## 3 Control Law Design and Stability Analysis
### Stability Analysis  
Based on the above analysis, the main results can be summarized as Theorem 1. **Theorem 1**. : _Under Assumptions 1-3, consider the uncertain multi-agent systems with unknown control gains (5), the virtual control laws (25) and (43), the actual control law (61), and the adaptive laws (26)-(29), (44)-(48), and (62)-(66). Then, it can be guaranteed that: (1) all signals of the closed-loop system SGPFS and (2) the tracking error converges to a small neighborhood of zero in a finite time._  
**Proof. **_Construct the following Lyapunov function candidate_  
\[ \begin{array}{l}V=\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{n}V_{i,m}=\sum \limits_{i=1}^{N}\sum\limits_{m=1}^{n}\underline{1}\tilde{e}_{i,m}^{2}+\sum \limits_{i=1}^{N}\sum\limits_{m=2}^{n}\underline{1}\tilde{2}_{i,m-1}^{2}+\sum \limits_{i=1}^{N}\sum\limits_{m=1}^{n}\underline{1}\widetilde{\Sigma}_{i,m}^{ 2}\\ +\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{n}\frac{1}{2\Upsilon_{i,m}^{T}} \widetilde{\Phi}_{i,m}+\sum\limits_{i=1}^{N}\sum\limits_{m=2}^{n}\frac{1}{2 \mathbb{N}_{i,m-1}}\widetilde{\Pi}_{i,m-1}^{2}\end{array}  \tag{68}\]  
Define the compact sets as \(\Omega_{i}=\Omega_{i,1}\lor\Omega_{i,2}\cdots\lor\Omega_{i,n-1}\), and \(\Omega=\Omega_{1}\lor\Omega_{1}\cdots\lor\Omega_{N}=\{V(t)\leq p\}\), where \(p>0\), thus, there are positive constants \(\Pi_{i,m}\), such that \(|M_{i,m}(\cdot)|\leq\Pi_{i,m}\) on \(\Omega\) for all \(i=1,\cdots,N\) and \(m=1,\cdots,n-1\). Taking the derivative of (68) and applying (30), (49) and (67), we obtain  
\[ \begin{array}{l}\dot{V}\leq\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{n}(\dot{h} _{i,m}\mathcal{N}(\varsigma_{i,m})+1)\dot{\varsigma}_{i,m}-\sum\limits_{i=1}^{N }\sum\limits_{m=1}^{n}c_{i,m}e_{i,m}^{2u_{i}}-\sum\limits_{i=1}^{N}\sum\limits_ {m=2}^{n}\frac{z_{i,m-1}^{2}}{\tau_{i,m-1}}-\sum\limits_{i=1}^{N}\sum\limits_{m= 1}^{n}\frac{\gamma_{i,m}}{\Gamma_{i,m}}\underline{\widetilde{\Sigma}}_{i,m} \underline{\widetilde{\Sigma}}_{i,m}\\ -\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{n}\frac{\dot{\varphi}_{i,m}}{\Upsilon _{i,m}^{T}}\widetilde{\Phi}_{i,m}^{T}\widetilde{\Phi}_{i,m}-\sum\limits_{i=1}^{N }\sum\limits_{m=2}^{n}\frac{\omega_{i,m-1}}{\mathbb{N}_{i,m-1}}\widetilde{\Pi}_ {i,m-1}\hat{\Pi}_{i,m-1}+\eta\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{n} \Xi_{i,m}+\eta(n-1)\end{array}  \tag{69}\]  
From (82), because \(\int_{0}^{t}B_{0}d\tau\) and \(V(0)\) are bounded and \(-C_{1}\int_{0}^{t}V(\tau)^{\mu}d\tau<0\), there exists a positive constant \(A_{0}\), such that \(\int_{0}^{t}B_{0}d\tau+V(0)-C_{1}\int_{0}^{t}V(\tau)^{\mu}d\tau\leq A_{0}\). Further, according to (82) and applying Lemma 3, it is obtained that \(V(t)\), \(\xi_{i,m}(t)\) and \(\int_{0}^{t}(h_{i,m}\mathcal{N}(\xi_{i,m})+1)\dot{\xi}_{i,m}(\tau)d\tau\), \(i=1,\cdots,N\), \(m=1,\cdots,n\), are bounded over the interval \([0,t]\) when \(t<+\infty\), which implies that \(\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{n}\int_{0}^{t}(h_{i,m}\mathcal{N}( \xi_{i,m})+1)\dot{\xi}_{i,m}(\tau)d\tau\) is bounded over \([0,t]\). Thus, we have  
\[\dot{V}\leq-C_{1}V^{\mu}+C_{0} \tag{83}\]  
where \(C_{0}=\max_{1\leq i\leq N,\;1\leq m\leq n}(h_{i,m}\mathcal{N}(\xi_{i,m})+1) \dot{\xi}_{i,m}+B_{0}\). Considering (83) and Lemma 2, we can easily obtain that the inequality \(V^{\mu}\leq C_{0}/(1-\lambda)C_{1}\) for \(\lambda\in(0,1)\) and \(\forall t\geq T_{r}\) holds, where the form of \(T_{r}\) is described as  
\[T_{r}=\frac{1}{(1-\mu)\lambda C_{1}}\left[V^{1-\mu}(\sigma(0))-\left(\frac{C_ {0}}{(1-\lambda)C_{1}}\right)^{\frac{1-\mu}{\mu}}\right]\]  
where \(V(\sigma(0))\) is the initial value of \(V\). Therefore, all signals of the closed-loop system are SGPFS. Moreover, observing (68) and \(V^{\mu}\leq C_{0}/(1-\lambda)C_{1}\), one has  
\[\frac{1}{2}e_{1}^{T}e_{1}=\sum\limits_{i=1}^{N}\frac{1}{2}e_{i,1}^{2}\leq \left(\frac{C_{0}}{(1-\lambda)C_{1}}\right)^{\frac{1}{\mu}} \tag{84}\]  
where \(e_{1}=[e_{1,1},\cdots,e_{N,1}]^{T}\).