## 3 Control Design and Stability Analysis
### Stability Analysis
 \tag{50}\]  
According to Theorem 1, the time derivative of \(V\) can be expressed as:  
\[\dot{V}\leq-\lambda_{1}V+\lambda_{2} \tag{51}\]  
where  
\[\lambda_{1}=\min\left\{k_{x},k_{y},k_{u},k_{v},k_{r},\sigma_{u}\lambda_{\min} \left(\Gamma_{u}\right),\sigma_{v}\lambda_{\min}\left(\Gamma_{v}\right), \sigma_{r}\lambda_{\min}\left(\Gamma_{r}\right)\right\} \tag{52}\]  
\[\lambda_{2}=\frac{\sigma_{u}}{2}{W_{u}^{*}}^{T}\tilde{W}_{u}^{*}+\frac{\sigma_{v} }{2}{W_{v}^{*}}^{T}\tilde{W}_{v}^{*}+\frac{\sigma_{r}}{2}{W_{r}^{*}}^{T}\tilde{W}_ {r}^{*} \tag{53}\]Solving the inequality (51) yields:  
\[0\leq V\leq\left(V\left(0\right)-\frac{\lambda_{2}}{\lambda_{1}}\right)e^{- \lambda_{1}t}+\frac{\lambda_{2}}{\lambda_{1}} \tag{54}\]  
Invoking Theorem 1 again yields:  
\[\frac{i_{e}^{2}}{2}\leq V\leq\left(V\left(0\right)-\frac{\lambda_{2}}{\lambda_{ 1}}\right)e^{-\lambda_{1}t}+\frac{\lambda_{2}}{\lambda_{1}},\quad i=x,y,u,v,r \tag{55}\]  
\[\frac{1}{2}\tilde{W}_{j}^{T}\Gamma_{j}^{-1}\tilde{W}_{j}\leq\left(V\left(0 \right)-\frac{\lambda_{2}}{\lambda_{1}}\right)e^{-\lambda_{1}t}+\frac{\lambda_ {2}}{\lambda_{1}},\quad j=u,v,r \tag{56}\]  
For given \(\Delta_{\lambda}\geq\sqrt{2\lambda_{2}/\lambda_{1}}\), there is a constant \(T_{\lambda}>0\) such that the tracking errors \(|i_{e}|\leq\Delta_{\lambda}\), \(i=x,y,u,v,r\), and \(\left\|\tilde{W}_{j}\right\|\leq\Delta_{\lambda}\bigg{/}\sqrt{\lambda_{\min} \left(\Gamma_{j}^{-1}\right)}\), \(j=u,v,r\) for all \(t>T_{\lambda}\). Therefore, all the tracking errors are ultimately uniformly bounded. We know that the decrease of \(\left(V\left(0\right)-\lambda_{2}/\lambda_{1}\right)e^{-\lambda_{1}t}+\lambda _{2}/\lambda_{1}\) can be achieved by increasing \(\lambda_{1}\) or decreasing \(\lambda_{2}\). Hence, the tracking errors can be adjusted to arbitrarily small through selecting appropriately the design parameters \(k_{x},k_{y},k_{u},k_{v},k_{r}\) and \(\sigma_{j}\lambda_{\min}\left(\Gamma_{j}\right)\). According to the boundedness of \(V\), we know that there is a positive constant \(k_{m}\) such that \(V<k_{m}\). Then, in light of the definition of the proposed new integral barrier Lyapunov function, we have \(V\rightarrow\infty\) only if \(|x_{e}|\to k_{xu}(t)\) or \(|y_{e}|\to k_{yu}(t)\) or \(|u_{e}|\to k_{ua}(t)\) or \(|v_{e}|\to k_{ua}(t)\) or \(|r_{e}|\to k_{ra}(t)\). Therefore, we have \(|x_{e}|\
eq k_{xa}(t)\), \(|y_{e}|\
eq k_{yu}(t)\), \(|u_{e}|\
eq k_{ua}(t)\), \(|v_{e}|\
eq k_{ua}(t)\) and \(|r_{e}|\
eq k_{ra}(t)\). In virtue of Theorem 2 and the initial conditions in Assumption 1 in this paper, it is straightforward to obtain that \(|x_{e}|<k_{xa}\left(t\right)\), \(|y_{e}|<k_{yu}\left(t\right)\), \(|u_{e}|<k_{ua}\left(t\right)\), \(|v_{e}|<k_{pa}\left(t\right)\) and \(|r_{e}|<k_{ra}\left(t\right)\). The proof of Theorem 3 is completed. **Remark 9**. : _Compared with the existing control method based on the integral barrier Lyapunov function technique in [23; 24; 25; 26; 27], the improved time-varying integral barrier Lyapunov function proposed in this paper is capable of handling the time-varying constraint on the system tracking error--for instance, the time-varying constraints on the tracking errors of the underactuated hovercraft. The time-varying constraint technique is more universal and complex than the time-invariant constraint technique in addition to having a wider application range._  
**Remark 10**. : _Compared with the previous integral barrier Lyapunov function technique in [23; 24; 25; 26; 27], an appointed-time constraint function is proposed to incorporate into the integral barrier Lyapunov function, which can guarantee that all the tracking errors converge to a small range containing zero while ensuring that all the tracking errors remain within the pre-specified transient and steady performance. In addition, unlike the traditional finite/fixed-time control in [21; 22], the tracking errors under the proposed controller can converge to a small range around zero within the appointed time, which can be user-appointed arbitrarily._  
**Remark 11**.