# Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control
## 2 Preliminaries and Problems Formulation
### Controller Design and Stability Analysis
As no edges are lost \(\forall t\geq 0\), one has \(m_{k}\leq N(N-1)/2-m_{0}\), the number of switching times of the system (1) is finite, thus the topology of graph \(\mathcal{G}\) becomes fixed eventually. Denote the last topology switching instant as \(\hat{t}_{k^{\prime}}\), then \(Q(t)\) is continuous and monotonously decreasing for \(t\in[\hat{t}_{k^{\prime}},\infty)\). Hence, the set \(\Omega:=\{\tilde{x}(t)\in D,\tilde{v}(t)\in\mathbb{R}^{Nn}|Q(\tilde{x}(t), \tilde{v}(t))\leq Q^{*}\}\) is positive invariant, where  
\[t_{k+1}^{i}=\inf\left\{t>t_{k}^{i}:g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i} ^{y}(t),\tilde{v}_{i}(t)\right)\geq 0\right\}, \tag{9}\]3 \[t_{k+1}^{i}=\inf\left\{t>t_{k}^{i}:g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i} ^{y}(t),\tilde{v}_{i}(t)\right)\geq 0\right\}, \tag{9}\]4  
\(\tilde{x}=(\tilde{x}_{11},\cdots,\tilde{x}_{1N},\cdots,\tilde{x}_{N1},\cdots,\tilde{x}_{NN})^{T}\in\mathbb{R}^{N^{2}n}\) and \(\tilde{v}=(\tilde{v}_{1},\cdots,\tilde{v}_{N})^{T}\in\mathbb{R}^{Nn}\). Since \(\mathcal{G}(t)\) is connected for \(t\geq 0\), one has \(\left\|\tilde{x}_{ij}(t)\right\|\leq(N-1)r,\forall i,j\in\mathcal{E}(t)\), besides, one has \(\tilde{v}_{i}^{T}(t)\tilde{v}_{i}(t)\leq 2Q^{*}\Rightarrow\left\|\tilde{v}_{i}(t) \right\|\leq\sqrt{2Q^{*}}\), and thus the set \(\Omega\) is compact. It follows from [PERSON]'s invariance principle that if the initial condition lies in \(\Omega\), its trajectories will converge to the largest invariant set inside the region \(S=\{\tilde{x}(t)\in\mathbb{R}^{N^{2}n},\tilde{v}(t)\in\mathbb{R}^{Nn}|\dot{Q} (\tilde{x}(t),\tilde{v}(t))=0\}\). Thus, the MAS converges asymptotically to a fixed configuration corresponding to the extreme of the global potential \(Q(t)\). Please note that not all solutions of (1) converge to local minima. However, anything but the local minima is an unstable equilibrium [25]. Thus, every final configuration locally minimizes each agent's global potential, which implies \(\left\|x_{ij}\right\|\in[\underline{d},\underline{d}],\forall(i,j)\in\mathcal{E}\).Besides, from (12), one has \(\dot{Q}(t)=0\) if and only if \(\tilde{v}_{1}(t)=\tilde{v}_{2}(t)=\cdots=\tilde{v}_{N}(t)\), which is equivalent to \(\left\|v_{i}(t)-v_{j}(t)\right\|=0,\forall(i,j)\in\mathcal{E}\). Thus, the flocking motion is achieved. \(\square\)  
**Remark 2**. : _Since the potential function \(\psi(\cdot)\) is bounded, we can also have that the action function \(\phi(\cdot)\) is bounded. Thus, there exists a constant \(M>0\), such that \(\left\|\phi(z)\right\|\leq M,\forall z\in(0,\infty)\)._  
_Let \(\tilde{e}_{i}(t)=1/2a_{1}\big{\|}\tilde{e}_{i}^{x}(t)\big{\|}^{2}+c/2a_{2} \big{\|}\tilde{e}_{i}^{v}(t)\big{\|}^{2}\). Combining (7) with inequality (10), we can get that \(\left\|\tilde{e}_{i}(t)\right\|\) decreasing to \(\theta_{1}=1/(1+\sqrt{\eta})\left\|\tilde{v}_{i}\left(\hat{t}_{k}^{i}\right)\right\|\) and \(\left\|\tilde{e}_{i}(t)\right\|\) increasing to \(\theta_{2}=1/(1-\sqrt{\eta})\left\|\tilde{v}_{i}\left(\hat{t}_{k}^{i}\right)\right\|\) are the two \"worst\" cases when event occurs. Thus, \(\tilde{t}_{k}^{i}\) exists when \(\tilde{v}_{i}(\hat{t}_{k}^{i})\
eq 0\)._  
_From (7), one has \(\hat{e}_{i}^{x}\left(t\right)=-\hat{q}_{i}^{x}(t),\hat{e}_{i}^{v}\left(t\right) =-\hat{q}_{i}^{x}(t)\).