## 4 Controller Design
\[u=-c_{n}\hat{z}_{n}-3\hat{z}_{n}-\hat{\theta}_{n}^{T}\varphi_{n}(\widetilde{ \widetilde{\mathbf{x}}}_{n})-\hat{\phi}^{T}\phi(\widetilde{\widetilde{\mathbf{ x}}}_{n})+\dot{v}_{n} \tag{26}\]  
\[\dot{\theta}_{n}=p_{n}\hat{z}_{n}\varphi_{n}(\widetilde{\widetilde{\mathbf{x} }}_{n})+\sigma_{n}\hat{\theta}_{n} \tag{27}\]  
\[\dot{\hat{w}}_{n}=q_{n}\hat{z}_{n}\varphi_{n}(\varsigma,\widetilde{\widetilde{ \mathbf{x}}}_{n})+\overline{\sigma}_{n}\hat{w}_{n} \tag{28}\]  
\[\dot{\hat{v}}=m\hat{z}_{n}\varphi_{n}(\varsigma,\widetilde{\widetilde{\mathbf{ x}}}_{n})+\overline{\sigma}\hat{v} \tag{29}\]  
where \(c_{n}>0\), \(p_{n}>0\), \(q_{n}>0\), \(\sigma_{n}>0\), \(\overline{\sigma}_{n}>0\) and \(\overline{\sigma}>0\) are design parameters. Additionally, \(\hat{z}_{n}=\widetilde{\widetilde{\mathbf{x}}}_{n}-\hat{w}_{n}^{T}\varphi_{n}( \varsigma,\widetilde{\widetilde{\mathbf{x}}}_{n})\). The configuration of the NN-adaptive state-feedback control scheme is shown in Figure 1.  
_._
2. _The above secure control process reaches the expected trajectory and all the closed-loop signals are bounded._
3. _The output error of the system converges to a small neighborhood of zero._  
Proof. : According to (19) and (20), we can obtain  
\[ \begin{array}{ll}\dot{V}_{1}&=z_{1}\dot{z}_{1}+\frac{1}{p_{1}}\widetilde{ \theta}_{1}^{T}(\dot{\theta}_{1}^{*}-\dot{\theta}_{1})+\frac{1}{q_{1}}\widetilde {w}_{1}^{T}(\dot{w}_{1}^{*}-\dot{\Phi}_{1})\\ &=z_{1}(z_{2}+\xi_{2}+\theta_{1}^{*T}\varphi_{1}(x_{1})+\varepsilon_{1})-\frac{ 1}{p_{1}}\widetilde{\theta}_{1}^{T}\hat{\theta}_{1}-\frac{1}{q_{1}}\widetilde {w}_{1}^{T}\hat{w}_{1}\\ &=z_{1}(z_{2}+\xi_{2}+\theta_{1}^{*T}\varphi_{1}(x_{1})-\theta_{1}^{*T}\varphi _{1}(\widetilde{x}_{1})+\theta_{1}^{*T}\varphi_{1}(\widetilde{x}_{1})+ \varepsilon_{1}+\widetilde{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{x}_{1})\\ &-\widetilde{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{x}_{1}))-\frac{1}{p_{1}} \widetilde{\theta}_{1}^{T}\hat{\theta}_{1}-\frac{1}{q_{1}}\widetilde{\omega} _{1}^{T}\hat{\psi}_{1}\\ &=z_{1}(z_{2}+\xi_{2}+\theta_{1}^{*T}\varphi_{1}(x_{1})-\theta_{1}^{*T} \varphi_{1}(\widetilde{x}_{1})+\delta_{1}^{T}\varphi_{1}(\widetilde{x}_{1})+ \varepsilon_{1}+\hat{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{x}_{1})\\ &-w_{1}^{*T}\varphi_{1}(\xi,\widetilde{x}_{1})+\beta_{1})+(z_{1}-\hat{z}_{1}) \widetilde{\theta}_{1}^{T}\varphi_{1}(\widetilde{x}_{1})+\frac{1}{p_{1}} \widetilde{\theta}_{1}^{T}(p_{1}\hat{z}_{1}\varphi_{1}(\widetilde{x}_{1})- \dot{\theta}_{1})\\ &+(z_{1}-\hat{z}_{1})\widetilde{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{x}_{1})+ \frac{1}{q_{1}}\widetilde{w}_{1}^{T}(q_{1}\hat{z}_{1}\varphi_{1}(\xi, \widetilde{x}_{1})-\dot{\theta}_{1})\end{array}  \tag{30}\]  
According to (1) and (10), we have  
\[ \begin{array}{ll}z_{1}-\hat{z}_{1}&=-a_{1}(t,x_{1})+\vartheta_{1}^{T}\varphi _{1}(\xi,\widetilde{x}_{1})\\ &=-(w_{1}^{*T}\varphi_{1}(\xi,x_{1})+\varepsilon_{10})+\vartheta_{1}^{T} \varphi_{1}(\xi,\widetilde{x}_{1})\end{array}  \tag{31}\]  
From (14), by applying the inequality \(2 ab\leq a^{2}+b^{2}\), we can derive the following expression/relationship, which will help in simplifying the analysis and in providing a clearer understanding of the system's behavior. \[z_{1}(z_{2}+\xi_{2}+\epsilon_{1})\leq\frac{3}{2}z_{1}^{2}+\frac{1}{2}z_{2}^{2 }+\frac{1}{2}\|\varepsilon_{1}^{*}\|^{2}+\frac{1}{2}\xi_{2}^{2} \tag{32}\]  
\[