# Stabilization of Underactuated Surface Vessels: A Continuous Fractional Power Control Method
## 3 Main Results
### A Continuous Fractional Power Control Framework
Together with Lemma 3, we can have  
\[\lim\limits_{t\rightarrow+\infty}\dot{\varGamma_{1}}=\lim\limits_{t \rightarrow+\infty}\left[\kappa_{1}\theta_{2}^{K_{1}}\theta_{6}-(K_{1}-\xi_{ 1})\theta_{2}^{K_{1}-\xi_{1}-1}\theta_{5}^{2}\theta_{6}+\kappa_{2}\theta_{5} \theta_{2}^{K_{1}-\xi_{1}}\right]=0. \tag{16}\]  
Since \(\lim\limits_{t\rightarrow+\infty}\theta_{5}=0\) and states \(\theta_{2}\) and \(\theta_{6}\) are bounded, Equation (16) indicates that  
\[\lim\limits_{t\rightarrow+\infty}\theta_{2}^{K_{1}}\dot{\theta}_{6}=0. \tag{17}\]  
Define even \(p_{4}\) and odd \(q_{4}\) as \(\dfrac{p_{4}}{q_{4}}-\xi_{2}>K_{1}>2\). Due to \(\dot{\theta}_{2}\) is bounded, Equation (17) indicates that there must be a positive real number \(K_{2}=\xi_{2}+\dfrac{p_{4}}{q_{4}}\) satisfying that  
\[\lim\limits_{t\rightarrow+\infty}\theta_{2}^{K_{2}-\xi_{2}}\dot{\theta}_{6}= \lim\limits_{t\rightarrow+\infty}\theta_{2}^{K_{2}-K_{1}-\xi_{2}}\cdot\theta _{2}^{K_{1}}\dot{\theta}_{6}=0. \tag{18}\]  
Define \(\varGamma_{2}=\theta_{2}^{K_{1}}\theta_{6}=\), according to Appendix B, \(\dot{\varGamma_{2}}\) is uniformly continuous. We can then have  
\[\lim\limits_{t\rightarrow+\infty}\dot{\varGamma_{2}}=\lim\limits_{t \rightarrow+\infty}\left[-\kappa_{3}\theta_{3}\theta_{2}^{K_{2}-\xi_{2}}- \kappa_{4}\theta_{6}\theta_{2}^{K_{2}-\xi_{2}}+\lambda\theta_{2}^{K_{2}}\sin t +(K_{2}-\xi_{2})\theta_{2}^{K_{2}-\xi_{2}-1}\theta_{5}\theta_{6}^{2}\right]=0. \tag{19}\]  
Due to the fact that \(\lim\limits_{t\rightarrow+\infty}\theta_{5}=0\) and \(\theta_{3}\), \(\theta_{2}\), \(\theta_{6}\) are bounded. The Equation (19) means that  
\[\lim\limits_{t\rightarrow+\infty}\left[\kappa_{3}\theta_{3}\theta_{2}^{K_{2} -\xi_{2}}-\lambda\theta_{2}^{K_{2}}\sin t\right]=0. \tag{20}\]  
Define \(\varGamma=\left[\kappa_{3}\theta_{3}\theta_{2}^{K_{2}-\xi_{2}}-\lambda\theta_ {2}^{K_{3}}\sin t\right]\), according to the Appendix C, we can have that \(\dot{\varGamma}\) is uniformly continuous, and hence  
\[\lim\limits_{t\rightarrow+\infty}\dot{\varGamma}(t) \tag{21}\] \[=\lim\limits_{t\rightarrow+\infty}\left[\kappa_{3}\theta_{6} \theta_{2}^{K_{2}-\xi_{2}}+\kappa_{3}(K_{2}-\xi_{2})\theta_{3}\theta_{2}^{K_{ 2}-\xi_{2}-1}\theta_{5}\theta_{6}-\lambda K_{2}\theta_{2}^{K_{2}-1}\theta_{5} \theta_{6}\sin t-\lambda\theta_{2}^{K_{2}}\cos t\right]=0.\]  
Together with the fact that \(\lim\limits_{t\rightarrow+\infty}\theta_{2}^{K_{2}-\xi_{2}}=0\), \(\lim\limits_{t\rightarrow+\infty}\theta_{5}=0\) and states \(\dot{\theta}_{2}\), \(\dot{\theta}_{3}\) and \(\dot{\theta}_{6}\) are bounded, we can have  
\[\lim\limits_{t\rightarrow+\infty}\lambda\theta_{2}^{K_{2}}\cos t=0. \tag{22}\]  
Obviously, the state \(\dot{\theta}_{2}\) converges to zero, according to (11), we know that \(\dot{\theta}_{3}\) and \(\dot{\theta}_{6}\) can all converge to zero globally and asymptotically. The proof is completed. **Remark 1**. : _The presented continuous control law can globally asymptotically stabilize USVs with off-diagonal dynamics, which is the first of its kind. In comparison with the backstepping method in [10; 12], the new controller also has a simpler structure._  
_Please note that there exists a trade-off in selecting parameters like \(\kappa_{3}\) and \(\kappa_{4}\). Generally, larger positive values of these parameters can result in faster convergence of the system state. However, too large \(\kappa_{3}\) and \(\kappa_{4}\) may lead to slowdown of the convergence of \(\dot{\theta}_{2}\), while too small \(\kappa_{3}\) and \(\kappa_{4}\) may result in slowdown of the convergence of \(\phi_{3}\) and \(\phi_{6}\). Moreover, a large ratio of \(\kappa_{1}\) and \(\kappa_{2}\) is good for quick convergence of \(\phi_{2}\). Similarly, a large ratio of \(\kappa_{3}\), \(\kappa_{4}\) and \(\lambda\) is also preferable.