## 2 Preliminaries
### Formation Description  
In this note, the multi-agent system is considered as the undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\), in which each agent is a vertex in \(\mathcal{V}\). Additionally, a pair \((i,j)\in\mathcal{E}\) implies that the \(i\)th agent can simultaneously receive/transmit its state information from/to the \(j\)th agent. From our control objectives, the group of \(N\) agents is navigated by a proposed control algorithm to reach an anticipated formation. Generally, a formation of the multiple agents is considered as a geometric shape in three-dimensional space that satisfies some prescribed constraints achieved and preserved by the group of agents. The formation control survey in [9] reported various formation representations, such as distance-based, displacement-based, and position-based formations. In this paper, let us define a reference trajectory of the formation \(r(t):\mathbb{R}_{+}\rightarrow\mathbb{R}^{n}\) and a formation shape vector of the \(i\)th agent \(f_{p,i}(t):\mathbb{R}_{+}\rightarrow\mathbb{R}^{n}\), as seen in Figure 1. Additionally, \(f_{p,i}(t)\) are continuously twice differentiable functions, i.e., \(f_{p,i}(t)=\dot{f}_{p,i}(t)\), \(f_{a,i}(t)=\ddot{f}_{p,i}(t)\), and \(\|f_{a,i}(t)\|<\varpi_{fa}\), where \(\varpi_{fa}\) is a positive constant. Furthermore, in the group, at least one agent knows the reference trajectory of the formation \(r(t)\), and the \(i\)th agent only knows its position in the formation via \(f_{p,i}(t)\). Let \(\delta_{i}=1\) if the \(i\)th agent knows \(r(t)\) and \(\delta_{i}=0\) otherwise. Let us consider that \(r(t)\) satisfies the following:  
\[\dot{r}(t)=v_{0}, \tag{1}\]  
where \(v_{0}\in\mathbb{R}^{n}\) is the constant reference velocity of the formation. Let us consider time-varying formation control for \(N\) second-order agents, each of which has the following dynamics:  
\[ \begin{cases}\dot{p}_{i}(t)=v_{i}(t),\\ \dot{v}_{i}(t)=u_{i}(t).\end{cases}  \tag{2}\]  
where \(i\in\mathcal{V}=\{1,2,\ldots,N\}\) begin the set of all agents and \(p_{i}(t),v_{i}(t)\in\mathbb{R}^{n}\) stand for the state of agent \(i\)th with control input \(u_{i}(t)\). **Definition 1**. : _The multiple agents (2) are said to achieve the state formation specified by the vectors \(f_{p,i}(t)\), \(i\in\mathcal{V}\) for any given bounded initial states if:_  
\[\lim_{t\rightarrow\infty}\big{(}p_{i}(t)-f_{p,i}(t)-r(t)\big{)}=0,\ i\in \mathcal{V}. \tag{3}\]  
**Lemma 2**. : _By letting \(\Delta=\textbf{diag}\,(\delta_{1},\,\delta_{2},\,\ldots,\,\delta_{N})\in \mathbb{R}^{N\times N}\), if the graph \(\mathcal{G}\) is undirected and strongly connected, the matrices \(\mathcal{L}+\Delta\) and \(\mathcal{M}=(\mathcal{L}+\Delta)\otimes I_{n}\) are symmetric positive-definite._  
Proof of Lemma 2.: Let us recall Lemma 1. The Laplacian symmetric matrix \(\mathcal{L}\) is semi-positive definite, in that there is only one zero eigenvalue and the rest are positive. \(x^{T}\mathcal{L}x=0\) iff \(x=a\ \mathbf{1}_{n}\), \(a\in\mathbb{R}\), while \(x^{T}\mathcal{L}x>0\) for any \(x\
eq 0\in\mathbb{R}^{N}\). Moreover, from the previously-defined matrix \(\Delta\) with \(\delta_{i}\in\{0,1\}\), \((i\in\mathcal{V})\), because it is assumed that there is at least one \(\delta_{i}=1\), \(x^{T}\Delta x>0\) for any \(x=a\mathbf{1}_{n}\), \(a\
eq 0\). Therefore, it is clear that \(x^{T}\left(\mathcal{L}+\Delta\right)x>0\) for any \(x\
eq 0\in\mathbb{R}^{N}\), leading to \((\mathcal{L}+\Delta)>0\). Furthermore, it is obvious that the matrix \(\mathcal{M}\) is symmetric by taking the Kronecker product of the symmetric \((\mathcal{L}+\Delta)\) and the identity matrix. Let us consider the vector \(\xi=\big{[}\xi_{1}^{T},\,\xi_{2}^{T},\ldots,\xi_{n}^{T}\big{]}^{T}\in\mathbb{ R}^{nN}\), in which \(\xi_{1}\), \(\xi_{2}\), \(\ldots\), \(\xi_{n}\in\mathbb{R}^{N}\), and \(\xi\
eq 0\):  
\[\xi^{T}\mathcal{M}\xi =\left[\xi_{1}^{T}\left(\mathcal{L}+\Delta\right),\xi_{2}^{T} \left(\  
**Remark 1**.