# Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control
## 3 Main Results
\begin{split} V_{2}^{\frac{2\alpha}{\alpha+1}}(t)=& \Big{(}\sum_{j=1}^{n_{2}}\omega_{j}\Big{(}\frac{\beta}{1+\alpha}|y_{j}|^{1+ \alpha}+\frac{\gamma}{2}y_{j}^{2}\Big{)}\Big{)}^{\frac{2\alpha}{\alpha+1}}\\ \leq&\sum_{j=1}^{n_{2}}\omega_{j}^{\frac{2\alpha}{ \alpha+1}}\Big{(}\frac{\beta}{1+\alpha}|y_{j}|^{1+\alpha}+\frac{\gamma}{2}y_{ j}^{2}\Big{)}^{\frac{2\alpha}{\alpha+1}}\\ \leq&\sum_{j=1}^{n_{2}}\omega_{j}^{\frac{2\alpha}{ \alpha+1}}\Big{(}(\frac{\beta}{1+\alpha})^{\frac{2\alpha}{\alpha+1}}|y_{j}|^{2 \alpha}+(\frac{\gamma}{2})^{\frac{2\alpha}{\alpha+1}}|y_{j}|^{\frac{4\alpha}{ \alpha+1}}\Big{)}.\end{split}  \tag{13}\]  
Consider that \(\frac{v^{T}\,\big{(}\text{diag}(\omega)L+L^{T}\text{diag}(\omega)\big{)}\,v}{ 2}\) is a non-negative continuous function defined on \(\mathbb{R}^{N}\), and \(\frac{v^{T}\,\big{(}\text{diag}(\omega)L+L^{T}\text{diag}(\omega)\big{)}\,v}{ 2}\
eq 0\) for \(v\in\mathcal{U}=\{\,\beta\text{sig}^{\alpha}(v)+\gamma\,v\,|\,v\perp\omega\,v \in\mathbb{R}^{N}\}\cap S^{N-1}\). In addition, \(\mathcal{U}\) is a bounded closed set; therefore, the minimal value  
\[M_{1}=\min_{v\in\mathcal{U}}(\frac{v^{T}\,\big{(}\text{diag}(\omega)L+L^{T} \text{diag}(\omega)\big{)}\,v}{2})\
eq 0\]  
exists. Then, for \(y\) belonging to \(\mathcal{U}\), there is  
\[\frac{\big{(}\beta\text{sig}^{\alpha}(y)+\gamma\,y\big{)}^{T}\,\text{diag}( \omega_{1},\dots,\omega_{n_{2}})\,L\,\big{(}\beta\text{sig}^{\alpha}(y)+\gamma \,y\big{)}}{\big{(}\beta\text{sig}^{\alpha}(y)+\gamma\,y\big{)}^{T}\big{(} \beta\text{sig}^{\alpha}(y)+\gamma\,y\big{)}}\geq M_{1}. \tag{14}\]  
This leads to  
\[-\dot{V}_{2}(t)\geq M_{1}\,\sum_{j=1}^{n_{2}}\Big{(}\beta\text{sig}^{\alpha}(y _{j})+\gamma\,y_{j}\Big{)}^{2}. \tag{15}\]  
Consider the case when \(0<|y|\leq 1\), there are  
\[ \begin{split} V_{2}^{\frac{2\alpha}{\alpha+1}}(t)\leq& \max_{\ell\in\mathcal{I}}(\omega_{\ell}^{\frac{2\alpha}{\alpha+1}}) \Big{(}(\frac{\beta}{1+\alpha})^{\frac{2\alpha}{\alpha+1}}+(\frac{\gamma}{2}) ^{\frac{2\alpha}{\alpha+1}}\Big{)}\sum_{j=1}^{n_{2}}|y_{j}|^{2\alpha}\\ \leq&\max_{\ell\in\mathcal{I}}(\omega_{\ell}^{\frac {2\alpha}{\alpha+1}})\Big{(}(\frac{\beta}{1+\alpha})^{\frac{2\alpha}{\alpha+1} }+(\frac{\gamma}{2})^{\frac{2\alpha}{\alpha+1}}\Big{)}\\ &-\dot{V}_{2}(t)\geq M_{1}(\beta+\gamma).\end{split}  \tag{16}\]  
Thus,  
\[-\frac{\dot{V}_{2}(t)}{V_{2}^{\frac{2\alpha}{\alpha+1}}(t)}\geq\frac{M_{1}( \beta+\gamma)}{\frac{\text{max}(\omega_{\ell}^{\frac{2\alpha}{\alpha+1}}) \Big{(}(\frac{\beta}{1+\alpha})^{\frac{2\alpha}{\alpha+1}}+(\frac{\gamma}{2}) ^{\frac{2\alpha}{\alpha+1}}\Big{)}}}\doteq C_{1}. \tag{18}\]For the other case \(1<|y|\leq\delta\), consider the function \(\frac{\sum\limits_{i=1}^{n_{2}}\sum\limits_{t=1}^{n_{2}}l_{j}\omega_{j}\left( \beta isg^{a}(v_{t})+\gamma\,v_{t}\right)\left(\beta isg^{a}(v_{j})+\gamma\,v_{j} \right)}{\left(\sum\limits_{j=1}^{n_{2}}\omega_{j}\left(\frac{\beta}{1+\pi|y_{ j}|^{1+a}+\frac{2}{2}v_{j}^{2}}\right)\right)^{\frac{2}{\delta+1}}}\) with respect to the variable \(v=\left(v_{1},\ldots,v_{n_{2}}\right)\), which is continuous on the compact set \(\mathcal{O}=\{v:1\leq v^{T}v\leq\delta\}\); therefore, there exists a minimal value \(C_{2}=\min\limits_{1<|v|\leq\delta}\left(\frac{\sum\limits_{i=1}^{n_{2}}\sum \limits_{t=1}^{n_{2}}l_{j}\omega_{j}\left(\beta isg^{a}(v_{t})+\gamma\,v_{t} \right)\left(\beta isg^{a}(v_{j})+\gamma\,v_{j}\right)}{\left(\sum\limits_{j=1}^ {n_{2}}\omega_{j}\left(\frac{\beta}{1+\pi|y_{j}|^{1+a}+\frac{2}{2}v_{j}^{2}} \right)\right)^{\frac{2}{\delta+1}}}\right)\) such that \[-\frac{\dot{V}_{2}(t)}{V_{2}^{2\pi+1}(t)}\geq C_{2}.\] (19) As a result of the above, we can obtain \(-\frac{\dot{V}_{2}(t)}{V_{2}^{2\pi+1}(t)}\geq C\), where \(C=\max\{C_{1},C_{2}\}\). Based on the _Differential Comparison Principle_, \(V_{2}(t)\) will reach \(0\) in finite time \(t=\frac{(1+\omega)V_{2}^{2\pi+1}(0)}{(1-\alpha)C}\). The above proof process is also applicable when the case of \(p\,>\,2\) is considered.