Advanced Observation-Based Bipartite Containment Control of Fractional-Order Multi-Agent Systems Considering Hostile Environments, Nonlinear Delayed Dynamics, and Disturbance Compensation

Abstract

This paper introduces an advanced observer-based control strategy designed for fractional multi-agent systems operating in hostile environments. We take into account the dynamic nature of the agents with nonlinear delayed dynamics and consider external disturbances affecting the system. The manuscript presents an improved observation-based control approach tailored for fractional-order multi-agent systems functioning in challenging conditions. We also establish various applicable conditions governing the creation of observers and disturbance compensation controllers using the fractional Razmikhin technique, signed graph theory, and matrix transformation. Furthermore, our investigation includes observation-based control on switching networks by employing a typical Lyapunov function approach. Finally, the effectiveness of the proposed strategy is demonstrated through the analysis of two simulation examples.

1. Introduction

Multi-agent systems (MASs) are composed of multiple interacting autonomous subsystems that utilize shared information to tackle challenging or complex tasks. Consensus is a critical aspect of MASs’ collective actions and ensures that through effective distributed control based solely on local knowledge, each agent’s state converges to a desired common state. The existing literature has proposed several significant results addressing consensus problems under conditions of limited communication, dynamically changing topologies, uniform or non-uniform time delays, and external disturbances [1,2,3,4,5]. Additionally, the exploration of output containment control strategies has been a focal point of research efforts [6]. Various techniques, ranging from those leveraging the principles of Laplace transforms to sophisticated frequency domain analysis approaches, have been employed to address this intricate problem space [7]. In contrast to the cooperative frameworks typically studied in academia [8], real-world MASs frequently operate within environments characterized by a delicate interplay of cooperative and hostile connections among agents [9]. This detailed dynamic is enclosed within the framework of signed networks, where positive and negative edge weights encode the nature of the connection between agents [10]. These signed networks find widespread application across different domains, including social networks and multi-robot systems, where understanding and navigating the complex inter-dependencies between agents is preeminent [11,12]. The appreciation of these antagonistic–cooperative dynamics underscores the need for adaptive and robust control strategies capable of accommodating the intricate interplay between the agents within such networks. Numerous significant investigations have focused on bipartite containment control. References [13,14,15] present fascinating works in this field. In reference [16], the authors convert bipartite containment control into a general form using matrix transformation techniques. An observer-based strategy to tackle bipartite containment control is introduced in [17]. Reference [18] explores the bipartite output containment issue by developing various control protocols. Moreover, other approaches have been employed, such as an output feedback method [19], a fuzzy observer method [20], and a delayed event-triggered mechanism [21]. It is important to note that the majority of existing research focuses on bipartite containment control for integer-order multi-agent systems (MASs). However, fractional calculus has demonstrated substantial benefits in modeling complex phenomena, including the transient response of elastomers and the motion of objects in viscoelastic materials [22,23]. In recent years, there has been significant research on control systems for connected vehicle platooning. Niazi et al. have made important contributions in this area [24]. The latest advancements in multi-agent system control have addressed various challenges, including communication delays and nonlinear couplings. Given the inherent uncertainties and nonlinearities in physical applications, substantial research has focused on the robust formation and consensus control problems in uncertain multi-agent systems (MASs). Studies such as those in [25,26] have introduced a fully distributed adaptive consensus protocol for higher-order systems with general linear and Lipschitz nonlinear dynamics. This protocol effectively reduces the consensus error to a small neighborhood around the origin and implements exponential convergence. In a similar way, adaptive techniques have been used to deal with the formation tracking problem in teams of uncertain systems with nonlinear terms satisfying Lipschitz conditions [27,28]. The motivation for studying fractional-order multi-agent systems (FMASs) with delays is grounded in both practical applications and theoretical advancements. Practically, FMASs can model complex systems in engineering, biology, and social networks and capture memory effects and long-term dependencies more accurately than traditional models. Communication delays are inherent in distributed systems and smart devices, necessitating robust control strategies. Theoretically, this research fills a gap in existing studies by providing new insights into the stability and control of FMASs with delays, leading to improved performance and resilience in real-world applications.

Inspired by these findings, this paper proposes advanced observer-based bipartite containment control of fractional multi-agent systems considering hostile environments, nonlinear delayed dynamics, and disturbance compensation. The specific contributions of this paper are given below.

• As compared to [29], we use a fractional-order multi-agent system instead of an integer-order derivative; our work contributes to the field by providing a more general and flexible framework for modeling and controlling dynamic behaviors in complex networks.
• By incorporating disturbances into the controller, our work enhances the robustness and practical applicability of the control strategy for fractional-order multi-agent systems, addressing real-world uncertainties.
• By incorporating a nonlinear delayed function in the dynamics, our work provides a more accurate and comprehensive framework for modeling and controlling fractional-order multi-agent systems, capturing complex temporal interactions and delays that have not been addressed in previous studies [29].

1.

Section 1 is the introduction of the paper.

2.

Section 2 presents the foundational knowledge that is useful for our main results.

3.

Observation-based analyses of bipartite containment control for fixed and switching signed digraphs are discussed in Section 3.2 and Section 4.

4.

Numerical examples are presented in Section 5.

5.

The paper is summarized in Section 6.

Figure 1 represents the framework of this paper.

2. Foundational Knowledge

2.1. Graph Theory

Consider a weighted signed digraph $\mathcal{G}=(\mathtt{N},\mathtt{E})$, where $\mathtt{N}=\{n_1,\dots ,n_{\mathcal{N}}\}$ is the set of nodes and $\mathtt{E}\subseteq \mathtt{N}\times \mathtt{N}$ is the set of directed edges. The adjacency matrix $\mathtt{F}={\left[f_{ij}\right]}_{\mathcal{N}\times \mathcal{N}}$ is defined such that $f_{ij}\ne 0$ if $(n_i,n_j)\in \mathtt{E}$ and $f_{ij}=0$ otherwise. An edge $(n_i,n_j)$ means that node $n_i$ is capable of receiving information from node $n_j$; $f_{ij}>0$ indicates a cooperative relationship, and $f_{ij}<0$ indicates a hostile connection between nodes $n_i$ and $n_j$. The Laplacian matrix $L={\left[{\ell }_{ij}\right]}_{\mathcal{N}\times \mathcal{N}}$ for $\mathcal{G}$ is defined such that ${\ell }_{ij}=-f_{ij}$ for $i\ne j$ and ${\ell }_{ii}={\sum }_{j=1,j\ne i}^{\mathcal{N}}|f_{ij}|$. Additionally, $\mathcal{G}$ is considered to be structurally balanced if the node set $\mathtt{N}$ of the signed graph $\mathcal{G}$ can be divided into two disjoint subsets ${\mathtt{N}}_1$ and ${\mathtt{N}}_2$ such that $f_{ij}>0$ for any $n_i,n_j\in {\mathtt{N}}_1$ or $n_i,n_j\in {\mathtt{N}}_2$ and $f_{ij}<0$ for any $n_i\in {\mathtt{N}}_1$ and $n_j\in {\mathtt{N}}_2$.

2.2. Basic Lemmas and Fractional Operators

In this subsection, various valuable definitions and lemmas are revisited.

Definition 1 

([30]). The Caputo derivative of order γ for a function $\psi \left(s\right)$ that is c-differentiable is expressed as:

$$D^{\gamma }\psi \left(s\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(c-\gamma )}}{\int }_{s_0}^s{\displaystyle \frac{{\psi }^{\left(c\right)}\left(t\right)}{{(s-t)}^{\mu -c+1}}}dt,0\le c-1<\gamma \le c,c\in {\mathbb{Z}}^{+}.$$

Lemma 1 

([31]). If $\mathcal{Q}\left(s\right)\in {\mathbb{R}}^n$ is an absolutely continuous and differentiable function, then the following property holds:

$${\displaystyle \frac{1}{2}}{D}^{\gamma }\left({\mathcal{Q}}^T\left(s\right)\mathcal{Q}\left(s\right)\right)\le {\mathcal{Q}}^T\left(s\right)\left[D^{\gamma }\mathcal{Q}\left(s\right)\right]\le 0,\gamma \in (0,1).$$

Remark 1. 

It is significant to observe that the Leibniz rule for the Caputo fractional derivative does not have a simple form. The fractional Leibniz rule, which is appropriate for the Caputo derivative, includes complex integral expressions that account for the non-local properties of the fractional derivative. For detailed discussions on the Leibniz rule for fractional derivatives, refer to foundational works on fractional calculus like [32].

Lemma 2 

([31]). Let ${\beta }_1,{\beta }_2\in {\mathbb{R}}^n$ and $\rho >0$; then the following inequality holds:

$$2{\beta }_1^T{\beta }_2\le \rho {\beta }_1^T{\beta }_1+{\displaystyle \frac{1}{\rho }}{\beta }_2^T{\beta }_2$$

Consider a delayed fractional-order system for $0<\gamma <1$ that can present both switching and non-switching behaviors:

$$\begin{array}{c}\hfill D^{\gamma }z\left(s\right)=k^{\sigma \left(s\right)}(s,z_s),s\ge s_0\\ \hfill z_{s_0}\left(\theta \right)=\varphi \left(\theta \right),\theta \in [-e_2,0]\end{array}$$

(1)

where $z_s\left(\theta \right)=z(s+\theta )$, $\theta \in [-e_2,0]$, for any $s\ge s_0$ and time delay $e_2>0$, $\varphi \in \mathcal{P}$. The function $k^{\sigma \left(s\right)}$ maps $\mathbb{R}\times$ (bounded sets of $\mathcal{P}$) into bounded sets of ${\mathbb{R}}^n$ and $k^{\sigma \left(s\right)}(s,0)=0$. Here, $\sigma \left(s\right)$ is a piecewise constant switching signal that can be constant in both cases (switching or non-switching).

Lemma 3. 

The equilibrium point at zero of system (1), whether it presents switching or non-switching behavior, is asymptotically stable provided there exist positive numbers $c_1,c_2,c_3$ and a quadratic Lyapunov function $\mathcal{V}:{\mathbb{R}}^n\to \mathbb{R}$ such that

$$c_1{\parallel z\left(s\right)\parallel }^2\le \mathcal{V}\left(z\left(s\right)\right)\le c_2{\parallel z\left(s\right)\parallel }^2$$

and whenever

$$\begin{array}{cc}\hfill & \mathcal{V}\left(z(s+\theta )\right)\le \kappa \mathcal{V}\left(z\left(s\right)\right),\theta \in [-e_2,0],\hfill \\ \hfill & D^{\gamma }\mathcal{V}\left(z\left(s\right)\right)\le -c_3{\parallel z\left(s\right)\parallel }^2,\hfill \end{array}$$

where $\kappa >1$; for more details see [33].

3. Observation-Based Bipartite Containment Control Analysis

In this section, we examine observation-based bipartite containment control of nonlinear FMASs with delays and disturbance in controllers for fixed signed directed networks as well as switching signed directed networks employing the common Lyapunov function approach. This is done by using the fractional Razmikhin technique, signed graph theory, and matrix transformation.

3.1. Model Building

In a fractional-order multi-agent system (FMAS) containing $\mathcal{F}$ followers and $\mathcal{N}-\mathcal{F}$ leaders, represented as $\mathbb{F}=\{1,2,\dots ,\mathcal{F}\}$ and $\mathbb{B}=\{\mathcal{F}+1,\mathcal{F}+2,\dots ,\mathcal{N}\}$, respectively, the behavior of the followers is governed by:

$$\begin{array}{cc}\hfill D^{\gamma }{y}_i\left(s\right)& =D{y}_i\left(s\right)+p(s,y_i\left(s\right))+p_1(s,y_i(s-e_2))+E{\int }_{-e_1}^0{y}_i(s+\varphi )d\varphi +F{v}_i\left(s\right)+\Delta v_i\left(s\right)\hfill \\ \hfill z_i& =G{y}_i\left(s\right)i\in \mathbb{F}\hfill \end{array}$$

(2)

and the behavior of the leaders is described by

$$\begin{array}{cc}\hfill D^{\gamma }{y}_i\left(s\right)& =D{y}_i\left(s\right)+p(s,y_i\left(s\right))+p_1(s,y_i(s-e_2))+E{\int }_{-e_1}^0{y}_i(s+\varphi )d\varphi \hfill \\ \hfill z_i& =G{y}_i\left(s\right)i\in \mathbb{B}\hfill \end{array}$$

(3)

Each agent i in the system is described by its state $y_i\in {\mathbb{R}}^n$, output $z_i\in {\mathbb{R}}^a$, and control input $v_i\in {\mathbb{R}}^b$; also, $e_2$ and $e_1$ are positive constants representing the communication and distributed delay, respectively. The function $p:\mathbb{R}\to {\mathbb{R}}^n$ is continuous and odd with respect to $y_i$ and satisfyies $p(s,0)=0$ and $p(s,-y_i)=-p(s,y_i)$. The term $p_1$ is the nonlinear known function satisfying the Lipschitz condition. Matrices D, $E\in {\mathbb{R}}^{n\times n}$, $F\in {\mathbb{R}}^{n\times b}$, and $G\in {\mathbb{R}}^{a\times n}$ are constant. Furthermore, we assume that every agent can access relative output measurements of its neighboring agents and that the $i-th$ agent belongs to the set $\mathbb{B}$; the control input $v_i$ is equal to zero.

Definition 2. 

