## 3 Theoretical Analysis
### Formation Analysis
\right.^{\prime} \tag{27}\]  
_We can thus prove the necessary and sufficient conditions for the multi-AUV system in Equation (17) to obtain \(m(t)\) by the following theorem._  
**Theorem 1**. : _The multi-AUV system in Equation (17) establishes the time-varying formation \(m(t)\) if, and only if, the following conditions hold simultaneously:_  
(1) \[\lim\limits_{i\to\infty}\bigl{(}\bigl{(}m_{iv}(t)-m_{jv}(t)\bigr{)}-\bigl{(}\dot{ m}_{ix}(t)-\dot{m}_{jx}(t)\bigr{)}\bigr{)}=0,\,(i\in\{1,2\ldots,N\}j\in N_{i}),\] (28)  
(2) _if_ \(i\in\{1,2\ldots,N\}\) _and_ \(\psi_{i}=k_{12}k_{11}-\text{Re}(\lambda_{i})(k_{12}k_{21}+k_{11}k_{22})+\Bigl{(} \text{Re}(\lambda_{i})^{2}+\text{Im}(\lambda_{i})^{2}\Bigr{)}k_{21}k_{22}\)_, then:_  
\[-k_{12}+\text{Re}(\lambda_{i})k_{22}>0, \tag{29}\] \[(-k_{12}+\text{Re}(\lambda_{i})k_{22})\psi_{i}-\text{Im}(\lambda_{ i})^{2}k_{21}^{2}>0, \tag{30}\]  
The proof process is as follows:  
Define the following:  
\[\left\{ \begin{array}{l}\widetilde{\theta}_{\mathbb{C}}(t)=(U\otimes I_{2}) \bigl{[}\xi^{H}(t),0\bigr{]}^{H}\\ \widetilde{\theta}_{\mathbb{C}}(t)=(U\otimes I_{2})\bigl{[}0,\varsigma^{H}(t )\bigr{]}^{H}\\ e_{i}\in R^{N}\end{array} \right., \tag{31}\]  
where \(e_{i}\in R^{N}\) is a vector with 1 as its \(i\)th component and 0 as other components. Note that \(c_{1}\) and \(c_{2}\) refer to the linearly independent vectors. \(\alpha_{1}(t)\), \(\alpha_{2}(t)\), and \(\alpha_{2k+j}(t)(k=1,2,\ldots,N-1;j=1,2)\) must exist to meet \(\xi(t)=\alpha_{1}(t)c_{1}+\alpha_{2}(t)c_{2}\) and \(\varsigma(t)=\bigl{[}\alpha_{3}(t)c_{1}^{H}+\alpha_{4}(t)c_{2}^{H},\ldots, \alpha_{2N-1}(t)c_{1}^{H}+\alpha_{2N}(t)c_{2}^{H}\bigr{]}^{H}\). Because \(\bigl{[}\xi^{H}(t),0\bigr{]}^{H}=e_{1}\otimes\xi(t)\),  
\[\widetilde{\theta}_{\mathbb{C}}(t) =(U\otimes I_{2})(e_{1}\otimes\xi(t)) \tag{32}\] \[=\overline{u}_{1}\otimes\xi(t)\] \[=\alpha_{1}(t)p_{1}+\alpha_{2}(t)p_{2}\in\mathbb{C}(\text{U})\]  
Because \(p_{j}(j=3,4,\ldots,2N)\), Equation (32) can be expressed as follows:  
\[\widetilde{\theta}_{\mathbb{C}}(t) =\sum\limits_{i=2}^{N}(\alpha_{2i-1}(t)(\overline{u}_{i}\otimes c _{1})+\alpha_{2i}(t)(\overline{u}_{i}\otimes c_{2})) \tag{33}\] \[=\sum\limits_{j=3}^{2N}(\alpha_{j}(t)p_{j})\in\overline{\mathbb{C }}(\text{U})\]  
\[\bigl{[}\xi^{H}(t),\varsigma^{H}(t)\bigr{]}^{H}=\bigl{(}U^{-1}\otimes I_{2} \bigr{)}\widetilde{\theta}(t)\text{ then }\widetilde{\theta}(t)=\widetilde{\theta}_{\mathbb{C}}(t)+ \widetilde{\theta}_{\mathbb{C}}(t).\]  
According to Lemmas 2 and 3, the only condition is that when \(\lim_{t\to\infty}\widetilde{\theta}_{\mathbb{C}}(t)=0\), the multi-AUV system in Equation (17) obtains \(m(t)\), so:  
\[\lim\limits_{t\to\infty}\varsigma(t)=0, \tag{34}\]  
\[\dot{\widetilde{\varsigma}}(t)=I_{N-1}\otimes\Bigl{(}B_{2}K_{1}+B_{1}B_{2}^{T }\Bigr{)}-\overline{J}\otimes(B_{2}K_{2})\widetilde{\varsigma}(t), \tag{35}\]  
Now, the multi-AUV system in Equation (17) can establish formation \(m(t)\) because Equation (25) is under the condition from Equations (26) and (34). So:  
\[\lim\limits_{t\to\infty}\Bigl{(}\overline{\Omega}\otimes B_{1}\Bigr{)}\bigl{(} m_{v}(t)-\dot{m}_{x}(t)\bigr{)}=0, \tag{36}\]  
Based on Equation (28), then:  
\[\lim\limits_{t\to\infty}(L\otimes B_{1})\bigl{(}m_{v}(t)-\dot{m}_{x}(t)\bigr{)} =0, \tag{37}\]Based on Equation (16), then:  
\[=\overline{u}_{1}\otimes\xi(t)\]0  
Now multiplying both sides of Equation (37) with \(U^{-1}\otimes I\),  
\[=\overline{u}_{1}\otimes\xi(t)\]1  
Note that \(\overline{J}\) is nonsingular by Lemma 1. Next, multiplying both sides of Equation (39) with \(\overline{J}^{-1}\otimes I_{2}\) results in Equation (36). It shows Equation (12) being sufficient for Equation (36). Then, define \(\widetilde{U}=[\hat{U},\hat{u}]\), \(\hat{U}\in\mathbb{C}^{(N-1)\times(N-1)}\), where \(\hat{u}\in\mathbb{C}^{(N-1)\times 1}\) means that \(\hat{u}\) is the last column vector of \(\widetilde{U}\). Then, we \(\text{rank}\Big{(}\widetilde{U}\Big{)}=N-1\) and \(\text{rank}\big{(}\hat{U}\big{)}=N-1\).