

$$\begin{array}{c}\hfill \sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(\mathfrak{s},t)\left(H\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{v}_{\rho }(\mathfrak{s},t)\right)=0.\end{array}$$

(9)

Furthermore, on the basis of Assumption 2 and the inequality $2x^{T}y\le x^{T}x+y^{T}y$, one can obtain

$$\begin{array}{cc}\hfill & 2{\epsilon }_i^T(\mathfrak{s},t)E\left(f\left(z_i(\mathfrak{s},t)\right)-f\left(\widehat{z}(\mathfrak{s},t)\right)\right)\hfill \\ \hfill \le & {\epsilon }_i^T(\mathfrak{s},t)E{E}^T{\epsilon }_i(\mathfrak{s},t)+{\left(f\left(z_i(\mathfrak{s},t)\right)-f\left(\widehat{z}(\mathfrak{s},t)\right)\right)}^T\left(f\left(z_i(\mathfrak{s},t)\right)-f\left(\widehat{z}(\mathfrak{s},t)\right)\right)\hfill \\ \hfill \le & {\epsilon }_i^T(\mathfrak{s},t)(E{E}^T+{\omega }^2{I}_m){\epsilon }_i(\mathfrak{s},t).\hfill \end{array}$$

(10)

Taking advantage of the Neumann boundary condition (4) and integration by parts, under boundary controller (6), it can be deduced that

$$\begin{array}{cc}\hfill & {\displaystyle 2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)R\frac{{\partial }^2{\epsilon }_i(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}d\mathfrak{s}}\hfill \\ \hfill =& {\displaystyle 2\sum _{i=1}^n{\mathrm{\xi }}_i\left({\epsilon }_i^T(\mathfrak{s},t)R\frac{\partial {\epsilon }_i(\mathfrak{s},t)}{\partial \mathfrak{s}}{|}_{\mathfrak{s}=0}^{\mathfrak{s}=l}-{\int }_0^l\frac{\partial {\epsilon }_i^T(\mathfrak{s},t)}{\partial \mathfrak{s}}R\frac{\partial {\epsilon }_i(\mathfrak{s},t)}{\partial \mathfrak{s}}d\mathfrak{s}\right)}\hfill \\ \hfill =& {\displaystyle 2\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)R\left(u_i\left(t\right)-\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{u}_{\rho }\left(t\right)\right)-2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l\frac{\partial {\epsilon }_i^T(\mathfrak{s},t)}{\partial \mathfrak{s}}R\frac{\partial {\epsilon }_i(\mathfrak{s},t)}{\partial \mathfrak{s}}d\mathfrak{s}}\hfill \\ \hfill =& {\displaystyle 2\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)R\left(-K{\epsilon }_i(l,t)-\frac{\mathcal{l}}{2\tilde{\mathrm{\xi }}}{R}^{-1}\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)\right)-2\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)R\left(\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{u}_{\rho }\left(t\right)\right)}\hfill \\ \hfill & {\displaystyle -2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l\frac{\partial {\epsilon }_i^T(\mathfrak{s},t)}{\partial \mathfrak{s}}R\frac{\partial {\epsilon }_i(\mathfrak{s},t)}{\partial \mathfrak{s}}d\mathfrak{s}}\hfill \\ \hfill =& {\displaystyle -2\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)RK{\epsilon }_i(l,t)-\frac{\mathcal{l}}{\tilde{\mathrm{\xi }}}\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)}\hfill \\ \hfill & {\displaystyle -2\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\left(\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{u}_{\rho }\left(t\right)\right)-2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l\frac{\partial {\tilde{\epsilon }}_i^T(\mathfrak{s},t)}{\partial \mathfrak{s}}R\frac{\partial {\tilde{\epsilon }}_i(\mathfrak{s},t)}{\partial \mathfrak{s}}d\mathfrak{s},}\hfill \end{array}$$

(11)

where ${\tilde{\epsilon }}_i(\mathfrak{s},t)={\epsilon }_i(\mathfrak{s},t)-{\epsilon }_i(l,t)$.

If $\epsilon (l,t)=0_{nm}$, then $\phi \left(\epsilon \right(l,t\left)\right)=0$ by the definition (5). It follows from the definition of $\mathrm{\Psi }$ that

$${\displaystyle -\frac{\mathcal{l}}{\tilde{\mathrm{\xi }}}\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)=0=-\mathcal{l}\phi \left(\epsilon (l,t)\right).}$$

If $\epsilon (l,t)\ne 0_{nm}$, then there exists a subset $\mathcal{F}\subseteq \mathcal{I}$ such that ${\epsilon }_i(l,t)\ne 0_m$ for $i\in \mathcal{F}$ and ${\epsilon }_i(l,t)=0_m$ for $i\in \mathcal{I}\setminus \mathcal{F}$. So

