## 3 Controller Design
Note that \(\zeta(t)\in\Omega^{\prime}\subset\Omega,\forall t\in[0,\tau_{\max})\), with \(\Omega^{\prime}\coloneqq\Omega_{\bar{\xi}_{i}}\times\Omega_{p_{i}},\ i=1, \ldots,6\). Thus, assuming that \(\tau_{\max}<\infty\), then according to Proposition C.3.6 in [34] (p. 481) there exists a time instant \(\tau^{\prime}\) such that \(\zeta(\tau^{\prime})\
otin\Omega^{\prime}\), which is a contradiction. As a consequence, \(\tau_{\max}=\infty\), which implies the stability of the initial system (7). Finally, invoking the fact that \(\xi_{i}(t)\in\Omega_{\bar{\xi}_{i}},\ i=1,\ldots,6\) for all \(t\geq 0\) we conclude that:  
\[\|x_{i}(t)-x_{d_{i}}(t)\|<p_{i}(t)<\bar{p}_{i}\]  
for all \(t\geq 0\) and \(i=1,\ldots,6\) with \(x_{d}\coloneqq[h_{d}(t),V_{d}(t),\gamma_{d}(t),\sigma(u_{d}(t),\bar{\xi}_{T}),\theta_{d}(t),\sigma(q_{d}(t),\bar{q})]^{T}\). Note that both the state and output errors evolve strictly within the adaptive performance envelope, which completes the proof. **Remark 1**. : _Note that the constraint regarding the elevator deflection \(\delta_{\epsilon}(t)\) is imposed directly by saturating the control signal (20), whereas the throttle limit is achieved by exploiting two nested saturation constraints in combination with the APPC approach that guarantees \(\|\delta_{\mathcal{T}}(t)\|\leq\bar{\delta}_{\mathcal{T}}\) and \(\|\delta_{\mathcal{T}}(t)\|\leq r\). Furthermore, state constraints on \(\gamma(t),\theta(t),q(t)\) are guaranteed via saturating the reference state trajectories \(\gamma_{d}(t),\theta_{d}(t),q_{d}(t)\), while imposing tracking with adaptive prescribed transient and steady-state specifications. Thus, the system will operate in a compact set \(\mathcal{X}\), the size of which is determined by the desired state constraints, i.e., the saturation levels of the state reference trajectories in combination with the adaptive performance specifications affected by input limitations and the couplings among system dynamics._  
**Remark 2**. : _The proposed control scheme (8)-(22) does not require either knowledge of the system nonlinearities or any approximating structures exhibiting robustness against unknown external disturbances. Additionally, both the intermediate control signals and the adaptive performance functions are obtained through simple calculations preventing the explosion of complexity. Finally, the simple gain selection facilitates the practical application of the proposed algorithm._  
**Remark 3**. : _The appropriate saturation level \(\bar{\sigma}\) of the control signals and the states that prevents the finite-time blow-up of the closed-loop system is affected by various factors. Such factors include the unknown system dynamics and external disturbances as well as the strong couplings among the states of the system and the reference trajectory. For example, small control inputs, i.e., small input saturation levels, may be insufficient to prevent the system from becoming unstable in case of large airspeed and wind gusts. Hence, the proper selection of saturation levels in accordance to the reference trajectory constitutes a pivotal task so that the stability of the closed-loop system is ensured._