# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 4 Stability Analysis Criteria for Interval Time-Delay Systems
**_Given scalars \(d_{2}>d_{1}=0\) and \(\mu>0\), the system (1) with a time delay \(d(t)\) subject to (2) is asymptotically stable if there exist matrices \(\overline{P}=\left[\overline{P}_{ij}\right]_{2\times 2}>0\), \(\overline{Q}_{i}>0\), \(\overline{W}_{i}>0\) (\(i=1\), \(2\); \(j=1\), \(2\)), \(\overline{U}>0\) and any matrices \(\overline{Z}_{l}\) (\(l=1,2,3,4\)) of appropriate dimensions such that the following LMIs are feasible:_  
\[\left[ \begin{array}{cc}\overline{U}+\overline{W}_{1}&\overline{Z}_{l}\\ *&\overline{U}+\overline{W}_{2}\end{array} \right]>0(l=1,2,3,4), \tag{41}\]  
\[\overline{\Pi}_{i}=\text{sym}\Big{\{}\overline{\Omega}_{i}^{T}\overline{P \Lambda}\Big{\}}+\overline{\Phi}_{1}+\overline{\Phi}_{2}+\overline{\Gamma}+ \sum_{l=1}^{4}\overline{\Theta}_{l}<0\ (i=1,2), \tag{42}\]  
_where_  
\[\overline{\Omega}_{1}=\left[ \begin{array}{c}c_{1}\\ d_{25}\end{array} \right],\,\overline{\Omega}_{2}=\left[ \begin{array}{c}c_{ 1}\\ d_{24}\end{array} \right],\,\overline{\Phi}_{1}=\left[ \begin{array}{c}c_{1} \\ c_{3}\end{array} \right]^{T}\!\!\left[ \begin{array}{cc}\overline{\Omega}_{1}&0 \\ *&-(1-\mu)\overline{\Omega}_{1}\end{array} \right]\left[ \begin{array}{cc}c_{1} \\ c_{3}\end{array} \right]+\left[ \begin{array}{cc}c_{1}\\ c_{2}\end{array} \right]^{T}\!\!\left[ \begin{array}{cc}\overline{\Omega}_{2}&0 \\ *&-\overline{\Omega}_{2}\end{array} \right]\left[ \begin{array}{c}c_{1}\\ c_{2}\end{array} \right]\!\!,\]  
\[\overline{\Lambda}=\left[ \begin{array}{c}Ae_{1}+B_{5}\\ e_{1}-e_{2}\end{array} \right],\,\overline{\Phi}_{2}=(Ae_{1}+B_{5})^{T}\Big{(}d \frac{d}{2}\overline{U}+\frac{d}{2}(\overline{W}_{1}+\overline{W}_{2})\Big{)} (Ae_{1}+B_{5}),\]  
\[\bar{\Gamma}=\text{col}\{\overline{\Gamma}_{3},\,\overline{\Gamma}_{4},\, \overline{\Gamma}_{5},\,\overline{\Gamma}_{6}\},\,\overline{\Gamma}=-\bar{ \Gamma}^{T}diag\{\sigma_{2}(\overline{W}_{1}),\,\sigma_{2}(\overline{W}_{1}),\, \sigma_{2}(\overline{W}_{2}),\,\sigma_{2}(\overline{W}_{2})\}\bar{\Gamma},\]  
\[\overline{\Gamma}_{1}=\text{col}\{e_{1}-e_{3},\,e_{1}+c_{3}-2e_{4},\,e_{1}-e_{ 3}+e_{4}-12e_{6},\,e_{1}+c_{3}-12e_{4}+60e_{6}-120\kappa\},\]  
\[\overline{\Gamma}_{2}=\text{col}\{e_{3}-e_{2},\,e_{3}+e_{2}-2e_{5},\,e_{3}-e_{2}+ e_{5}-12e_{7},\,e_{3}+e_{2}-12e_{5}+60e_{7}-120\kappa\},\]  
\[\overline{\Gamma}_{3}=\text{col}\{e_{1}-e_{4},\,e_{1}+2e_{4}-6e_{6},\,e_{1}-3e_{4 }+24e_{6}-60\kappa\},\]  
\[\overline{\Gamma}_{4}=\text{col}\{e_{3}-e_{5},\,e_{3}+2e_{5}-6e_{7},\,e_{3}-3e_{ 5}+24e_{7}-60\kappa_{9}\},\]  
\[\overline{\Gamma}_{5}=\text{col}\{e_{3}-e_{4},\,e_{3}-4e_{4}+6e_{6},\,e_{3}-9e_{ 4}+36e_{6}-60\kappa_{8}\},\]  
\[\overline{\Gamma}_{6}=\text{col}\{e_{2}-e_{5},\,e_{2}-4e_{5}+6e_{7},\,e_{2}-9e_{ 5}+36e_{7}-60\kappa_{9}\},\]  
\[\overline{\Theta}_{l}=-(2l-1)\left[ \begin{array}{cc}e_{1}\overline{\Gamma}_{1} \\ e_{1}\overline{\Gamma}_{2}\end{array} \right]^{T}\!\!\left[ \begin{array}{cc} \overline{\Omega}&\overline{Z}_{l}\\ *&\overline{U}\end{array} \right]\left[ \begin{array}{c}e_{l}\overline{\Gamma}_{1} \\ *\overline{\Gamma}_{2}\end{array} \right]\,(l=1,2,3,4).\]  
**Proof. ** Choose the following LKF candidate  
\[\overline{V}(t)=\overline{V}_{1}(t)+\overline{V}_{2}(t)+\overline{V}_{3}(t)+ \overline{V}_{4}(t), \tag{43}\]where  
\[\overline{V}_{1}(t)=\left[ \begin{array}{cc}x(t)\\ \int_{t-d_{2}}^{t}x(s)ds\end{array} \right]^{T}\left[ \begin{array}{cc}\overline {P}_{11}&\overline{P}_{12}\\ *&\overline{P}_{22}\end{array} \right]\left[ \begin{array}{c}x(t)\\ \int_{t-d_{2}}^{t}x(s)ds\end{array}