([1]). The set $\delta \in {\mathbb{R}}^n$ is convex if, for any pair of points $x_1,x_2$ in δ and any λ, where $0<\lambda <1$, the point $\lambda x_1+(1-\lambda )x_2$ is also in δ. The convex hull formed by a finite set of points $x_1,x_2,\dots ,x_n\in {\mathbb{R}}^n$ is the set of all possible convex combinations of these points and is defined as follows: it consists of all points of the form

$$co\{x_1,x_2,...x_n\}=\left\{\sum _{i=1}^n{\lambda }_i{x}_i|{\lambda }_i\ge 0,\sum _{i=1}^n{\lambda }_i=1\right\}$$

(4)

Definition 3. 

Observation-based bipartite containment control of the FMAS described in Equations (2) and (3) is achieved if certain followers converge to the convex hull of $y_q$ for $q\in \mathbb{B}$, while others converge to $-co\{y_q,q\in \mathbb{B}\}$.

Assumption 1. 

If the functions $p_i$ are Lipschitz continuous, then for each $p_i$, there exists a constant $k_i>0$ such that for any vectors θ and α in ${\mathbb{R}}^n$, the condition below is satisfied:

$$||p_i(s,\theta )-p_i(\theta ,\alpha )||\le k_i||\theta -\alpha ||,i=1,2.$$

(5)

Assumption 2. 

The nonlinear function p complies with the following inequality: for any $z,z_i\in {\mathbb{R}}^n$, where ${\lambda }_i\ge 0$ and ${\sum }_{i=1}^{\mathcal{N}-\mathcal{F}}{\lambda }_i=1$, and for a given $r>0$,

$$\parallel p(s,z)-\sum _{i=1}^{\mathcal{N}-\mathcal{F}}{\lambda }_{i}p(s,z_i)\parallel \le r\parallel z-\sum _{i=1}^{\mathcal{N}-\mathcal{F}}{\lambda }_i{z}_i\parallel .$$

Assumption 3. 

The signed digraph $\mathcal{G}$ exhibits structural balance.

Assumption 4. 

Each follower within the signed digraph $\mathcal{G}$ is connected by a directed link to at least one leader.

Remark 2. 

If Assumption 2 holds, this implies that the Lipschitz condition is fulfilled; however, the converse is not true. The condition outlined in Assumption 2 guarantees both the existence and uniqueness of the solution for a nonlinear fractional-order system under delays [34]. Assumption 4 is crucial to ensure the absence of isolated vertices, ensuring that every agent can receive information from its neighboring agent. This configuration is commonly encountered in various containment control problems [35].

Remark 3. 

If Assumption 3 holds, then the followers may be split into two separate non-overlapping subsets: ${\mathbb{F}}_1$ and ${\mathbb{F}}_2$. Within each subset, agents connect cooperatively, while connections between agents from different subsets are hostile in behavior. By using Definition 3, observation-based bipartite containment control of the system defined by (2) and (3) is accomplished if the following conditions are satisfied:

$$\begin{array}{cc}\hfill \underset{s\to \infty }{lim}\parallel y_i-co\{y_q,q\in \mathbb{B}\}\parallel & =0,\forall i\in {\mathbb{F}}_1,\hfill \\ \hfill \underset{s\to \infty }{lim}\parallel y_i+co\{y_q,q\in \mathbb{B}\}\parallel & =0,\forall i\in {\mathbb{F}}_2.\hfill \end{array}$$

3.2. Observation-Based Analysis of Bipartite Multi-Agent System for Fixed Signed Digraph

Based on the idea from the containment control protocols presented by [9,29], we introduce an observation-based bipartite containment controller for every $i\in \mathbb{F}$. This study leverages the state, which is observed information, to enhance the system’s performance:

$$\begin{array}{cc}\hfill D^{\gamma }{\widehat{y}}_i\left(s\right)& =D{\widehat{y}}_i\left(s\right)+p(s,{\widehat{y}}_i\left(s\right))+p_1(s,{\widehat{y}}_i(s-e_2))+E{\int }_{-e_1}^0{\widehat{y}}_i(s+\varphi )d\varphi +F{v}_i\left(s\right)+\Delta v_i\left(s\right)\hfill \\ & +\mu A\left(\sum _{j=1}^{\mathcal{F}}|f_{ij}|(z_i-sgn\left(f_{ij}\right)z_j)-\sum _{j=1}^{\mathcal{F}}|f_{ij}|({\widehat{z}}_i-sgn\left(f_{ij}\right){\widehat{z}}_j)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}|(z_i-{\widehat{z}}_i))\right),\hfill \\ \hfill {\widehat{z}}_i& =G{\widehat{y}}_i\left(s\right)i\in \mathbb{F}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill v_i& =-\beta T\left(\sum _{j=1}^{\mathcal{N}}|f_{ij}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right){\widehat{y}}_j(s-e_2)\right)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right)y_j(s-e_2)\right)\right)\hfill \\ & +\Delta v_i\left(s\right)\hfill \end{array}$$

(7)

Here, ${\widehat{y}}_i$ denotes the state estimate of $y_i$ for each $i\in \mathbb{F}$, and ${\widehat{z}}_i$ is the output of the observer. The parameters $\mu$ and $A\in {\mathbb{R}}^{n\times a}$ are the coupling strength and the gain matrix of the observer, respectively, both of which are designed; $e_2>0$ represents the input delay. Moreover, $T\in {\mathbb{R}}^{b\times n}$ and $\beta$ represent the gain matrix and coupling constant for the controller, respectively. The term $f_{ij}$ refers to entries of the adjacency matrix of digraph $\mathcal{G}$, and $\Delta v_i\left(s\right)$ is the disturbance in the controller. Since $v_i=0$ for all $i\in \mathbb{B}$, the Laplacian matrix L of the signed digraph $\mathcal{G}$ is defined as:

$$L=\left(\begin{array}{cc}{L}_{\mathcal{F}}& L_{\mathcal{B}}\\ 0_{(\mathcal{N}-\mathcal{F})\times \mathcal{F}}& 0_{(\mathcal{N}-\mathcal{F})\times (\mathcal{N}-\mathcal{F})}\end{array}\right)$$

where $L_{\mathcal{F}}\in {\mathbb{R}}^{\mathcal{F}\times \mathcal{F}}$ and $L_{\mathcal{B}}\in {\mathbb{R}}^{\mathcal{F}\times (\mathcal{N}-\mathcal{F})}$. If Assumption 3 is met, a diagonal matrix exists such that $\psi =\mathrm{diag}({\psi }_1,{\psi }_2,\dots ,{\psi }_{\mathcal{F}},{\psi }_{\mathcal{F}+1},\dots ,{\psi }_{\mathcal{N}})$; the matrix is defined by its elements as:

$$\left\{\begin{array}{c}{\psi }_i=1ifi\in {\mathbb{F}}_1\cup \mathbb{B}\hfill \\ {\psi }_i=-1ifi\in {\mathbb{F}}_2\hfill \end{array}\right.$$

such that

$$L=\psi L\psi =\left(\begin{array}{cc}{\overline{L}}_{\mathcal{F}}& {\overline{L}}_{\mathcal{B}}\\ 0_{(\mathcal{N}-\mathcal{F})\times \mathcal{F}}& 0_{(\mathcal{N}-\mathcal{F})\times (\mathcal{N}-\mathcal{F})}\end{array}\right)$$

Here, ${\overline{L}}_{\mathcal{F}}={\left[{\overline{\ell }}_{ij}\right]}_{\mathcal{F}\times \mathcal{F}}$ if $i\ne j$; then ${\left[{\overline{\ell }}_{ij}\right]}_{\mathcal{F}\times \mathcal{F}}$ is equal to $-|f_{ij}|$; otherwise, it is equal to ${\sum }_{j=1}^{\mathcal{N}}|f_{ij}|$. Also, ${\overline{L}}_{\mathcal{B}}={\left[{\overline{\ell }}_{ij}\right]}_{\mathcal{F}\times (\mathcal{N}-\mathcal{F})}$${\overline{\ell }}_{ij}=-|f_{ij}|\le 0$ for $i=1,2,3,...\mathcal{F}$, $j=\mathcal{F}+1,...,\mathcal{N}$.

Lemma 4 

([1]). If Assumption 4 holds, ${\overline{L}}_{\mathcal{F}}$ is a non-singular M-matrix, and each component of $-{\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}$ is nonnegative, with each row summing to 1.

Remark 4. 

Dealing with fractional-order systems is significantly more complex than handling integer-order systems [15]. Fractional-order systems have memory effects and non-locality, meaning their future states depend on their entire past history rather than just the current state. This complexity requires more advanced mathematical tools, such as fractional derivatives and integrals, which often lack closed-form solutions and require numerical methods and approximations. Analyzing the stability of fractional-order systems is more intricate and requires extensions or adaptations of traditional methods like Lyapunov functions. Controller design is also more challenging due to the need to account for fractional dynamics, demanding more sophisticated techniques. Additionally, the application of fractional-order Laplace transforms and the formulation of linear matrix inequalities (LMIs) introduce further complexities. These factors collectively make the analysis and control of fractional-order systems significantly more challenging than that of integer-order systems [21].

Theorem 1. 

Observer-based bipartite containment control of the FMAS defined by Equations (2) and (3), with the controller given in Equation (7) and under Assumptions 1, 2, 3, and 4, can be successfully realized if there exists a diagonal matrix $\mathsf{\Phi }=\mathit{diag}({\varphi }_1,\dots ,{\varphi }_{\mathcal{F}})>0$ such that $\mathsf{\Phi }{L}_{\mathcal{F}}+L_{\mathcal{F}}^T\mathsf{\Phi }>0$ along with positive definite matrices $S_1$ and $S_2$ satisfying the following set of inequalities:

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}(1-r)S_1& S_{1}E\\ E^T{S}_1& -S_1\end{array}\right)<0\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}{S}_{1}D+D^T{S}_1+S_1^2+(k^2+{\ell }^2)I_n+(1+r{e}_1)S_1-\rho G^{T}G& S_1\\ S_1& -I_n\end{array}\right)<0\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}(1-b)S_2^{-1}& E{S}_2^{-1}\\ S_2^{-1}{E}^T& -S_2^{-1}\end{array}\right)<0\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}D{S}_2^{-1}+S_2^{-1}{D}^T+(1+b{e}_1)S_2^{-1}+(2+{\displaystyle \frac{q}{\nu }})I_n+\omega F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T& k{S}_2^{-1}\\ k{S}_2^{-1}& -I_n\end{array}\right)<0\hfill \end{array}$$

(11)

Given the parameters $\rho ,\omega >0$ and $r,b>1$, the observer and controller are designed with the following specifications:

• $A=S_1^{-1}{G}^T$
• $\mu \ge {\displaystyle \frac{\rho {\varphi }_{max}}{{\lambda }_1}}$, where:

  -

  ${\lambda }_1={\lambda }_{min}\{\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}+{\overline{L}}_{\mathcal{F}}\}$

  -

  ${\varphi }_{max}={max}_{1\le j\le \mathcal{F}}\left\{{\varphi }_j\right\}$

• $T=F^T{S}_2$
• $0<\beta \le \sqrt{{\displaystyle \frac{\omega {\varphi }_{min}}{{\lambda }_2}}}$, where:

  -

  ${\lambda }_2={\lambda }_{max}\left\{{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\right\}$

  -

  ${\varphi }_{min}={min}_{1\le j\le \mathcal{F}}\left\{{\varphi }_j\right\}$

Proof. 

First, we define the observed state $\widehat{y_i}$ and the state $y_i$ for each $i\in \mathbb{F}$ using Equations (2), (6) and (7):

$$\begin{array}{cc}\hfill & D^{\gamma }{y}_i\left(s\right)=D{y}_i\left(s\right)+p(s,y_i\left(s\right))+p_1(s,y_i(s-e_2))+E{\int }_{-e_1}^0{y}_i(s+\varphi )d\varphi \hfill \\ \hfill & -\beta FT\left(\sum _{j=1}^{\mathcal{N}}|f_{ij}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right){\widehat{y}}_j(s-e_2)\right)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right)y_j(s-e_2)\right)\right)\hfill \\ & +2\Delta v_i\left(s\right)\hfill \\ \hfill & D^{\gamma }{\widehat{y}}_i\left(s\right)=D{\widehat{y}}_i\left(s\right)+p(s,{\widehat{y}}_i\left(s\right))+p_1(s,{\widehat{y}}_i(s-e_2))+E{\int }_{-e_1}^0{\widehat{y}}_i(s+\varphi )d\varphi \hfill \\ \hfill & -\beta FT\left(\sum _{j=1}^{\mathcal{N}}|f_{ij}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right){\widehat{y}}_j(s-e_2)\right)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right)y_j(s-e_2)\right)\right)\hfill \\ & +2\Delta v_i\left(s\right)+\mu A\left(\sum _{j=1}^{\mathcal{F}}|f_{ij}|(z_i-sgn\left(f_{ij}\right)z_j)-\sum _{j=1}^{\mathcal{F}}|f_{ij}|({\widehat{z}}_i-sgn\left(f_{ij}\right){\widehat{z}}_j)+\sum _{j=\mathcal{F}+1}^{\mathcal{F}}|f_{ij}|(z_i-\widehat{z_j})\right)\hfill \end{array}$$