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathcal{l}}{\tilde{\mathrm{\xi }}}}\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)\hfill \\ \hfill =& {\displaystyle \frac{\mathcal{l}}{\tilde{\mathrm{\xi }}}}\left(\sum _{i\in \mathcal{F}}{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)+\sum _{i\in \mathcal{I}\setminus \mathcal{F}}{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)\right)\hfill \\ \hfill =& {\displaystyle \frac{\mathcal{l}}{\tilde{\mathrm{\xi }}}}\sum _{i\in \mathcal{F}}{\mathrm{\xi }}_i\hfill \\ \hfill \ge & \mathcal{l},\hfill \end{array}$$

which shows that

$${\displaystyle -\frac{\mathcal{l}}{\tilde{\mathrm{\xi }}}\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)\le -\mathcal{l}=-\mathcal{l}\phi \left(\epsilon (l,t)\right).}$$

By the above analysis, one has

$${\displaystyle -\frac{\mathcal{l}}{\tilde{\mathrm{\xi }}}\sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)\mathrm{\Psi }\left({\epsilon }_i(l,t)\right)\le -\mathcal{l}\phi \left(\epsilon (l,t)\right).}$$

(12)

According to Lemma 2 and ${\tilde{\epsilon }}_i(l,t)={\epsilon }_i(l,t)-{\epsilon }_i(l,t)=0_m$, it obtains

$$\begin{array}{c}\hfill {\displaystyle -{\int }_0^l\frac{\partial {\tilde{\epsilon }}_i^T(\mathfrak{s},t)}{\partial \mathfrak{s}}R\frac{\partial {\tilde{\epsilon }}_i(\mathfrak{s},t)}{\partial \mathfrak{s}}d\mathfrak{s}\le -\frac{{\pi }^2}{4l^2}{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)R{\tilde{\epsilon }}_i(\mathfrak{s},t)d\mathfrak{s}.}\end{array}$$

(13)

Moreover, on account of ${\sum }_{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i(l,t)=0$, one further derives

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^n{\mathrm{\xi }}_i{\epsilon }_i^T(l,t)R\left(\sum _{\rho =1}^n{\mathrm{\xi }}_{\rho }{u}_{\rho }\left(t\right)\right)=0.}\end{array}$$

(14)

From (11)–(14),

$$\begin{array}{cc}\hfill & {\displaystyle 2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)R\frac{{\partial }^2{\epsilon }_i(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}d\mathfrak{s}}\hfill \\ \hfill \le & \sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\left({\epsilon }_i(\mathfrak{s},t)-{\tilde{\epsilon }}_i(\mathfrak{s},t)\right)}^T\left(-{\displaystyle \frac{2}{l}}RK\right)({\epsilon }_i(\mathfrak{s},t)-{\tilde{\epsilon }}_i(\mathfrak{s},t))d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right)\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)\left(-\frac{{\pi }^2}{2l^2}R\right){\tilde{\epsilon }}_i(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill =& \sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)\left(-{\displaystyle \frac{2}{l}}RK\right){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}+\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)\left({\displaystyle \frac{4}{l}}RK\right){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}\hfill \\ \hfill & +\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)\left(-{\displaystyle \frac{2}{l}}RK-{\displaystyle \frac{{\pi }^2}{2l^2}}R\right){\tilde{\epsilon }}_i(\mathfrak{s},t)d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right).\hfill \end{array}$$

(15)

Combining (8)–(10) and (15), one obtains

$$\begin{array}{cc}\hfill & \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)\hfill \\ \hfill \le & \sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)\left(-{\displaystyle \frac{2}{l}}RK\right){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}+\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)\left({\displaystyle \frac{4}{l}}RK\right){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}\hfill \\ \hfill & +\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)\left(-{\displaystyle \frac{2}{l}}RK-{\displaystyle \frac{{\pi }^2}{2l^2}}R\right){\tilde{\epsilon }}_i(\mathfrak{s},t)d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right)\hfill \\ \hfill & {\displaystyle +2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)A{\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}+\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)(E{E}^T+{\omega }^2{I}_m){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +2\sum _{s=1}^{\sigma }\sum _{i=1}^n\sum _{j=1}^n{c}^s{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)G_{ij}{\mathrm{\Gamma }}^s{\epsilon }_j(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)H{v}_i(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)\left(-\frac{1}{l}(RK+K^{T}R)+A+A^T+E{E}^T+{\omega }^2{I}_m\right){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +2\sum _{s=1}^{\sigma }\sum _{i=1}^n\sum _{j=1}^n{c}^s{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)G_{ij}{\mathrm{\Gamma }}^s{\epsilon }_j(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)\left(\frac{2}{l}(RK+K^{T}R)\right){\epsilon }_i(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\tilde{\epsilon }}_i^T(\mathfrak{s},t)\left(-\frac{1}{l}(RK+K^{T}R)-\frac{{\pi }^2}{2l^2}R\right){\tilde{\epsilon }}_i(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +2\sum _{i=1}^n{\mathrm{\xi }}_i{\int }_0^l{\epsilon }_i^T(\mathfrak{s},t)H{v}_i(\mathfrak{s},t)d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right).}\hfill \end{array}$$

