## 4 Stability Analysis
: _Consider the underactuated USV model (3) and (4) in the presence of unmodeled dynamics and unknown disturbances, suppose that Assumption 1 and Assumption 2 are satisfied, under the surge control law (21) and the yaw motion law (33), together with the adaptive laws (17), (19), (29) and (31), from the appropriately chosen parameters \(\lambda\), \(M\), \(\gamma_{1}\), \(\gamma_{2}\), \(\rho_{1}\), \(\rho_{2}\), \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\Gamma_{1}\), \(\Gamma_{2}\), \(\kappa_{1}\), \(\zeta_{1}\), \(\zeta_{2}\), \(\zeta_{1}\), \(\dot{\phi}_{2}\), \(K_{\zeta\mu}\), \(K_{\zeta r}\), \(k_{s1}\) and \(k_{s2}\), all signals in the closed-loop system are guaranteed to be semi-globally uniformly ultimately bounded (LUIB)._  
Proof of Theorem 1.: Consider the following Lyapunov function  
\[V=\frac{1}{2}m_{11}s_{1}^{2}+\frac{1}{2}m_{33}s_{2}^{2}+\frac{1}{2}\Gamma_{1}^ {-1}\tilde{d}_{u}^{*2}+\frac{1}{2}\Gamma_{2}^{-1}\tilde{d}_{v}^{*2}+\frac{1}{ 2}\xi_{1}^{-1}\tilde{\theta}_{1}^{2}+\frac{1}{2}\xi_{2}^{-1}\tilde{\theta}_{2} ^{2}+\frac{1}{2}\sum_{i=u,r}\left(\Phi_{i}^{2}+\xi_{i}^{2}\right) \tag{39}\]The time derivative of (39) is  
\[\dot{V}=m_{11}s_{1}\dot{s}_{1}+m_{33}s_{2}\dot{s}_{2}+\Gamma_{1}^{-1}\tilde{d}_{u }^{*}\hat{d}_{u}^{*}+\Gamma_{2}^{-1}\tilde{d}_{v}^{*}\hat{d}_{v}^{*}+\varepsilon _{1}^{-1}\tilde{\theta}_{1}\dot{\tilde{\theta}}_{1}+\varepsilon_{2}^{-1}\tilde{ \theta}_{2}\dot{\tilde{\theta}}_{2}+\sum_{i=u,r}\left(\Phi_{i}\dot{\Phi}_{i}+ \zeta_{i}\dot{\tilde{\xi}}_{i}\right) \tag{40}\]  
Substituting (15), (17), (27), (29) into (40) yields  
\[\dot{V} =s_{1}\left(W_{u}^{T}\Theta+\varepsilon_{u}-\frac{1}{2}s_{1}\hat{ \theta}_{1}\Theta^{T}\Theta-\rho_{1}s_{1}+d_{u}^{*}-\hat{d}_{u}^{*}\tanh\left( s_{1}/\varepsilon_{1}\right)+k_{s1}\zeta_{u}+\Delta\tau_{u}\right)\] \[+s_{2}\left(W_{m}^{T}\Theta+\varepsilon_{m}-\frac{1}{2}s_{2}\hat{ \theta}_{2}\Theta^{T}\Theta-\rho_{2}s_{2}+d_{r}^{*}-\hat{d}_{r}^{*}\tanh\left( s_{2}/\varepsilon_{2}\right)+k_{s2}\zeta_{r}+\Delta\tau_{r}\right)\] \[-\tilde{d}_{u}^{*}\left(\tanh\left(s_{1}/\varepsilon_{1}\right)- \kappa_{1}\left(\hat{d}_{u}^{*}-d_{u0}^{*}\right)\right)-\tilde{d}_{r}^{*} \left(\tanh\left(s_{2}/\varepsilon_{2}\right)-\kappa_{2}\left(\hat{d}_{r}^{*} -d_{r0}^{*}\right)\right)\] \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]0 \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]1 \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]2 \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]3 \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]4  
According to [PERSON]'s inequality, we have  
\[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]5 \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]6 \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]7 (43) \[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]8  
From Lemma 1, (42) and (43), we obtain  
\[=-\left(A_{u}+f\left(\alpha_{u}\right)+g\left(\alpha_{u}\right) \right)\beta_{u}+\left(B_{u}f\left(\alpha_{u}\right)-D_{u}g\left(\alpha_{u} \right)\right)-X_{u}\]9 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]0 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]1 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]2 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]3 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]4Define \(\theta_{i}=B_{i}\), substituting (34), (36) and (37) into (44) yields  
\[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]5 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]6 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]7 \[\dot{\Phi}_{u}=-\theta_{u}\beta_{u}+B_{u}\alpha_{u}-X_{u} \tag{36}\]8  
It is clear that \(\zeta_{u}\zeta_{u}=-K_{\zeta u}\zeta_{u}^{2}-\frac{|s_{1}\Delta\tau_{u}|+0.5 \Delta\tau_{u}^{2}}{\|\zeta_{u}^{2}\|}\cdot\zeta_{u}^{2}+\Delta\tau_{u}\zeta_{ u}\), \(2\Delta\tau_{u}\zeta_{u}\leq\Delta\tau_{u}^{2}+\zeta_{u}^{2}\).