Consider the coordinate transformation $x_i={\psi }_i{y}_i$ and ${\widehat{x}}_i={\psi }_i{\widehat{y}}_i$, where ${\psi }_i=1$ for $i\in {\mathbb{F}}_1\cup \mathbb{B}$, and ${\psi }_i=-1$ for $i\in {\mathbb{B}}_2$. Hence, we can obtain

$$\begin{array}{cc}\hfill & D^{\gamma }{x}_i\left(s\right)=D{x}_i\left(s\right)+p(s,x_i\left(s\right))+p_1(s,x_i(s-e_2))+E{\int }_{-e_1}^0{x}_i(s+\varphi )d\varphi \hfill \\ \hfill & -\beta FT{\psi }_i(\sum _{j=1}^{\mathcal{N}}|f_{ij}|\left({\psi }_i{\widehat{x}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right){\psi }_j{\widehat{x}}_j(s-e_2)\right)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}|({\psi }_i{\widehat{x}}_i(s-e_2)\hfill \\ \hfill & -\mathrm{sgn}\left(f_{ij}\right){\psi }_j{x}_j(s-e_2)\left)\right)+2\Delta v_i\left(s\right),i\in \mathbb{F}\hfill \end{array}$$

(12)

$$\begin{array}{c}{D}^{\gamma }{\widehat{x}}_i\left(s\right)=D{\widehat{x}}_i\left(s\right)+p(s,{\widehat{x}}_i\left(s\right))+p_1(s,{\widehat{x}}_i(s-e_2))+E{\int }_{-e_1}^0{\widehat{x}}_i(s+\varphi )d\varphi \hfill \\ -\beta FT{\psi }_i\left(\sum _{j=1}^{\mathcal{N}}|f_{ij}|\left({\psi }_i{\widehat{x}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right){\psi }_j{\widehat{x}}_j(s-e_2)\right)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}|\left({\psi }_i{\widehat{x}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}\right){\psi }_j{x}_j(s-e_2)\right)\right)\hfill \\ +\mu AG{\psi }_i\left(\sum _{j=1}^{\mathcal{F}}|f_{ij}|{\psi }_i(x_i-sgn\left(f_{ij}\right){\psi }_j{x}_j)-\sum _{j=1}^{\mathcal{F}}|f_{ij}|({\psi }_i{\widehat{x}}_i-sgn\left(f_{ij}\right){\psi }_j{\widehat{x}}_j)+\sum _{j=\mathcal{F}+1}^{\mathcal{F}}|f_{ij}|({\psi }_i{x}_i-{\psi }_j\widehat{x_j})\right)\hfill \\ +2\Delta v_i\left(s\right),i\in \mathbb{F}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & D^{\gamma }{x}_i\left(s\right)=D{x}_i\left(s\right)+p(s,x_i\left(s\right))+p_1(s,x_i(s-e_2))+E{\int }_{-e_1}^0{x}_i(s+\varphi )d\varphi i\in \mathbb{B}\hfill \end{array}$$

(14)

Let $X_{\mathcal{F}}\left(s\right)={(x_1^T\left(s\right),x_2^T\left(s\right),\dots ,x_{\mathcal{F}}^T\left(s\right))}^T$; $X_{\mathcal{B}}\left(s\right)={(x_{\mathcal{F}+1}^T\left(s\right),x_{\mathcal{F}+2}^T\left(s\right),\dots ,x_{\mathcal{N}}^T\left(s\right))}^T$;

$P(s,X_{\mathcal{F}})={(p^T(s,x_1),p^T(s,x_2),\dots ,p^T(s,x_{\mathcal{F}}))}^T$; $P(s,X_{\mathcal{B}})={(p^T(s,x_{\mathcal{F}+1}),p^T(s,x_{\mathcal{F}+2}),\dots ,p^T(s,x_{\mathcal{N}}))}^T$

${\widehat{X}}_{\mathcal{F}}\left(s\right)={({\widehat{x}}_1^T\left(s\right),{\widehat{x}}_2^T\left(s\right),\dots ,{\widehat{x}}_{\mathcal{F}}^T\left(s\right))}^T$; $P(s,{\widehat{X}}_{\mathcal{F}})={(p^T(s,{\widehat{x}}_1),p^T(s,{\widehat{x}}_2),\dots ,p^T(s,{\widehat{x}}_{\mathcal{F}}))}^T$;

$P_1(s,X_{\mathcal{F}}(s-e_2)=(p_1^T(s,x_1(s-e_2)),p_1^T(s,x_2(s-e_2),\dots ,p^T{(s,x_{\mathcal{F}}(s-e_2))}^T$;

$P_1(s,X_{\mathcal{B}}(s-e_2)=(p_1^T(s,x_{\mathcal{F}+1}(s-e_2)),p_1^T(s,x_{\mathcal{F}+1}(s-e_2),\dots ,p^T{(s,x_{\mathcal{N}}(s-e_2))}^T$;

$P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2)=(p_1^T(s,{\widehat{x}}_1(s-e_2)),p_1^T(s,{\widehat{x}}_2(s-e_2),\dots ,p^T{(s,{\widehat{x}}_{\mathcal{F}}(s-e_2))}^T$;

From Equations (12)–(14), we derive the following results:

$$\begin{array}{c}{D}^{\gamma }{X}_{\mathcal{F}}=(I_{\mathcal{F}}\otimes D)X_{\mathcal{F}}+P(s,X_{\mathcal{F}})+P_1(s,X_{\mathcal{F}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E)X_{\mathcal{F}}(s+\varphi )d\varphi \hfill \\ -({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\widehat{X}}_{\mathcal{F}}(s-e_2)-({\overline{L}}_{\mathcal{B}}\otimes \beta FT)X_{\mathcal{B}}(s-e_2)+2\Delta v_i\left(s\right)\hfill \end{array}$$

(15)

$$\begin{array}{c}{D}^{\gamma }{\widehat{X}}_{\mathcal{F}}=(I_{\mathcal{F}}\otimes D){\widehat{X}}_{\mathcal{F}}+P(s,{\widehat{X}}_{\mathcal{F}})+P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E){\widehat{X}}_{\mathcal{F}}(s+\varphi )d\varphi \hfill \\ -({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\widehat{X}}_{\mathcal{F}}(s-e_2)-({\overline{L}}_{\mathcal{B}}\otimes \beta FT)X_{\mathcal{B}}(s-e_2)+({\overline{L}}_{\mathcal{F}}\otimes \mu AG)X_{\mathcal{F}}\hfill \\ -({\overline{L}}_{\mathcal{F}}\otimes \mu AG){\widehat{X}}_{\mathcal{F}}+2\Delta v_i\left(s\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & D^{\gamma }{X}_{\mathcal{B}}=(I_{\mathcal{N}}-\mathcal{F}\otimes D)X_{\mathcal{B}}+P(s,X_{\mathcal{B}})+P_1(s,X_{\mathcal{B}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{N}}-\mathcal{F}\otimes E)X_{\mathcal{B}}(s+\varphi )d\varphi \hfill \end{array}$$

(17)

Let observation error ${\alpha }_1(.)={\widehat{X}}_{\mathcal{F}}(.)-X_{\mathcal{F}}(.)$ and bipartite containment control error ${\alpha }_2(.)=X_{\mathcal{F}}(.)-(-{\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)X_{\mathcal{B}}(.)$; one has

$$\begin{array}{cc}\hfill & D^{\gamma }{\alpha }_1=(I_{\mathcal{F}}\otimes D){\alpha }_1+P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})+P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2))-P_1(s,X_{\mathcal{F}}(s-e_2))\hfill \\ \hfill & {\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E){\alpha }_1(s+\varphi )d\varphi -({\overline{L}}_{\mathcal{F}}\otimes \mu AG){\alpha }_1.\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & D^{\gamma }{\alpha }_2=(I_{\mathcal{F}}\otimes D)X_{\mathcal{F}}+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes D)X_{\mathcal{B}}+P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})+P_1(s,X_{\mathcal{F}}(s-e_2))\hfill \\ \hfill & +({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P_1(s,X_{\mathcal{B}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E)X_{\mathcal{F}}(s+\varphi )d\varphi +{\int }_{-e_1}^0({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes E)X_{\mathcal{B}}(s+\varphi )d\varphi \hfill \\ \hfill & -({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\widehat{X}}_{\mathcal{F}}(s-e_2)+({\overline{L}}_{\mathcal{F}}\otimes \beta FT)X_{\mathcal{F}}(s-e_2)-({\overline{L}}_{\mathcal{F}}\otimes \beta FT)X_{\mathcal{F}}(s-e_2)\hfill \\ \hfill & -({\overline{L}}_{\mathcal{B}}\otimes \beta FT)X_{\mathcal{F}}(s-e_2)+2\Delta v_i\left(s\right).\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & D^{\gamma }{\alpha }_2=(I_{\mathcal{F}}\otimes D){\alpha }_2++P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})+P_1(s,X_{\mathcal{F}}(s-e_2))\hfill \\ \hfill & +({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P_1(s,X_{\mathcal{B}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E){\alpha }_2(s+\varphi )d\varphi \hfill \\ \hfill & -({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_1(s-e_2)-({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_2(s-e_2)+2\Delta v_i\left(s\right)\hfill \end{array}$$

(20)

Let $\alpha (.)={[{\alpha }_1^T(.),{\alpha }_2^T(.)]}^T$ represent the error system and

$$D^{\gamma }\alpha =R_1\alpha +\overline{P}(s,\alpha )+{\overline{P}}_1(s,\alpha (s-e_2))+R_2\alpha (s-e_2)+{\int }_{-e_1}^0{R}_3\alpha (s+\varphi )d\varphi +2\Delta v_i\left(s\right),$$

(21)

where

$$R_1=\left(\begin{array}{cc}(I_F\otimes D)-({\overline{L}}_{\mathcal{F}}\otimes \mu AG)& 0\\ 0& I_F\otimes D\end{array}\right);R_2=\left(\begin{array}{cc}0& 0\\ -{\overline{L}}_{\mathcal{F}}\otimes \beta FT& -{\overline{L}}_{\mathcal{F}}\otimes \beta FT\end{array}\right);$$

$$R_3=\left(\begin{array}{cc}{I}_F\otimes E& 0\\ 0& I_F\otimes E\end{array}\right);$$

$$\overline{P}(s,\alpha )=\left[P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})\right]\left[P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})\right],$$

$${\overline{P}}_1(s,\alpha (s-e_2)=\left[P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2)-P_1(s,X_{\mathcal{F}}(s-e_2)\right]\left[P_1(s,X_{\mathcal{F}}(s-e_2)+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P_1(s,X_{\mathcal{B}}(s-e_2)\right].$$

Let us delve into the asymptotic stability analysis of the error system (21). Given that ${\overline{L}}_{\mathcal{F}}$ is a non-singular M-matrix according to Lemma 4, then there exists a positive definite diagonal matrix $\mathsf{\Phi }=\mathrm{diag}({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_{\mathcal{F}})$ such that $\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}+{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }>0$. With this in mind, let us formulate the following Lyapunov function:

$$\begin{array}{cc}\hfill & \mathcal{V}\left(\alpha \left(s\right)\right)={\mathcal{V}}_1\left({\alpha }_1\left(s\right)\right)+{\mathcal{V}}_2\left({\alpha }_2\left(s\right)\right)\hfill \\ \hfill & ={\alpha }_1^T(\mathsf{\Phi }\otimes S_1){\alpha }_1+{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2){\alpha }_2\hfill \end{array}$$

(22)

where $\eta >0$ is very small scalar.

Remark 5. 

Our method extends traditional Lyapunov techniques, which typically rely on a single positive definite matrix P. Instead of using P, we introduce a matrix Φ with a structured diagonal form, offering greater versatility and adaptability to various system dynamics. Recent research on adaptive and robust control also supports the use of structured Lyapunov functions [29]. Our approach aligns with these recent advancements, as it provides a structured yet adaptable framework to guarantee stability even when dealing with changing system parameters.

