## 3 Boundary Control Design
### Stability Analysis
: From Equation (34) after substituting \(\varphi>0\), the dynamics Equations (37)-(39) are said to be quadratic stabilize with \(H_{\infty}\) norm bound \(\varphi\), if there \(U^{k}\) be existent such that the closed-loop string system is quadratic stabilize with an \(H_{\infty}\) norm bound \(\varphi\). **Lemma 1**[44]. : If the uncontrolled string system Equations (37)-(39) is quadratic stabilize with an \(H_{\infty}\) norm bound \(\varphi>0\), therefore it has stability with an \(H_{\infty}\) norm bound \(\varphi\) for any allowable uncertainties. **Lemma 2**[44]. : If the string system Equations (37)-(39) is quadratic stabilize with an \(H_{\infty}\) norm bound \(\varphi\), therefore it has stability with an \(H_{\infty}\) norm bound \(\varphi\) for all allowable uncertainties. Then, the dynamic Equations (37)-(39) are expressed as:  
\[\left[ \begin{array}{cc}\Delta A_{d}&\Delta B_{d1}\\ \Delta B_{d2}&0\end{array} \right]=\left[ \begin{array}{c}O_{1}\\ O_{2}\end{array} \right]K\left[ \begin{array}{cc}R_{1}&R_{2}\end{array} \right] \tag{48}\]  
where \(O_{1},O_{2},R_{1},\text{ and }R_{2}\) are constant matrices and \(KK^{T}\leq 1\). By introducing the following auxiliary system, the uncertain string system is stabilizable with \(\|G(\omega)\|<1\) if the system in Equations (37)-(39) is stabilizable with \(\|G_{a}(\omega)\|_{\infty^{\omega}}<1\). \[Z_{a}^{k+1}=A_{d}Z_{a}^{k}+H_{1}F_{a}^{k}+B_{d2}U^{k} \tag{49}\]  
\[X_{a}^{k}=J_{1}Z_{a}^{k}+D_{11}F_{a}^{k}+J_{2}U^{k} \tag{50}\]  
\[Y_{a}^{k}=C_{d2}Z_{a}^{k}+H_{2}F_{a}^{k} \tag{51}\]  
where  
\[H_{1}=\left[ \begin{array}{cc}\sqrt{\varphi}O_{1}I&B_{d1}\\ 0&0\end{array} \right],\,H_{2}=\left[ \begin{array}{cc}\sqrt{\varphi}O_{2}I&D _{21}\\ 0&0\end{array} \right],\,J_{1}=\left[ \begin{array}{cc}\frac{R_{1}}{\sqrt{ \varphi}}I&0\\ C_{d1}&0\end{array} \right],\,J_{2}=\left[ \begin{array}{cc}\frac{R_{2}}{ \sqrt{\varphi}}I&0\\ C_{d2}&0\end{array} \right].\]  
**Theorem 1**[44]. : From Equation (34) after substituting \(\varphi>0\), the uncontrolled string system in Equations (37)-(39) is said to be quadratic stabilize with \(H_{\infty}\) norm bound \(\varphi>0\), if just the unregulated auxiliary string system of Equations (53) and (54) is stabilizable within unitary \(H_{\infty}\) norm bound. Moreover, the controlled string system in Equations (37)-(39) is said to be quadratic stabilize with \(H_{\infty}\) norm bound \(\varphi>0\), if only the auxiliary string system of Equations (37)-(39) is stabilizable within unitary \(H_{\infty}\) norm bound. Applying Theorem 1 to Equations (37)-(39), one obtains  
\[\left[ \begin{array}{cc}0&0\\ \Delta H&0\end{array} \right]=\left[ \begin{array}{cc}0\\ \Delta H\end{array} \right]N_{H}I\left[ \begin{array}{cc}N_{H}I&0\end{array} \right]. \tag{52}\]  
Therefore, the resulting \(H_{\infty}\) system integrating with ILBC expressed as  
\[Z_{a}^{k+1}=A_{d}^{a}Z_{a}^{k}+B_{d1}^{a}F_{a}^{k}+B_{d2}^{a}U^{k}, \tag{53}\]  
\[X_{a}^{k}=C_{d1}^{a}Z_{a}^{k}+D_{11}^{a}F_{a}^{k}+D_{12}^{a}U^{k}, \tag{54}\]  
\[Y_{a}^{k}=C_{d2}^{a}Z_{a}^{k}+D_{21}^{a}F_{a}^{k}, \tag{55}\]  
where  
\[ \begin{array}{cc}A_{d}^{a}=I,\ B_{d1}^{a}=\left[\begin{array}{cc}0&\psi I &0\end{array}\right],\ B_{d2}^{a}=I,\ C_{d1}^{a}=\left[\begin{array}{cc}I& \sqrt{\varphi}&0&I\end{array}\right],\ C_{d2}^{a}=H,\ D_{12}^{a}=\left[ \begin{array}{cc}0&I&0\end{array}\right],\\ D_{21}^{a}=\sqrt{\varphi}\Delta H\psi,\ D_{22}^{a}=0.\end{array} \]  
**Theorem 2**[44].