## 3 Main Results
### Consensus Analysis
Moreover, by (6), we have  
\[+(c+k_{2})(\overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))+( \overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))^{\frac{p}{q}})-b_{i}(w_{i}g_{i}( x_{i}(t),\]2  
\[+(c+k_{2})(\overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))+( \overline{\psi}_{j}(t)+\overline{\xi}_{j}(t))^{\frac{p}{q}})-b_{i}(w_{i}g_{i}( x_{i}(t),\]3  
where \(\check{I}_{m}= \begin{bmatrix}I_{m}&0\\ 0&0_{N-m}\end{bmatrix} \), and \(\lambda_{\min}(M^{\rm T}\check{I}_{m}M)\) and \(\lambda_{\min}(\tilde{M}^{\rm T}\tilde{M})\) denote the non-zero eigenvalues of matrices \(M^{\rm T}\check{I}_{m}M\) and \(\tilde{M}^{\rm T}\tilde{M}\), respectively. Therefore,  
\[ \begin{split} e(t)^{\rm T}e(t)&\leq\frac{1}{\lambda_{ \min}(M^{\rm T}\check{I}_{m}M)}\overline{\psi}(t)^{\rm T}\overline{\psi}(t)\\ v^{\rm T}(t)v(t)&\leq\frac{1}{\lambda_{\min}(\tilde{M}^{ \rm T}\tilde{M})}\overline{\xi}(t)^{\rm T}\overline{\xi}(t)\end{split}  \tag{23}\]  
\(\qed\)  
Substituting Equations (17), (21)-(23) into (15), we have  
\[ \begin{split}\dot{V}(t)\leq&(\lambda_{\max}(MM^{ \rm T})+\frac{\gamma_{1}}{2}+\frac{1}{2}\lambda_{\max}(LL^{\rm T})+\frac{ \sigma_{1}}{2\lambda_{min}(M^{\rm T}\check{I}_{m}M)}-\frac{k_{1}\mu_{\min}}{2} )\overline{\varphi}(t)^{\rm T}\overline{\varphi}(t)\\ &+\frac{1}{2}((\varepsilon+1)\lambda_{\max}(\tilde{M}\tilde{M}^{ \rm T})+\gamma_{2}-k_{2}\check{\mu}_{\min})(\overline{\psi}(t)^{\rm T}+ \overline{\xi}(t)^{\rm T})(\overline{\psi}(t)+\overline{\xi}(t))\\ &+\frac{1}{2}(2+(1+\frac{\sigma_{2}}{\varepsilon})\frac{1}{ \lambda_{\min}(\tilde{M}^{\rm T}\tilde{M})}-c\check{\mu}_{\min})\overline{ \xi}(t)^{\rm T}\overline{\xi}(t)-\frac{c}{2}\check{\mu}_{\min}\overline{\psi} (t)^{\rm T}\overline{\psi}(t)\\ &-\overline{\varphi}(t)^{\rm T}(M\otimes I_{n})\overline{\varphi} (t)^{\frac{p}{q}}-(\overline{\psi}(t)^{\rm T}+\overline{\xi}(t)^{\rm T})( \tilde{M}\otimes I_{n})(\overline{\psi}(t)+\overline{\xi}(t))^{\frac{p}{q}} \end{split}  \tag{24}\]  
Thus, if conditions (11)-(13) in Theorem 1 are guaranteed, \(\lambda_{\max}(MM^{\rm T})+\frac{\gamma_{1}}{2}+\frac{1}{2}\lambda_{\max}( LL^{\rm T})+\frac{\sigma_{1}}{2\lambda_{\min}(M^{\rm T}\mathbf{1}_{m}M)}-\frac{k_{1}\mu_{ \min}}{2}\leq 0\), \((\varepsilon+1)\lambda_{\max}(\tilde{M}\tilde{M}^{\rm T})+\gamma_{2}-k_{2} \check{\mu}_{\min}\leq 0\), \(2+(1+\frac{\sigma_{2}}{\varepsilon})\frac{1}{\lambda_{\min}(\tilde{M}^{\rm T} \tilde{M})}-c\check{\mu}_{\min}\leq 0\). At this point, we have  
\[\dot{V}(t)\leq-\overline{\varphi}(t)^{\rm T}(M\otimes I_{n})\overline{\varphi }(t)^{\frac{p}{q}}-(\overline{\psi}(t)^{\rm T}+\overline{\xi}(t)^{\rm T})( \tilde{M}\otimes I_{n})(\overline{\psi}(t)+\overline{\xi}(t))^{\frac{p}{q}} \tag{25}\]  
From Lemmas 3 and 4, it can be obtained that  
\[ \begin{split}\dot{V}(t)&\leq-\mathcal{K}_{1}\sum_{i=1 }^{m}\|\overline{\varphi}_{i}(t)\|^{\frac{p}{q}+1}-\mathcal{K}_{2}\sum_{i=m+1}^ {N}\|\overline{\psi}_{i}(t)+\overline{\xi}_{i}(t)\|^{\frac{p}{q}+1}\\ &\leq-\mathcal{K}_{1}(\sum_{i=1}^{m}\overline{\varphi}_{i}(t)^{2 }\}^{\frac{1}{2}(\frac{p}{q}+1)}-\mathcal{K}_{2}(\sum_{i=m+1}^{N}(\overline{ \psi}_{i}(t)+\overline{\xi}_{i}(t))^{2})^{\frac{1}{2}(\frac{p}{q}+1)}\end{split}  \tag{26}\]  
where \(\mathcal{K}_{1}\), \(\mathcal{K}_{2}>0\) are constants. Let \(\zeta(t)=(\overline{\psi}(t)^{\rm T},\overline{\xi}(t)^{\rm T})^{\rm T}\) and \(C= \begin{bmatrix}I_{Nn}&I_{Nn}\\ I_{Nn}&I_{Nn}\end{bmatrix} \). We have \(\sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)+\overline{\xi}_{i}(t))^{2}=\zeta(t)^{ \rm T}C\zeta(t)>0\) before consensus is achieved, and \(\frac{\zeta(t)}{\|\zeta(t)\|}\in\Theta\), where \(\Theta=\{\chi\in R^{2 Nn},\chi^{\rm T}\chi=1\}\) is a bounded and closed set. Thus, there exists a positive constant \(m=\min\frac{\zeta(t)}{\|\zeta(t)\|}(\frac{\zeta(t)}{\|\zeta(t)\|})^{\rm T}C\frac{ \zeta(t)}{\|\zeta(t)\|}\) such that \(\sum_{i=m+1}^{N}(\overline{\psi}_{i}(t)+\overline{\xi}_{i}(t))^{2}\geq m\|\zeta(t )\|^{2}\). Therefore,  
\[