## 3 Main Results
\begin{split}\dot{V}(t)&\leq\sum_{i=1}^{N}\xi_{i}^{ \mathrm{T}}(t)(AP+PA^{\mathrm{T}})\hat{\zeta}_{i}(t)+\sum_{i=1}^{N}\xi_{i}^{ \mathrm{T}}(t)[\gamma^{2}I+P^{2}]\hat{\zeta}_{i}(t)\\ &\quad-\sum_{i-1}^{N}\sum_{-1,-1,j=i}^{N}2\alpha w_{ij}(t)\xi_{i} ^{\mathrm{T}}(t)BB^{\mathrm{T}}(\hat{\zeta}_{i}(t)-\hat{\zeta}_{j}(t))-\sum_{i =1}^{N}2\alpha d_{i0}(t)\xi_{i}^{\mathrm{T}}(t)BB^{\mathrm{T}}\xi_{i}(t)\end{split}  \tag{17}\]  
If \(H_{\sigma(t)}=L_{\sigma(t)}+D_{\sigma(t)}\) is satisfied, Formula (17) can be rewritten as follows:  
\[\dot{V}(t)=\sum_{i=1}^{N}\xi_{i}^{\mathrm{T}}(t)([AP+PA^{\mathrm{T}}+\gamma^{ 2}I+P^{2}]-2\alpha\sum_{i=1}^{N}\sum_{j=1,j\
eq i}^{N}H_{ij}^{\sigma^{(t)}}BB^ {\mathrm{T}})\xi_{i}(t) \tag{18}\]  
Writing Equation (18) in a compact form gives the following:  
\[\dot{V}(t)=\xi^{\mathrm{T}}(t)(I_{N}\otimes[AP+PA^{\mathrm{T}}+\gamma^{2}I+P^ {2}]-2\alpha H_{\sigma(t)}\otimes BB^{\mathrm{T}})\hat{\zeta}(t) \tag{19}\]  
Because the matrix \(H_{\sigma(t)}\) is symmetric, an orthogonal matrix \(T_{\sigma(t)}\) can always be found at any non-switching moment, such that \(H_{\sigma(t)}\) is transformed into a diagonal form:  
\[T_{p}H_{\sigma(t)}T_{p}^{\mathrm{T}}=\Lambda_{\sigma(t)}=diag\Big{(}\lambda_{ \sigma(t)}^{\pi_{p}(1)},\lambda_{\sigma(t)}^{\pi_{p}(2)},\cdots,\lambda_{ \sigma(t)}^{\pi_{p}(n)}\Big{)} \tag{20}\]  
Among them, \(\pi_{p}\) is some permutation of the set \(\{1,2,\ldots,N\}\). Let \(\xi(t)=(T_{\sigma(t)}\otimes I_{n})\hat{\zeta}(t)\). After substitution, the following expression is obtained:  
\[ \begin{split}\dot{V}(t)&\leq\tilde{\xi}^{\mathrm{T} }(t)\Bigg{(}I_{N}\otimes\begin{bmatrix}AP+PA^{\mathrm{T}}+\gamma^{2}I&-P\\ P&I\end{bmatrix}-2\alpha\Lambda_{\sigma(t)}\otimes BB^{\mathrm{T}}\Bigg{)} \tilde{\xi}(t)\\ &\leq\sum_{i=1}^{N}\xi_{i}^{\mathrm{T}}(t)(\begin{bmatrix}AP+PA^{ \mathrm{T}}+\gamma^{2}I&-P\\ P&I\end{bmatrix}-2\alpha\lambda_{1}BB^{\mathrm{T}})\xi_{i}(t)\\ &\leq\sum_{i=1}^{N}\xi_{i}^{\mathrm{T}}(t)\Bigg{[}\begin{array}{cc}AP+PA^{ \mathrm{T}}-2\alpha\lambda_{1}BB^{\mathrm{T}}+\gamma^{2}I&P\\ P&-I\end{array}\Bigg{]}\xi_{i}(t)\end{split}  \tag{21}\]  
As long as an appropriate \(\alpha\) is selected, satisfying \(2\alpha\lambda_{i}>\theta,i=1,2...N\), (21) is converted into the following:  
\[ \begin{split}\dot{V}(t)&\leq\sum_{i=1}^{N}\xi_{i}^{\mathrm{T} }(t)\Bigg{[}\begin{array}{cc}AP+PA^{\mathrm{T}}-\theta BB^{\mathrm{T}}+ \gamma^{2}I&P\\ P&-I\end{array}\Bigg{]}\xi_{i}(t)\\ &\leq-\sum_{i=1}^{N}\xi_{i}^{\mathrm{T}}(t)\hat{\zeta}_{i}(t)\leq 0\end{split}  \tag{22}\]  
For any \(i\in l(\sigma(t))\), from the above, it becomes apparent that \(\dot{V}\leq 0\) exists, and then \(\lim_{t\to\infty}V_{1}(t)\) exists. In light of [PERSON]'s convergence criterion, for any \(\mathrm{E}>0\), there always exists a positive number \(M_{\mu}\), so that for any \(\mathrm{m}>M_{\mu}\), we have the following:  
\[\big{|}V(\xi(t_{k+1}))-V(\xi(t_{k}))\big{|}=\bigg{|}\int_{t_{k}}^{t_{k+1}}\dot{ V}(\xi(t))dt\bigg{|}<\Xi \tag{23}\]  
We can rewrite Equation (23) as follows:  
\[-\Xi<\sum_{j=0}^{m_{\mathrm{m}}-1}\int_{t_{k}}^{t_{k+1}}\dot{V}(\xi(t))dt\leq- \sum_{j=0}^{m_{\mathrm{m}}-1}\int_{t_{k}}^{t_{k}+1}\xi^{\mathrm{T}}(t)\xi(t) dt\leq-\sum_{j=0}^{m_{\mathrm{m}}-1}\int_{t_{k}}^{t_{k}+\tau}\xi^{ \mathrm{T}}(t)\xi(t)dt \tag{24}\]Therefore, for any \(j=0,1,...,m_{k}-1\); we reformulate Equation (24) as follows:  
\[ \begin{split}&\int_{\ell_{k}^{2}}^{\ell_{k}^{2}+\tau}\sum_{i\in l( \sigma(\ell_{k}^{2}))}\tilde{\xi}_{i}^{\mathsf{T}}(s)\tilde{\xi}_{i}(s)ds+\int _{t_{k}^{2}}^{t_{k}^{2}+\tau}\sum_{i\in l(\sigma(\ell_{k}^{2}))}\tilde{\xi}_{i }^{\mathsf{T}}(s)\tilde{\xi}_{i}(s)ds+\cdots\\ &+\int_{t_{k}^{2}-1}^{t_{k}^{2}-1}\sum_{i\in l(\sigma(\ell_{k}^{ \mathsf{T}}))}\xi_{i}^{\mathsf{T}}(s)\tilde{\xi}_{i}(s)ds<\Xi\end{split}