## 3 Main Results
### Adaptive Rules for Parameters  
Referring to (4), we have \(v_{-}(t)\leq v_{i}(t)\leq v_{+}(t)\ \forall i\in\mathcal{N}_{a}\), where \(v_{-}(t)\) indicates the lower limit of \(v_{i}(t)\), and \(v_{+}(t)>0\) is the upper limit. In addition, \(v_{-}(t)\) and \(v_{+}(t)\) are time varying but satisfy the constraint \(-v_{m}\leq v_{-}(t)\leq v_{+}(t)\leq v_{m}\). **Theorem 1**. : _If \(V(t)\) is the quadratic scalar function with the derivative \(\dot{V}(t)=-\pi_{1}V(t)\pm\sqrt{V(t)}\pi(t)\), where \(\pi_{1}>0\) and \(\pi(t)\) satisfies the inequation \(-\pi_{m}\leq\pi_{-}(t)\leq\pi(t)\leq\pi_{+}(t)\leq\pi_{m}\), then \(V(t)\) is bounded by \(\sqrt{V(t)}\leq\pi_{m}\)._  
**Proof. ** Handling the derivative, we have \(\dot{V}(t)=\sqrt{V(t)}\Big{(}-\pi_{1}\sqrt{V(t)}\pm\pi(t)\Big{)}\). Using the condition \(-\pi_{m}\leq\pi_{-}(t)\leq\pi(t)\leq\pi_{+}(t)\leq\pi_{m}\) yields \(\dot{V}(t)\leq\sqrt{V(t)}\Big{(}-\pi_{1}\sqrt{V(t)}+\pi_{m}\Big{)}\), we obtain the result \(\sqrt{V(t)}\leq\pi_{m}\) referring to Lemma 4. \(\square\)  
**Proposition 2. **_Define the max acceleration, angular rate, and velocity as \(a_{m}\), \(\omega_{m}\), and \(v_{m}\), respectively, with \(\big{|}a_{t}^{i}\big{|}\leq a_{i}\leq a_{m}\big{|}\omega_{t}^{i}\big{|}\leq \omega_{i}\leq\omega_{m}\), \(\big{|}v_{t}^{i}\big{|}\leq v_{i}\leq v_{m}\). Then, the velocity of agents must be bounded if \(k_{ij_{k}}^{1}=\frac{a_{i_{k}}^{2}}{d_{k_{k}}(t)}\) and \(k_{ij_{k}}^{2}=\beta_{j_{k}}k_{ij_{k}}^{1}\) hold with \(\alpha_{i_{k}}\),\(\beta_{j_{k}}\), and \(d_{j_{k}}\) satisfying the following equations._  
\[\alpha_{i_{k}}+\alpha_{i_{k}}\beta_{j_{k}}\big{|}\omega_{i_{k}} \big{|}\] \[\leq\sqrt{0.5\Big{(}\Big{(}2k_{ij_{k}}^{2}v_{m}\Big{)}^{2}-0.5 \Big{(}k_{ij_{k}}^{2}v_{j_{k}}+a_{j_{k}}\Big{)}^{2}\Big{)}} \tag{25}\] \[2k_{ij_{k}}^{2}v_{m}\geq\big{|}k_{ij_{k}}^{2}v_{j_{k}}+a_{j_{k}} \big{|} \tag{26}\]  
**Remark 1. **_\(d_{j_{k}}\) is correlated to \(\widetilde{\mathbf{X}}_{ij}^{t}\), because \(V(t)\) is asymptotically convergent, which means \(V_{j_{k}}(t)=\eta_{j_{k}}\Big{(}\mathbf{\mathcal{T}}_{0j_{k}}\widetilde{\mathbf{X}}_{ ij_{k}}^{t}\Big{)}^{T}\mathbf{\mathcal{T}}_{0j_{k}}\widetilde{\mathbf{X}}_{ij_{k}}^{t}+ \eta_{0}\eta_{j_{k}}\Big{(}\widetilde{\mathbf{q}}_{ij_{k}}^{t}\Big{)}^{T}\widetilde {\mathbf{q}}_{ij_{k}}^{t}\) is decreasing. Here, we can choose the positive variable \(d_{j_{k}}(t)=d_{j_{k}}(t_{0})\), where \(\sqrt{V_{j_{k}}(t_{0})}\leq d_{j_{k}}(t_{0})\), so we can obtain \(\sqrt{V_{j_{k}}(t)}\leq d_{j_{k}}(t_{0})\), thus \(\left|\widetilde{\mathbf{X}}_{ij}^{t}\right|\leq d_{j_{k}}(t_{0})\mathbf{1}_{1\times 2}\)._  
**Remark 2. **_Since \(\big{|}v_{j_{k}}\big{|}\leq v_{m}k_{ij_{k}}^{2}\big{(}2v_{m}-\big{|}v_{j_{k}} \big{|}\big{)}\geq\big{|}a_{j_{k}}\big{|}\) can be realized, which means we can obtain positive \(\alpha_{k}\) and \(\beta_{j_{k}}\) for bounded controller design._  
**Proof of Proposition 2. ** Based on the law (3), we can obtain:  
\[\dot{v}_{t}^{i}=-\sum_{k=1}^{L}\bigg{[} \begin{array}{cc}k_{ij_{k}}^{1}&k_{ ij_{k}}^{2}&\dot{\psi}_{t}^{j_{k}}\end{array} \bigg{]}\mathbf{\mathcal{T}}_{0i} \widetilde{\mathbf{X}}_{ij_{k}}^{t} \tag{27}\] \[-\sum_{k=1}^{L}k_{ij_{k}}^{2}\Big{(}v_{t}^{i}-v_{t}^{j_{k}}cos \Big{(}\psi_{r}^{j_{k}}\Big{)}\Big{)}-cos\Big{(}\psi_{r}^{j_{k}}\Big{)}a_{t}^{j _{k}}\]  
Introducing Lyapunov function \(V_{3}\):  
\[V_{3}(t)=\frac{1}{2}\big{(}v_{t}^{i}\big{)}^{2} \tag{28}\]  
Based on Equation (30), the derivative of \(V_{3}\) can be obtained:  
\[\dot{V}_{3}(t) =-2k_{p}^{\varphi}\big{(}v_{t}^{i}\big{)}^{2}+v_{t}^{i}\sum_{k=1} ^{L}cos\Big{(}\psi_{r}^{j_{k}}\Big{)}a_{t}^{j_{k}} \tag{29}\] \[+v_{t}^{i}\sum_{k=1}^{L}k_{ij_{k}}^{2}v_{t}^{i}\cos\Big{(}\psi_{ r}^{j_{k}}\Big{)}\] \[+v_{t}^{i}\sum_{k=1}^{L}-\bigg{[} \begin{array}{cc}k_{ij_{k}}^{1} &k_{ij_{k}}^{2}\dot{\psi}_{t}^{i_{k}}\end{array}