# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 4 Stability Analysis Criteria for Interval Time-Delay Systems  
In this section, in order to demonstrate the merits of the new integral inequalities, the proposed ones are applied to the stability analysis of the interval time-delay systems. For simplicity of presentation, for a matrix \(Q\), define \(\varphi_{1}(Q)=diag\{Q\,3Q\,5Q\,7Q\}\) and \(\varphi_{2}(Q)=diag\{2Q\,4Q\,6Q\}\). **Theorem 1**. : _Given scalars \(d_{2}>d_{1}>0\) and \(\mu>0\), the system (1) with an interval time-delay \(d(t)\) subject to (2) is asymptotically stable if there exist matrices \(P=\left[P_{ij}\right]_{2\times 2}>0\), \(R=\left[R_{ij}\right]_{2\times 2}>0\), \(Q_{i}>0\), \(U_{i}>0\), \(S_{i}>0\), \(W_{i}>0\)(\(i=1,2;j=1,2\)) and any matrices \(Z_{l}\)\((l=1,2,3,4)\) of appropriate dimensions such that the following LMIs hold:_  
\[\left[ \begin{array}{cc}U_{2}+W_{1}&Z_{l}\\ *&U_{2}+W_{2}\end{array} \right]>0\ (l=1,2,3,4), \tag{25}\]  
\[\Pi_{i}=sym\Big{\{}\Omega_{l}^{T}P\Lambda\Big{\}}+\sum_{j=1}^{5}\Phi_{j}+ \Gamma+\sum_{l=1}^{4}\Theta_{l}<0\ (i=1,2), \tag{26}\]  
_where_  
\[\Omega_{1}=\left[ \begin{array}{c}e_{1}\\ de_{5}\end{array} \right]\!,\ \Omega_{2}=\left[ \begin{array}{c}e_{1}\\ de_{6}\end{array} \right]\!,\ \Phi_{1}=\left[ \begin{array}{c}e_{1}\\ e_{4}\end{array} \right]^{T}\!\left[ \begin{array}{cc}Q_{1}&0\\ *&-(1-\mu)Q_{1}\end{array} \right]\left[ \begin{array}{cc}e_{1}\\ e_{4}\end{array} \right]+\left[ \begin{array}{cc}e_{2}\\ e_{3}\end{array} \right]^{T}\!\left[ \begin{array}{cc}Q_{2}&0\\ *&-Q_{2}\end{array} \right]\!\left[ \begin{array}{c}e_{2}\\ e_{3}\end{array} \right]\!,\]  
\[\Lambda=\left[ \begin{array}{c}Ae_{1}+Be_{4}\\ e_{2}-e_{3}\end{array} \right]\!,\ \[PERSON]_{2}=\frac{1}{2}sym\[PERSON]{\{}\left[ \begin{array}{cc}2d_{1}(e_{1}-e_{7}) \\ d_{1}^{2}(e_{1}-2e_{10})\end{array} \right]^{T}\!\left[ \begin{array}{cc}R_{1 }&R_{12}\\ *&R_{22}\end{array} \right]\left[ \begin{array}{c}d_{2}^{2}e_{10}\\ d_{1}^{2}e_{10}\end{array} \right]\Bigg{\}},\]  
\[\Phi_{3}=(Ae_{1}+Be_{4})^{T}(d_{1}^{2}U_{1}+A^{2}U_{2})(Ae_{1}+Be_{4}),\ \Phi_{4}=(d_{1}^{2}/2)(Ae_{1}+Be_{4})^{T}(S_{1}+S_{2})(Ae_{1}+Be_{4}),\]  
\[\Phi_{5}=(d^{2}/2)(Ae_{1}+Be_{4})^{T}(W_{1}+W_{2})(Ae_{1}+Be_{4}),\ \widetilde{ \Gamma}=col\{\Gamma_{1},\ \Gamma_{4},\ \Gamma_{5},\ \Gamma_{6},\ \Gamma_{7},\ \Gamma_{8},\ \Gamma_{9}\},\]  
\[\Gamma=-\widetilde{\Gamma}^{T}diag\{\varphi_{1}(U_{1}),\ \varphi_{2}(S_{1}),\ \varphi_{2}(S_{2}),\ \varphi_{2}(W_{1}),\ \varphi_{2}(W_{1}),\ \varphi_{2}(W_{2}),\ \varphi_{2}(W_{2})\}\widetilde{\Gamma},\]  
\[\Gamma_{1}=col\{e_{1}-e_{2},\ \epsilon_{1}+e_{2}-2\sigma_{1},\ \epsilon_{1}-e_{2}+ 6\sigma_{7}-12\epsilon_{10},\ \epsilon_{1}+e_{2}-12\sigma_{7}+60\epsilon_{10}-120\epsilon_{13}\},\]  
\[\Gamma_{2}=col\{e_{2}-e_{4},\ e_{2}+e_{4}-2e_{5},\ \epsilon_{2}-e_{4}+6e_{5}-12\kappa_{ 8},\ \epsilon_{2}+\epsilon_{4}-12\kappa_{5}+60\kappa_{8}-120\epsilon_{11}\},\]  
\[\Gamma_{3}=col\{e_{4}-e_{3},\ \epsilon_{4}+e_{3}-2\epsilon_{6},\ \epsilon_{4}-e_{3}+6\epsilon_{6}-12\epsilon_{9},\ \epsilon_{4}+e_{3}-12\epsilon_{6}+60\epsilon_{9}-120\epsilon_{12}\},\]  
\[\Gamma_{4}=col\{e_{1}-e_{7},\ \epsilon_{1}+2\sigma_{7}-6\epsilon_{10},\ \epsilon_{1}-3 \sigma_{7}+24\epsilon_{10}-60\epsilon_{13}\},\]  
\[\Gamma_{5}=col\{e_{2}-e_{7},\ \epsilon_{2}-4\epsilon_{7}+6\epsilon_{10},\ \epsilon_{2}-9 \sigma_{7}+36\epsilon_{10}-60\epsilon_{13}\},\]  
\[\Gamma_{6}=col\{e_{2}-e_{5},\ \epsilon_{2}+2\epsilon_{5}-6\kappa_{8},\ \epsilon_{2}-3 \sigma_{5}+24\kappa_{8}-60\epsilon_{11}\},\]  
\[\Gamma_{7}=col\{\epsilon_{4}-e_{6},\ \epsilon_{4}+2\epsilon_{6}-6\kappa_{9},\ \epsilon_{4}-3 \sigma_{6}+24\kappa_{9}-60\epsilon_{12}\},\]  
\[\Phi_{5}=(d^{2}/2)(Ae_{1}+Be_{4})^{T}(W_{1}+W_{2})(Ae_{1}+Be_{4}),\ \widetilde{ \Gamma}=col\{\Gamma_{1},\ \Gamma_{4},\ \Gamma_{5},\ \Gamma_{6},\ \Gamma_{7},\ \Gamma_{8},\ \Gamma_{9}\},\]0  
\[\Phi_{5}=(d^{2}/2)(Ae_{1}+Be_{4})^{T}(W_{1}+W_{2})(Ae_{1}+Be_{4}),\ \widetilde{ \Gamma}=col\{\Gamma_{1},\ \Gamma_{4},\ \Gamma_{5},\ \Gamma_{6},\ \Gamma_{7},\ \Gamma_{8},\ \Gamma_{9}\},\]1  
\[\Theta_{l}=-(2l-1)\left[