## 3 Main Results
### Leader-Following Tracking Control under Switching Topology
Let feedback gain matrix \(K=B^{T}P\), then for any initial conditions, the swarm systems could track the leader agent and achieve the desired formation under protocol (19)._  
Proof of Theorem 2.: Consider the following Lyapunov function candidate:  
\[V_{2}(t)=\sum\limits_{i=1}^{N}e_{i}^{T}Pe_{i}+\sum\limits_{i=1}^{N}\sum\limits _{j\in N_{i}(t)}\frac{\big{(}c_{ij}(t)-\beta\big{)}^{2}}{2}+\sum\limits_{i=1}^ {N}\sum\limits_{j\in N_{i}(t)}(c_{i}(t)-\beta)^{2} \tag{24}\]  
where \(\beta\) is a positive constant to be determined later. The Lyapunov function \(V_{2}(t)\) is continuous and differentiable at any time except the switching instants. Then, we take the derivative of the function along the solution of (22) as:  
\[ \begin{array}{ll}\dot{V}_{2}(t)&=2\sum\limits_{i=1}^{N}e_{i}^{T}P\dot{e}_{ i}+\sum\limits_{i=1}^{N}\sum\limits_{j\in N_{i}(t)}a_{ij}(t)\big{(}c_{ij}(t)- \beta\big{)}\cdot\big{(}e_{i}-e_{j}\big{)}^{T}\Gamma\big{(}e_{i}-e_{j}\big{)} \\ &+2\sum\limits_{i=1}^{N}\sum\limits_{j\in N_{i}(t)}b_{i}(t)(c_{i}(t)-\beta) \cdot{e_{i}}^{T}\Gamma{e}_{i}\\ &=e_{i}^{T}\Big{[}I_{N}\otimes\big{(}A^{T}P+PA\big{)}-2\beta H_{\sigma(t)} \otimes\Gamma\Big{]}e_{i}\#\end{array}  \tag{25}\]  
Matrix \(H_{\sigma(t)}\) is symmetric, and we can always find an orthogonal matrix \(T_{\sigma(t)}\) at any non-switching instants to translate the \(H_{\sigma(t)}\) into diagonal forms. \[T_{p}H_{\sigma(t)}T_{p}^{T}=\Lambda_{\sigma(t)}=diag\Big{(}\lambda_{\sigma(t )}^{\pi_{p}(1)},\lambda_{\sigma(t)}^{\pi_{p}(2)},\cdots,\lambda_{\sigma(t)}^{ \pi_{p}(n)}\Big{)} \tag{26}\]  
where \(\pi_{p}\) is some sort of permutation for the set \(\{1,2,\cdots n\}\). Let \(\widetilde{e}=\Big{(}T_{\sigma(t)}\otimes I_{n}\Big{)}e\), we can obtain:  
\[ \begin{array}{ll}\dot{V}_{2}(t)&=\widetilde{e}^{T}\Big{[}I_{n}\otimes\big{(} A^{T}P+PA\big{)}-2\beta\Lambda_{\sigma(t)}\otimes\Gamma\Big{]}\widetilde{e}\\ &\leq\sum\limits_{i\in l(p)}\widetilde{e}_{i}^{T}\big{[}I_{n}\otimes\big{(}A^ {T}P+PA\big{)}-2\beta\lambda_{i}\otimes\Gamma\big{]}\widetilde{e}_{i}\\ &\leq-\sum\limits_{i\in l(p)}\widetilde{e}_{i}^{T}\widetilde{e}_{i}\leq 0 \end{array}  \tag{27}\]  
Because the coupling weights \(c_{ij}(t)\) and \(c_{i}(t)\) are incremental, the positive constant \(\beta\) could be large enough to make \(\beta\lambda_{i}>1\), for any \(i\in 1(p)\). From the above we can see that \(\dot{V}_{2}(t)\leq 0\), which means \(\lim\limits_{t\to\infty}V_{2}(t)\) exists. According to the [PERSON]'s convergence criteria, an infinite sequence \(V_{2}(t_{i}),\mathsf{i}=0,1,2,\cdots\), for any \(\mathsf{\mu}>0\), there always exists a positive number \(M_{\mu}\), such that for any \(\mathsf{m}>M_{\mu}\):  
\[\left|\int_{t_{m}}^{t_{m+1}}\dot{V}_{2}(t)dt\right|<\mu \tag{28}\]Rewrite the above inequation as the sum of multiple integrals:  
\[\int_{t_{m}^{1}}^{t_{m}^{1}}\bigl{|}\dot{V}_{2}(t)\Bigr{|}dt+\int_{t_{m}^{1}}^{t_ {m}^{2}}\bigl{|}\dot{V}_{2}(t)\Bigr{|}dt+\cdots+\int_{t_{m}^{n-1}}^{t_{m}^{n}} \bigl{|}\dot{V}_{2}(t)\Bigr{|}dt<\mu \tag{29}\]  
For each subitem:  
\[\int_{t_{m}^{1}}^{t_{m}^{1}+1}\bigl{|}\dot{V}_{2}(t)\Bigr{|}dt=\int_{t_{m}^{1} }^{t_{m}^{1}+1}\bigl{(}-\dot{V}_{2}(t)\Bigr{)}dt\geq\int_{t_{m}^{1}}^{t_{m}^{1 +1}}\sum_{i\in l(\sigma(t_{m}^{1}))}\tilde{e}_{i}^{T}\tilde{e}_{i}dt\geq\int_{ t_{m}^{1}}^{\tau}\sum_{i\in l(\sigma(t_{m}^{1}))}\tilde{e}_{i}^{T}\tilde{e}_{i}dt \tag{30}\]  
Thus, we have:  
\[ \begin{array}{l}\int_{t_{m}^{1}}^{t_{m}^{0}+\tau}\sum\limits_{i\in l(\sigma( t_{m}^{0}))}\tilde{e}_{i}^{T}(s)\tilde{e}_{i}(s)ds+\int_{t_{m}^{1}}^{t_{m} ^{1}+\tau}\sum\limits_{i\in l(\sigma(t_{m}^{1}))}\tilde{e}_{i}^{T}(s)\tilde{ e}_{i}(s)ds\\ +\cdots+\int_{t_{m}^{n-1}}^{t_{m}^{n-1}+\tau}\sum\limits_{i\in l(\sigma(t_{m}^{ 1}-1))}\tilde{e}_{i}^{T}(s)\tilde{e}_{i}(s)ds<\mu\end{array}  \tag{31}\]  
There are finite sequences between the time interval \([t_{m},t_{m+1}]\), so the value of the positive integer \(n_{k}\) is also finite.