## 3 Main Results
Since the heterogeneous nonlinear systems (1) and (2) need to satisfy the TVF feasibility conditions, the following equation can be obtained from Equation (6) to Equation (9):  
\[\dot{\zeta}_{i}(t)=A\zeta_{i}(t)+\vartheta(\varepsilon_{i}(t))-\vartheta(x_{ 0}(t))-BB^{T}P^{-1}[\sum_{j=1}^{N}w_{ij}\eta_{ij}(t)(\zeta_{i}(t)-\zeta_{j}(t) )+d_{i0}\eta_{i0}(t)\zeta_{i}(t)] \tag{10}\]  
Consider the alternative Lyapunov function:  
\[V(t)=\sum_{i=1}^{N}\zeta_{i}^{\intercal}(t)P^{-1}\zeta_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\tilde{\eta}_{ij}^{2}}{2\kappa}+\sum_{i=1}^{N} \frac{\tilde{\eta}_{i0}^{2}}{\partial} \tag{11}\]We obtain the derivative of Equation (11):  
\[\dot{V}(t)=2\sum_{i=1}^{N}\zeta_{i}^{1}(t)P^{-1}\dot{\zeta}_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\bar{\eta}_{ij}}{\kappa}\bar{\eta}_{ij}+2\sum_{i=1}^ {N}\frac{\bar{\eta}_{i0}}{\partial}\bar{\eta}_{i0} \tag{12}\]  
Substituting Equations (9) and (10) into (12) yields the following:  
\[\dot{V}(t) =2\sum_{i=1}^{N}\zeta_{i}^{\mathrm{T}}(t)P^{-1}(A\zeta_{i}(t)+ \vartheta(\epsilon_{i}(t))-\vartheta(x_{0}(t)) \tag{13}\] \[\quad-BB^{\mathrm{T}}P^{-1}\left[\sum_{j=1}^{N}w_{ij}\eta_{ij}(t) (\zeta_{i}(t)-\zeta_{j}(t))+d_{i0}\eta_{i0}(t)\zeta_{i}(t)\right]\right)\] \[\quad+\sum_{i=1}^{N}\sum_{j=1,j\
eq i}^{N}(\eta_{ij}-\alpha)w_{ij }(t)(\zeta_{i}(t)-\zeta_{j}(t))^{\mathrm{T}}\Gamma(\zeta_{i}(t)-\zeta_{j}(t))\] \[\quad+2\sum_{i=1}^{N}(\eta_{i0}-\alpha)d_{i0}(t)\zeta_{i}^{ \mathrm{T}}(t)\Gamma\zeta_{i}(t)\]  
Formula (13) is divided into the following:  
\[\dot{V}(t) =\sum_{i=1}^{N}\zeta_{i}^{\mathrm{T}}(t)(P^{-1}A+A^{\mathrm{T}}P^ {-1})\zeta_{i}(t)+2\sum_{i=1}^{N}\zeta_{i}^{\mathrm{T}}(t)P^{-1}(\vartheta( \epsilon_{i}(t))-\vartheta(x_{0}(t))) \tag{14}\] \[-2\sum_{i=1}^{N}\zeta_{i}^{\mathrm{T}}(t)P^{-1}BB^{\mathrm{T}}P^ {-1}[\sum_{j=1}^{N}w_{ij}\eta_{ij}(t)(\zeta_{i}(t)-\zeta_{j}(t))+d_{i0}\eta_ {i0}(t)\zeta_{i}(t)]\] \[+\sum_{i=1}^{N}\sum_{j=1,j=i}^{N}(\eta_{ij}-\alpha)w_{ij}(t)( \zeta_{i}(t)-\zeta_{j}(t))^{\mathrm{T}}\Gamma(\zeta_{i}(t)-\zeta_{j}(t))+2 \sum_{i=1}^{N}(\eta_{i0}-\alpha)d_{i0}(t)\zeta_{i}^{\mathrm{T}}(t)\Gamma \zeta_{i}(t)\]  
From Lipschitz condition (3), the following can be stated:  
\[V(t)=\sum_{i=1}^{N}\zeta_{i}^{\intercal}(t)P^{-1}\zeta_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\tilde{\eta}_{ij}^{2}}{2\kappa}+\sum_{i=1}^{N} \frac{\tilde{\eta}_{i0}^{2}}{\partial} \tag{11}\]0 \[V(t)=\sum_{i=1}^{N}\zeta_{i}^{\intercal}(t)P^{-1}\zeta_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\tilde{\eta}_{ij}^{2}}{2\kappa}+\sum_{i=1}^{N} \frac{\tilde{\eta}_{i0}^{2}}{\partial} \tag{11}\]1 \[V(t)=\sum_{i=1}^{N}\zeta_{i}^{\intercal}(t)P^{-1}\zeta_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\tilde{\eta}_{ij}^{2}}{2\kappa}+\sum_{i=1}^{N} \frac{\tilde{\eta}_{i0}^{2}}{\partial} \tag{11}\]2  
Looking at controller (5), it can be seen that \(\eta_{ij}(t)=\eta_{ji}(t),\forall t\geq 0\). And by inserting \(\Gamma=P^{-1}BB^{\mathrm{T}}P^{-1}\) into it, we get the following:  
\[V(t)=\sum_{i=1}^{N}\zeta_{i}^{\intercal}(t)P^{-1}\zeta_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\tilde{\eta}_{ij}^{2}}{2\kappa}+\sum_{i=1}^{N} \frac{\tilde{\eta}_{i0}^{2}}{\partial} \tag{11}\]3 \[V(t)=\sum_{i=1}^{N}\zeta_{i}^{\intercal}(t)P^{-1}\zeta_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\tilde{\eta}_{ij}^{2}}{2\kappa}+\sum_{i=1}^{N} \frac{\tilde{\eta}_{i0}^{2}}{\partial} \tag{11}\]4 \[V(t)=\sum_{i=1}^{N}\zeta_{i}^{\intercal}(t)P^{-1}\zeta_{i}(t)+\sum_{i=1}^{N} \sum_{j=1,j\
eq i}^{N}\frac{\tilde{\eta}_{ij}^{2}}{2\kappa}+\sum_{i=1}^{N} \frac{\tilde{\eta}_{i0}^{2}}{\partial} \tag{11}\]5Let \(\hat{\zeta}_{i}(t)=P^{-1}\zeta_{i}(t)\), \(\hat{\zeta}(t)=[\hat{\zeta}_{1}(t),\hat{\zeta}_{2}(t),\hat{\zeta}_{3}(t)\cdots \hat{\zeta}_{N}(t)]^{\mathrm{T}}\). Bringing (15) and (16) into (14) yields the following:  
\[