## 3 Controller Design
\begin{table}
\begin{tabular}{c} \hline \hline
**Adaptive performance functions** \\ \hline \(\dot{p}_{1}(t)=-\lambda_{b}(p_{1}(t)-p_{\infty}^{1})+\eta(t)\Big{(}\sin\gamma _{d}(t)+\frac{\bar{\eta}(t)}{\bar{\eta}(t)}\Big{)}\) see (9) \\ \(\dot{p}_{2}(t)=-\lambda_{v}(p_{2}(t)-p_{\infty}^{2})+\frac{V(t)-V_{d}(t)}{p_{1} (t)}\big{(}\sigma(F_{x}(t),|\delta_{T}\cos a(t)|)-F_{x}(t)\big{)}\) see (11) \\ \(\dot{p}_{3}(t)=-\lambda_{\gamma}(p_{3}(t)-p_{\infty}^{3})+\frac{\gamma(t)- \gamma_{d}(t)}{p_{3}(t)}\big{(}\sigma(F_{h}(t),|\delta_{T}\sin a(t)|)-F_{h}(t) \big{)}\) see (13) \\ \(\dot{p}_{4}(t)=-\lambda_{r}(p_{4}(t)-p_{\infty}^{4})+\frac{\bar{\gamma}_{r}(t)- \sigma(u_{d}(t)\dot{x}_{T})}{p_{4}(t)}\big{(}\sigma(\dot{\delta}_{T}(t),r)- \dot{\delta}_{T}(t)\big{)}\) see (15) \\ \(\dot{p}_{5}(t)=-\lambda_{\theta}(p_{5}(t)-p_{\infty}^{5})+\frac{\theta(t)- \theta_{d}(t)}{p_{5}(t)}\big{(}\sigma(q_{d}(t),\bar{\theta})-q_{d}(t)\big{)}\) see (19) \\ \(\dot{p}_{6}(t)=-\lambda_{q}(p_{6}(t)-p_{\infty}^{6})-\frac{q(t)- \sigma(q_{d}(t)\dot{d})}{p_{6}(t)}\big{(}\sigma(\ddot{\delta}_{e}(t),\delta_{e })-\ddot{\delta}_{e}(t)\big{)}\) see (21) \\ \hline \hline
**Control inputs** \\ \(\dot{\delta}_{T}(t)=\sigma(\dot{\delta}_{e}(t),\bar{\delta}_{e})\) see (22) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. guarantees the boundedness of all closed-loop signals of (24) for all \(t\in[0,\tau_{\infty})\). Finally, we prove that \(\zeta\) remains strictly within a compact subset of \(\Omega\), which leads to \(\tau_{max}=\infty\) by contradiction. Consider the closed-loop dynamical system (24). By construction, it holds that \(|\xi_{i}(0)|<1,p_{i}(0)\in(0,\bar{p}_{i}),\ i=1,\ldots,6\) and therefore \(\zeta(0)\in\Omega\). Moreover, \(\zeta:\Omega\to\mathbb{R}^{12}\) is piece-wise continuous and locally integratable on \(t\) as well as locally Lipschitz on \(\zeta\), over the open set \(\Omega\). Hence, invoking Theorem 54 in [34] (p. 476) we conclude that there exists a unique maximal solution \(\zeta:[0,\tau_{\max})\to\Omega,\ \forall t\in[0,\tau_{\max})\). Consider the positive definite and radially unbounded Lyapunov function candidate \(V_{1}=\frac{1}{2}\epsilon_{1}^{2}(t)\). By differentiating \(V_{1}\) with respect to time, we obtain:  
\[\dot{V}_{1}= \frac{\epsilon_{1}(t)}{(1-\xi_{1}^{2}(t))p_{1}(t)}(h(t)-h_{d}(t)- \xi_{1}(t)\dot{p}_{1}(t)  
when the saturation level \(\bar{\gamma}\) satisfies the following inequality:  
\[\bar{\gamma}>-\frac{\lambda_{1}(\bar{p}_{1}-p_{\infty}^{1})-\frac{\bar{\gamma}^{2 }(t_{p})}{V_{d}(t_{p})}}{\text{sign}(\eta(t_{p}))\eta(t_{p})}. \tag{30}\]  
_Phase B.2-3_ Regarding the boundedness of \(\epsilon_{i}(t),p_{i}(t),\ i=2,3,4\), Phase B.2, consider the Lyapunov function candidate \(V_{2}=\frac{1}{2}\epsilon_{2}^{2}+\frac{1}{2}\epsilon_{3}^{2}\) (as in [30]) and \(V_{4}=\frac{1}{2}\epsilon_{4}^{2}\) as well as (Phase B.3) the functions \(V_{i}=\frac{1}{2}\epsilon_{i}^{2},i=5,\ldots,6\) in order to prove the boundedness of \(\epsilon_{i}(t),p_{i}(t),\ i=4,5\), respectively. Subsequently, it is easy to obtain stability sets \(\mathcal{E}_{i},\Omega_{p_{i}},\ i=2,\ldots,6\) for \(\epsilon_{i}(t),p_{i}(t),\ i=2,\ldots,6\) similar to (28) and (29), by following the analysis presented in [32] with minor modifications. _Phase C._ In Phase B, we proved the boundedness of the transformed errors \(\epsilon_{i}(t)=\mathcal{T}(\bar{\zeta}_{i}(t)),\ i=1,\ldots,6\) for all \(t\in[0,\tau_{\max})\) within the compact sets \(\mathcal{E}_{i}\ i=1,\ldots,6\) for sufficiently large saturation levels. Utilizing this with the inverse of function \(\mathcal{T}\), we conclude the boundedness of the normalized errors \(\bar{\xi}_{i}(t)\ i=1,\ldots,6\) within the compact sets \(\Omega_{\bar{\xi}_{i}}\coloneqq[\underline{\xi}_{i},\bar{\xi}_{i}]\subset(-1, 1),\ i=1,\ldots,6\). Moreover we showed that the adaptive performance functions \(p_{i}(t)\in\Omega_{p_{i}}\subset(0,\bar{p}_{i}),\ i=1,\ldots,6\).