(16)

Using Kronecker product, Equation (16) can be rewritten in the following form:

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)\le & {\displaystyle {\int }_0^l[{\epsilon }^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes {\mathrm{\Phi }}_1+\sum _{s=1}^{\sigma }{c}^s(\mathrm{\xi }G+G^T\mathrm{\xi })\otimes {\mathrm{\Gamma }}^s\right)\epsilon (\mathfrak{s},t)}\hfill \\ \hfill & +{\tilde{\epsilon }}^T(\mathfrak{s},t)\left({\displaystyle \frac{2}{l}}\left(\mathrm{\xi }\otimes (RK+K^{T}R)\right)\right)\epsilon (\mathfrak{s},t)\hfill \\ \hfill & +{\tilde{\epsilon }}^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes \left(-{\displaystyle \frac{1}{l}}(RK+K^{T}R)-{\displaystyle \frac{{\pi }^2}{2l^2}}R\right)\right)\tilde{\epsilon }(\mathfrak{s},t)\hfill \\ \hfill & +{\epsilon }^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes (H+H^T)\right)v(\mathfrak{s},t)]d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right),\hfill \end{array}$$

(17)

where $\mathrm{\xi }=\mathrm{diag}\{{\mathrm{\xi }}_1,{\mathrm{\xi }}_2,\dots ,{\mathrm{\xi }}_n\}\in {\mathbb{R}}^{n\times n}$, $\tilde{\epsilon }(\mathfrak{s},t)={\left({\tilde{\epsilon }}_1^T(\mathfrak{s},t),{\tilde{\epsilon }}_2^T(\mathfrak{s},t),\dots ,{\tilde{\epsilon }}_n^T(\mathfrak{s},t)\right)}^T\in {\mathbb{R}}^{nm}$.

Then,

$$\begin{array}{cc}\hfill & {\displaystyle \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-{\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill \le & {\displaystyle {\int }_0^l[{\epsilon }^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes {\mathrm{\Phi }}_1+\sum _{s=1}^{\sigma }{c}^s(\mathrm{\xi }G+G^T\mathrm{\xi })\otimes {\mathrm{\Gamma }}^s\right)\epsilon (\mathfrak{s},t)}\hfill \\ \hfill & +{\tilde{\epsilon }}^T(\mathfrak{s},t)\left({\displaystyle \frac{2}{l}}\left(\mathrm{\xi }\otimes (RK+K^{T}R)\right)\right)\epsilon (\mathfrak{s},t)\hfill \\ \hfill & +{\tilde{\epsilon }}^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes \left(-{\displaystyle \frac{1}{l}}(RK+K^{T}R)-{\displaystyle \frac{{\pi }^2}{2l^2}}R\right)\right)\tilde{\epsilon }(\mathfrak{s},t)\hfill \\ \hfill & +{\epsilon }^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes (H+H^T)\right)v(\mathfrak{s},t)]d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right)\hfill \\ \hfill & {\displaystyle -{\int }_0^l\left[{\epsilon }^T(\mathfrak{s},t)(I_n\otimes M^T)v(\mathfrak{s},t)+v^T(\mathfrak{s},t)(I_n\otimes N^T)v(\mathfrak{s},t)\right]d\mathfrak{s}}\hfill \\ \hfill =& {\displaystyle {\int }_0^l[{\epsilon }^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes {\mathrm{\Phi }}_1+\sum _{s=1}^{\sigma }{c}^s(\mathrm{\xi }G+G^T\mathrm{\xi })\otimes {\mathrm{\Gamma }}^s\right)\epsilon (\mathfrak{s},t)}\hfill \\ \hfill & +{\tilde{\epsilon }}^T(\mathfrak{s},t)\left({\displaystyle \frac{2}{l}}\left(\mathrm{\xi }\otimes (RK+K^{T}R)\right)\right)\epsilon (\mathfrak{s},t)\hfill \\ \hfill & +{\tilde{\epsilon }}^T(\mathfrak{s},t)\left(\mathrm{\xi }\otimes \left(-{\displaystyle \frac{1}{l}}(RK+K^{T}R)-{\displaystyle \frac{{\pi }^2}{2l^2}}R\right)\right)\tilde{\epsilon }(\mathfrak{s},t)\hfill \\ \hfill & +{\epsilon }^T(\mathfrak{s},t)\left(\left(\mathrm{\xi }\otimes (H+H^T)-I_n\otimes {\displaystyle \frac{M^T+M}{2}}\right)\right)v(\mathfrak{s},t)\hfill \\ \hfill & -v^T(\mathfrak{s},t)\left(I_n\otimes {\displaystyle \frac{N+N^T}{2}}\right)v(\mathfrak{s},t)]d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right)\hfill \\ \hfill =& {\displaystyle {\int }_0^l{\eta }^T(\mathfrak{s},t)\mathrm{\Phi }\eta (\mathfrak{s},t)d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right),}\hfill \end{array}$$