It follows from Lemma 1 that

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_1\left({\alpha }_1\left(s\right)\right)=D^{\gamma }\left[{\alpha }_1^T(\mathsf{\Phi }\otimes S_1){\alpha }_1\right]\le 2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)\left[D^{\gamma }{\alpha }_1\right]\hfill \\ \hfill & 2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)[(I_{\mathcal{F}}\otimes D-{\overline{L}}_{\mathcal{F}}\otimes \mu AG){\alpha }_1+P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})+P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2))\hfill \\ \hfill & -P_1(s,X_{\mathcal{F}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E){\alpha }_1(s+\varphi )d\varphi ]\hfill \\ \hfill & ={\alpha }_1^T\left[\mathsf{\Phi }\otimes (S_{1}D+D^T{S}_1)-(\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \mu S_{1}AG)-({\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }\otimes \mu G^T{A}^T{S}_1)\right]{\alpha }_1\hfill \\ \hfill & +2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)\left[P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})\right]+2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)\left[P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2))-P_1(s,X_{\mathcal{F}}(s-e_2))\right]\hfill \\ \hfill & +{\int }_{-e_1}^{0}2{\alpha }_1^T(\mathsf{\Phi }\otimes S_{1}E){\alpha }_1(s+\varphi )d\varphi \hfill \end{array}$$

(23)

Notice that

$$2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)\left[P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})\right]=\sum _{i=1}^{\mathcal{F}}2{\varphi }_i{\alpha }_{1i}^T{S}_1[p(s,{\widehat{x}}_i)-p(s,x_i)];$$

one has from Assumption 2 and Lemma 2

$$2{\varphi }_i{\alpha }_{1i}^T{S}_1[p(s,{\widehat{x}}_i)-p(s,x_i)]\le {\varphi }_i\left[{\alpha }_{1i}^T{S}_1^2{\alpha }_{1i}+{(p(s,{\widehat{x}}_i)-p(s,x_i))}^{T}p(s,{\widehat{x}}_i)-p(s,x_i)\right]\le {\varphi }_i\left[{\alpha }_{1i}^T{S}_1{\alpha }_{1i}^2+k^2{\alpha }_{1i}^T{\alpha }_{1i}\right]$$

Due to ${\alpha }_1^T\left[\mathsf{\Phi }\otimes (S_1^2+k^2{I}_n)\right]{\alpha }_1={\sum }_{i=1}^{\mathcal{F}}{\varphi }_i\left[{\alpha }_{1i}^T{S}_1^2{\alpha }_{1i}+k^2{\alpha }_{1i}^T{\alpha }_{1i}\right]$, we can derive that

$$2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)\left[P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})\right]\le {\alpha }_1^T\left[\mathsf{\Phi }\otimes (S_1^2+k^2{I}_n\right]{\alpha }_1$$

(24)

$$\begin{array}{cc}\hfill & 2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)\left[P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2)-P_1(s,X_{\mathcal{F}}(s-e_2)\right]=\sum _{i=1}^{\mathcal{F}}2{\varphi }_i{\alpha }_{1i}^T{S}_1\left[p_1\right(s,{\widehat{x}}_i(s-e_2)-p_1(s,x_i(s-e_2)],\hfill \end{array}$$

using Assumption 1 and Lemma 2, we have

$$\begin{array}{cc}\hfill & 2{\varphi }_i{\alpha }_{1i}^T{S}_1[p_1(s,{\widehat{x}}_i(s-e_2)-p_1(s,x_i(s-e_2)]\le {\varphi }_i[{\alpha }_{1i}^T{S}_1^2{\alpha }_{1i}\hfill \\ \hfill & +(p_1(s,{\widehat{x}}_i(s-e_2)-p_1{(s,x_i(s-e_2))}^T{p}_1(s,{\widehat{x}}_i(s-e_2)-p_1(s,x_i(s-e_2)]\hfill \\ \hfill & \le {\varphi }_i\left[{\alpha }_{1i}^T{S}_1{\alpha }_{1i}^2+{\ell }^2{\alpha }_{1i}{(s-e_2)}^T{\alpha }_{1i}(s-e_2)\right]\hfill \end{array}$$

Due to

$${\alpha }_1^T\left(s\right)\left[\mathsf{\Phi }\otimes S_1^2\right]{\alpha }_1\left(s\right)+{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)=\sum _{i=1}^{\mathcal{F}}{\varphi }_i\left[{\alpha }_{1i}^T{S}_1^2{\alpha }_{1i}+{\ell }^2{\alpha }_{1i}^T(s-e_2){\alpha }_{1i}(s-e_2)\right],$$

we can derive that

$$2{\alpha }_1^T(\mathsf{\Phi }\otimes S_1)\left[P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2)-P_1(s,X_{\mathcal{F}}(s-e_2)\right]\le {\alpha }_1^T\left(s\right)\left[\mathsf{\Phi }\otimes S_1^2\right]{\alpha }_1\left(s\right)+{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)$$

(25)

By (23)–(25), we have

$$\begin{array}{cc}\hfill D^{\gamma }{\mathcal{V}}_1& \le {\alpha }_1^T\left[\mathsf{\Phi }\otimes (S_{1}D+D^T{S}_1+2S_1^2+k^2{I}_n)-(\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \mu S_{1}AG)-({\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }\otimes \mu G^T{A}^T{S}_1)\right]{\alpha }_1\hfill \\ \hfill & +{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)+{\int }_{-e_1}^{0}2{\alpha }_1^T(\mathsf{\Phi }\otimes S_{1}E){\alpha }_1(s+\varphi )d\varphi \hfill \end{array}$$

Substituting the gain matrix $A=S_1^T{G}^T$ into the above equation yields

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_1\le {\alpha }_1^T\left[\mathsf{\Phi }\otimes (S_{1}D+D^T{S}_1+2S_1^2+k^2{I}_n)-\mu (\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}+{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi })\otimes G^{T}G\right]{\alpha }_1\hfill \\ \hfill & +{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)+{\int }_{-e_1}^{0}2{\alpha }_1^T(\mathsf{\Phi }\otimes S_{1}E){\alpha }_1(s+\varphi )d\varphi \hfill \end{array}$$

(26)

Now, we utilize the fractional Razumikhin technique on Equation (26). Notice that

$$\begin{array}{cc}\hfill & 2{\alpha }_1^T(\mathsf{\Phi }\otimes S_{1}E){\alpha }_1(s+\varphi )=2{\alpha }_1^T(I_{\mathcal{F}}\otimes S_{1}E)({\mathsf{\Phi }}^{{\displaystyle \frac{1}{2}}}\otimes S_1^{-{\displaystyle \frac{1}{2}}})({\mathsf{\Phi }}^{{\displaystyle \frac{1}{2}}}\otimes S_1^{{\displaystyle \frac{1}{2}}}){\alpha }_1(s+\varphi )\hfill \\ \hfill & \le {\alpha }_1^T(I_{\mathcal{F}}\otimes S_{1}E)({\mathsf{\Phi }}^{{\displaystyle \frac{1}{2}}}\otimes S_1^{-{\displaystyle \frac{1}{2}}})({\mathsf{\Phi }}^{{\displaystyle \frac{1}{2}}}\otimes S_1^{-{\displaystyle \frac{1}{2}}})(I_{\mathcal{F}}\otimes E^T{S}_1){\alpha }_1\hfill \\ \hfill & {\alpha }_1^T(s+\varphi )({\mathsf{\Phi }}^{{\displaystyle \frac{1}{2}}}\otimes S_1^{{\displaystyle \frac{1}{2}}})({\mathsf{\Phi }}^{{\displaystyle \frac{1}{2}}}\otimes S_1^{{\displaystyle \frac{1}{2}}}){\alpha }_1(s+\varphi )\hfill \\ \hfill & ={\alpha }_1^T\left(\mathsf{\Phi }\otimes S_{1}E{S}_1^{-1}{E}^T{S}_1\right){\alpha }_1+{\alpha }_1^T(s+\varphi )(\mathsf{\Phi }\otimes S_1){\alpha }_1(s+\varphi ),\varphi \in [-e_1,0]\hfill \end{array}$$

(27)

For some $\kappa >1$, whenever ${\mathcal{V}}_1(s+h)\le \kappa {\mathcal{V}}_1\left(s\right),h\in [-e_1,0]$. That is,

$${\alpha }_1^T(s+h)(\mathsf{\Phi }\otimes S_1){\alpha }_1(s+h)\le \kappa {\alpha }_1^T(\mathsf{\Phi }\otimes S_1){\alpha }_1;h\in [-e_1,0].$$

(28)

When we substitute (27) and (28) into Equation (26), then for a suitably small $\delta >0$, the value of $\kappa$ equals $1+\delta$, and we have following result:

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_1\le {\alpha }_1^T\left[\mathsf{\Phi }\otimes \left(S_{1}D+D^T{S}_1+2S_1^2+k^2{I}_n)+\left(S_{1}E{S}_1^{-1}{E}^T{S}_1+S_1\right)e_1\right)-\mu (\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}+{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi })\otimes G^{T}G\right]{\alpha }_1\hfill \\ & +{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)\hfill \end{array}$$

Denote ${\lambda }_1={\lambda }_{min}\{\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}+{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }\},{\varphi }_{max}=ma{x}_{1\le \mathcal{F}}\left\{{\varphi }_j\right\}$; the following inequality with $\mu \ge {\displaystyle \frac{\rho {\varphi }_{max}}{{\lambda }_1}}$ and $\rho >0$ holds:

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_1\le {\alpha }_1^T\left[\mathsf{\Phi }\otimes \left(S_{1}D+D^T{S}_1+2S_1^2+k^2{I}_n)+\left(S_{1}E{S}_1^{-1}{E}^T{S}_1+S_1\right)e_1\right)-\rho G^{T}G\right]{\alpha }_1\hfill \\ \hfill & +{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)\hfill \\ \hfill & ={\alpha }_1^T(\mathsf{\Phi }\otimes {\mathsf{\Lambda }}_1){\alpha }_1+{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)\hfill \end{array}$$

(29)

Now, again from Lemma 1,

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_2\left(s\right)=D^{\gamma }\left[{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2){\alpha }_2\right]\le 2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)\left[D^{\gamma }{\alpha }_2\right]\hfill \\ \hfill & =2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)[(I_{\mathcal{F}}\otimes D){\alpha }_2+P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})+P_1(s,X_{\mathcal{F}}(s-e_2))\hfill \\ \hfill & +({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P_1(s,X_{\mathcal{B}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E){\alpha }_2(s+\varphi )d\varphi -({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_1(s-e_2)\hfill \\ \hfill & -({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_2(s-e_2)+2\Delta v_i\left(s\right)]\hfill \end{array}$$

By simplifying the above equation, we have

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_2\left({\alpha }_2\left(s\right)\right)={\alpha }_2^T\left[\eta \mathsf{\Phi }\otimes (S_{2}D+D^T{S}_2)\right]{\alpha }_2+2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)[P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})]\hfill \\ \hfill & +2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)[P_1(s,X_{\mathcal{F}}(s-e_2))+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P_1(s,X_{\mathcal{B}}(s-e_2))]+{\int }_{-e_1}^{0}2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_{2}E){\alpha }_2(s+\varphi )d\varphi \hfill \\ \hfill & -2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_1(s-e_2)-2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_2(s-e_2)+4{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)\Delta v_i\left(s\right)\hfill \end{array}$$

(30)

Let $-{\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}={\left[{\overline{\ell }}_{ij}\right]}_{\mathcal{F}\times (\mathcal{N}-\mathcal{F}}$; then

$$2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)[P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})]=\sum _{i=1}^{\mathcal{F}}2{\varphi }_i{\alpha }_{2i}^T{S}_2[p(s,x_i)-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}p(s,x_{\mathcal{F}+j})].$$

From Assumption 2 and Lemma 2, we have

$$\begin{array}{cc}\hfill & 2\eta {\varphi }_i{\alpha }_{2i}^T{S}_2[p(s,x_i)-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}p(s,x_{\mathcal{F}+j})]\le \eta {\varphi }_i[{\alpha }_{2i}^T{S}_2^2{\alpha }_{2i}+{\left(p(s,x_i)-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}p(s,x_{\mathcal{F}+j})\right)}^T\hfill \\ \hfill & \left(p(s,x_i)-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}p(s,x_{\mathcal{F}+j})\right)]\le \eta {\varphi }_i[{\alpha }_{2i}^T{S}_2^2{\alpha }_{2i}+k^2{\alpha }_{2i}^T{\alpha }_{2i}]\hfill \end{array}$$

Due to

$${\alpha }_2^T[\eta \mathsf{\Phi }\otimes (S_2^2+K^2{I}_n)]{\alpha }_2=\sum _{i=1}^{\mathcal{F}}\eta {\varphi }_i[{\alpha }_{2i}^T{S}_2^2{\alpha }_{2i}+k^2{\alpha }_{2i}^T{\alpha }_{2i}]$$

we can derive that

$$2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)[P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})]\le {\alpha }_2^T[\eta \mathsf{\Phi }\otimes (S^2+k^2{I}_n)]{\alpha }_2$$

(31)

Also, we have

$$2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)[P(s,X_{\mathcal{F}})+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P(s,X_{\mathcal{B}})]=\sum _{i=1}^{\mathcal{F}}2{\varphi }_i{\alpha }_{2i}^T{S}_2[p(s,x_i)-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}p(s,x_{\mathcal{F}+j})],$$

Again, by using Assumption 1 and Lemma 2, we have

$$\begin{array}{cc}\hfill & 2\eta {\varphi }_i{\alpha }_{2i}^T{S}_2[p_1(s,x_i(s-e_2))-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}{p}_1(s,x_{\mathcal{F}+j}(s-e_2))]\hfill \\ \hfill & \le \eta {\varphi }_i[{\alpha }_{2i}^T{S}_2^2{\alpha }_{2i}+{\left(p_1(s,x_i(s-e_2))-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}{p}_1(s,x_{\mathcal{F}+j}(s-e_2))\right)}^T\times \hfill \\ \hfill & \left(p_1(s,x_i(s-e_2))-\sum _{j=1}^{\mathcal{N}}-\mathcal{F}{\overline{\ell }}_{ij}{p}_1(s,x_{\mathcal{F}+j}(s-e_2))\right)\hfill \\ \hfill & \le \eta {\varphi }_i[{\alpha }_{2i}^T{S}_2^2{\alpha }_{2i}+{\ell }^2{\alpha }_{2i}{(s-e_2)}^T{\alpha }_{2i}(s-e_2)]\hfill \end{array}$$

(32)

Due to

$${\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2^2){\alpha }_2+{\alpha }_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{I}_n){\alpha }_2(s-e_2)=\sum _{i=1}^{\mathcal{F}}\eta {\varphi }_i[{\alpha }_{2i}^T{S}_2^2{\alpha }_{2i}+{\ell }^2{\alpha }_{2i}^T(s-e_2){\alpha }_{2i}(s-e_2)]$$

we can derive that

$$2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)[P_1(s,X_{\mathcal{F}}(s-e_2))+({\overline{L}}_{\mathcal{F}}^{-1}{\overline{L}}_{\mathcal{B}}\otimes I_n)P_1(s,X_{\mathcal{B}}(s-e_2))]\le {\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2^2){\alpha }_2+{\alpha }_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{I}_n){\alpha }_2(s-e_2)$$

(33)

