## 3 Control Design
### Stability Analysis of the Closed-Loop Control System
\begin{array}{ll}\dot{V}&\leq-\frac{\varepsilon\lambda_{\min}^{1/2}(\mathbf{ P})}{\lambda_{\max}(\mathbf{P})}V_{0}^{1/2}+\frac{1}{\gamma_{1}}\varepsilon_{ \alpha}\dot{\alpha}_{k}+\frac{1}{\gamma_{2}}\varepsilon_{\beta}\dot{\beta}_{k} \\ &=-\frac{\varepsilon\lambda_{\min}^{1/2}(\mathbf{P})}{\lambda_{\max}(\mathbf{ P})}V_{0}^{1/2}-\frac{\varepsilon_{1}}{\sqrt{2\gamma_{1}}}|\varepsilon_{ \alpha}|-\frac{\varepsilon_{2}}{\sqrt{2\gamma_{2}}}|\varepsilon_{\beta}|+ \frac{1}{\gamma_{1}}\varepsilon_{\alpha}\dot{\alpha}_{k}+\frac{1}{\gamma_{2} }\varepsilon_{\beta}\dot{\beta}_{k}+\frac{\varepsilon_{1}}{\sqrt{2\gamma_{1} }}|\varepsilon_{\alpha}|+\frac{\varepsilon_{2}}{\sqrt{2\gamma_{2}}}| \varepsilon_{\beta}|\end{array}  \tag{40}\]  
Here, \(\varepsilon_{\alpha}=\alpha_{k}-\alpha^{*}\) and \(\varepsilon_{\beta}=\beta_{k}-\beta^{*}\). It is apparent that both \(\varepsilon_{\alpha}\) and \(\varepsilon_{\beta}\) are negative or equal to zero according to Assumptions 3 and 4. Since \((\overline{a}^{2}+\overline{b}^{2}+\overline{c}^{2})^{\frac{1}{2}}\leq| \overline{a}|+|\overline{b}|+|\overline{c}|\), it is concluded that  
\[-\frac{\varepsilon\lambda_{\min}^{1/2}(\mathbf{P})}{\lambda_{\max}(\mathbf{ P})}V_{0}^{1/2}-\frac{\varepsilon_{1}}{\sqrt{2\gamma_{1}}}|\varepsilon_{ \alpha}|-\frac{\varepsilon_{2}}{\sqrt{2\gamma_{2}}}|\varepsilon_{\beta}|\leq -\eta_{0}V^{1/2}\]  
Here, \(\eta_{0}=\min(\frac{\varepsilon\lambda_{\min}^{1/2}(\mathbf{P})}{\lambda_{ \max}(\mathbf{P})},\ \ \varepsilon_{1},\ \varepsilon_{2})\). Then, (40) can have the form of  
\[\dot{V}\leq-\eta_{0}V^{1/2}+\frac{1}{\gamma_{1}}\varepsilon_{\alpha}\dot{ \alpha}_{k}+\frac{1}{\gamma_{2}}\varepsilon_{\beta}\dot{\beta}_{k}+\frac{ \xi_{1}}{\sqrt{2\gamma_{1}}}|\varepsilon_{\alpha}|+\frac{\xi_{2}}{\sqrt{2 \gamma_{2}}}|\varepsilon_{\beta}| \tag{41}\]  
Since both \(\varepsilon_{\alpha}\) and \(\varepsilon_{\beta}\) in (41) are equal to or less than zero, we can obtain  
\[\dot{V}\leq-\eta_{0}V^{1/2}-|\varepsilon_{\alpha}|(\frac{1}{\gamma_{1}}\dot{ \alpha}_{k}-\frac{\xi_{1}}{\sqrt{2\gamma_{1}}})-|\varepsilon_{\beta}|(\frac{1 }{\gamma_{2}}\dot{\beta}_{k}-\frac{\xi_{2}}{\sqrt{2\gamma_{2}}}) \tag{42}\]  
The motivation of designing the adaptive gains is to investigate a domain. The domain acts as a flag. The gains \(\alpha_{k}\) and \(\beta_{k}\) can start dynamically reducing when the system trajectories come to the domain in finite time. Once the trajectories leave the domain, the gains start dynamically increasing in order to draw the trajectories back. Inspired by the methodology of sliding mode, we picked up the domain \(\|\mathbf{s}_{ik}\|_{2}\leq\mu\) as this flag. Thereafter, (42) is deduced by different cases in accordance with such a flag. **Case 1. **: \(\|\mathbf{s}_{ik}\|_{2}>\mu\) _and \(\alpha_{k}>\alpha_{m}\) for all \(t\geq 0\). Take (24) into account. With regard to this case, (43) can be deduced from (24)._  
\[\dot{\alpha}_{k}=\xi_{1}\sqrt{\frac{\gamma_{1}}{2}} \tag{43}\]  
Therefore, (42) becomes  
\[\dot{V}\leq-\eta_{0}V^{1/2}-|\varepsilon_{\beta}|(\frac{1}{\gamma_{2}}\dot{ \beta}_{k}-\frac{\xi_{2}}{\sqrt{2\gamma_{2}}}) \tag{44}\]  
Picking up \(\varepsilon=\frac{\xi_{2}}{2\xi_{1}}\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}\) in (14) and substituting it into the time derivative of \(\beta_{k}=2\varepsilon\alpha_{k}\) yields  
\[\dot{\beta}_{k}=\xi_{2}\sqrt{\frac{\gamma_{2}}{2}} \tag{45}\]  
Finally, (44) becomes  
\[\dot{V}\leq-\eta_{0}V^{1/2} \tag{46}\]Associated with the closed-loop system stability, (33) must be held true in order to have the positive definite matrix \(\overline{\mathbf{Q}}\), meaning that \(\alpha\) should be increased as its adaptive law (24) until (33) is satisfied in the finite time \(t_{F}\). From this time \(t_{F}\) on, (46) will guarantee the convergence of this closed-loop system to the domain \(\|\mathbf{s}_{ik}\|_{2}\leq\mu\). **Case 2.**\(\|\mathbf{s}_{ik}\|_{2}\leq\mu\). Take (24) into account. According to the motivation, \(\alpha_{k}\) needs to be reduced in light of the adaptive law (24) so that it has a form of  
\[\dot{\alpha}_{k}=