## 5 Formulation of the LMI Stability Criterion
### Design of the LMI Condition  
By setting \(\mathcal{S}=\mathcal{P}^{-1}\) and multiplying the obtained nonlinear matrix condition described by (53) on the right side and on the left side by \(\mathcal{Q}=\left[ \begin{array}{cc}\mathcal{S}&\mathcal{O}\\ \star&\mathcal{I}\end{array} \right]\), we obtain the the following inequality:  
\[\left[ \begin{array}{cc}\Gamma_{11}&\Gamma_{12}\\ \star&\Gamma_{22}\end{array} \right]<0 \tag{54}\]  
where  
\[\Gamma_{11} =(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+(\mathcal{A} \mathcal{S}+\mathcal{B}\mathcal{  
Therefore, using the two expressions (63b) and (63a) in the condition (62), it is easy to demonstrate the subsequent relation:  
\[\left(\mathcal{AS}+\mathcal{BR}\right)+\left(\mathcal{AS}+\mathcal{BR}\right)^{ \text{T}}+\mathcal{M}_{1}+\mathcal{D}_{2}m_{2}\mathcal{D}_{2}^{\text{T}}+ \mathcal{M}_{3}+\zeta\mathcal{S}+\xi_{1}\mathcal{S}^{2}+\mathcal{JWJ}^{\text{T} }<0 \tag{64}\]  
where the quantity \(\mathcal{W}\) is defined as follows:  
\[\mathcal{W}=\sigma^{-1}\mathcal{M}^{-1}-\mathcal{QH}^{-1}\mathcal{Q} \tag{65}\]  
where \(\sigma^{-1}=\frac{\mu_{1}}{\mu_{2}^{2}}-\frac{1}{\mu_{3}}\). Through the application of the Schur complement to condition (57a), and given that \(\mathcal{MQ}>0\), we arrive at the subsequent expression:  
\[\left[ \begin{array}{cc}\beta^{-1}&\mathcal{D}^{\text{T}}\\ \star&\mathcal{MQ}\end{array} \right]>0 \tag{66}\]  
Moreover, applying the Schur complement to the previous expression (66), we obtain the subsequent relation:  
\[\mathcal{MQ}-\beta\mathcal{DD}^{\text{T}}>0 \tag{67}\]  
therefore, using expression (59), it follows that:  
\[\mathcal{H}<0 \tag{68}\]  
Moreover, using the expressions of the three variables \(\mu_{1}\), \(\mu_{2}\), and \(\mu_{3}\) defined in (45), it is easy to show that \(\sigma^{-1}<0\). Therefore, based on the expression of the quantity \(\mathcal{W}\) defined in Expression (65), it is obvious that since \(\mathcal{M}>0\), we can easily obtain \(\mathcal{W}>0\). Then, by setting \(\gamma=\xi_{1}^{-1}\) and applying the Schur complement to Equation (64), we establish the following criterion:  
\[\left[ \begin{array}{ccc}\Xi_{11}&\mathcal{S}&\mathcal{J}\\ \star&-\gamma\mathcal{I}&\mathcal{O}\\ \star&\star&-\mathcal{W}^{-1}\end{array} \right]<0 \tag{69}\]  
with  
\[\Xi_{11}=\left(\mathcal{AS}+\mathcal{BR}\right)+\left(\mathcal{AS}+\mathcal{ BR}\right)^{\text{T}}+\mathcal{M}_{1}+m_{2}\mathcal{D}_{2}\mathcal{D}_{2}^{ \text{T}}+\mathcal{M}_{3}+\zeta\mathcal{S} \tag{70}\]  
Furthermore, applying the matrix inversion lemma (Lemma 4) to Expression (65), we obtain the following:  
\[\mathcal{W}^{-1}=\sigma\mathcal{M}-\sigma^{2}\mathcal{MQ}(\sigma\mathcal{QM} \mathcal{Q}-\mathcal{H})^{-1}\mathcal{QM} \tag{71}\]  
Notably, we have the following:  
\[-\mathcal{W}^{-1}=-\sigma\mathcal{M}+\sigma^{2}\mathcal{MQ}(\sigma\mathcal{QM} \mathcal{Q}-\mathcal{H})^{-1}\mathcal{QM}<0 \tag{72}\]  
Moreover, as \(\sigma<0\) then \(\sigma\mathcal{M}<0\). Therefore, relying on condition (72), it is obvious that the inequality \(\sigma^{2}\mathcal{MQ}(\sigma\mathcal{QM}\mathcal{Q}-\mathcal{H})^{-1} \mathcal{QM}<0\) holds. Hence, we conclude the following:  
\[\sigma\mathcal{QM}\mathcal{Q}-\mathcal{H}<0 \tag{73}\]The Young inequality (Lemma 1) enables us to derive the following:  
\[\sigma^{2}\mathcal{MQ}(\sigma\mathcal{QM}\mathcal{Q}-\mathcal{H})^{-1}\mathcal{QM }\leq-\sigma\mathcal{QM}\mathcal{Q}+\mathcal{H}+2\sigma\mathcal{MQ} \tag{74}\]  
Therefore, by using Expression (74) in (72), we obtain the following:  
\[-\mathcal{W}^{-1}\leq-\sigma\mathcal{M}(\mathcal{Q}-\mathcal{I})^{2}+\mathcal{H} \tag{75}\]  
Hence, using this previous relation (75), Expression (69) becomes as follows:  
\[\left[ \begin{array}{ccc}\Xi_{11}&\mathcal{S}&\mathcal{J}\\ \star&-\gamma\mathcal{I}&\mathcal{O}\\ \star&\star&-\sigma\mathcal{M}(\mathcal{Q}-\mathcal{I})^{2}+\mathcal{H}\\ \end{array}