## 4 Controller Design
\begin{array}{ll}\dot{V}_{n}&=\dot{V}_{n-1}+z_{n}(u+v^{\top T}\phi(\xi, \overline{x}_{n})-v^{\top T}\phi(\xi,\widetilde{\overline{x}}_{n})+\delta^{T} \phi(\xi,\widetilde{\overline{x}}_{n})+\theta_{n}^{\top T}\vartheta_{n}( \widetilde{\overline{x}}_{n})-\theta_{n}^{\top T}\vartheta_{n}(\widetilde{ \overline{x}}_{n})\\ &+\theta_{n}^{T}\varphi_{n}(\widetilde{\overline{x}}_{n})+v_{0}+\varepsilon_{ n}-w^{\top T}\varphi(\xi,\widetilde{\overline{x}}_{n})-w^{\top T}\varphi(\xi, \widetilde{\overline{x}}_{n}))+(z_{n}-\hat{z}_{n})\overset{\sim}{\sim}_{n} \varphi_{n}(\widetilde{\overline{x}}_{n})\\ &+(z_{n}-\hat{z}_{n})\overset{\sim}{\sim}^{T}\varphi(\xi,\widetilde{ \overline{x}}_{n})+(z_{n}-\hat{z}_{n})\overset{\sim}{\sim}^{T}\varphi(\xi, \widetilde{\overline{x}}_{n})+\frac{1}{p_{n}}\overset{\sim}{\sim}_{n}(p_{n} \hat{z}_{n}\hat{\overline{x}}_{n}(\widetilde{\overline{x}}_{n})-\hat{\theta} _{n})\\ &+\frac{1}{q_{n}}\widetilde{w}_{n}^{\top}(q_{n}z_{n}\hat{\varphi}_{n}(\xi, \widetilde{\overline{x}}_{n})-\hat{w}_{n})+\frac{1}{m}\overset{\sim}{\sim} ^{T}(m\sharp\varphi(\xi,\widetilde{\overline{x}}_{n})-\hat{\vartheta})\\ \end{array}  \tag{48}\]  
Similary to (40)-(46), we can gather that  
\[ \begin{array}{ll}\dot{V}_{n}&\leq-\sum\limits_{s=1}^{n}{(c_{s}-3)z_{s}^{2}- \sum\limits_{s=2}^{n}{(\frac{1}{\tau_{s}}-N_{s}^{2}-\frac{1}{2})\xi_{s}^{2}}}+ \sum\limits_{s=1}^{n}{(\frac{\sigma_{s}}{p_{s}}\overset{\sim}{\theta}_{s}^{ \top}\delta_{s}+\frac{\overline{\sigma}_{s}}{q_{s}}\widetilde{w}_{s}^{\top} \vartheta_{s})}\\ &+\frac{\sigma}{m}\overset{\sim}{\sim}^{T}\vartheta+2n\left\|\widetilde{ \theta}_{1}\right\|^{2}+\sum\limits_{s=1}^{n}{(\frac{9+4c_{k}^{2}}{2}\left\| \widetilde{w}_{1}\right\|^{2}+\left\|\widetilde{w}_{s}\right\|^{2})}+D_{n}\\ \end{array}  \tag{49}\]  
where \(D_{n}^{\prime}=D_{n-1}+\left\|\theta_{n}^{*}\right\|^{2}+(\frac{41}{2}+2c_{n} ^{2})\left\|w_{n}^{*}\right\|^{2}+\frac{1}{2}\left\|e_{n}^{*}\right\|^{2}+6 \left\|e_{n0}^{*}\right\|^{2}+\frac{1}{2}\left\|v_{0}^{*}\right\|^{2}\). By employing the inequality \(2 ab\leq a^{2}+b^{2}\), we can obtain  
\[ \begin{array}{ll}\widetilde{\theta}_{s}^{\top}\theta_{s}+\widetilde{w}_{s}^{ \top}w_{s}^{\top}\frac{\sigma}{m}\overset{\sim}{v}^{T}\vartheta\leq-\frac{1} {2}\left\|\widetilde{w}_{s}\right\|^{2}-\frac{1}{2}\left\|\widetilde{v} \right\|^{2}-\frac{1}{2}\left\|\widetilde{\theta}_{s}\right\|^{2}+\frac{1}{2} \left\|\theta_{s}^{*}\right\|^{2}+\frac{1}{2}\left\|w_{s}^{*}\right\|^{2}+ \frac{1}{2}\left\|v^{*}\right\|^{2}\end{array}  \tag{50}\]  
According to (49) and (50), we have  
\[ \begin{array}{ll}\dot{V}_{n}&\leq-\sum\limits_{s=1}^{n}{(c_{s}-3)z_{s}^{2}- \sum\limits_{s=2}^{n}{(\frac{1}{\tau_{s}}-N_{s}^{2}-\frac{1}{2})\xi_{s}^{2}- \sum\limits_{s=1}^{n}{(\frac{\sigma_{s}}{2p_{s}}\left\|\widetilde{\theta}_{s} \right\|^{2}-2n\left\|\widetilde{\theta}_{1}\right\|^{2})}}}\\ &-\sum\limits_{s=1}^{n}{(\frac{\overline{\sigma}_{s}}{2q_{s}}\left\| \widetilde{w}_{s}\right\|^{2}-\frac{9+4c_{k}^{2}}{2}\left\|\widetilde{w}_{1} \right\|^{2}-\left\|\widetilde{w}_{s}\right\|^{2})}-\frac{1}{2}\left\| \widetilde{v}\right\|^{2}+D_{n}\\ \end{array}  \tag{51}\]  
where \(D_{n}=D_{n}^{\prime}+\sum\limits_{s=1}^{n}{(\frac{\sigma_{s}}{2p_{s}}\left\| \theta_{s}^{*}\right\|^{2}+\frac{\overline{\sigma}_{s}}{2q_{s}}\left\|w_{s}^{*} \right\|^{2})}+\frac{1}{2}\left\|v^{*}\right\|^{2}\). Then, (51) can be written as  
\[\dot{V}_{n}\leq-C_{n}V_{n}+D_{n} \tag{52}\]  
where \(C_{n}=\min\{c_{s}-3,\frac{1}{\tau_{s}}-N_{s}^{2}-\frac{1}{2},\frac{c_{s}}{2p_{s} },\frac{\overline{\sigma}_{s}}{2q_{s}},2n,-\frac{9+4c_{k}^{2}}{2},-1\}\), \(s=1,\cdots,n\). This means that the system exhibits stability and boundedness over time, ensuring that the state variables remain within a certain range, even in the presence of potential disturbances or uncertainties.