## 3 Control Design and Stability Analysis
### Stability Analysis  
In this paper, the major results are summarized as:  
**Theorem 3**. : _Considering the hovercraft models (12) in the presence of model uncertainties and multiple performance constraints, if Assumption 1 is satisfied, the appointed-time integral barrier Lyapunov function-based trajectory tracking controllers are obtained by (22), (29), (37), and (45), and the adaptive updating laws are expressed by (30), (38), and (46), then all the tracking errors can converge to a small region containing zero within a user-appointed time while guaranteeing that the appointed-time performance constraint conditions on the tracking errors are never violated._  
Proof of Theorem 3.: Assign the complete Lyapunov function as:  
\[ \begin{split} V=& V_{p}+V_{u}+V_{v}+V_{r}\\ =&\int_{0}^{x_{e}}\frac{k_{xa}^{2}\left(t\right)\xi}{ k_{xa}^{2}\left(t\right)-\xi^{2}}d\xi+\int_{0}^{y_{e}}\frac{k_{ya}^{2}\left(t \right)\xi}{k_{ya}^{2}\left(t\right)-\xi^{2}}d\xi+\int_{0}^{u_{e}}\frac{k_{ua }^{2}\left(t\right)\xi}{k_{ua}^{2}\left(t\right)-\xi^{2}}d\xi+\frac{1}{2} \tilde{W}_{u}^{\mathrm{T}}\Gamma_{u}^{-1}\tilde{W}_{u}\\ &+\int_{0}^{v_{e}}\frac{k_{ya}^{2}\left(t\right)\xi}{k_{xa}^{2} \left(t\right)-\xi^{2}}d\xi+\frac{1}{2}\tilde{W}_{v}^{\mathrm{T}}\Gamma_{v}^ {-1}\tilde{W}_{v}+\int_{0}^{r_{e}}\frac{k_{ra}^{2}\left(t\right)\xi}{k_{ra}^{ 2}\left(t\right)-\xi^{2}}d\xi+\frac{1}{2}\tilde{W}_{r}^{\mathrm{T}}\Gamma_{r}^ {-1}\tilde{W}_{r}\end{split}  \tag{49}\]  
In virtue of (24), (32), (40), and (48), the time derivative of \(V\) is:  
\[ \begin{split}\dot{V}\leq&-k_{x}\frac{k_{xa}^{2} \left(t\right)x_{e}^{2}}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}-k_{y}\frac{k_{ya }^{2}\left(t\right)y_{e}^{2}}{k_{ya}^{2}\left(t\right)-y_{e}^{2}}\\ &+u_{e}\left(\cos\psi\frac{k_{xa}^{2}\left(t\right)x_{e}}{k_{xa} ^{2}\left(t\right)-x_{e}^{2}}+\sin\psi\frac{k_{ya}^{2}\left(t\right)y_{e}}{k_{ ya}^{2}\left(t\right)-y_{e}^{2}}\right)\\ &+v_{e}\left(-\sin\psi\cos\phi\frac{k_{xa}^{2}\left(t\right)x_{e} }{k_{xa}^{2}\left(t\right)-x_{e}^{2}}+\cos\psi\cos\phi\frac{k_{ya}^{2}\left(t \right)y_{e}}{k_{ya}^{2}\left(t\right)-y_{e}^{2}}\right)\\ &-k_{u}\frac{k_{ua}^{2}\left(t\right)u_{e}^{2}}{k_{ua}^{2}\left( t\right)-u_{e}^{2}}+\sigma_{u}\tilde{W}_{u}^{T}\hat{W}_{u}-u_{e}\left(\cos\psi \frac{k_{xa}^{2}\left(t\right)x_{e}}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}+ \sin\psi\frac{k_{ya}^{2}\left(t\right)y_{e}}{k_{ya}^{2}\left(t\right)-y_{e}^{ 2}}\right)\\ &-k_{v}\frac{k_{va}^{2}\left(t\right)v_{e}^{2}}{k_{va}^{2}\left( t\right)-v_{e}^{2}}-\frac{k_{va}^{2}\left(t\right)v_{e}}{k_{va}^{2}\left(t \right)-v_{e}^{2}}u_{e}+\sigma_{v}\tilde{W}_{v}^{\mathrm{T}}\hat{W}_{v}\\ &-v_{e}\left(-\sin\psi\cos\phi\frac{k_{xa}^{2}\left(t\right)x_{e }}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}+\cos\psi\cos\phi\frac{k_{ya}^{2}\left( t\right)y_{e}}{k_{ya}^{2}\left(t\right)-y_{e}^{2}}\right)\\ &-k_{r}\frac{k_{ra}^{2}\left(t\right)r_{e}^{2}}{k_{ra}^{2}\left( t\right)-r_{e}^{2}}+\frac{k_{qa}^{2}\left(t\right)v_{e}}{k_{ua}^{2}\left(t \right)-v_{e}^{2}}u_{e}+\sigma_{r}\tilde{W}_{r}^{\mathrm{T}}\hat{W}_{r}\\ \leq&-k_{x}\frac{k_{xa}^{2}\left(t\right)x_{e}^{2}} {k_{xa}^{2}\left(t\right)-x_{e}^{2}}-k_{y}\frac{k_{ya}^{2}\left(t\right)y_{e} ^{2}}{k_{ya}^{2}\left(t\right)-y_{e}^{2}}-k_{u}\frac{k_{ua}^{2}\left(t\right)u _{e}^{2}}{k_{ua}^{2}\left(t\right)-u_{e}^{2}}-k_{v}\frac{k_{va}^{2}\left(t \right)v_{e}^{2}}{k_{va}^{2}\left(t\right)-v_{e}^{2}}\\ &-k_{r}\frac{k_{ra}^{2}\left(t\right)r_{e}^{2}}{k_{ra}^{2}\left(t \right)-r_{e}^{2}}-\frac{\sigma_{u}}{2}\tilde{W}_{u}^{\mathrm{T}}\hat{W}_{u}- \frac{\sigma_{v}}{2}\tilde{W}_{v}^{\mathrm{T}}\hat{W}_{v}-\frac{\sigma_{r}}{2} \tilde{W}_{r}^{\mathrm{T}}\hat{W}_{r}+\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}^{*}\end{split}