(18)

where $\eta (\mathfrak{s},t)={({\epsilon }^T(\mathfrak{s},t),{\tilde{\epsilon }}^T(\mathfrak{s},t),v^T(\mathfrak{s},t))}^T.$

In light of (7) and (18), one can derive

$${\displaystyle {\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}\ge \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)+\mathcal{l}\phi \left(\epsilon (l,t)\right).}$$

Therefore, system (3) under boundary controller (6) is FT strictly passive. □

Theorem 2.

Based on Assumptions 1 and 2, system (3) under the boundary controller (6) is FT input strictly passive if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\tilde{\mathrm{\Phi }}=\left(\begin{array}{ccc}{\mathrm{\Phi }}_2& {\displaystyle \frac{2}{l}\mathrm{\xi }\otimes \left(RK\right)}& {\displaystyle \mathrm{\xi }\otimes H-I_n\otimes \frac{M}{2}}\\ \ast & {\mathrm{\Phi }}_3& 0\\ \ast & \ast & {\mathrm{\Phi }}_4\end{array}\right)\le 0,$$

(19)

where ${\displaystyle {\mathrm{\Phi }}_4=I_n\otimes \left({\varrho }_1{I}_m-\frac{N+N^T}{2}\right).}$

Proof. 

From (17) and (19), one has

$$\begin{array}{cc}\hfill & {\displaystyle \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-{\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}+{\varrho }_1{\int }_0^l{v}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill =& {\displaystyle \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-{\int }_0^l\left[{\epsilon }^T(\mathfrak{s},t)(I_n\otimes M^T)v(\mathfrak{s},t)+v^T(\mathfrak{s},t)(I_n\otimes N^T)v(\mathfrak{s},t)\right]d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +{\varrho }_1{\int }_0^l{v}^T(\mathfrak{s},t)(I_n\otimes I_m)v(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill \le & {\displaystyle {\int }_0^l{\eta }^T(\mathfrak{s},t)\tilde{\mathrm{\Phi }}\eta (\mathfrak{s},t)d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right).}\hfill \end{array}$$

(20)

It follows from (19) and (20) that

$${\displaystyle {\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_1{\int }_0^l{v}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}\ge \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)+\mathcal{l}\phi \left(\epsilon (l,t)\right).}$$

Therefore, system (3) under boundary controller (6) is FT input strictly passive. □

Theorem 3.

Based on Assumptions 1 and 2, system (3) under the boundary controller (6) is FT output strictly passive if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\overline{\mathrm{\Phi }}=\left(\begin{array}{ccc}{\mathrm{\Phi }}_5& {\displaystyle \frac{2}{l}\mathrm{\xi }\otimes \left(RK\right)}& {\mathrm{\Phi }}_6\\ \ast & {\mathrm{\Phi }}_3& 0\\ \ast & \ast & {\mathrm{\Phi }}_7\end{array}\right)\le 0,$$

(21)

where ${\mathrm{\Phi }}_5={\mathrm{\Phi }}_2+{\varrho }_2{I}_n\otimes \left(M^{T}M\right)$, ${\displaystyle {\mathrm{\Phi }}_6=\mathrm{\xi }\otimes H-I_n\otimes \left(\frac{M}{2}-{\varrho }_2{M}^{T}N\right)}$, ${\displaystyle {\mathrm{\Phi }}_7=I_n\otimes \left({\varrho }_2{N}^{T}N-\frac{N+N^T}{2}\right)}$.

Proof. 

From (17) and (21), one obtains

$$\begin{array}{cc}\hfill & {\displaystyle \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-{\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}+{\varrho }_2{\int }_0^l{y}^T(\mathfrak{s},t)y(\mathfrak{s},t)d\mathfrak{s}}\hfill \\ \hfill =& {\displaystyle \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-{\int }_0^l[{\epsilon }^T(\mathfrak{s},t)(I_n\otimes M^T)v(\mathfrak{s},t)+v^T(\mathfrak{s},t)(I_n\otimes N^T)v(\mathfrak{s},t)]d\mathfrak{s}}\hfill \\ \hfill & {\displaystyle +{\varrho }_2{\int }_0^l[{\epsilon }^T(\mathfrak{s},t)\left(I_n\otimes \left(M^{T}M\right)\right)\epsilon (\mathfrak{s},t)+2{\epsilon }^T(\mathfrak{s},t)\left(I_n\otimes \left(M^{T}N\right)\right)v(\mathfrak{s},t)}\hfill \\ \hfill & +v^T(\mathfrak{s},t)\left(I_n\otimes \left(N^{T}N\right)\right)v(\mathfrak{s},t)]d\mathfrak{s}\hfill \\ \hfill \le & {\displaystyle {\int }_0^l{\eta }^T(\mathfrak{s},t)\overline{\mathrm{\Phi }}\eta (\mathfrak{s},t)d\mathfrak{s}-\mathcal{l}\phi \left(\epsilon (l,t)\right).}\hfill \end{array}$$