By (30), (31), and (33),

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_2\left({\alpha }_2\left(s\right)\right)\le {\alpha }_2^T\left[\eta \mathsf{\Phi }\otimes (S_{2}D+D^T{S}_2+2S_2^2+k^2{I}_n)\right]{\alpha }_2+{\int }_{-e_1}^{0}2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_{2}E){\alpha }_2(s+\varphi )d\varphi \hfill \\ \hfill & -2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_1(s-e_2)-2{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)({\overline{L}}_{\mathcal{F}}\otimes \beta FT){\alpha }_2(s-e_2)+\hfill \\ \hfill & {\alpha }_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{I}_n){\alpha }_2(s-e_2)+4{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2)\Delta v_i\left(s\right)\hfill \end{array}$$

(34)

To simplify (29) and (34), we define

${\mathcal{Q}}_1(.)=(I_{\mathcal{F}}\otimes S_2){\alpha }_1(.)$

${\mathcal{Q}}_2(.)=(I_{\mathcal{F}}\otimes S_2){\alpha }_2(.)$

$\mathcal{Q}(.)=(I_{2\mathcal{F}}\otimes S_2)\alpha (.)$

By applying the above condition on Equation (29), we have

$$D^{\gamma }{\mathcal{V}}_1\left({\alpha }_1\left(s\right)\right)\le {\mathcal{Q}}_1\left(\mathsf{\Phi }\otimes S_2^{-1}{\mathsf{\Lambda }}_1{S}_2^{-1}\right){\mathcal{Q}}_1+{\mathcal{Q}}_1^T(s-e_2)(\mathsf{\Phi }\otimes S_2^{-1}{\ell }^2{I}_n{S}_2^{-1}){\mathcal{Q}}_1(s-e_2)$$

(35)

and also, from (34), we have

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_2\le {\mathcal{Q}}_2^T\left[\eta \mathsf{\Phi }\otimes (D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2})\right]{\mathcal{Q}}_2\hfill \\ \hfill & {\int }_{-e_1}^0{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes E{S}_2^{-1}){\mathcal{Q}}_2(s+\varphi )d\varphi -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta FT{S}_2^{-1}){\mathcal{Q}}_2(s-e_2)\hfill \\ \hfill & -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta FT{S}_2^{-1}){\mathcal{Q}}_1(s-e_2)+{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{S}_2^{-2}){\mathcal{Q}}_2(s-e_2)+4{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes I_n)\Delta v_i\left(s\right)\hfill \end{array}$$

(36)

Substituting the gain matrix $T=F^T{S}_2$ into (36), we have

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_2\le {\mathcal{Q}}_2^T\left[\eta \mathsf{\Phi }\otimes (D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2})\right]{\mathcal{Q}}_2\hfill \\ \hfill & {\int }_{-e_1}^0{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes E{S}_2^{-1}){\mathcal{Q}}_2(s+\varphi )d\varphi -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T{S}_2^{-1}){\mathcal{Q}}_2(s-e_2)\hfill \\ \hfill & -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T{S}_2^{-1}){\mathcal{Q}}_1(s-e_2)+{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{S}_2^{-2}){\mathcal{Q}}_2(s-e_2)+4{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes I_n)\Delta v_i\left(s\right)\hfill \end{array}$$

(37)

Now, from (35) and (37), we have

$$\begin{array}{cc}\hfill & D^{\gamma }\mathcal{V}\left(\alpha \left(s\right)\right)=D^{\gamma }{\mathcal{V}}_1\left({\alpha }_1\left(s\right)\right)+D^{\gamma }{\mathcal{V}}_2\left({\alpha }_2\left(s\right)\right)\hfill \\ \hfill & \le {\mathcal{Q}}_1\left(\mathsf{\Phi }\otimes S_2^{-1}{\mathsf{\Lambda }}_1{S}_2^{-1}\right){\mathcal{Q}}_1+{\mathcal{Q}}_1^T(s-e_2)(\mathsf{\Phi }\otimes {\ell }^2{S}_2^{-2}{\mathcal{Q}}_1(s-e_2)\hfill \\ \hfill & +{\mathcal{Q}}_2^T\left[\eta \mathsf{\Phi }\otimes (D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2})\right]{\mathcal{Q}}_2\hfill \\ \hfill & {\int }_{-e_1}^0{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes E{S}_2^{-1}){\mathcal{Q}}_2(s+\varphi )d\varphi -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_2(s-e_2)\hfill \\ \hfill & -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_1(s-e_2)+{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{S}_2^{-2}){\mathcal{Q}}_2(s-e_2)+4{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes I_n)\Delta v_i\left(s\right)\hfill \end{array}$$

(38)

For some $\kappa >0$, whenever

$${\mathcal{V}}_1(s+h)\le \kappa {\mathcal{V}}_1\left(s\right),{\mathcal{V}}_2(s+h)\le \kappa {\mathcal{V}}_2\left(s\right),h\in [-e_3,0],$$

(39)

where $e_3=max\{e_1,e_2\}$, then we can derive

$$\begin{array}{cc}\hfill & D^{\gamma }\mathcal{V}\left(\alpha \left(s\right)\right)\le {\mathcal{Q}}_1\left(\mathsf{\Phi }\otimes S_2^{-1}{\mathsf{\Lambda }}_1{S}_2^{-1}\right){\mathcal{Q}}_1+{\mathcal{Q}}_1^T(s-e_2)(\mathsf{\Phi }\otimes {\ell }^2{S}_2^{-1}{S}_2^{-1}){\mathcal{Q}}_1(s-e_2)\hfill \\ \hfill & +{\mathcal{Q}}_2^T\left[\eta \mathsf{\Phi }\otimes (D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2})\right]{\mathcal{Q}}_2\hfill \\ \hfill & {\int }_{-e_1}^0\left[2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes E{S}_2^{-1}){\mathcal{Q}}_2(s+\varphi )+\kappa {\mathcal{Q}}_2^T(\eta \varphi \otimes S_2^{-1}){\mathcal{Q}}_2-{\mathcal{Q}}_2^T(s+\varphi )(\eta \mathsf{\Phi }\otimes S_2^{-1}){\mathcal{Q}}_2(s+\varphi )\right]d\varphi \hfill \\ \hfill & -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_2(s-e_2)+\kappa {\mathcal{Q}}_1^T(\varphi \otimes S_2^{-1}{S}_1{S}_2^{-1}){\mathcal{Q}}_1+\kappa {\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes S_2^{-1}){\mathcal{Q}}_2\hfill \\ \hfill & -{\mathcal{Q}}_1^T(s-e_2)(\varphi \otimes S_2^{-1}{S}_1{S}_2^{-1}){\mathcal{Q}}_1(s-e_2)-{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes S_2^{-1}){\mathcal{Q}}_2(s-e_2)\hfill \\ \hfill & -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_1(s-e_2)+{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{S}_2^{-2}){\mathcal{Q}}_2(s-e_2)+4{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes I_n)\Delta v_i\left(s\right)\hfill \end{array}$$

(40)

Let $\kappa \to 1^{+}$; one has for some $b>1$

$$\begin{array}{cc}\hfill & D^{\gamma }\mathcal{V}\left(\alpha \left(s\right)\right)\le {\mathcal{Q}}_1\left(\mathsf{\Phi }\otimes S_2^{-1}({\mathsf{\Lambda }}_1+S_1)S_2^{-1}\right){\mathcal{Q}}_1+{\mathcal{Q}}_1^T(s-e_2)(\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}){\mathcal{Q}}_1(s-e_2)\hfill \\ \hfill & +{\mathcal{Q}}_2^T\left[\eta \mathsf{\Phi }\otimes (D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2}+(1+b{e}_1)S_2^{-1}\right]{\mathcal{Q}}_2\hfill \\ \hfill & {\int }_{-e_1}^0\left[2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes E{S}_2^{-1}){\mathcal{Q}}_2(s+\varphi )+{\mathcal{Q}}_2^T(\eta \varphi \otimes (1-b)S_2^{-1}){\mathcal{Q}}_2-{\mathcal{Q}}_2^T(s+\varphi )(\eta \mathsf{\Phi }\otimes S_2^{-1}){\mathcal{Q}}_2(s+\varphi )\right]d\varphi \hfill \\ \hfill & -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_2(s-e_2)-{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2+S_2^{-1})){\mathcal{Q}}_2(s-e_2)\hfill \\ \hfill & -2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_1(s-e_2)+{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes {\ell }^2{S}_2^{-2}){\mathcal{Q}}_2(s-e_2)+4{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes I_n)\Delta v_i\left(s\right)\hfill \end{array}$$

(41)

Inequality (10) implies

$$\left(\begin{array}{cc}{\mathcal{Q}}_2^T& {\mathcal{Q}}_2^T(s+\varphi )\end{array}\right)\left(\begin{array}{cc}\eta \mathsf{\Phi }\otimes (1-b)S_2^{-1}& \eta \mathsf{\Phi }\otimes E{S}_2^{-1}\\ \eta \mathsf{\Phi }\otimes S_2^{-1}{E}^T& -\eta \mathsf{\Phi }\otimes S_2^{-1}\end{array}\right)\left(\begin{array}{c}{\mathcal{Q}}_2\\ {\mathcal{Q}}_2(s+\varphi )\end{array}\right)<0$$

Then

$$\begin{array}{cc}\hfill & {\int }_{-e_1}^0\left[2{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes E{S}_2^{-1}){\mathcal{Q}}_2(s+\varphi )+{\mathcal{Q}}_2^T(\eta \varphi \otimes (1-b)S_2^{-1}){\mathcal{Q}}_2-{\mathcal{Q}}_2^T(s+\varphi )(\eta \mathsf{\Phi }\otimes S_2^{-1}){\mathcal{Q}}_2(s+\varphi )\right]d\varphi \hfill \\ \hfill & <0\hfill \end{array}$$

(42)

Substituting (42) into (41), we have

$$\begin{array}{cc}\hfill & D^{\gamma }\mathcal{V}\left(\alpha \left(s\right)\right)\le {\mathcal{Q}}_1\left(\mathsf{\Phi }\otimes S_2^{-1}({\mathsf{\Lambda }}_1+S_1)S_2^{-1}\right){\mathcal{Q}}_1+{\mathcal{Q}}_1^T(s-e_2)(\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}){\mathcal{Q}}_1(s-e_2)\hfill \\ \hfill & +{\mathcal{Q}}_2^T\left[\eta \mathsf{\Phi }\otimes (D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2}+(1+b{e}_1)S_2^{-1}\right]{\mathcal{Q}}_2\hfill \\ \hfill & -{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_2(s-e_2)-{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_2\hfill \\ \hfill & -{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})){\mathcal{Q}}_2(s-e_2)-{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_1(s-e_2)\hfill \\ \hfill & -{\mathcal{Q}}_1{(s-e_2)}^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T){\mathcal{Q}}_2+4{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes I_n)\Delta v_i\left(s\right)\hfill \end{array}$$

(43)

Now, let $\mathcal{J}=({\mathcal{Q}}^T,{\mathcal{Q}}^T(s-e_2)),$ where $\mathcal{Q}(.)=(I_{2\mathcal{F}}\otimes S_2)\alpha (.)$. Also let ${\mathsf{\Lambda }}_2=D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2}+(1+b{e}_1)S_2^{-1}$, $\nu ={\mathcal{Q}}_2$, and $q=4\Delta v_i\left(s\right)$. We can reduce the above equation in the following matrix:

$$\begin{array}{cc}\hfill & {\mathcal{J}}^T\left(\begin{array}{cccc}\mathsf{\Phi }\otimes S_2^{-1}({\mathsf{\Lambda }}_1+S_1)S_2^{-1}& 0& 0& 0\\ 0& \eta \mathsf{\Phi }\otimes ({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n)& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T\\ 0& -\eta {\overline{L}}_{\mathcal{F}}\mathsf{\Phi }\otimes \beta F{F}^T& -\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}& 0\\ 0& -\eta {\overline{L}}_{\mathcal{F}}\mathsf{\Phi }\otimes \beta F{F}^T& 0& -\eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})\end{array}\right)\mathcal{J}\hfill \\ \hfill & \triangleq {\mathcal{J}}^T{\lambda }_3\mathcal{J}\hfill \\ \hfill & =\left(\begin{array}{cc}{\alpha }^T\left(s\right)& {\alpha }^T(s-e_2)\end{array}\right)(I_{4\mathcal{F}}\otimes S_2){\mathsf{\Lambda }}_3(I_{4\mathcal{F}}\otimes S_2)\left(\begin{array}{c}\alpha \left(s\right)\\ \alpha (s-e_2)\end{array}\right)\hfill \end{array}$$

Let

$${\mathcal{U}}_1\triangleq \left(\begin{array}{ccc}\eta \mathsf{\Phi }\otimes ({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n)& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T\\ -\eta {\overline{L}}_{\mathcal{F}}\mathsf{\Phi }\otimes \beta F{F}^T& -\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}& 0\\ -\eta {\overline{L}}_{\mathcal{F}}\mathsf{\Phi }\otimes \beta F{F}^T& 0& -\eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})\end{array}\right)$$

(44)

Then there is an invertible matrix

$$\mathcal{W}=\left(\begin{array}{ccc}0& I_{\mathcal{F}}\otimes I_n& 0\\ I_{\mathcal{F}}\otimes I_n& 0& 0\\ 0& 0& I_{\mathcal{F}}\otimes I_n\end{array}\right)$$

that is used as

$${\mathcal{W}}^{-1}{\mathcal{U}}_1\mathcal{W}=\left(\begin{array}{ccc}-\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}& -\eta {\overline{L}}_{\mathcal{F}}\mathsf{\Phi }\otimes \beta F{F}^T& 0\\ -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T& \eta \mathsf{\Phi }\otimes ({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n)& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T\\ 0& -\eta {\overline{L}}_{\mathcal{F}}\mathsf{\Phi }\otimes \beta F{F}^T& -\eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})\end{array}\right)\triangleq {\mathcal{U}}_2.$$

