## 4 Model Stability Analysis
\begin{split}& V_{2}(k+1)-e^{-a}V_{2}(k)=\delta^{T}(k)\hat{Q} \delta(k)-e^{-a\bar{\tau}_{1}}\delta^{T}(k-\overline{\tau}_{1})\hat{Q}\delta(k -\overline{\tau}_{1})\\ &+\tau_{1}y^{T}(k)\hat{Z}y(k)-\sum_{\theta=k-\tau_{1}}^{k-1}e^{a( \theta-k)}y^{T}(\theta)\hat{Z}y(\theta)+\delta^{T}(0)\delta(0)\\ &=\xi^{T}(k)\psi_{2}\bar{\varsigma}(k)+\bar{\tau}_{1}y^{T}(k)\hat{Z}y(k)- \sum_{\theta=k-\tau_{1}}^{k-1}e^{a(\theta-k)}y^{T}(\theta)\hat{Z}y(\theta) \end{split}  \tag{30}\]where \(\psi_{2}=diag\{\hat{Q},-e^{-\alpha\tau_{1}}\hat{Q},0,0,I\}\)  
An equation of the same kind obtains  
\[ \begin{split}& V_{3}(k+1)-e^{-\alpha}V_{3}(k)=\delta^{\mathrm{T}}(k) \overline{Q}\delta(k)-e^{-\alpha\overline{\tau}_{2}}\delta^{\mathrm{T}}(k- \overline{\tau}_{2})\overline{Q}\delta(k-\overline{\tau}_{2})\\ &+\overline{\tau}_{2}y^{\mathrm{T}}(k)\overline{Z}y(k)-\sum_{ \theta=\bar{k}-\overline{\tau}_{2}}^{k-1}e^{\alpha(\theta-\bar{k})}y^{\mathrm{ T}}(\theta)\overline{Z}y(\theta)\\ &=\xi^{\mathrm{T}}(k)\psi_{3}\xi(k)+\overline{\tau}_{2}y^{ \mathrm{T}}(k)\overline{Z}y(k)-\sum_{\theta=k-\overline{\tau}_{2}}^{k-1}e^{ \alpha(\theta-k)}y^{\mathrm{T}}(\theta)\overline{Z}y(\theta)\end{split}  \tag{30}\]  
where \(\psi_{3}=diag\{\overline{Q},0,-e^{-\alpha\tau_{2}}\overline{Q},0,0\}\). From (19) and (28)-(30), we obtain  
\[\Delta V(k)=V(k+1)-e^{-\alpha}V(k)\leq\xi^{\mathrm{T}}(k)\psi\xi(k)\]  
\[\psi= \begin{bmatrix}\Phi_{1}&\hat{\mathrm{Y}}-P\hat{B}&\overline{\mathrm{Y}} -P\overline{B}_{r}&PB_{\omega}&-P\hat{B}\\ &-e^{-\alpha\tau_{1}}\hat{Q}&0&0&0\\ &*&-e^{-\alpha\tau_{2}}\hat{Q}&0&0\\ &*&*&0&0\\ &*&*&*&I\end{bmatrix} + \begin{bmatrix}(\hat{A}-I)^{T}\\ \hat{B}^{T}\\ \overline{B}_{r}\\ -\hat{B}^{T}\end{bmatrix} \]  
Using the Schur complement lemma, it can be shown that LMI (20) is equivalent to \(\psi<0\). The system (18) is stable by [PERSON]'s stability theory. Next, under zero initial conditions, consider the performance metrics  
\[J(\omega)=\sum_{k=0}^{\infty}\left(z^{\mathrm{T}}(t)z(t)-\gamma^{2}\omega^{ \mathrm{T}}(t)\omega(t)\right) \tag{31}\]  
Then, for any \(\omega(t)\in l_{2}[0,\infty)\), it follows from the properties of the Lyapunov generalized function and the zero initial condition that  
\[ \begin{split} J(\omega)&=\sum_{k=0}^{\infty}\left(z^{ \mathrm{T}}(k)z(k)-\gamma^{2}\omega^{\mathrm{T}}(k)\omega(k)+V(k+1)-e^{-\alpha }V(k)\right)-(1-e^{-\alpha})\sum_{k=1}^{\infty}V(k)-V(\infty)\\ &\leq\sum_{k=0}^{\infty}\left(z^{\mathrm{T}}(k)z(k)-\gamma^{2} \omega^{\mathrm{T}}(k)co(k)+V(k+1)-e^{-\alpha}V(k)\right)\\ &=\sum_{k=0}^{\infty}\left(\xi^{T}(k)\Psi\xi(k)+\delta^{\mathrm{T} }(k)\delta(k)-\gamma^{2}\omega^{\mathrm{T}}(k)\omega(k)\right)\\ &=\sum_{k=0}^{\infty}\left(\xi^{T}(k)\Xi\xi(k)\right)\\ &\qquad\qquad\text{where }\Xi=\psi-diag(I,0,0,\gamma^{2},0)\\ &\qquad\qquad\text{By [PERSON] complement lemma, (\ref{eq:LMII}) is equivalent to $\Xi<0$, and thus $J(\omega)$<$0, i.e.,}\\ &\sum_{k=0}^{\infty}z^{\mathrm{T}}(k)z(k)<\gamma^{2}\sum_{k=0}^{ \infty}\omega^{\mathrm{T}}(k)\omega(k)\\ &\qquad\text{Thus, the system (\ref{eq:LMII}) has $H_{\infty}$ performance $\gamma$.}\end{split} \]  
In the operation of a supply chain system, major accidents can lead to the failure of nodes, resulting in what we term \"isolated nodes.\" When the supply chain cannot meet the supply demand, the system proactively replaces and upgrades nodes to ensure overall security and efficiency. However, this process may temporarily result in isolated nodes. We define these failed nodes as the isolated nodes of the system, denoted by \(\sigma(h)=j(j=0,1\cdots,n)\), where \(\sigma(h)=0\) indicates that there are no isolated nodes, while \(\sigma(h)=1\) signifies that the first node is isolated. For simplicity, we assume that only one node fails at any given time, and the failed node can be restored to normal operation after adjusting and replacing the supply facility. For systems with isolated nodes, there are  
\[