## 3 Control Design
### Stability Analysis of the Closed-Loop Control System
Consider a Lyapunov candidate as  
\[V_{0}=\mathbf{Z}^{T}\mathbf{P}\mathbf{Z} \tag{32}\]  
Here, \(\mathbf{P}=\left[ \begin{array}{cc}(\lambda+4\varepsilon^{2})\mathbf{I}_{2}& -2\varepsilon\mathbf{I}_{2}\\ -2\varepsilon\mathbf{I}_{2}&\mathbf{I}_{2}\end{array} \right]\) is a \(4\times 4\) positive definite matrix, \(\lambda>0\) and \(\varepsilon>0\). From (31), the time derivative of \(V_{0}\) in (32) can be written by  
\[\dot{V}_{0}=\dot{\mathbf{Z}}^{T}\mathbf{P}\mathbf{Z}+\mathbf{Z}^{T}\mathbf{P} \dot{\mathbf{Z}}=\mathbf{Z}^{T}\Big{(}\overline{\mathbf{A}}^{T}\mathbf{P}+ \mathbf{P}\overline{\mathbf{A}}^{T}\Big{)}\mathbf{Z}\leq-\frac{1}{2\|\mathbf{ Z}_{1}\|_{\infty}}\mathbf{Z}^{T}\overline{\mathbf{Q}}\mathbf{Z} \tag{33}\]  
Here, \(\overline{\mathbf{Q}}=\left[ \begin{array}{cc}\overline{\mathbf{Q}}_{11}& \overline{\mathbf{Q}}_{12}\\ \overline{\mathbf{Q}}_{21}&4\varepsilon\end{array} \right]\) is a \(4\times 4\) symmetric matrix, \(\overline{\mathbf{Q}}_{11}\), \(\overline{\mathbf{Q}}_{12}\) and \(\overline{\mathbf{Q}}_{21}\) are given by  
\[\overline{\mathbf{Q}}_{11}=2\lambda\alpha_{k}\mathbf{b}_{1}+4\varepsilon(2 \varepsilon\alpha_{k}-\beta_{k})\mathbf{b}_{1}-[2(\lambda+4\varepsilon^{2}) \rho_{1}(\mathbf{x}_{ik},t)-4\varepsilon\rho_{2}(\mathbf{x}_{ik},t)]\mathbf{I }_{2}\]  
\[\overline{\mathbf{Q}}_{12}=\overline{\mathbf{Q}}_{21}=(\beta_{k}-2\varepsilon \alpha_{k})\mathbf{b}_{1}+[2\varepsilon\rho_{1}(\mathbf{x}_{ik},t)-\rho_{2}( \mathbf{x}_{ik},t)]\mathbf{I}_{2}-(\lambda+4\varepsilon^{2})\mathbf{I}_{2}\]  
From the aspect of the stability of \(V_{0}\), \(\overline{\mathbf{Q}}\) is not only symmetric but also positive definite in the sense of [PERSON]. Therefore, we selected \(\beta_{k}=2\varepsilon\alpha_{k}\) in \(\overline{\mathbf{Q}}_{11}\). Then, \(\overline{\mathbf{Q}}\) can be positive definite and its minimal eigenvalue is \(\lambda_{\min}(\overline{\mathbf{Q}})\geq 2\varepsilon\) if \(\alpha_{k}\) satisfies (25). From (32), we have  
\[\lambda_{\min}(\mathbf{P})\|\mathbf{Z}\|_{2}^{2}\leq\mathbf{Z}^{T}\mathbf{P} \mathbf{Z}=V_{0}\leq\lambda_{\max}(\mathbf{P})\|\mathbf{Z}\|_{2}^{2} \tag{34}\]  
Further, (35) can be deduced from (33) if (25) holds true. \[\dot{V}_{0}\leq-\frac{1}{2\|\mathbf{Z}_{1}\|_{\infty}}\mathbf{Z}^{T}\overline {\mathbf{Q}}\mathbf{Z}\leq-\frac{2\varepsilon}{2\|\mathbf{Z}_{1}\|_{\infty}} \mathbf{Z}^{T}\mathbf{Z}=-\frac{\varepsilon\|\mathbf{Z}\|_{2}^{2}}{\|\mathbf{ Z}_{1}\|_{\infty}} \tag{35}\]  
In (26), the following inequality exists. \[\|\mathbf{Z}_{1}\|_{\infty}=\sqrt{\|\mathbf{s}_{ik}\|_{2}}\leq\|\mathbf{Z}_{1} \|_{2}\leq\|\mathbf{Z}\|_{2}\leq\left(\frac{V_{0}}{\lambda_{\min}(\mathbf{P}) }\right)^{1/2} \tag{36}\]  
With regard to (34) and (36), (35) can be re-written by  
\[\dot{V}_{0}\leq-\frac{\varepsilon\lambda_{\min}^{1/2}(\mathbf{P})}{\lambda_{ \max}(\mathbf{P})}V_{0}^{1/2} \tag{37}\]  
_Analysis._  
Now, the closed-looped control system stability will be presented in the sense of Lyapunov. Define a Lyapunov candidate as  
\[V=V_{0}+\frac{1}{2\gamma_{1}}(\alpha_{k}-\alpha^{*})^{2}+\frac{1}{2\gamma_{2}} (\beta_{k}-\beta^{*})^{2} \tag{38}\]Here, \(\alpha^{*}\) and \(\beta^{*}\) are given in Assumptions 3 and 4. Concerning (33) and (37), the time derivative of \(V\) can have the form of  
\[\dot{V}=\dot{V}_{0}+\frac{1}{\gamma_{1}}(\alpha_{k}-\alpha^{*})\dot{\alpha}+ \frac{1}{\gamma_{2}}(\beta_{k}-\beta^{*})\dot{\beta} \tag{39}\]  
From (37), (39) can be written as  
\[