## 2 Preliminaries  
### Preliminaries on Graph Theory  
Consider an undirected graph \(\mathcal{G}\) with \(\mathcal{N}\) edges and \(N\) nodes, which is denoted by \(\mathcal{G}=(\mathcal{V},\mathcal{E})\) with node set \(\mathcal{V}=\{v_{1},v_{2},\cdots,v_{n}\}\) and edge set \(\mathcal{E}=\{(v_{i},v_{j}):v_{i},v_{j}\in\mathcal{V}\}\). The set of neighbours of node \(i\) is denoted by \(N_{i}=\{v_{j}\in\mathcal{V}:(v_{j},v_{i})\in\mathcal{E}\}\), and \(|N_{i}|\) denotes the cardinality of \(N_{i}\), ie. \(\mathcal{N}=|N_{1}|+|N_{2}|+\cdots+|N_{N}|\). Then the weighted adjacency matrix \(A\in\mathbb{R}^{N\times N}\) is used to describe the topology of system (2), in which self-loops and multiple information links are not allowed, where \(a_{ij}>0\) if \((v_{i},v_{j})\in\mathcal{E}\), while \(a_{ij}=0\), otherwise. The graph Laplacian of \(\mathcal{G}\) is defined as \(L=\{l_{ij}\}\in\mathbb{R}^{N\times N}\), in which \(l_{ij}=-a_{ij}\) if \(i\
eq j\), and \(l_{ij}=\sum_{j\in\mathcal{V}_{i}}a_{ij}\) otherwise. We now introduce several important matrices in the graph theory. The matrix which relates the nodes to the edges is called incidence matrix, denoted by \(H=\{h_{ki}\}\in\mathbb{R}^{N\times N}\)[23]. By choosing arbitrary orientation for all edges for an undirected graph, the entries of its incidence matrix are defined as \(h_{ki}=1\) if the \(k\)th edge sinks at node \(i\), or \(h_{ki}=-1\) if the \(k\)th edge leaves node \(i\), or \(h_{ki}=0\) otherwise. Therefore, the corresponding Laplacian matrix is represented as \(L=H^{T}H\)[4; 43]. For the edge \(\epsilon_{p}\in\mathcal{E}\) with the orientation from agents \(v_{j}\) to \(v_{i}\), let \(r_{p}(t)=r_{(i,j)}(t)=x_{i}(t)-x_{j}(t)\) be the corresponding edge state. We say that system (2) reaches the state synchronization if and only if \(\lim_{t\to\infty}\|r_{p}(t)\|=0\), \(p=1,2,\cdots,\mathcal{N}\), for any initial values \(r_{p}(t_{0})\). The following assumption and lemma are presented:  
**Definition 1**. : _[_44_]_ _Let \(S_{1},S_{2}\in\mathbb{R}^{s\times s}\) be real symmetric matrices such that \(S_{1}\) is positive semi-definite and \(S_{1}-S_{2}\) is positive definite, i.e., \(0\leq S_{1}<S_{2}\). Then \(\phi(\cdot)\) is called sector bounded within the sector \([S_{1},S_{2}]\) if it satisfies_  
\[(\phi(y)-S_{1}y)^{T}(\phi(y)-S_{2}y)\leq 0, \tag{1}\]  
_for all \(y\in\mathbb{R}^{m}\)._  
**Assumption 1**. : _The interaction topology \(G\) of system (2) is connected._  
**Lemma 1**. : _There is \(\phi(\cdot)\) called sector bounded within the sector \([S_{1},S_{2}]\), such that_  
\[\|\phi(y_{i})-\phi(y_{j})\|^{2}\leq\eta\|y_{i}-y_{j}\|^{2}\leq\eta\|C\|^{2} \|x_{i}-x_{j}\|^{2}.\]  
_ie. \([(H\otimes I)\Phi(y)]^{T}[(H\otimes I)\Phi(y)]\leq\eta\|C\|^{2}R(t)^{T}R(t)\)._  
Proof. : By (1), we have  
\[\phi^{T}(y)\phi(y)<\phi^{T}(y)(S_{1}+S_{2})y-\frac{1}{2}\Big{(}y^{T}(S_{1}S_{2 }+S_{2}S_{1})y\Big{)}.\]  
Using the fact \(-2\phi^{T}(y_{i})\phi(y_{j})\leq\phi^{T}(y_{i})\phi(y_{i})+\phi^{T}(y_{j})\phi( y_{j})\), one gets\[\|\phi(y_{i})-\phi(y_{j})\|^{2}= \big{(}\phi(y_{i})-\phi(y_{j})\big{)}^{T}\big{(}\phi(y_{i})-\phi(y _{j})\big{)}\] \[= \Phi^{T}(y_{i})\phi(y_{i})-2\phi^{T}(y_{i})\phi(y_{j})+\phi^{T}(y_{ j})\phi(y_{j})\] \[\leq 2\Big{(}\phi^{T}(y_{i})\phi(y_{i})+\phi^{T}(y_{j})\phi(y_{j}) \Big{)}\] \[\leq 2\phi^{T}(y_{i})(S_{1}+S_{2})y_{i}-y_{i}^{T}(S_{1}S_{2}+S_{2}S_{ 1})y_{i}+2\phi^{T}(y_{j})(S_{1}+S_{2})y_{j}-y_{j}^{T}(S_{1}S_{2}+S_{2}S_{1})y_{ j}.\]  
Then we derived that  
\[\|\phi(y_{i})-\phi(y_{j})\|^{2}=Y^{T} \begin{bmatrix}D_{3}&0&-D_{2}&0\\ 0&D_{3}&0&-D_{2}\\ -D_{2}&0&0&0\\ 0&-D_{2}&0&0\end{bmatrix} Y,\]  
where \(D_{2}=-(S_{1}+S_{2})\), \(D_{3}=-(S_{1}S_{2}+S_{2}S_{1})\). \(Y=(y_{i}^{T},y_{j}^{T},\phi^{T}(y_{i}),\phi^{T}(y_{j}))^{T}\). Then in order to prove that \(\|\phi(y_{i})-\phi(y_{j})\|^{2}\leq\eta\|y_{i}-y_{j}\|^{2}\), we just need to find the positive constant \(\eta\) that satisfies that  
\[ \begin{bmatrix}D_{3}&0&-D_{2}&0\\ 0&D_{3}&0&-D_{2}\\ -D_{2}&0&0&0\\ 0&-D_{2}&0&0\end{bmatrix} \leq\eta