## 3 Main Results
### Nonlinear Controller Design and Stability Analysis
Proof: Firstly, we introduce the combined vector \(\overset{\sim t}{\mathbf{X}_{i}},\overset{\sim t}{\mathbf{Z}_{i}}\), and permutation matrix \(\mathbf{P}_{e}^{t}\), where \(\overset{\sim t}{\mathbf{X}_{i}}=\left(\left[\overset{\sim t}{\mathbf{X}_{i_{\overline {i}}}}^{t}\right]_{1}^{nN}\right)^{\mathsf{T}},\overset{\sim t}{\mathbf{Z}_{i}}= \left[\overset{\sim t}{\mathbf{X}_{i_{\overline{i}}}}\right]_{n}^{N}\), and \(\mathbf{P}_{e}^{t}\) satisfies:  
\[\overset{\sim t}{\mathbf{Z}_{i}}\mathbf{P}_{e}^{t}=\left[\overset{\sim t}{\mathbf{Z}_{i}} \quad\overset{\sim\mathbf{O}_{t}}{\mathbf{Z}_{i}}\right] \tag{10}\]  
where \(\overset{\sim t}{\mathbf{Z}_{i}}=\left[\overset{\sim t}{\mathbf{X}_{i_{\overline{i}} }}^{t}\right]_{n}^{L},\overset{\sim\mathbf{O}_{t}}{\mathbf{Z}_{i}}=\left[\overset{\sim t }{\mathbf{X}_{i_{\overline{i}}}}^{t}\right]_{n}^{N-L}\), \(k=1,2,\cdots,N\). \(\overset{\sim\mathbf{L}_{i}}{\mathbf{Z}_{i}}\) represents the interacted information from nearby agents, and \(\overset{\sim}{\mathbf{Z}_{i}}\) denotes the information of agents without interaction which is impossible to obtain. Reshaping \(\overset{\sim t}{\mathbf{Z}_{i}}\mathbf{P}_{e}^{t}\) yields:  
\[\phi\left(\overset{\sim t}{\mathbf{Z}_{i}}\mathbf{P}_{e}^{t}\right)=\left[\overset{ \sim t}{\mathbf{X}_{i}}\quad\overset{\sim\mathbf{O}_{t}}{\mathbf{X}_{i}}\right]^{\mathsf{T}} \tag{11}\]  
where \(\overset{\sim t}{\mathbf{X}_{i}}=\left[\overset{\sim t}{\mathbf{X}_{i_{\overline{i}} }}^{t}\right]_{1}^{nL},\overset{\sim\mathbf{O}_{t}}{\mathbf{X}_{i}}=\left[\overset{ \sim t}{\mathbf{X}_{i_{\overline{i}}}}^{t}\right]_{n}^{N}\). Define the rotation matrices \(\mathbf{Q}_{i}^{t}\) with \(\mathbf{\mathcal{T}}_{0i}^{\ell}=\mathbf{\Lambda}_{L}\big{(}\mathbf{\mathcal{T}}_{\mathbf{0}_{i} \mathbf{1}}\big{)}\). \[\mathbf{Q}_{i}^{t}= \begin{bmatrix}\mathbf{\mathcal{T}}_{0i}^{\ell}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}  \tag{12}\]  
Define the Lyapunov function \(V^{i}(t)=V^{i}_{1}(t)+V^{i}_{2}(t)\) with  
\[\left\{ \begin{array}{l}V^{i}_{1}(t)=\frac{1}{2}\Big{(}\mathbf{Q}_{i}^{t}\mathbf{ \phi}\left(\widetilde{\widetilde{\mathbf{Z}}_{i}}\mathbf{P}_{\epsilon}^{t}\right) \Big{)}^{\mathsf{T}}\mathbf{R}_{1}\mathbf{Q}_{i}^{t}\mathbf{\phi}\left(\widetilde{\widetilde {\mathbf{Z}}_{i}}\mathbf{P}_{\epsilon}^{t}\right)\\ V^{i}_{2}(t)=\frac{1}{2}\Big{(}\widetilde{\widetilde{\mathbf{q}}_{i}}^{t}\Big{)}^{ \mathsf{T}}\mathbf{R}_{2}\widetilde{\widetilde{\mathbf{q}}_{i}}^{t}\end{array} \right. \tag{13}\]  
where \(\widetilde{\mathbf{q}}_{i}^{t}=D_{1}\Big{(}\mathbf{Q}_{i}^{t}\mathbf{\phi}\left(\widetilde {\widetilde{\mathbf{Z}}_{i}}\mathbf{P}_{\epsilon}^{t}\right)\Big{)}\), and matrices \((\mathbf{R}_{1},\;\mathbf{R}_{2})\) satisfy:  
\[\left\{ \begin{array}{l}\mathbf{R}_{1}=\text{diag}\big{(}\mathbf{\Lambda}_{L}\big{(} \eta_{j_{k}}a_{ij_{k}}\big{)},\mathbf{0}_{N-L}\big{)}\otimes\mathbf{I}_{2}\\ \mathbf{R}_{2}=\eta_{0}\text{diag}\big{(}\mathbf{\Lambda}_{L}\big{(}\eta_{j_{k}}a_{ij_ {k}}\big{)},\mathbf{0}_{N-L}\big{)}\otimes\mathbf{I}_{2}\end{array} \right. \tag{14}\]  
With \(\text{diag}(\cdot)\) being a diagonal matrix, \(\mathbf{0}_{N-L}\) being a zero matrix, and \(\left\{\eta_{i}\right\}_{0}^{L+1}\) being positive real numbers. Taking the derivative of (12), we have:  
\[\left\{ \begin{array}{l}\dot{V}^{i}_{1}(t)=\Big{(}\mathbf{Q}_{i}^{t}\mathbf{\phi} \left(\widetilde{\widetilde{\mathbf{Z}}_{i}}\mathbf{P}_{\epsilon}^{t}\right)\Big{)}^{ \mathsf{T}}\mathbf{R}_{1}\widetilde{\mathbf{q}}_{i}^{t}\\ =\Big{(}\mathbf{\mathcal{T}}_{0i}^{\ell}\widetilde{\mathbf{X}}_{i}^{t}\Big{)}^{ \mathsf{T}}\big{(}\mathbf{\Lambda}_{L}\big{(}\eta_{j_{k}}a_{ij_{k}}\big{)}\otimes \mathbf{I}_{2}\big{)}\widetilde{\widetilde{\mathbf{q}}_{i}}^{\ell_{t}}\\ \dot{V}^{i}_{2}(t)=\eta_{0}\bigg{(}\widetilde{\widetilde{\mathbf{q}}_{i}}^{\ell_{t} }\Big{)}^{\mathsf{T}}\big{(}\mathbf{\Lambda}_{L}\big{(}\eta_{j_{k}}a_{ij_{k}}\big{)} \otimes\mathbf{I}_{2}\big{)}D_{1}\bigg{(}\widetilde{\widetilde{\mathbf{q}}_{i}}^{\ell_ {t}}\bigg{)}\end{array} \right.