## 3 Main Results  
In this section, the problem of adaptive TVF tracking control for heterogeneous nonlinear MASs with a jointly connected topology will be discussed. Therefore, the controller is designed as follows:  
\[u_{i}(t) =K\sum_{j=1}^{N}w_{ij}(t)\eta_{ij}(t)(x_{i}(t)-h_{i}(t) \tag{5}\] \[-(x_{j}(t)-h_{j}(t)))+Kd_{i0}(t)\eta_{i0}(t)(x_{i}(t)-h_{i}(t)-x_ {0}(t))+\xi_{i}(t)\] \[\eta_{ij}(t) =\kappa w_{ij}(t)\big{(}x_{i}(t)-h_{i}(t)-(x_{j}(t)-h_{j}(t)) \big{)}^{\top}\Gamma\big{(}x_{i}(t)-h_{i}(t)-(x_{j}(t)-h_{j}(t))\big{)}\] \[\dot{\eta}_{i0}(t) =\partial d_{i0}(t)(x_{i}(t)-h_{i}(t)-x_{0}(t))^{\top}\Gamma(x_{i }(t)-h_{i}(t)-x_{0}(t))\]  
Among them, \(\kappa\) and \(\partial\) denote adaptive parameter convergence factors, and \(\kappa>0\), \(\partial>0\), and \(\eta_{ij}(t),\eta_{i0}(t)\) denote adaptive coupling weights between neighboring agents and adaptive coupling weights between following agents and leaders, respectively. Since the communication topology network structure is an undirected graph, \(\eta_{ij}(t)=\eta_{ji}(t)\). \(K\in\mathbb{R}^{m\times n}\) is the feedback gain matrix, \(\Gamma\in\mathbb{R}^{n\times n}\) is a continuous gain matrix, and \(\xi_{i}(t)\) is the input compensation in the extended feasible formation set, which can be determined by the formation feasibility condition. Given the formation vector \(h(t)\) and the leader's control input \(u_{0}(t)\), the TVF feasibility condition for the compensating input \(\xi_{i}(t)\) is derived as follows:  
\[Ah_{i}(t)-h_{i}(t)-Bu_{0}(t)+B\xi_{i}(t)=0,\quad i=1,\cdots,N \tag{6}\]  
**Remark 3**. : _For the heterogeneous nonlinear MASs made up of (1) and (2), if the TVF control objective needs to be satisfied, then the TVF feasibility condition of Formula (6) is very important. The coefficient matrix and the TVF vector function \(h_{i}(t)\) in the nonlinear dynamic equation should satisfy Formula (6); otherwise, they need to be redesigned._  
**Theorem 1**. : _Consider that when a heterogeneous nonlinear MAS consists of (1) and (2), Assumptions 1-5 hold, and the required TVF configuration vector satisfies the TVF tracking feasibility condition (6), then, under jointly connected topologies, the fully distributed adaptive TVF tracking control problem can be resolved. The following LMI represents the feedback gain matrix and the continuous gain matrix:_  
\[ \begin{bmatrix}AP+PA^{\rm T}-\theta BB^{\rm T}+\gamma^{2}I&P\\ P&-I\end{bmatrix} <0 \tag{7}\]  
A matrix \(P>0\) can be obtained, and the parameter \(\theta>0\). And feedback gain matrix is given by \(K=-B^{T}P^{-1}\), and the continuous gain matrix is \(\Gamma=P^{-1}BB^{T}P^{-1}\). Proof of Theorem 1.: Bringing each follower agent (1) into the controller (5):  
\[ \begin{split}\dot{x}_{i}(t)&=Ax_{i}(t)+Bu_{i}(t)\\ &=Ax_{i}(t)+\vartheta(\varepsilon_{i}(t))+B\xi_{i}(t)\\ &\quad-BB^{T}P^{-1}\left[\sum_{l=1}^{N}w_{ij}\eta_{ij}(t)\big{(}x_{ i}(t)-h_{i}(t)-(x_{j}(t)-h_{j}(t))\big{)}\right.\\ &\quad\left.+d_{i0}\eta_{i0}(t)(x_{i}(t)-h_{i}(t)-x_{0}(t))\right] \end{split}  \tag{8}\]  
Let \(\zeta_{i}(t)=x_{i}(t)-h_{i}(t)-x_{0}(t)\), \(\zeta_{i}(t)\) denote the TVF tracking error for each agent, \(\zeta(t)=[\zeta_{1}(t),\zeta_{2}(t),\zeta_{3}(t)\cdots\zeta_{N}(t)]^{T}\), and make \(\dot{\zeta}_{i}(t)=\dot{x}_{i}(t)-h_{i}(t)-\dot{x}_{0}(t)\) available:  
\[ \begin{split}\dot{\zeta}_{i}(t)&=A\zeta_{i}(t)+ \vartheta(\varepsilon_{i}(t))-\vartheta(x_{0}(t))\\ &\quad-BB^{T}P^{-1}\left[\sum_{l=1}^{N}w_{ij}\eta_{ij}(t)(\zeta_{i }(t)-\zeta_{j}(t))+d_{i0}\eta_{i0}(t)\zeta_{i}(t)\right]\\ &\quad+Ah_{i}(t)-h_{i}(t)-Bu_{0}(t)+B\xi_{i}(t)\\ \dot{\eta}_{ij}&=\kappa w_{ij}(t)(\zeta_{i}(t)-\zeta_{ j}(t))^{\rm T}\Gamma(\zeta_{i}(t)-\zeta_{j}(t))\\ \dot{\tilde{\eta}}_{i0}&=\partial d_{i0}(t)\zeta_{i} ^{\intercal}(t)\Gamma\zeta_{i}(t)\end{split}  \tag{9}\]  
Among them, \(\eta_{ij}=\tilde{\eta}_{ij}+\alpha\), \(\eta_{i0}=\tilde{\eta}_{i0}+\alpha\), \(\alpha\) is a positive number.