# Event-Triggered MFAILC Bipartite Formation Control for Multi-Agent Systems Under DoS Attacks
## 4 Proof of Convergence
### Boundedness of the \(\hat{\Phi}_{p}(k,i)\)  
In column vector form, \(\hat{\Phi}_{p}(k,i)\) is first rewritten as follows:  
\[\hat{\Phi}_{p}(k,i)=\begin{bmatrix}\hat{\phi}_{p,11}(k,i)&\hat{\phi}_{p,12}( k,i)&\cdots&\hat{\phi}_{p,1m}(k,i)\\ \hat{\phi}_{p,21}(k,i)&\hat{\phi}_{p,22}(k,i)&\cdots&\hat{\phi}_{p,2m}(k,i)\\ \vdots&\vdots&\ddots&\vdots\\ \hat{\phi}_{p,m1}(k,i)&\hat{\phi}_{p,m2}(k,i)&\cdots&\hat{\phi}_{p,mm}(k,i) \end{bmatrix}=[\hat{\phi}_{p,1}^{T}(k,i),\ldots,\hat{\phi}_{p,m}^{T}(k,i)]^{T}\]  
where \(\hat{\phi}_{p,l}=\left[\hat{\phi}_{p,1l}(k,i),\ldots,\hat{\phi}_{p,lm}(k,i) \right]\), \(l=1,2,\ldots m\). Rewriting Equation (17) as below:  
\[\hat{\phi}_{p,l}(k,i)=\hat{\phi}_{p,l}(k,i-1)+\frac{\eta(\Delta y_{p}(k+1,i-1)- \hat{\phi}_{p,l}(k,i-1)\Delta u_{p}(k,i-1))}{\mu+\left\|\Delta u_{p}(k,i-1) \right\|^{2}}\Delta u_{p}^{T}(k,i-1)) \tag{20}\]Defining \(\tilde{\phi}_{p,l}(k,i)=\hat{\phi}_{p,l}(k,i)-\phi_{p,l}(k,i)\), and subtracting \(\phi_{p,l}(k,i)\) from both sides of (20), and combining with Equation (4), we obtain:  
\[\begin{split}\tilde{\phi}_{p,l}(k,i)&=\tilde{\phi} _{p,l}(k,i-1)+\phi_{p,l}(k,i-1)-\phi_{p,l}(k,i)+\\ &\frac{\eta(\Delta y_{p}(k+1,i-1)-\hat{\phi}_{p,l}(k,i-1)\Delta u _{p}(k,i-1))}{\mu+\big{\|}\Delta u_{p}(k,i-1)\big{\|}^{2}}\Delta u_{p}^{T}(k,i- 1))\\ &=\tilde{\phi}_{p,l}(k,i-1)+\phi_{p,l}(k,i)-\phi_{p,l}(k,i-1)\\ &+\frac{\eta(\Delta y_{p}(k+1,i-1)-\hat{\phi}_{p,l}(k,i-1)\Delta u _{p}(k,i-1))}{\mu+\big{\|}\Delta u_{p}(k,i-1)\big{\|}^{2}}\Delta u_{p}^{T}(k,i -1))\end{split} \tag{21}\]  
Taking the norm of both sides of Equation (21), it follows \(\big{\|}\Phi_{p}(k,i)\big{\|}\leq\overline{b}\) from Theorem 1 that:  
\[\begin{split}\big{\|}\tilde{\phi}_{p,l}(k,i)\big{\|}& \leq\bigg{\|}\tilde{\phi}_{p,l}(k,i-1)(I-\frac{\eta\Delta u_{p}(k,i-1) \Delta u_{p}^{T}(k,i-1)}{\mu+\big{\|}\Delta u_{p}(k,i-1)\big{\|}^{2}})\bigg{\|} +\big{\|}\phi_{p,l}(k,i)-\phi_{p,l}(k,i-1)\big{\|}\\ &\leq\bigg{\|}\tilde{\phi}_{p,l}(k,i-1)(I-\frac{\eta\Delta u_{p} (k,i-1)\Delta u_{p}^{T}(k,i-1)}{\mu+\big{\|}\Delta u_{p}(k,i-1)\big{\|}^{2}}) \bigg{\|}+2b\end{split} \tag{22}\]  
Squaring both sides of the norm in the above equation, we obtain:  
\[\begin{split}&\bigg{\|}\tilde{\phi}_{p,l}(k,i-1)(I-\frac{\eta \Delta u_{p}(k,i-1)\Delta u_{p}^{T}(k,i-1)}{\mu+\big{\|}\Delta u_{p}(k,i-1) \big{\|}^{2}})\bigg{\|}^{2}\\ &=\big{\|}\tilde{\phi}_{p,l}(k,i-1)\big{\|}^{2}+(-2+\frac{\eta \big{\|}\Delta u_{p}(k,i-1)\big{\|}^{2}}{\mu+\big{\|}\Delta u_{p}(k,i-1) \big{\|}^{2}})\times\frac{\eta\big{\|}\hat{\phi}_{p,j}(k,i-1)\Delta u_{p}(k,i- 1)\big{\|}^{2}}{\mu+\big{\|}\Delta u_{p}(k,i-1)\big{\|}^{2}}\end{split} \tag{23}\]  
Selecting \(\mu>0\) and \(\eta\in(0,2)\), then  
\[-2+\frac{\eta\big{\|}\Delta u_{p}(k,i-1)\big{\|}^{2}}{\mu+\big{\|}\Delta u_{p} (k,i-1)\big{\|}^{2}}<0 \tag{24}\]  
From Equations (23) and (24), it follows that there exists a \(c_{1}\in(0,1)\) such that:  
\[\begin{split}\big{\|}\tilde{\phi}_{p,l}(k,i)\big{\|}& \leq c_{1}\big{\|}\tilde{\phi}_{p,l}(k,i-1)\big{\|}+2b\\ &\leq c_{1}^{2}\big{\|}\tilde{\phi}_{p,l}(k,i-2)\big{\|}+2 bc_{1}+ 2b\\ &\leq c_{1}^{3}\big{\|}\tilde{\phi}_{p,l}(k,i-3)\big{\|}+2 bc_{1}^{2 }+2 bc_{1}+2b\\ &\leq\cdots\leq c_{1}^{i-1}\big{\|}\tilde{\phi}_{p,l}(k,1)\big{\|} +\frac{2b(1-c_{1}^{i-1})}{1-c_{1}}\end{split} \tag{25}\]  
From the above, \(\tilde{\phi}_{p,l}(k,i)\) is bounded; since \(\phi_{p,l}(k,i)\) is bounded, we obtain a conclusion that \(\hat{\phi}_{p,l}(k,i)\) and \(\hat{\Phi}_{p}(k,i)\) are bounded.