(45)

From Equations (44) and (45), one can easily say that matrices ${\mathcal{U}}_1\mathrm{and}{\mathcal{U}}_2$ are similar matrices and have the same eigenvalues. Now, we divide matrix ${\mathcal{U}}_2$ into matrix

$$\left(\begin{array}{cc}-\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}& -\eta {\mathcal{C}}_1^T\\ -\eta {\mathcal{C}}_1& -\eta {\mathcal{C}}_2\end{array}\right)$$

where matrices ${\mathcal{C}}_1\mathrm{and}{\mathcal{C}}_2$ are defined as

$${\mathcal{C}}_1=\left(\begin{array}{c}\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T\\ 0\end{array}\right),{\mathcal{C}}_2\left(\begin{array}{cc}-\mathsf{\Phi }\otimes ({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n)& \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes \beta F{F}^T\\ {\overline{L}}_{\mathcal{F}}\mathsf{\Phi }\otimes \beta F{F}^T& \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})\end{array}\right)$$

Next, we have to prove that ${\mathcal{U}}_2$ is less than zero. Since by Lemma 4, ${\overline{L}}_{\mathcal{F}}$ is invertible,

$$\begin{array}{cc}\hfill & \Rightarrow \mathrm{for}\mathrm{any}\mathrm{non}-\mathrm{zero}\mathrm{vector}\mathrm{x}{\left({\overline{L}}_{\mathcal{F}}x\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}x=x^T\left({\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\right)x>0\hfill \\ \hfill & \Rightarrow {\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}>0\hfill \end{array}$$

Suppose that $0<\beta \le \sqrt{{\displaystyle \frac{\omega {\varphi }_{min}}{{\lambda }_2}}},$ where ${\lambda }_2={\lambda }_{max}\left\{{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^T\right\},{\varphi }_{min}=mi{n}_{1\le j\le \mathcal{F}}\left\{{\varphi }_j\right\},$

$$\begin{array}{cc}\hfill & -\mathsf{\Phi }\otimes {\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n-{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes {\beta }^{2}F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T\hfill \\ \hfill & \ge -\mathsf{\Phi }\otimes \left({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n+{\displaystyle \frac{{\lambda }_2}{{\varphi }_{min}}}{\beta }^{2}F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T\right)\hfill \\ \hfill & \ge -\mathsf{\Phi }\otimes \left({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n+\omega F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T\right)\hfill \end{array}$$

From (11) ${\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n+\omega F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T<0$; hence, $-\mathsf{\Phi }\otimes {\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n-{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}\otimes {\beta }^{2}F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T>0$, which implies that ${\mathcal{C}}_2>0$.

Now, by choosing a small scalar $\eta >0$ such that

$$-\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}+\eta {\mathcal{C}}_1{\mathcal{C}}_2^{-1}{\mathcal{C}}_1<0$$

$\Rightarrow {\mathcal{U}}_2<0$; hence, we proved that ${\mathcal{U}}_1<0$. As a result ${\mathsf{\Lambda }}_1+S_1<0$ and ${\mathcal{U}}_1<0$ from (8)–(11). Hence,

$${\mathsf{\Lambda }}_3=\left(\begin{array}{cc}\mathsf{\Phi }\otimes S_2^{-1}({\mathsf{\Lambda }}_1+S_1)S_2^{-1}& 0\\ 0& {\mathcal{U}}_1\end{array}\right)<0$$

Then, from (43), $D^{\gamma }\mathcal{V}\left(\alpha \left(s\right)\right)<0$. Hence, there exists a scalar $\kappa >0$ such that

$$D^{\gamma }\mathcal{V}\left(\alpha \left(s\right)\right)\le -{\kappa ||\alpha \left(s\right)||}^2.$$

(46)

Based on Lemma 3, it can be concluded that ${lim}_{s\to \infty }\alpha =0$, indicating that the zero solutions of systems (18) and (21) are asymptotically stable. As a result, observer-based bipartite containment control of the FMAS described by Equations (2) and (3) with the delayed controller (7) is achieved. □

4. Observation-Based Analysis of Bipartite Containment Control for Switching Signed Digraph

Firstly, some important preliminaries on switching signed digraphs are introduced. Let $\sigma :[s_0,\infty )\to \mathcal{T}=\{1,2,\dots ,\mathcal{M}\}$ be a piecewise switching signal, and let ${\mathcal{G}}^{\sigma \left(s\right)}=(\mathit{N},{\mathit{E}}^{\sigma \left(s\right)})$ show a switching signed digraph. When $\sigma \left(s\right)=t\in \mathcal{T}$ for $s\in [s_g,s_{g+1})$, where $s_g$ is a switching instant, the t-th topology is activated. Suppose that there is an arbitrarily small $\Xi$ such that $s_{g+1}-s_g\ge \Xi$ for all $s_g$ and $qg\in {\mathbb{Z}}^{+}\cup \left\{0\right\}$ to avoid Zeno behavior. The matrices ${\mathtt{F}}^{\sigma \left(s\right)}={\left[f_{ij}^{\sigma \left(s\right)}\right]}_{\mathcal{N}\times \mathcal{N}}$ and $L^{\sigma \left(s\right)}={\left[{\ell }_{ij}^{\sigma \left(s\right)}\right]}_{\mathcal{N}\times \mathcal{N}}$ are called the adjacency matrices and Laplacian matrices of ${\mathcal{G}}^{\sigma \left(s\right)}$, respectively. For any $\sigma \left(s\right)=w\in \mathcal{T}$, it holds that ${\mathcal{G}}^w=\mathcal{G}$. Next, we propose the following state observer and observation-based switching controller for each $i\in \mathbb{F}$.

$$\begin{array}{cc}\hfill D^{\gamma }{\widehat{y}}_i\left(s\right)& =D{\widehat{y}}_i\left(s\right)+p(s,{\widehat{y}}_i\left(s\right))+p_1(s,{\widehat{y}}_i(s-e_2))+E{\int }_{-e_1}^0{\widehat{y}}_i(s+\varphi )d\varphi +F{v}_i\left(s\right)+\Delta v_i\left(s\right)\hfill \\ & +\mu A\left(\sum _{j=1}^{\mathcal{F}}|f_{ij}^{\sigma \left(s\right)}|(z_i-sgn\left(f_{ij}^{\sigma \left(s\right)}\right)z_j)-\sum _{j=1}^{\mathcal{F}}|f_{ij}^{\sigma \left(s\right)}|({\widehat{z}}_i-sgn\left(f_{ij}^{\sigma \left(s\right)}\right){\widehat{z}}_j)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}^{\sigma \left(s\right)}|(z_i-{\widehat{z}}_i))\right),\hfill \\ \hfill {\widehat{z}}_i& =G{\widehat{y}}_i\left(s\right)i\in \mathbb{F}\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill v_i& =-\beta T\left(\sum _{j=1}^{\mathcal{N}}|f_{ij}^{\sigma \left(s\right)}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}^{\sigma \left(s\right)}\right){\widehat{y}}_j(s-e_2)\right)+\sum _{j=\mathcal{F}+1}^{\mathcal{N}}|f_{ij}^{\sigma \left(s\right)}|\left({\widehat{y}}_i(s-e_2)-\mathrm{sgn}\left(f_{ij}^{\sigma \left(s\right)}\right)y_j(s-e_2)\right)\right)\hfill \\ & +\Delta v_i\left(s\right)\hfill \end{array}$$

(48)

Assumption 5. 

The signed digraph ${\mathit{G}}^{\sigma \left(s\right)}$ exhibits structural balance for any $\sigma \left(s\right)\in \mathcal{T}$.

Assumption 6. 

Every follower in the signed digraph ${\mathcal{G}}^{\sigma \left(s\right)}$ has a directed link from at least one leader for any $\sigma \left(s\right)\in \mathcal{T}$.

Remark 6. 

Since $v_i=0$ for all $i\in \mathbb{B}$, the Laplacian matrix L of the signed digraph ${\mathcal{G}}^{\sigma \left(s\right)}$ for some $\sigma \left(s\right)\in \mathcal{T}$ is defined as:

$$L^{\sigma \left(s\right)}=\left(\begin{array}{cc}{L}_{\mathcal{F}}^{\sigma \left(s\right)}& L_{\mathcal{B}}^{\sigma \left(s\right)}\\ 0_{(\mathcal{N}-\mathcal{F})\times \mathcal{F}}& 0_{(\mathcal{N}-\mathcal{F})\times (\mathcal{N}-\mathcal{F})}\end{array}\right)$$

Here, $L_{\mathcal{F}}^{\sigma \left(s\right)}\in {\mathbb{R}}^{\mathcal{F}\times \mathcal{F}}$ and $L_{\mathcal{B}}^{\sigma \left(s\right)}\in {\mathbb{R}}^{\mathcal{F}\times (\mathcal{N}-\mathcal{F})}$. If Assumption 5 is met, a diagonal matrix exists such that $\psi =\mathit{diag}({\psi }_1,{\psi }_2,\dots ,{\psi }_{\mathcal{F}},{\psi }_{\mathcal{F}+1},\dots ,{\psi }_{\mathcal{N}})$ for which the elements are defined as

$$\left\{\begin{array}{c}{\psi }_i=1ifi\in {\mathbb{F}}_1\cup \mathbb{B}\hfill \\ {\psi }_i=-1ifi\in {\mathbb{F}}_2\hfill \end{array}\right.$$

such that

$${\overline{L}}^{\sigma \left(s\right)}=\psi L^{\sigma \left(s\right)}\psi =\left(\begin{array}{cc}{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}& {\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}\\ 0_{(\mathcal{N}-\mathcal{F})\times \mathcal{F}}& 0_{(\mathcal{N}-\mathcal{F})\times (\mathcal{N}-\mathcal{F})}\end{array}\right)$$

where ${\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}={\left[{\overline{\ell }}_{ij}^{\sigma \left(s\right)}\right]}_{\mathcal{F}\times \mathcal{F}}$ if $i\ne j$ then ${\left[{\overline{\ell }}_{ij}^{\sigma \left(s\right)}\right]}_{\mathcal{F}\times \mathcal{F}}$ is equal to $-|f_{ij}^{\sigma \left(s\right)}|$ otherwise it is equal to ${\sum }_{j=1}^{\mathcal{N}}|f_{ij}^{\sigma \left(s\right)}|$, also ${\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}={\left[{\overline{\ell }}_{ij}^{\sigma \left(s\right)}\right]}_{\mathcal{F}\times (\mathcal{N}-\mathcal{F})}$${\overline{\ell }}_{ij}^{\sigma \left(s\right)}=-|f_{ij}^{\sigma \left(s\right)}|\le 0$ for $i=1,2,3,...\mathcal{F}$, $j=\mathcal{F}+1,...,\mathcal{N}$

Lemma 5. 

([1]). If Assumption 6 holds, then for any $\sigma \left(s\right)\in \mathcal{T}$${\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}$, is a non-singular M-matrix, and every element of -${\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^{-1}{\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}$ is nonnegative, with each row summing to 1.

Theorem 2. 

Observer-based bipartite containment control of FMAS (2) and (3) with controller (48) under Assumptions 1, 2, 5, and 6 is successfully realized if there exists a diagonal matrix $\mathsf{\Phi }=\mathit{diag}({\varphi }_1,\dots ,{\varphi }_{\mathcal{F}})>0$ such that $\mathsf{\Phi }{L}_{\mathcal{F}}^{\sigma \left(s\right)}+{\left(L_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }>0$ along with positive definite matrices $S_1$ and $S_2$ that satisfy the following set of inequalities:

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}(1-r)S_1^{-1}& E{S}_1^{-1}\\ S_1^{-1}{E}^T& -S_1^{-1}\end{array}\right)<0\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}{S}_{1}D+D^T{S}_1+S_1^2+(k^2+{\ell }^2)I_n+(1+r{e}_1)S_1-\rho G^{T}G& S_1\\ S_1& -I_n\end{array}\right)<0\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}(1-b)S_2^{-1}& E{S}_2^{-1}\\ S_2^{-1}{E}^T& -S_2^{-1}\end{array}\right)<0\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & \left(\begin{array}{cc}D{S}_2^{-1}+S_2^{-1}{D}^T+(1+b{e}_1)S_2^{-1}+(2+{\displaystyle \frac{q}{\nu }})I_n+\omega F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T& k{S}_2^{-1}\\ k{S}_2^{-1}& -I_n\end{array}\right)<0\hfill \end{array}$$

(52)

Given the parameters $\rho ,\omega >0$ and $r,b>1$, the observer and controller are designed with the following specifications:

• $A=S_1^{-1}{G}^T$
• $\mu \ge {\displaystyle \frac{\rho {\varphi }_{max}}{{\lambda }_1}}$, where:

  -

  ${\lambda }_3=mi{n}_{\sigma \left(s\right)\in \mathcal{T}}\left\{{\lambda }_{min}\left\{\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}+{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }\right\}\right\}$

  -

  ${\varphi }_{max}={max}_{1\le j\le \mathcal{F}}\left\{{\varphi }_j\right\}$

• $T=F^T{S}_2$
• $0<\beta \le \sqrt{{\displaystyle \frac{\omega {\varphi }_{min}}{{\lambda }_2}}}$, where:

  -

  ${\lambda }_4=ma{x}_{\sigma \left(s\right)\in \mathcal{T}}\left\{{\lambda }_{max}\left\{{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right\}\right\}$

  -

  ${\varphi }_{min}={min}_{1\le j\le \mathcal{F}}\left\{{\varphi }_j\right\}$

Proof. 