(22)

In view of (21) and (22), one can obtain

$${\displaystyle {\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_2{\int }_0^l{y}^T(\mathfrak{s},t)y(\mathfrak{s},t)d\mathfrak{s}\ge \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)+\mathcal{l}\phi \left(\epsilon (l,t)\right).}$$

Therefore, system (3) under boundary controller (6) is FT output strictly passive. □

Remark 5.

Recently, the problem of FT passivity for fractional-order systems has been discussed [29,30,33,40,41,42]. Unfortunately, models in [29,33,40,41,42] did not take into account the reaction diffusion phenomenon. Furthermore, the topology of network models in [29,30,33,40,41,42] was undirected. Practically, the model with spatial diffusion and directed topology in this study more aligns with the actual situation. Under directed topology, the Lyapunov function in Theorem 1 is constructed by means of the left eigenvector corresponding to the zero eigenvalue of the outer coupling matrix. Based on this Lyapunov function, sufficient conditions of FT passivity for system (3) under boundary control are established.

Remark 6.

There were many results on the dynamical behaviors of spatiotemporal networks under boundary control with different boundary conditions [44,45,46]. Neumann boundary conditions can naturally describe the flow or gradient on the boundary, which is conducive to maintaining the system stable. Therefore, the system (1) with Neumann conditions is considered in this paper. Since Lemma 2 is not directly usable for Neumann boundary conditions, variable transformation is used in the proof of Theorem 1, where the derivative term is transformed to a state term using the relationship between the states and their derivatives in Lemma 2.

Next, we consider the network (1) with undirected topology, that is to say, if there is a link between vertices i and j, then $G_{ij}=G_{ji}>0$, otherwise $G_{ij}=G_{ji}=0(i\ne j)$, and $G_{ii}=-{\sum }_{\stackrel{j=1}{j\ne i}}^n{G}_{ij},i=1,2,\dots ,n.$ Additionally, $\alpha ={(1,1,\dots ,1)}^T$, ${\mathrm{\xi }}_i={\displaystyle \frac{1}{n}}$, $\widehat{z}(\mathfrak{s},t)={\displaystyle \frac{1}{n}}{\sum }_{\rho =1}^n{z}_i(\mathfrak{s},t)$. The following observations can be made:

Corollary 1.

System (3) with undirected topology under the boundary controller (6) is FT strictly passive if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\mathrm{\Xi }=\left(\begin{array}{ccc}{\mathrm{\Xi }}_1& {\displaystyle \frac{2}{l}{I}_n\otimes \left(RK\right)}& {\displaystyle I_n\otimes \left(H-\frac{M}{2}\right)}\\ \ast & {\mathrm{\Xi }}_2& 0\\ \ast & \ast & {\displaystyle -I_n\otimes \frac{N+N^T}{2}}\end{array}\right)\le 0,$$

where ${\displaystyle {\mathrm{\Xi }}_1=I_n\otimes {\mathrm{\Phi }}_1+2\sum _{s=1}^{\sigma }{c}^{s}G\otimes {\mathrm{\Gamma }}^s}$, ${\displaystyle {\mathrm{\Xi }}_2=I_n\otimes \left(-\frac{1}{l}(RK+K^{T}R)-\frac{{\pi }^2}{2l^2}R\right)}$.

Corollary 2.

System (3) with undirected topology under the boundary controller (6) is FT input strictly passive if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\begin{array}{c}\hfill \tilde{\mathrm{\Xi }}=\left(\begin{array}{ccc}{\mathrm{\Xi }}_1& {\displaystyle \frac{2}{l}{I}_n\otimes \left(RK\right)}& {\displaystyle I_n\otimes \left(H-\frac{M}{2}\right)}\\ \ast & {\mathrm{\Xi }}_2& 0\\ \ast & \ast & {\mathrm{\Phi }}_4\end{array}\right)\le 0.\end{array}$$

Corollary 3.

System (3) with undirected topology under the boundary controller (6) is FT output strictly passive if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\begin{array}{c}\hfill \overline{\mathrm{\Xi }}=\left(\begin{array}{ccc}{\mathrm{\Xi }}_3& {\displaystyle \frac{2}{l}{I}_n\otimes \left(RK\right)}& {\mathrm{\Xi }}_4\\ \ast & {\mathrm{\Xi }}_2& 0\\ \ast & \ast & {\mathrm{\Phi }}_7\end{array}\right)\le 0,\end{array}$$

where ${\mathrm{\Xi }}_3={\mathrm{\Xi }}_1+{\varrho }_2{I}_n\otimes \left(M^{T}M\right)$, ${\displaystyle {\mathrm{\Xi }}_4=I_n\otimes \left(H-\frac{M}{2}+{\varrho }_2{M}^{T}N\right)}$.

