## 3 Control Design and Stability Analysis
### Surge Control  
The surge control law will be designed to guarantee that the surge speed tracking error converges to a small range around zero within the appointed time while ensuring that the surge speed tracking error remains within the predefined appointed-time constraint performance. Consider the following new integral barrier Lyapunov function:  
\[V_{u}=\int_{0}^{u_{e}}\frac{k_{uu}^{2}\left(t\right)\varsigma}{k_{uu}^{2}\left( t\right)-\varsigma^{2}}d\varsigma+\frac{1}{2}\tilde{W}_{u}^{T}\Gamma_{u}^{-1} \tilde{W}_{u} \tag{25}\]  
where \(\tilde{W}_{u}=W_{u}^{*}-\hat{W}_{u}\) represents the estimation error with \(\hat{W}_{u}\) representing the estimated value of \(W_{u}^{*}\) and \(\Gamma_{u}\) is a positive definite diagonal matrix. \(k_{uu}\left(t\right)\) is given by:  
\[k_{uu}\left(t\right)=\left\{ \begin{array}{l}\frac{\tilde{k}_{uu}-k_{uu}}{T_ {uu}^{2}}t^{2}-2\frac{\tilde{k}_{uu}-k_{uu}}{T_{uu}}t+\tilde{k}_{ub},t\in[0,T_{ uu})\\ k_{ub},t\in[T_{uu},+\infty)\end{array} \right. \tag{26}\]  
We know that the candidate Lyapunov function \(V_{u}\) is positive definite, continuously differentiable, and radially unbounded in the set \(\Omega_{u}=\left\{u_{e}:\left|u_{e}\right|<k_{uu}\left(t\right)\right\}\); therefore, \(V_{u}\) is a valid candidate Lyapunov function. **Remark 5**: _According to [30], we know that, for any unknown nonlinear function \(f(x)\): \(R^{m}\to R\), it can be approximated by the neural networks over a compact set \(\Omega\subseteq R^{m}\) as \(f\left(x\right)={W^{*}}^{T}H\left(x\right)+\delta\), where \(x\in R^{m}\) stands for the input of the neural networks, and \(W^{*}=\left[w_{1}^{*},\cdots,w_{n}^{*}\right]^{T}\in R^{n}\) denotes the ideal weight vector. \(n\) is the number of hidden nodes, and \(\delta\) represents the minimum approximation error. The bounded ideal weight value \(W^{*}\) is given by \(W^{*}=\arg\min_{\hat{W}}\left\{\sup_{x\in\Omega}\left|f\left(x\right)-\hat{W}^{T} H\left(x\right)\right|\right\}\); where \(\hat{W}\) denotes the estimation of \(W^{*}\). \(H\left(x\right)=\left[h_{1}\left(x\right),\cdot\cdot\cdot,h_{n}\left(x\right) \right]^{T}\): \(\Omega\to R^{n}\), \(h_{i}(x)\) can be selected as \(h_{i}=\exp\left(-\left\|x-\mu_{i}\right\|/\eta_{i}^{2}\right)\) with \(\mu_{i}\in R^{m}\) and \(\eta_{i}\in R\) being the center and width of the radial basis function, respectively. In this paper, since the model uncertainties \(f_{u},f_{v},f_{r}\) in (12) are unknown, the model-based designed control law is not feasible. To solve this problem, the neural networks \(\hat{W}_{j}^{T}H_{j}(Z_{j})\), \(j=u,v,r\) are used to approximate \(\hat{W}_{j}^{*T}H_{j}(Z_{j})\) such that \(f_{j}\approx\hat{W}_{j}^{T}H_{j}\left(Z_{j}\right)\) with \(H_{j}(Z_{j})\) being the basis functions and \(Z_{j}=[u,v,r]^{T}\) being the input of the neural networks. Calculating the time derivative of \(V_{u}\) yields:  
\[ \begin{split}\hat{V}_{u}=&\frac{\partial V_{u}}{ \partial u_{e}}\dot{u}_{e}+\frac{\partial V_{u}}{\partial k_{ua}(t)}k_{ua}(t)- \hat{W}_{u}^{T}\Gamma_{u}^{-1}\dot{\hat{W}}_{u}\\ =&\frac{k_{ua}^{2}\left(t\right)u_{e}}{k_{ua}^{2} \left(t\right)-u_{e}^{2}}\dot{u}_{e}+u_{e}\left[\frac{k_{ua}}{u_{e}}\log\frac{ k_{ua}^{2}\left(t\right)}{k_{ua}^{2}\left(t\right)-u_{e}^{2}}-\frac{k_{ua}u_{e}}{k_{ua}^{ 2}\left(t\right)-u_{e}^{2}}\right]\dot{k}_{ua}-\hat{W}_{u}^{T}\Gamma_{u}^{-1} \dot{\hat{W}}_{u}\end{split}  \tag{27}\]  
Substituting \(\dot{u}_{e}\) into (27), we have:  
\[ \begin{split}\hat{V}_{u}=&\frac{k_{ua}^{2}\left(t \right)u_{e}}{k_{ua}^{2}\left(t\right)-u_{e}^{2}}\left(vr+\frac{F_{xD0}}{m_{0} }+f_{u}+\frac{\tau_{u}}{m_{0}}-\dot{\alpha}_{u}\right)\\ &+u_{e}\left[\frac{k_{ua}}{u_{e}}\log\frac{k_{ua}^{2}\left(t \right)}{k_{ua}^{2}\left(t\right)-u_{e}^{2}}-\frac{k_{ua}u_{e}}{k_{ua}^{2} \left(t\right)-u_{e}^{2}}\right]\dot{k}_{ua}-\hat{W}_{u}^{T}\Gamma_{u}^{-1} \dot{\hat{W}}_{u}\end{split}  \tag{28}\]  
According to [PERSON]'s direct method, the surge control law and the adaptive updating law are designed as:  
\[