## 3 Main Results
### Leader-Following Tracking Control under Switching Topology
Rewrite the Laplacian matrix of \(\overline{G}_{p}\) in the form \(L_{p}=diag(L^{1}_{\sigma(t)},L^{2}_{\sigma(t)},\cdots,L^{I}_{\sigma(t)})\); \(L^{i}_{\sigma(t)}\in\mathbb{R}^{t^{i}_{\sigma(t)}\times t^{i}_{\sigma(t)}}\) (\(t^{i}_{\sigma(t)}=|\mathcal{V}(S^{i}_{\sigma(t)})|\))is the counterpart of component \(S^{i}_{\sigma(t)}\), where \(\mathcal{V}\Big{(}S^{i}_{\sigma(t)}\Big{)}=\left\{v_{1},v_{2},\cdots v_{i^{i}_{ \sigma(t)}}\right\}\), which is the subset of \(\{1,2,\cdots,n\}\) and \(v_{1}<v_{2}<\cdots<v_{i^{i}_{\sigma(t)}}\). Arrange the eigenvalues of the matrix \(H^{i}_{\sigma(t)}\) in order from small to large, \(\delta^{i}_{\sigma(t),1}<\delta^{i}_{\sigma(t),2}<\cdots<\delta^{i}_{\sigma(t),i^{i}_{\sigma(t)}}\) and label them \(\lambda^{v_{1}}_{\sigma(t)}\Big{(}\leftrightarrow\delta^{i}_{\sigma(t),1} \Big{)},\lambda^{v_{2}}_{\sigma(t)}\Big{(}\leftrightarrow\delta^{i}_{\sigma(t ),2}\Big{)},\cdots\lambda^{v_{l_{\sigma}}}_{\sigma(t)}\Big{(}\leftrightarrow \delta^{i}_{\sigma(t),t^{i}_{\sigma(t)}}\Big{)}\). The eigenvalue of \(H_{\sigma(t)}\) is obtained by combining the eigenvalues of matrix \(H^{1}_{\sigma(t)},H^{2}_{\sigma(t)},\cdots H^{r}_{\sigma(t)}\) and labeling all of them. **Example 1**. : _To illustrate the rules, consider a communication graph \(\overline{G}_{p}\) in Figure 2. There are three components in \(G_{p}\), which are \(S^{1}_{p}\), \(S^{2}_{p}\) and \(S^{3}_{p}\), respectively. According to the connection situation, the matrixes \(H^{1}_{p}\), \(H^{2}_{p}\) and \(H^{3}_{p}\) are:_  
\[ \begin{array}{l}H^{1}_{p}=\left[\begin{array}{cc}1&-1\\ -1&1\\ 1&-1\\ -1&1\\ \end{array}\right]+\left[\begin{array}{cc}0&0\\ 0&0\\ 1&0\\ 0&0\\ \end{array}\right]=\left[\begin{array}{cc}1&-1\\ -1&1\\ 2&-1\\ -1&1\\ \end{array}\right]\\ \end{array} \]  
The eigenvalues of \(H^{1}_{p}\) are 0 and 2, then we label them as \(\lambda^{1}_{p}(\leftrightarrow 0)\) and \(\lambda^{3}_{p}(\leftrightarrow 2)\) by the method above. The vertices set \(\mathcal{V}\Big{(}S^{1}_{p}\Big{)}=\{1,3\}\). The eigenvalues of \(H^{2}_{p}\) are 0.382 and 2.618, then we label them as \(\lambda^{2}_{p}(\leftrightarrow 0.382)\) and \(\lambda^{4}_{p}(\leftrightarrow 2.618)\). The corresponding vertices set \(\mathcal{V}\Big{(}S^{2}_{p}\Big{)}=\{2,4\}\). As for \(H^{3}_{p}\), obviously the eigenvalue and vertices set are \(\lambda^{5}_{p}(\leftrightarrow 0)\) and \(\mathcal{V}\Big{(}S^{3}_{p}\Big{)}=\{5\}\), respectively (in a sense, \(H^{3}_{p}\) can be seen as a \(1\times 1\) matrix). Therefore, all of the eigenvalues of \(H_{p}\) have been labeled as \(\lambda^{1}_{p}(\leftrightarrow 0)\), \(\lambda^{2}_{p}(\leftrightarrow 0.382)\), \(\lambda^{3}_{p}(\leftrightarrow 2)\), \(\lambda^{4}_{p}(\leftrightarrow 2.618)\) and \(\lambda^{5}_{p}(\leftrightarrow 2.618)\). According to the above labeling method, for \(\forall\), there are always n eigenvalue sequences of \(H_{p}\) as: \(\left\{\lambda^{1}_{p},\lambda^{2}_{p},\cdots,\lambda^{n}_{p}\right\}\). We define set  
\[1(p)=\left\{ \begin{array}{c}k,\text{\it corresponding eigenvalue }\lambda^{k}_{p} \text{\it is nonzero},\\ k=1,2,\cdots\end{array} \right\}\]  
Figure 2: A graph to illustrate the labeling rules. **Lemma 4**. : _If the graphs are jointly connected in the time interval \([t_{m},t_{m+1})\), then we have:_  
\[\mathop{\cup}\limits_{t\in[t_{m},t_{m+1})}l(\sigma(t))=\{1,2,\cdots,N\}\]  
**Theorem 2**. : _If the multi-agent systems (1) and (2) satisfy Assumption 1, and the interaction topology which is described by the switching signal \(\sigma(t)\) satisfies Assumption 3, there exists a symmetric positive definite matrix \(P\), which is the solution of the following algebraic Riccati inequation:_  
\[A^{T}P+PA-2 PBB^{T}P+\theta I<0 \tag{23}\]  
_where \(\theta\) is a positive number.