# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 3 New Integral Inequalities
**_For a given matrix \(Q>0\), the following inequalities hold for all continuously differentiable function \(x\) in \([\alpha,\beta]\rightarrow\Re^{n}:\)_  
\[\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)dsd\theta\geq 2 \left(\psi_{5}^{T}Q\psi_{5}+2\psi_{6}^{T}Q\psi_{6}+3\psi_{7}^{T}Q\psi_{7}\right), \tag{17}\]  
\[\int_{\alpha}^{\beta}\int_{\alpha}^{\theta}\dot{x}^{T}(s)Q\dot{x}(s)dsd\theta \geq 2\left(\psi_{8}^{T}Q\psi_{8}+2\psi_{9}^{T}Q\psi_{9}+3\psi_{10}^{T}Q\psi_{10} \right), \tag{18}\]_where_  
\[\psi_{5}=x(\beta)-\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}x(s)ds, \ \psi_{6}=x(\beta)+\frac{2}{\beta-\alpha}\int_{\alpha}^{\beta}x(s)ds-\frac{6}{( \beta-\alpha)^{2}}\int_{\alpha}^{\beta}\int_{\beta}^{\beta}x(s)dsd\theta,\] \[\psi_{7}=x(\beta)-\frac{3}{\beta-\alpha}\int_{\alpha}^{\beta}x(s) ds+\frac{24}{(\beta-\alpha)^{2}}\int_{\alpha}^{\beta}\int_{\beta}^{\beta}x(s)dsd \theta-\frac{60}{(\beta-\alpha)^{3}}\int_{\alpha}^{\beta}\int_{\alpha}^{\beta} \int_{\beta}^{\beta}\int_{\alpha}x(s)dsd\theta d\sigma,\] \[\psi_{8}=x(\alpha)-\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}x(s) ds,\ \psi_{9}=x(\alpha)-\frac{4}{\beta-\alpha}\int_{\alpha}^{\beta}x(s)ds+\frac{6}{( \beta-\alpha)^{2}}\int_{\alpha}^{\beta}\int_{\beta}^{\beta}x(s)dsd\theta,\] \[\psi_{10}=x(\alpha)-\frac{9}{\beta-\alpha}\int_{\alpha}^{\beta}x( s)ds+\frac{36}{(\beta-\alpha)^{2}}\int_{\alpha}^{\beta}\int_{\beta}^{\beta}x(s) dsd\theta-\frac{60}{(\beta-\alpha)^{3}}\int_{\alpha}^{\beta}\int_{\alpha}^{\beta} \int_{\beta}^{\beta}\int_{\beta}^{\beta}x(s)dsd\theta d\sigma.\]  
Proof. : In order to prove Equation (17), consider a function \(\omega_{2}(s)\) for all \(s\in[\alpha,\ \beta]\) defined as  
\[\omega_{2}(s)=\dot{x}(s)-\frac{2}{(\beta-\alpha)^{2}}\int_{\alpha}^{\beta}\int _{\theta}^{\beta}\dot{x}(s)dsd\theta-\delta_{4}(s)\vartheta_{4}-\delta_{5}(s) \vartheta_{5}, \tag{19}\]  
where \(\vartheta_{4}\) and \(\vartheta_{5}\) are constant vectors to be determined and  
\[\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds\geq\frac{1}{\beta-\alpha} \left(\psi_{1}^{T}Q\psi_{1}+5\psi_{3}^{T}Q\psi_{3}\right), \tag{15}\]0  
satisfying \(\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\delta_{4}(s)dsd\theta=0\), \(\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\delta_{5}(s)dsd\theta=0\) and \(\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\delta_{4}(s)\delta_{5}(s)dsd\theta=0\). For a matrix \(Q>0\), one has  
\[ \begin{array}{rl}\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\omega_{2}^{\rm T }(s)Q\omega_{2}(s)dsd\theta&=\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\dot{x }^{\rm T}(s)Q\dot{x}(s)dsd\theta-\frac{2}{(\beta-\alpha)^{2}}\int_{\alpha}^{ \beta}\int_{\theta}^{\beta}\dot{x}^{\rm T}(s)dsd\theta Q\int_{\alpha}^{\beta} \int_{\theta}^{\beta}\dot{x}(s)dsd\theta\\ &-2\sum\limits_{i=4}^{5}\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\dot{x}^{\rm T }(s)Q\dot{x}_{i}(s)dsd\theta_{i}+\sum\limits_{i=4}^{5}\int_{\alpha}^{\beta} \int_{\theta}^{\beta}\delta_{i}^{2}(s)dsd\theta_{i}^{\rm T}Q\vartheta_{i}\geq 0.\end{array}  \tag{20}\]  
From Equation (20), one can obtain  
\[ \begin{array}{rl}\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\dot{x}^{\rm T}( s)Q\dot{x}(s)dsd\theta\geq\frac{2}{(\beta-\alpha)^{2}}\int_{\alpha}^{\beta} \int_{\theta}^{\beta}\dot{x}^{\rm T}(s)dsd\theta Q\int_{\alpha}^{\beta}\int_{ \theta}^{\beta}\dot{x}(s)dsd\theta+\sum\limits_{i=4}^{5}\left(\int_{\alpha}^{ \beta}\int_{\theta}^{\beta}\delta_{i}^{2}(s)dsd\theta\right)^{-1}\\ \times\left(\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\delta_{i}(s)\dot{x}(s) dsd\theta\right)^{T}\!Q\!\left(\int_{\alpha}^{\beta}\int_{\theta}^{\beta} \delta_{i}(s)\dot{x}(s)dsd\theta\right)-\sum\limits_{i=4}^{5}\int_{\alpha}^{ \beta}\int_{\theta}^{\beta}\delta_{i}^{2}(s)dsd\theta(\vartheta_{i}-\lambda_{i} )^{T}\!Q(\vartheta_{i}-\lambda_{i}).\end{array}  \tag{21}\]  
where \(\lambda_{i}=\left(\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\delta_{i}^{2}(s)dsd \theta\right)^{-1}\int_{\alpha}^{\beta}\int_{\theta}^{\beta}\delta_{i}(s)\dot {x}(s)dsd\theta\ (i=4,5)\).