3.2. FT Synchronization

The FT synchronization of the network (1) will be further studied based on the FT passivity result obtained above.

Definition 5

([54]). The multi-weighted FSN (1) is said to realize FT synchronization if for any initial state ${\epsilon }_0\triangleq \epsilon (\mathfrak{s},t_0)$, there exists a settling time $T\left({\epsilon }_0\right)\ge 0$ such that

$${\displaystyle \underset{t\to T\left({\epsilon }_0\right)}{lim}{\parallel \epsilon (\mathfrak{s},t)\parallel }_{{\mathcal{L}}_2}=0,}$$

$${\parallel \epsilon (\mathfrak{s},t)\parallel }_{{\mathcal{L}}_2}=0,t\ge T\left({\epsilon }_0\right),$$

where $(\mathfrak{s},t)\in [0,l]\times [t_0,+\infty ).$

Theorem 4.

Based on Assumptions 1 and 2, the multi-weighted FSN (1) with $v(\mathfrak{s},t)\equiv 0$ under the boundary controller (6) realizes FT synchronization if system (3) is FT output strictly passive. Besides, the settling time T is estimated as

$$T\le T^{\ast }=t_0+{\left({\displaystyle \frac{V\left(t_0\right)\mathrm{\Gamma }(\kappa +1)}{\mathcal{l}}}\right)}^{{\displaystyle \frac{1}{\kappa }}}.$$

Proof. 

In light of the FT output strict passivity for system (3), one has

$${\displaystyle {\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_2{\int }_0^l{y}^T(\mathfrak{s},t)y(\mathfrak{s},t)d\mathfrak{s}\ge \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)+\mathcal{l}\phi \left(\epsilon (l,t)\right).}$$

Let $v(\mathfrak{s},t)=0$, when $\epsilon (\mathfrak{s},t)\in {\mathbb{R}}^{nm}\setminus \left\{0_{nm}\right\}$,

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)& {\displaystyle \le -{\varrho }_2{\int }_0^l{\epsilon }^T(\mathfrak{s},t)(I_n\otimes M^{T}M)\epsilon (\mathfrak{s},t)d\mathfrak{s}-\mathcal{l}}\hfill \\ & \le -\mathcal{l}.\hfill \end{array}$$

According to Lemma 3, $V\left(\epsilon \right(\mathfrak{s},t\left)\right)=0$ for $t\ge T^{\ast }$, it means that the network (1) under boundary controller (6) realizes FT synchronization within the settling time T. □

By exploiting Theorems 3 and 4, the following conclusion can be drawn:

Corollary 4.

Based on Assumptions 1 and 2, the network (1) with $v(\mathfrak{s},t)\equiv 0$ under the boundary controller (6) realizes FT synchronization if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\begin{array}{c}\hfill \overline{\mathrm{\Phi }}=\left(\begin{array}{ccc}{\mathrm{\Phi }}_5& {\displaystyle \frac{2}{l}\mathrm{\xi }\otimes \left(RK\right)}& {\mathrm{\Phi }}_6\\ \ast & {\mathrm{\Phi }}_3& 0\\ \ast & \ast & {\mathrm{\Phi }}_7\end{array}\right)\le 0.\end{array}$$

Besides, the settling time T is estimated as

$$T\le T^{\ast }=t_0+{\left({\displaystyle \frac{V\left(t_0\right)\mathrm{\Gamma }(\kappa +1)}{\mathcal{l}}}\right)}^{{\displaystyle \frac{1}{\kappa }}}.$$

Similarly, the following statement about FT synchronization of the multi-weighted FSN (1) with undirected topology can be easily derived:

Corollary 5.

Based on Assumptions 1 and 2, the multi-weighted FSN (1) with undirected topology and $v(\mathfrak{s},t)\equiv 0$ realizes FT synchronization under the boundary controller (6) if there exists a matrix $0<K\in {\mathbb{R}}^{m\times m}$ such that

$$\begin{array}{c}\hfill \overline{\mathrm{\Xi }}=\left(\begin{array}{ccc}{\mathrm{\Xi }}_3& {\displaystyle \frac{2}{l}{I}_n\otimes \left(RK\right)}& {\mathrm{\Xi }}_4\\ \ast & {\mathrm{\Xi }}_2& 0\\ \ast & \ast & {\mathrm{\Phi }}_7\end{array}\right)\le 0.\end{array}$$

Besides, the settling time T is estimated as

$$T\le T^{\ast }=t_0+{\left({\displaystyle \frac{V\left(t_0\right)\mathrm{\Gamma }(\kappa +1)}{\mathcal{l}}}\right)}^{{\displaystyle \frac{1}{\kappa }}}.$$

Remark 7.

