## 2 Graph Theory and Problem Description
### Problem Description  
A collection of nonlinear MASs is considered to be composed of the \(N+1\) agent, \(N\) identical nonlinear followers, and a leader who satisfies the nonlinear condition. For each follower, the dynamic equation is depicted as follows:\[\hat{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t)+\vartheta(\varepsilon_{i}(t)) \tag{1}\]  
Among them, \(x_{i}\in\mathbb{R}^{n}\) is the state information of the \(i\)-th agent; \(u_{i}\in\mathbb{R}^{m}\) is the control input of the multi-agent system; \(A\in\mathbb{R}^{n\times n}\) and \(B\in\mathbb{R}^{n\times m}\) are system matrices with corresponding dimensions; \(\vartheta:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) represents an unknown nonlinear term in the follower, \(\varepsilon_{i}(t)=x_{i}(t)-h_{i}(t)\), \(i=1,2,\cdots,N\); and \(h_{i}(t)\), the vector representing the desired position configuration, will be defined later. The nonlinear leader's agent is described as follows:  
\[\dot{x}_{0}(t)=Ax_{0}(t)+Bu_{0}(t)+\vartheta(\varepsilon_{0}(t)) \tag{2}\]  
Among them, \(x_{0}(t)\) indicates the leader's status; \(u_{0}(t)\) represents the leader's control input; and \(\vartheta:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) represents the unknown nonlinear term of the leader. **Assumption 1**. : \(\vartheta(\cdot)\) _satisfies Lipschitz, as follows:_  
\[|\vartheta(f_{1})-\vartheta(f_{2})|\leq\omega|f_{1}-f_{2}|,\forall f_{1},f_ {2}\in\mathbb{R}^{n} \tag{3}\]  
**Remark 1**. : _Both the follower and the leader contain nonlinear terms. However, this paper selects different nonlinear terms for the leader and follower, so (1) and (2) constitute heterogeneous nonlinear MASs._  
In order to analyze changing topology \(\overline{G}\) of the leader-follower MAS, the following general assumption is given: there is a switching signal \(\sigma:[t_{0},\infty)\rightarrow\mathcal{P}\), which is a piecewise constant. \(\mathcal{P}\) is the limited collection of all potential interconnection topologies of the MAS, and \(t_{0}\) is the initial time. Defining the vertices \(\{0,1,2,\cdots,N\}\) as representing all possible nodes on the graph, the set of all possible topological connection diagrams is denoted by \(\{\overline{G}:p\in\mathcal{P}\}\), and \(\{\overline{G}:p\in\mathcal{P}\}\) is the sub-atlas defined at point \(\{0,1,2,\cdots,N\}\). **Remark 2**. : _In order to ensure that each follower tracks the leader in completing the TVF, by defining the \(h(t)\) required for a MAS, it is clear that \(h(t)\) is a function of time, meaning that the relative offset between all followers and leaders is constantly changing over time. And this paper also needs to consider that the topology is switched over time._  
**Assumption 2**. : _The pair \((A,B)\) is stabilizable._  
**Assumption 3**. : _The leader-related vertex 0 is the global reachable point in the undirected graph \(\overline{G}\)._  
**Assumption 4**. : _The child graph associated with the follower is undirected, and in the union of all such subgraphs within the graph, the leader has a directed path to all followers._  
**Definition 1**. : _In the case of any arbitrarily provided initial state, if the closed-loop system satisfies the following condition:_  
\[\lim_{t\rightarrow\infty}\lvert x_{i}(t)-x_{0}(t)-h_{i}(t)\rvert=0,\quad i=1,2,\cdots,N \tag{4}\]  
_then the heterogeneous nonlinear MASs defined by (1) and (2) satisfy the above conditions, indicating that the MASs complete the TVF control under the jointly connected topology._  
The temporal interval \([t_{0},\infty)\) is made up of bounded, non-intersecting, continuous time intervals \([t_{j},t_{j+1})\) for \(j=0,1,\cdots\) and \(t_{0}=0\). For each separate interval \([t_{j},t_{j+1})\), there is a non-overlapping interval sequence \([t_{k}^{0},t_{k}^{1}),[t_{k}^{1},t_{k}^{2}),\cdots,[t_{k}^{m_{k}-1},t_{k}^{m_ {k}})\), \(t_{k}=t_{k}^{0},t_{k+1}=t_{k}^{m_{k}}\)satisfying \(t_{k}^{j+1}-t_{k}^{j}\geq\tau\), \(0\leq j\leq m_{k}-1\).