# Affine Formation Maneuver Control for Multi-Agent Based on Optimal Flight System
## 4 Formation Control with Virtual Leader
### Proof of Main Results
Then, obtaining the derivative to time \(t\) for \(Z(t)\) as follows:  
\[\dot{Z}(t)={\sum_{i=1,i\in O}^{N}}\left(\
abla_{\mathbf{p}_{i}}U_{i}\right)+C_{2} \widetilde{\mathbf{v}}_{i}^{\mathsf{T}}\widetilde{\mathbf{p}}_{i}+\widetilde{\mathbf{v}}_{ i}^{\mathsf{T}}\ddot{\widetilde{\mathbf{v}}}_{i} \tag{29}\]  
thus, via substituting (20) to (30), we might have:  
\[\dot{Z}(t)={\sum_{i=1}^{N}}\widetilde{\mathbf{v}}_{i}^{\mathsf{T}}{\sum_{j=1,j\
eq i }^{N}}C_{1}a_{ij}(\mathbf{v}_{ij}-\mathbf{v}_{iv})+{\sum_{i=1}^{N}}C_{3} \widetilde{\mathbf{v}}_{i}^{\mathsf{T}}\ddot{\widetilde{\mathbf{v}}}_{i} \tag{30}\]  
and Equation (31) can be expressed in matrix form as follows:  
\[\dot{Z}(t)=-\widetilde{\mathbf{v}}^{\mathsf{T}}[(C_{1}Y(t)+C_{3}\Omega(t))\otimes I _{n}]\widetilde{\mathbf{v}} \tag{31}\]  
where \(\Omega(t)\) is the stress matrix, \(Y(t)=diag[\gamma_{1},\gamma_{2},\dots,\gamma_{N}]\), \(\widetilde{\mathbf{v}}=\left(\widetilde{\mathbf{v}}_{1}^{\mathsf{T}},\widetilde{\mathbf{v }}_{2}^{\mathsf{T}},\dots,\widetilde{\mathbf{v}}_{N}^{\mathsf{T}}\right)^{\mathsf{T}}\). Apparently, \(\Omega(t)\) and \(Y(t)\) are positive semi-definite, these two factors would explain that \(Z(t\geq 0)\) is non-increasing function with \(\dot{Z}(t)\leq 0\). Thus, the level set \(\psi=\left\{\left[\widetilde{\mathbf{p}}^{\mathsf{T}},\widetilde{\mathbf{v}}^{ \mathsf{T}}\right]^{\mathsf{T}}\in R^{2 nN}:Z(t)\leq Z(0),t\geq 0\right\}\) can been seen as a positively invariant set. Then, supposing that \(Z(0)<U_{r}\), the equation can possibly meet \(U_{i}(||\widetilde{\mathbf{p}}_{ij^{\prime}}||)\leq Z(t)\leq Z(0)<U_{r}\) (\(i\in O\), \(t\geq 0\)) in each edge \([e_{ij}]\). Therefore, the connectivity of graph \(G\) of all agents \(\mathbf{p}_{i}\) for swarm is shown within \(t\geq 0\) and \(i\in O\); that is to say, the part 1 of Theorem 1 is proved. As for part 2 of Theorem 1, it is obvious that \(U_{i}(||\widetilde{\mathbf{p}}_{ij^{\prime}}||)\leq Z(t)\leq Z(0)<U_{r}<U_{0}\) from above; thus, the obstacles collision is proved. The part 3 of the theorem, \(||\widetilde{\mathbf{p}}_{i}||\) and \(||\widetilde{\mathbf{v}}_{i}||\) are all bounded and level set \(\psi\) is tight from the above analysis. Moreover, all trajectories of agents would finally converge and satisfy the invariant set \(\psi^{\prime}=\left\{\left[\widetilde{\mathbf{p}}^{\mathsf{T}},\widetilde{\mathbf{v }}^{\mathsf{T}}\right]^{\mathsf{T}}:\dot{Z}(t)=0,t\geq 0\right\}\). As \(Y(t)\) and \(\Omega(t)\) are positive semi-definite from the definition, based on Formulation (31) and as \(\dot{Z}(t)=0\), we could obtain \(\widetilde{\mathbf{v}}^{\mathsf{T}}[Y(t)\otimes I_{n}]\widetilde{\mathbf{v}}=0\) and \(\widetilde{\mathbf{v}}^{\mathsf{T}}[\Omega(t)\otimes I_{n}]\widetilde{\mathbf{v}}=0\). Since the stress matrix \(\Omega(t)\) can be considered as generalized graph Laplacian matrices with both positive and negative edge weights, \(\Omega(t)\)can thus also meet the sum of squares property [30]; therefore, \(\widetilde{v}^{T}[(C_{1}Y(t)+C_{3}\Omega(t))\otimes I_{n}]\widetilde{v}\) can be equivalent to \(\widetilde{v}_{1}=\widetilde{v}_{2}=\ldots=\widetilde{v}_{N}\). To be more specific, this result can demonstrate that the errors of all agents' velocity would converge to 0 asymptotically. Thus, it can imply the \(\widetilde{v}_{1}=\widetilde{v}_{2}=\ldots=\widetilde{v}_{N}=0\), such that part 3 of the Theorem 1 is proved. \(\square\)  
After examining the proof process, it has been determined that parts 1-3 of the theorem align with the fundamental principles of swarm algorithms, namely cohesion, separation, and alignment as proposed by [PERSON]. Furthermore, these essential criteria have been employed to validate the algorithm presented in this paper. Despite the similarities between certain terms in Equation (20) and existing protocols, the swarm behavior guided by Equation (20) would be distinctively unique. This is due to the presence of a virtual agent serving as an evaluation index through the local trajectory.