Suppose that ${\alpha }_1(.)={\widehat{X}}_{\mathcal{F}}(.)-X_{\mathcal{F}}(.)$ denotes the observation error and

${\alpha }_2(.)=X_{\mathcal{F}}(.)-\left(-{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^{-1}{\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}\otimes I_n\right)X_{\mathcal{B}}(.)$ is the bipartite containment control error; then under controller (48),

$$\begin{array}{cc}\hfill & D^{\gamma }{\alpha }_1=(I_{\mathcal{F}}\otimes D){\alpha }_1+P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})+P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2))-P_1(s,X_{\mathcal{F}}(s-e_2))\hfill \\ \hfill & {\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E){\alpha }_1(s+\varphi )d\varphi -({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \mu AG){\alpha }_1.\hfill \end{array}$$

(53)

and

$$\begin{array}{cc}\hfill & D^{\gamma }{\alpha }_2=(I_{\mathcal{F}}\otimes D){\alpha }_2++P(s,X_{\mathcal{F}})+\left({\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^{-1}{\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}\otimes I_n\right)P(s,X_{\mathcal{B}})+P_1(s,X_{\mathcal{F}}(s-e_2))\hfill \\ \hfill & +\left({\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^{-1}{\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}\otimes I_n\right)P_1(s,X_{\mathcal{B}}(s-e_2))+{\int }_{-e_1}^0(I_{\mathcal{F}}\otimes E){\alpha }_2(s+\varphi )d\varphi \hfill \\ & -({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta FT){\alpha }_1(s-e_2)-({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta FT){\alpha }_2(s-e_2)+2\Delta v_i\left(s\right)\hfill \end{array}$$

(54)

Let $\alpha (.)={[{\alpha }_1^T(.),{\alpha }_2^T(.)]}^T$ represent the error system, and

$$D^{\gamma }\alpha ={\overline{R}}_1\alpha +\overline{P}(s,\alpha )+{\overline{P}}_1(s,\alpha (s-e_2))+{\overline{R}}_2\alpha (s-e_2)+{\int }_{-e_1}^0{\overline{R}}_3\alpha (s+\varphi )d\varphi +2\Delta v_i\left(s\right),$$

(55)

where ${\overline{R}}_1=\left(\begin{array}{cc}(I_F\otimes D)-({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \mu AG)& 0\\ 0& I_F\otimes D\end{array}\right);{\overline{R}}_2=\left(\begin{array}{cc}0& 0\\ -{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta FT& -{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta FT\end{array}\right);$ ${\overline{R}}_3=\left(\begin{array}{cc}{I}_F\otimes E& 0\\ 0& I_F\otimes E\end{array}\right);$ $\overline{P}(s,\alpha )=\left[P(s,{\widehat{X}}_{\mathcal{F}})-P(s,X_{\mathcal{F}})\right]\left[P(s,X_{\mathcal{F}})+\left({\left({\overline{L}}_{\mathcal{F}}^{\sigma \left)s\right)}\right)}^{-1}{\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}\otimes I_n\right)P(s,X_{\mathcal{B}})\right],$

$$\begin{array}{cc}\hfill & {\overline{P}}_1(s,\alpha (s-e_2)=\left[P_1(s,{\widehat{X}}_{\mathcal{F}}(s-e_2)-P_1(s,X_{\mathcal{F}}(s-e_2)\right]\times \hfill \\ \hfill & \left[P_1(s,X_{\mathcal{F}}(s-e_2)+\left({\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^{-1}{\overline{L}}_{\mathcal{B}}^{\sigma \left(s\right)}\otimes I_n\right)P_1(s,X_{\mathcal{B}}(s-e_2)\right].\hfill \end{array}$$

Let us delve into the asymptotic stability analysis of the error system in Equation (55). Given that ${\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}$ for any $\sigma \left(s\right)\in \mathcal{T}$ is a non-singular M-matrix according to Lemma 4, then there exists a positive definite diagonal matrix $\mathsf{\Phi }=\mathrm{diag}({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_{\mathcal{F}})$ such that $\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}+{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }>0$. In this context, the following Lyapunov function is proposed for the stability analysis:

$$\begin{array}{cc}\hfill & \mathcal{V}\left(s\right)={\mathcal{V}}_1\left(s\right)+{\mathcal{V}}_2\left(s\right)\hfill \\ \hfill & ={\alpha }_1^T(\mathsf{\Phi }\otimes S_1){\alpha }_1+{\alpha }_2^T(\eta \mathsf{\Phi }\otimes S_2){\alpha }_2\hfill \end{array}$$

(56)

where $\eta >0$ is very small scalar.

By using Theorem 1,

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_1\le {\alpha }_1^T\left[\mathsf{\Phi }\otimes \left(S_{1}D+D^T{S}_1+2S_2^{2_1}+k^2{I}_n)+\left(S_{1}E{S}_1^{-1}{E}^{T{S}_1}+S_1\right)e_1\right)-\mu (\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}+{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi })\otimes G^{T}G\right]{\alpha }_1\hfill \\ & +{\alpha }_1^{T(s-e_2)}\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)\hfill \end{array}$$

Denote ${\lambda }_3={\lambda }_{min}\{\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}+{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }\},{\varphi }_{max}=ma{x}_{1\le \mathcal{F}}\left\{{\varphi }_j\right\}$. The following inequality with $\mu \ge {\displaystyle \frac{\rho {\varphi }_{max}}{{\lambda }_3}}$ and $\rho >0$ holds:

$$\begin{array}{cc}\hfill & D^{\gamma }{\mathcal{V}}_1\le {\alpha }_1^T\left[\mathsf{\Phi }\otimes \left(S_{1}D+D^T{S}_1+2S_2^{2_1}+k^{2I_n})+\left(S_{1}E{S}_1^{-1}{E}^{T{S}_1}+S_1\right)e_1\right)-\rho G^{T}G\right]{\alpha }_1\hfill \\ \hfill & +{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)\hfill \\ \hfill & ={\alpha }_1^T(\mathsf{\Phi }\otimes {\mathsf{\Lambda }}_1){\alpha }_1+{\alpha }_1^T(s-e_2)\left[\mathsf{\Phi }\otimes {\ell }^2{I}_n\right]{\alpha }_1(s-e_2)\hfill \end{array}$$

(57)

Now, by using Theorem 1 and Equation (48),

$$\begin{array}{cc}\hfill & D^{\gamma }\mathcal{V}\left(s\right)\le {\mathcal{Q}}_1\left(\mathsf{\Phi }\otimes S_2^{-1}({\mathsf{\Lambda }}_1+S_1)S_2^{-1}\right){\mathcal{Q}}_1+{\mathcal{Q}}_1^T(s-e_2)(\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}){\mathcal{Q}}_1(s-e_2)\hfill \\ \hfill & +{\mathcal{Q}}_2^T\left[\eta \mathsf{\Phi }\otimes (D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2}+(1+b{e}_1)S_2^{-1}\right]{\mathcal{Q}}_2\hfill \\ \hfill & -{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T){\mathcal{Q}}_2(s-e_2)-{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T){\mathcal{Q}}_2\hfill \\ \hfill & -{\mathcal{Q}}_2^T(s-e_2)(\eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})){\mathcal{Q}}_2(s-e_2)-{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T){\mathcal{Q}}_1(s-e_2)\hfill \\ \hfill & -{\mathcal{Q}}_1{(s-e_2)}^T(\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T){\mathcal{Q}}_2+4{\mathcal{Q}}_2^T(\eta \mathsf{\Phi }\otimes I_n)\Delta v_i\left(s\right)\hfill \\ \hfill & ={\mathcal{J}}^T\left(\begin{array}{cc}\mathsf{\Phi }\otimes S_2^{-1}({\mathsf{\Lambda }}_1+S_1)S_2^{-1}& 0\\ 0& {\mathcal{U}}_3\end{array}\right)\mathcal{J}\hfill \end{array}$$

(58)

where

$$\begin{array}{cc}\hfill & {\mathcal{U}}_3\triangleq \left(\begin{array}{ccc}\eta \mathsf{\Phi }\otimes ({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n)& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T\\ -\eta {\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\mathsf{\Phi }\otimes \beta F{F}^T& \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}& 0\\ -\eta {\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\mathsf{\Phi }\otimes \beta F{F}^T& 0& \eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})\end{array}\right)\hfill \\ \hfill & {\mathsf{\Lambda }}_2=D{S}_2^{-1}+S_2^{-1}{D}^T+2I_n+k^2{S}_2^{-2}+(1+b{e}_1)S_2^{-1}andq=4\Delta v_i\left(s\right)\hfill \end{array}$$

Then there is an invertible matrix

$$\mathcal{W}=\left(\begin{array}{ccc}0& I_{\mathcal{F}}\otimes I_n& 0\\ I_{\mathcal{F}}\otimes I_n& 0& 0\\ 0& 0& I_{\mathcal{F}}\otimes I_n\end{array}\right)$$

which is used as

$${\mathcal{W}}^{-1}{\mathcal{U}}_3\mathcal{W}=\left(\begin{array}{ccc}\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}& -\eta {\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\mathsf{\Phi }\otimes \beta F{F}^T& 0\\ -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T& \eta \mathsf{\Phi }\otimes ({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n)& -\eta \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T\\ 0& -\eta {\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\mathsf{\Phi }\otimes \beta F{F}^T& \eta \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})\end{array}\right)\triangleq {\mathcal{U}}_4.$$

(59)

One can easily say that matrices ${\mathcal{U}}_3\mathrm{and}{\mathcal{U}}_4$ are similar matrices and have the same eigenvalues. Now, we divide matrix ${\mathcal{U}}_4$ into matrix

$$\left(\begin{array}{cc}\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}& -\eta {\mathcal{C}}_1^T\\ -\eta {\mathcal{C}}_3& -\eta {\mathcal{C}}_4\end{array}\right)$$

where matrices ${\mathcal{C}}_3\mathrm{and}{\mathcal{C}}_4$ are defined as

$${\mathcal{C}}_3=\left(\begin{array}{c}\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T\\ 0\end{array}\right),{\mathcal{C}}_4\left(\begin{array}{cc}\mathsf{\Phi }\otimes ({\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n)& \mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes \beta F{F}^T\\ {\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\mathsf{\Phi }\otimes \beta F{F}^T& \mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_2^{-1})\end{array}\right)$$

Next, we have to prove that ${\mathcal{U}}_4$ is less than zero. Since, by Lemma 5, ${\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}$ is invertible,

$$\begin{array}{cc}\hfill & \Rightarrow \mathrm{for}\mathrm{any}\mathrm{non}-\mathrm{zero}\mathrm{vector}\mathrm{x}{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}x\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}x=x^T\left({\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)x>0\hfill \\ \hfill & \Rightarrow {\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\hfill \end{array}$$

Suppose that $0<\beta \le \sqrt{{\displaystyle \frac{\omega {\varphi }_{min}}{{\lambda }_4}}},$ where ${\lambda }_4={\lambda }_{max}\left\{{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right\},{\varphi }_{min}=mi{n}_{1\le j\le \mathcal{F}}\left\{{\varphi }_j\right\},$

$$\begin{array}{cc}\hfill & -\mathsf{\Phi }\otimes {\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n-{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes {\beta }^{2}F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T\hfill \\ \hfill & \ge -\mathsf{\Phi }\otimes \left({\mathsf{\Lambda }}_4+{\displaystyle \frac{q}{\nu }}{I}_n+{\displaystyle \frac{{\lambda }_2}{{\varphi }_{min}}}{\beta }^{2}F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T\right)\hfill \\ \hfill & \ge -\mathsf{\Phi }\otimes \left({\mathsf{\Lambda }}_4+{\displaystyle \frac{q}{\nu }}{I}_n+\omega F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T\right)\hfill \end{array}$$

From (52) ${\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n+\omega F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T<0$; hence

$$\begin{array}{c}\hfill -\mathsf{\Phi }\otimes {\mathsf{\Lambda }}_2+{\displaystyle \frac{q}{\nu }}{I}_n-{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\otimes {\beta }^{2}F{F}^T{({\ell }^2{I}_n+S_2^{-1})}^{-1}{S}_{2}F{F}^T>0\end{array}$$

⇒ that ${\mathcal{C}}_4>0$. Now, by choosing a small scalar $\eta >0$ such that

$$-\mathsf{\Phi }\otimes S_2^{-1}({\ell }^2{I}_n+S_1)S_2^{-1}+\eta {\mathcal{C}}_3{\mathcal{C}}_4^{-1}{\mathcal{C}}_3<0$$

$\Rightarrow {\mathcal{U}}_4<0$; hence, we proved that ${\mathcal{U}}_3<0$. As a result, ${\mathsf{\Lambda }}_1+S_1<0$, and ${\mathcal{U}}_3<0$ from (49)–(52). Hence,

$$=\left(\begin{array}{cc}\mathsf{\Phi }\otimes S_2^{-1}({\mathsf{\Lambda }}_1+S_1)S_2^{-1}& 0\\ 0& {\mathcal{U}}_3\end{array}\right)<0$$

Then from (58), $D^{\gamma }\mathcal{V}\left(s\right)<0$. Hence, there exists a scalar $\kappa >0$ such that

$$D^{\gamma }\mathcal{V}\left(\alpha \left(s\right)\right)\le -{\kappa ||\alpha \left(s\right)||}^2.$$

(60)

Based on Lemma 3, it can be concluded that ${lim}_{s\to \infty }\alpha =0$, indicating that the zero solution of the system (55) is asymptotically stable. As a result, observer-based bipartite containment control of the FMAS described by Equations (2) and (3) with the delayed controller (48) is achieved. □

5. Numerical Illustrations

Two examples will be offered to illustrate the practical significance of the conclusions drawn.

