# Event-Sampled Adaptive Neural Course Keeping Control for USVs Using Intermittent Course Data
## 4 Control Design
### Stability Analysis of Control Law  
According to the design above, one can summarize a theorem as follows. **Theorem 2**. : _Considering the CKC model described by (1), under Assumptions 1-3 and the ESM (3), the CLCS, consisting of the NN-SO (11), (12), filter (28), intermediate control law (30), control law (38), (39), and ETM (40), has the following features:_  
1. _All signals in the CLCS are bounded;_
2. _This can avoid the Zeno behavior._  
Proof. : Consider the following Lyapunov function for CLCS:  
\[L_{Y\epsilon}=\frac{1}{2}s_{1}^{2}+\frac{1}{2}s_{2}^{2}+\frac{1 }{2}\bar{\alpha}^{2}+\frac{1}{2\epsilon b}(\theta-b\hat{\theta})^{2} \tag{41}\]  
Figure 1: The control diagram of USVs under network environment. Differentiating \(L_{Vc}\) and using (28), (31), and (37)-(39), one can obtain  
\[\dot{L}_{Vc}\leq s_{1}(s_{2}+\tilde{\alpha}-k_{1}\tilde{s})+s_{2}\bigg{(}\big{(}( \theta-b\hat{\theta})\varphi(x)\tanh\bigg{(}\frac{s_{2}\varphi(x)}{\sigma} \bigg{)}-s_{1}+bz-bk_{2}s_{2}\bigg{)}\] \[+\tilde{\alpha}\dot{\tilde{\alpha}}-(\theta-b\hat{\theta})\bigg{(} s_{2}\varphi(x)\tanh\bigg{(}\frac{s_{2}\varphi(x)}{\sigma}\bigg{)}-\frac{c}{e} \hat{\theta}\bigg{)}+0.2785\sigma\] \[\leq s_{1}(\tilde{\alpha}-k_{1}\tilde{s})+s_{2}bz-bk_{2}s_{2}^{2}+ \tilde{\alpha}\dot{\tilde{\alpha}}+\frac{c}{e}\hat{\theta}(\theta-b\hat{ \theta})+0.2785\sigma \tag{42}\]  
Due to \(\tilde{\alpha}=\alpha-\alpha_{f}\), one can understand that \(\hat{\tilde{\alpha}}=\dot{\alpha}-\dot{\alpha_{f}}\) and \(\dot{\alpha}=-\frac{\tilde{\alpha}}{l_{f}}\). According to Assumptions 3 and [34], there exists a positive constant \(\tilde{\omega}\) satisfying \(|\dot{\alpha}_{f}|\leq\tilde{\omega}\). Furthermore, one can obtain  
\[\tilde{\alpha}\dot{\tilde{\alpha}}\leq-\bigg{(}\frac{1}{t_{f}}-\frac{1}{2} \bigg{)}\tilde{\alpha}^{2}+\frac{1}{2}\tilde{\omega}^{2} \tag{43}\]  
From \(s_{1}=\chi_{1}-\tilde{\chi}_{1}+\tilde{s}_{1}\) and the ESM, one can obtain  
\[-s_{1}\tilde{s}=-s_{1}^{2}+s_{1}(\chi_{1}-\hat{\chi}_{1})\leq-\frac{3}{4}s_{1 }^{2}+\beta^{2} \tag{44}\]  
In addition, according to the ETM (40), one can obtain \(|z|\leq\gamma\). Furthermore, one has  
\[s_{2}bz\leq\frac{k_{2}b}{4}s_{2}^{2}+\frac{b\gamma^{2}}{k_{2}} \tag{45}\]  
Using [PERSON]'s inequation, one obtains  
\[s_{1}\tilde{\alpha}\leq\frac{1}{2}s_{1}^{2}+\frac{1}{2}\tilde{ \alpha}^{2} \tag{46}\]  
\[\hat{\theta}(\theta-b\hat{\theta})\leq-\frac{(\theta-b\hat{\theta})^{2}}{2b}+ \frac{\theta^{2}}{2b} \tag{47}\]  
Substituting (43)-(47) into (42) yields  
Figure 1:0 Figure 1:1 Figure 1:2  
where \(\varphi=\min\Big{\{}\left(\frac{3k_{1}}{2}-1\right),\frac{3 bk_{1}}{2},\left( \frac{1}{t_{f}}-\frac{1}{2}\right),c\Big{\}}\) and \(\rho=\frac{1}{2}\tilde{\omega}^{2}+k_{1}\beta^{2}+\frac{b\gamma^{2}}{k_{2}}+ \frac{\phi^{2}}{2b\epsilon}+0.2785\sigma\). Solving (48), one can obtain \(L_{Vc}\leq\left(\frac{\rho}{\hat{\varphi}}-L_{Vc}(0)\right)\exp(-\phi t)+ \frac{\rho}{\hat{\varphi}}\), with \(L_{Vc}(0)\) being the initial value of \(L_{Vc}\), which means that \(L_{Vc}=\frac{\rho}{\hat{\varphi}}\) as \(t\longrightarrow\infty\). Furthermore, from (41), one can determine the boundedness of \(s_{1}\), \(s_{2}\), \(\tilde{\alpha}\) and \(\theta-b\hat{\theta}\). Furthermore, from Assumption 3, one determines the boundedness of \(\chi_{1}\) and \(s_{1}\), and then the boundedness of \(\dot{\alpha}_{f}\), \(\alpha_{f}\) and \(\alpha\) according to (26) and (28). In addition, one can understand that \(\chi_{2}\) is bounded, and \(\varphi(x)\) is also bounded due to the boundedness of \(s_{2}\), \(\chi_{2}\), and \(\hat{\alpha}_{f}\). Because of the boundedness of \(\varphi(x)\), \(s_{2}\), and \(\hat{\theta}\), one can determine the boundedness of \(\hat{\tau}\), that is, \(\tau\) is also bounded from the ETM (40). Therefore, all signals in the CLCS are bounded. According to (3) and (39), for \(\forall t\in[t_{\kappa},t_{\kappa+1})\) and \(\forall t\in[t_{\ell},t_{\ell+1})\), \(\ddot{\psi}(t)\) and \(\tau(t)\) are in the hold phase, which implies that they are constant.