By means of the Laplace transform, FT passivity-based FT synchronization of multi-weighted fractional-order networks was investigated in [42]. In contrast to this research method, this article directly obtains the desired conclusion using the inequality technique, which is simpler and more intuitive. Additionally, the FT synchronization of fractional-order networks was explored by constructing Lyapunov functions in [31,32]. The difference is that this paper first delves into the FT passivity of multi-weighted FSNs, and FT synchronization emerges as a natural conclusion from the FT passivity analysis.

Remark 8.

In [55], the FT synchronization for spatiotemporal networks was investigated via boundary control strategy, in which the considered network models were integer-order and undirected. Different from this, the FT synchronization for directed FSNs is discussed in this paper. Given the heritability and memory characteristics of fractional operators, it is inappropriate to directly generalize the results in [55] to FSNs, but the results of FT synchronization for FSNs can be directly extended to integer-order spatiotemporal networks with $\kappa =1$.

Remark 9.

Note that previous research [30,31,32] on the FT passivity or FT synchronization of FSNs has primarily concentrated on implementing control strategies within the spatial domain. However, taking into account the spatiotemporal distribution characteristics of the system, it is unrealistic to deploy controllers throughout the whole spatial domain. The control strategy is designed in this study only at the space boundary to realize FT passivity and FT synchronization. The design of the controller is more concise and more convenient to be implemented.

Remark 10.

The authors in [26,36,37] investigated the passivity or synchronization of FSNs in an asymptotic sense. Distinct from these results, by presenting boundary strategy in this article, the FT passivity and FT synchronization of FSNs are discussed, which have more robustness and high convergence speed.

4. Numerical Simulations

In this section, a numerical example is provided to demonstrate the effectiveness of the proposed theoretical results.

Consider the following multi-weighted FSN with directed topology:

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{D}_t^{\kappa }{z}_i(\mathfrak{s},t)=& R{\displaystyle \frac{{\partial }^2{z}_i(\mathfrak{s},t)}{\partial {\mathfrak{s}}^2}}+A{z}_i(\mathfrak{s},t)+Ef\left(z_i(\mathfrak{s},t)\right)\hfill \\ \hfill & +\sum _{s=1}^3\sum _{j=1}^6{c}^s{G}_{ij}{\mathrm{\Gamma }}^s{z}_j(\mathfrak{s},t)+H{v}_i(\mathfrak{s},t),\hfill \end{array}$$

(23)

where $i\in \mathcal{I}=\{1,2,\dots ,6\}$, $\mathfrak{s}\in [0,2]$, $\kappa =0.9$, $z_i(\mathfrak{s},t)={(z_{i1}(\mathfrak{s},t),z_{i2}(\mathfrak{s},t),z_{i3}(\mathfrak{s},t))}^T$, $R=0.1I_3$, $A=-1.7I_3$, $f\left(z_i\right)={\left(0.2\tanh \left(z_{i1}\right),0.2\tanh \left(z_{i2}\right),0.2\tanh \left(z_{i3}\right)\right)}^T$ with $\omega =0.2$, $c^1=0.3$, $c^2=0.2$, $c^3=0.4$, ${\mathrm{\Gamma }}^1=\mathrm{diag}\{1,0.9,1\}$, ${\mathrm{\Gamma }}^2=\mathrm{diag}\{1,1,1\}$, ${\mathrm{\Gamma }}^3=\mathrm{diag}\{1,1,0.9\}$, $H=\mathrm{diag}\{4,4.1,5.6\}$, $v_i(\mathfrak{s},t)={(i\sin \left(0.04\mathfrak{s}\right)\sqrt{0.02t},i\sin \left(0.04\mathfrak{s}\right)\sqrt{0.02t},i\sin \left(0.04\mathfrak{s}\right)\sqrt{0.01t})}^T$,

$$E=\left(\begin{array}{ccc}1& 0.5& -0.69\\ 0.22& -0.5& 0.5\\ -0.4& 0.2& -0.6\end{array}\right),$$

and

$$G=\left(\begin{array}{cccccc}-0.6& 0.2& 0& 0.1& 0.3& 0\\ 0.3& -0.6& 0.2& 0& 0.1& 0\\ 0& 0.4& -0.5& 0.1& 0& 0\\ 0.1& 0& 0.2& -0.4& 0.1& 0\\ 0.1& 0& 0& 0.3& -0.6& 0.2\\ 0& 0& 0.2& 0& 0.1& -0.3\end{array}\right).$$

A positive vector $\alpha ={(1.4396,1.7282,1.8725,1.9530,1.5000,1.0000)}^T$ can be found by utilizing the MATLAB R2024a NULL function, which satisfies $G^T\alpha =0$. According to the definition of $\mathrm{\xi }$, one can calculate $\mathrm{\xi }=\mathrm{diag}\{0.1516,0.1820,0.1972,0.2057,0.1580,0.1053\}$. Set the parameter of controller $\mathcal{l}=1.3$. The initial value ${\mathrm{\Theta }}_i\left(\mathfrak{s}\right)$ is chosen as random constants within the range $[-2,2]$ for $\mathfrak{s}\in [0,2]$ with $i\in \mathcal{I}$. The directed topology of network (23) is shown in Figure 1.