Example 1. 

In the context of observation-based bipartite containment control having five followers and two leaders, the signed digraph illustrated in Figure 2 is noted for its structural balance and division into bi-partitioned subgroups ${\mathbb{F}}_1$ and ${\mathbb{F}}_2$, where ${\mathbb{F}}_1=\{0,1\}$ and ${\mathbb{F}}_2=\{2,3,4\}$. First, we define the matrices that are used in FMAS Equations (2) and (3) as follows:

$$D=\left(\begin{array}{cc}1.25& 0.05\\ 0.05& 1.6\end{array}\right),G=\left(\begin{array}{cc}0.5& 0.25\\ 0.1& 0.3\end{array}\right)$$

$$E=\left(\begin{array}{cc}0.2& 0.1\\ 0.05& 0.07\end{array}\right),F=\left(\begin{array}{c}1\\ 0.5\end{array}\right)$$

and

$$p_1(s,y_i(s-e_2))=y_i\left(s\right){\displaystyle \frac{4}{7}}{y}_i\left(s\right)sin(s-e_2),$$

$$p(s,y_i\left(s\right)=cos(y_i\left(s\right)$$

⇒ we can take the value of constant $k=0.5$ and $\ell =0.4$. We take $\Delta v_i\left(s\right)=30cos\left(0.3\right)$ and $\nu =0.3$. Now, by using Figure 2,

$F=$$\left(\begin{array}{ccccccc}0& 0.9& 0& 0& 0& 0.7& 0\\ 0& 0& 0& 0& -1& 0& 0.5\\ -0.5& 0& 0& 0.2& 0& -0.5& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0.5& 0& 0& 0& -0.6\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right)$ then $L=$ $\left(\begin{array}{ccccccc}1.6& -0.9& 0& 0& 0& -0.7& 0\\ 0& 1.5& 0& 0& 1& 0& -0.5\\ 0.5& 0& 1.2& -0.2& 0& 0.5& 0\\ 0& 0& 0& 1& -1& 0& 0\\ 0& 0& -0.5& 0& 1.1& 0& 0.6\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right)$

$$\psi =\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right)$$

Now, since $\overline{L}=\varphi L\varphi$,

$$\Rightarrow \overline{L}=\left(\begin{array}{ccccccc}1.6& -0.9& 0& 0& 0& -0.7& 0\\ 0& 1.5& 0& 0& -1& 0& -0.5\\ -0.5& 0& 1.2& -0.2& 0& -0.5& 0\\ 0& 0& 0& 1& -1& 0& 0\\ 0& 0& -0.5& 0& 1.1& 0& -0.6\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right)$$

from the above matrix, one can obtain

${\overline{L}}_{\mathcal{F}}=\left(\begin{array}{ccccc}1.6& -0.9& 0& 0& 0\\ 0& 1.5& 0& 0& -1\\ -0.5& 0& 1.2& -0.2& 0\\ 0& 0& 0& 1& -1\\ 0& 0& -0.5& 0& 1.1\end{array}\right)$ and ${\overline{L}}_{\mathcal{B}}=\left(\begin{array}{cc}-0.7& 0\\ 0& -0.5\\ -0.5& 0\\ 0& 0\\ 0& -0.6\end{array}\right)$. Suppose that $e_1=0.4$; then by solving inequalities (8)–(11), one can have $\rho =0.0213$ and $r=0.2148$.

$$S_1=\left(\begin{array}{cc}0.3208& 0.0021\\ 0.0021& 0.2145\end{array}\right)$$

$\omega =0.0023,b=0.2312$

$$S_2^{-1}=\left(\begin{array}{cc}0.4521& 0.0123\\ 0.0123& 0.4325\end{array}\right)\Rightarrow S_2=\left(\begin{array}{cc}2.2136& -0.0630\\ -0.0630& 2.3139\end{array}\right)$$

We can select $\mathsf{\Phi }=I_5$ for $\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}+{\overline{L}}_{\mathcal{F}}^T\mathsf{\Phi }>0$; thus, ${\varphi }_{min}={\varphi }_{max}=1$.

$${\lambda }_1=0.6414$$

and

$${\lambda }_2=5.2776$$

$\mu \ge {\displaystyle \frac{\rho {\varphi }_{max}}{{\lambda }_1}}=0.0332$ and $0<\beta \le \sqrt{{\displaystyle \frac{\omega {\varphi }_{min}}{{\lambda }_2}}}=0.02085$ Now If we take

$$\beta =0.02\mu =0.1\mathit{gain}\mathit{matrices}A=\left(\begin{array}{cc}1.5511& 0.3026\\ 1.1503& 1.3956\end{array}\right)\mathit{and}T=\left(\begin{array}{cc}2.1821& 1.0939\end{array}\right)$$

Now, if $\gamma =0.7$ and $e_2=0.7$, then the state of the agents and the observation error of FMAS (2) and (3) are given in Figure 3 and Figure 4, respectively. Figure 3 shows that

$$\underset{s\to \infty }{lim}\parallel {\alpha }_1\parallel =0\mathit{and}\underset{s\to \infty }{lim}\parallel {\alpha }_1\parallel =0$$

where

$${\alpha }_1=\left\{\begin{array}{c}{\widehat{y}}_i-y_i,i=0,1\hfill \\ -{\widehat{y}}_i+y_i,i=2,3,4\hfill \end{array}\right.{\alpha }_2=\left\{\begin{array}{c}{y}_i-\mathrm{co}\{y_5,y_6\}i=0,1\hfill \\ -(y_i+\mathrm{co}\{y_5,y_6\})i=2,3,4.\hfill \end{array}\right.$$

Notice that

$$\begin{array}{cc}\hfill & \underset{s\to \infty }{lim}\parallel {\widehat{y}}_i-y_i\parallel =0,i=0,1,2,3,4\hfill \\ \hfill & \underset{s\to \infty }{lim}\parallel y_i-\mathrm{co}\{y_5,y_6\}\parallel =0,i=0,1,\hfill \\ \hfill & \underset{s\to \infty }{lim}\parallel y_i+\mathrm{co}\{y_5,y_6\}\parallel =0,i=2,3,4.\hfill \end{array}$$

Drawing from the simulation results, it is determined that the observation-based bipartite containment control for FMAS (2) and (3) using controller (7) is successfully implemented. Furthermore, the results confirm that the asymptotic stability of the observation error is assured.

Example 2. 

Consider the bipartite containment control for a system having five followers and two leaders. The system’s dynamics are governed by switching topology graphs ${\mathcal{G}}_1$, ${\mathcal{G}}_2$, and ${\mathcal{G}}_3$, which are illustrated in Figure 5 and Figure 6. These graphs, ${\mathcal{G}}_1$, ${\mathcal{G}}_2$, and ${\mathcal{G}}_3$, are signed digraphs that are structurally balanced, meaning that they can be split into bipartition subgroups where each link connecting nodes of the same subgroup is positively signed, and each edge connecting nodes of different subgroups is negatively signed. Specifically, the bipartition subgroups are $F_1=\{0,1\}$ and $F_2=\{2,3,4\}$. The coefficient matrices used in the finite multi-agent system (FMAS) Equations (2) and (3) are the same as those presented in Example 1, ensuring consistency in the system’s behavior and analysis.

From Figure 5 and Figure 6.

$F^1=\left(\begin{array}{ccccccc}0& 0.9& 0& 0& 0& 0.7& 0\\ 0& 0& 0& 0& 0& 0& 0.5\\ 0& 0& 0& 0.2& 0& -0.5& 0\\ 0& 0& 0& 0& 1& 0& 0\\ -0.5& 0& 0& 0& 0& 0& -0.6\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right)$  $F^2=\left(\begin{array}{ccccccc}0& 0.9& 0& 0& 0& 0.7& 0\\ 0& 0& 0& 0& -1& 0& 0.5\\ -0.5& 0& 0& 0.2& 0& -0.5& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& -0.6\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right)$

$$F^3=\left(\begin{array}{ccccccc}0& 0& -0.7& 0& 0& 1& 0\\ 1.2& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0.5& 0& 0& 0& 0\\ 0& -1.5& 0& 0.6& 0& 0& -0.2\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right)$$

$$\psi =\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right)$$

Then

$$\begin{array}{cc}\hfill & {\overline{L}}_{\mathcal{F}}^1=\left(\begin{array}{ccccc}1.6& -0.9& 0& 0& 0\\ 0& 0.5& 0& 0& 0\\ 0& 0& 0.7& -0.2& 0\\ 0& 0& 0& 0.5& -0.5\\ -0.5& 0& 0& 0& 1.1\end{array}\right){\overline{L}}_{\mathcal{B}}^1=\left(\begin{array}{cc}-0.7& 0\\ 0& -0.5\\ -0.5& 0\\ 0& 0\\ 0& -0.6\end{array}\right)\hfill \\ \hfill & {\overline{L}}_{\mathcal{F}}^2=\left(\begin{array}{ccccc}1.6& -0.9& 0& 0& 0\\ 0& 1.5& 0& 0& -1\\ -0.5& 0& 1.2& -0.2& 0\\ 0& 0& 0& 1& -1\\ 0& 0& 0& 0& 0.6\end{array}\right){\overline{L}}_{\mathcal{B}}^2=\left(\begin{array}{cc}-0.7& 0\\ 0& -0.5\\ -0.5& 0\\ 0& 0\\ 0& -0.6\end{array}\right)\hfill \\ \hfill & {\overline{L}}_{\mathcal{F}}^3=\left(\begin{array}{ccccc}1.7& 0& -0.7& 0& 0\\ -1.2& 2.2& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& -0.5& 0.5& 0\\ 0& -1.5& 0& -0.6& 2.3\end{array}\right){\overline{L}}_{\mathcal{B}}^3=\left(\begin{array}{cc}-1& 0\\ 0& -1\\ -1& 0\\ 0& 0\\ 0& -0.2\end{array}\right)\hfill \end{array}$$

In this scenario, the dynamical equations governing each agent remain unchanged; only the communication switching among the agents is different. Consequently, the feasible solutions to inequalities (49)–(52) are identical to those in Example 1. We can select $\mathsf{\Phi }=I_5$ for $\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}+{\left({\overline{L}}_{\mathcal{F}}^{\sigma \left(s\right)}\right)}^T\mathsf{\Phi }>0$ thus ${\varphi }_{min}={\varphi }_{max}=1$.

$${\lambda }_{min}\left\{\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^1+{\left({\overline{L}}_{\mathcal{F}}^1\right)}^T\mathsf{\Phi }\right\}=0.6372{\lambda }_{min}\left\{\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^2+{\left({\overline{L}}_{\mathcal{F}}^2\right)}^T\mathsf{\Phi }\right\}=0.2234{\lambda }_{min}\left\{\mathsf{\Phi }{\overline{L}}_{\mathcal{F}}^3+{\left({\overline{L}}_{\mathcal{F}}^3\right)}^T\mathsf{\Phi }\right\}=0.6467$$

Hence, ${\lambda }_3=0.2234$. Also,

$${\lambda }_{max}\left\{{\overline{L}}_{\mathcal{F}}^1\mathsf{\Phi }{\left({\overline{L}}_{\mathcal{F}}^1\right)}^T\right\}=3.7233{\lambda }_{max}\left\{{\overline{L}}_{\mathcal{F}}^2\mathsf{\Phi }{\left({\overline{L}}_{\mathcal{F}}^2\right)}^T\right\}=4.9827{\lambda }_{max}\left\{{\overline{L}}_{\mathcal{F}}^3\mathsf{\Phi }{\left({\overline{L}}_{\mathcal{F}}^3\right)}^T\right\}=10.7194$$

Hence, ${\lambda }_4=10.7194$, $\mu \ge {\displaystyle \frac{\rho {\varphi }_{min}}{{\lambda }_3}}=0.09534$, and $0<\beta \le \sqrt{{\displaystyle \frac{\omega {\varphi }_{max}}{{\lambda }_4}}}=0.0144$

$$\mathit{Now},\mathit{if}\mathit{we}\mathit{take}\beta =0.1\mathit{and}\mu =0.01,$$

and we choose $\gamma =0.7$ and $e_2=0.9$, then the states of the agents and the observation error under switching topologies of FMAS (2) and (3) are given in Figure 7 and Figure 8, respectively.

6. Conclusions

In networks that incorporate both fixed and switching attributes along with signed directed edges where both cooperative and hostile agent interactions occur, observation-based bipartite containment control of nonlinear FMASs has been investigated. Assuming the associated signed digraph is structurally balanced and at least one leader has a directed link to each follower, delayed and disturbance compensation control methods have been developed for both fixed and switching signed directed systems to address observation-based bipartite containment control networks. Using the Lyapunov function approach and the fractional Razumikhin technique, a reliable and practical solution has been proposed to tackle the issues resulting from switching topologies, fractional calculus, and delay. Basic matrix inequalities have been employed to guarantee multiple bipartite containment control. The validity and feasibility of the main findings are demonstrated through concrete numerical examples.