In the output vector, take

$$\begin{array}{c}\hfill M=\left(\begin{array}{ccc}1& 0.9& 0.8\\ 1.1& 0.8& 0.9\\ 0.9& 0.8& 1\end{array}\right),N=\left(\begin{array}{ccc}2.5& 0& 0\\ 0& 2.5& 0\\ 0& 0& 2.5\end{array}\right).\end{array}$$

Case 1: Using the LMI-Box in MATLAB, we can determine $K=7.4518I_3$ satisfying (7). From Theorem 1, FT strict passivity is achieved through boundary control strategy (6). The time evolution of ${\displaystyle {\chi }_1\left(t\right)={\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-\stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-\mathcal{l}\phi \left(\epsilon (\mathfrak{s},t)\right)}$ is displayed in Figure 2. It follows from Figure 2 that the value of ${\chi }_1\left(t\right)$ is always nonnegative, so the system is FT strictly passive by Definition 2.

We can find $K=14.6266I_3$ and ${\varrho }_1=0.0781$ to satisfy the LMI (19). According to Theorem 2, FT input strict passivity is achieved through boundary control strategy (6). The time evolution of ${\displaystyle {\chi }_2\left(t\right)={\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_1{\int }_0^l{v}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-\stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-\mathcal{l}\phi \left(\epsilon (\mathfrak{s},t)\right)}$ is displayed in Figure 3. It follows from Figure 3 that the value of ${\chi }_2\left(t\right)$ is always nonnegative, so the system is FT input strictly passive by Definition 3.

Similarly, by using MATLAB, we can find $K=13.0712I_3$ and ${\varrho }_2=0.0046$ to satisfy (21). From Theorem 3, through boundary control strategy (6), FT output strict passivity is achieved. The time evolution of ${\displaystyle {\chi }_3\left(t\right)={\int }_0^l{y}^T(\mathfrak{s},t)v(\mathfrak{s},t)d\mathfrak{s}-{\varrho }_2{\int }_0^l{y}^T(\mathfrak{s},t)y(\mathfrak{s},t)d\mathfrak{s}-\stackrel{C}{t_0}{D}_t^{\kappa }V\left(\epsilon (\mathfrak{s},t)\right)-\mathcal{l}\phi \left(\epsilon (\mathfrak{s},t)\right)}$ is displayed in Figure 4. It follows from Figure 4 that the value of ${\chi }_3\left(t\right)$ is always nonnegative, so the system is FT output strictly passive by Definition 4.

Case 2: Manifestly, the condition (21) holds if we take $K=13.0712I_3$ and ${\varrho }_2=0.0046$ as in Case 1, which means that FT output strict passivity is achieved. From Theorem 4, the multi-weighted FSN (23) under the boundary controller (6) realizes FT synchronization under external input $v(\mathfrak{s},t)=0$. Figure 5 depicts the dynamic evolutions of the synchronization error states ${\epsilon }_{ij}(\mathfrak{s},t)(i=1,2,\dots ,6,j=1,2,3)$. It clearly indicates that the multi-weighted FSN (23) is FT synchronized within $T^{\ast }=15.0345$.

5. Conclusions

In this article, the FT passivity and FT synchronization for multi-weighted FSNs with directed topology were investigated. Firstly, by making use of a fractional differential inequality in [50], several new definitions of FT passivity for multi-weighted FSNs were proposed. Furthermore, a type of boundary controller is presented, and some criteria were obtained to guarantee the FT passivity of multi-weighted FSNs by utilizing the inequality technique and finite-time convergence results of fractional systems. Different from the existing articles [44,45], the boundary controller designed is only dependent on the information at the spatial boundary. Moreover, the association between FT passivity and FT synchronization was addressed, and several FT synchronization criteria were developed by utilizing the derived FT output strict passivity results. A numerical example was ultimately performed to substantiate the validity of the proposed FT passivity and FT synchronization criteria.

Currently, the synchronization of FSNs with other boundary conditions (such as Dirichlet boundary condition and Robin boundary condition) [56,57,58] has received much attention, and Robin boundary is regarded as a more general boundary form. In addition, considering the diversity and complexity of connections between nodes in multi-weighted FSNs, the external coupling matrices of different coupling terms in the network should be different [26,36]. Accordingly, the FT passivity and FT synchronization for directed and multi-weighted FSNs with Robin boundary conditions and different external coupling matrices are worth investigating further.