# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 3 New Integral Inequalities
\begin{split}\int_{\alpha}^{\beta}\omega_{1}^{T}(s)Q\omega_{1}& (s)ds=\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds-\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta}\dot{x}^{T}(s)dsQ\int_{\alpha}^{\beta}\dot{x}(s)ds-2\sum \limits_{i=1}^{3}\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\delta_{i}(s)ds\theta_{i }\\ &+\frac{2}{\beta^{n}\alpha}\int_{\alpha}^{\beta}\dot{x}^{T}(s)dsQ \sum\limits_{i=1}^{3}\int_{\alpha}^{\beta}\delta_{i}(s)ds\theta_{i}+\sum \limits_{i=1}^{3}\int_{\alpha}^{\beta}\delta_{i}^{2}(s)ds\theta_{i}^{T}Q\theta_ {i}+2\int_{\alpha}^{\beta}\delta_{i}(s)ds\theta_{1}^{T}Q\theta_{2}\\ &+2\int_{\alpha}^{\beta}\delta_{1}(s)\delta_{3}(s)ds\theta_{1}^{ T}Q\theta_{3}+2\int_{\alpha}^{\beta}\delta_{2}(s)\delta_{3}(s)ds\theta_{2}^{T}Q \theta_{3}\geq 0.\end{split}  \tag{11}\]  
By noting \(\int_{\alpha}^{\beta}\delta_{i}(s)ds=0\)\((i=1,2,3)\), \(\int_{\alpha}^{\beta}\delta_{i}(s)\delta_{j}(s)ds=0\)\((i=1,2,3;j=1,2,3;i\
eq j)\), it yields  
\[ \begin{split}\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds& \geq\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}\dot{x}^{T}(s)dsQ\int_{ \alpha}^{\beta}\dot{x}(s)ds+\sum\limits_{i=1}^{3}\left(2\int_{\alpha}^{\beta} \dot{x}^{T}(s)Q\delta_{i}(s)ds\theta_{i}-\int_{\alpha}^{\beta}\delta_{i}^{2}(s )ds\theta_{i}^{T}Q\theta_{i}\right).\end{split}  \tag{12}\]  
Rewriting the last term of the right-hand side of Equation (12) as sums of squares, one has  
\[ \begin{split}\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds& \geq\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}\dot{x}^{T}(s)dsQ \int_{\alpha}^{\beta}\dot{x}(s)ds+\sum\limits_{i=1}^{3}\left(\int_{\alpha}^{ \beta}\delta_{i}^{2}(s)ds\right)^{-1}\left(\int_{\alpha}^{\beta}\delta_{i}(s) \dot{x}(s)ds\right)^{T}Q\left(\int_{\alpha}^{\beta}\delta_{i}(s)\dot{x}(s)ds \right)\\ &-\sum\limits_{i=1}^{3}\int_{\alpha}^{\beta}\delta_{i}^{2}(s)ds( \theta_{i}-\lambda_{i})^{T}Q(\theta_{i}-\lambda_{i}).\end{split}  \tag{13}\]  
where \(\lambda_{i}=\left(\int_{\alpha}^{\beta}\delta_{i}^{2}(s)ds\right)^{-1}\int_{ \alpha}^{\beta}\delta_{i}(s)\dot{x}(s)ds\)\((i=1,2,3)\). Equation (13) holds independently of the choice of constant vectors \(\theta_{i}\)\((i=1,2,3)\) and the last term of the right-hand side of Equation (13) is non-positive. Therefore, choosing \(\theta_{i}=\lambda_{i}\)\((i=1,2,3)\) leads to the maximum of the right-hand side of Equation (13). Rearranging Equation (13) leads to Equation (9). Thus, the proof is completed. **Remark 1. ** By the Wirtinger-based inequality [19], the new inequality [17] and the auxiliary function-based inequality [26] (the refined Jensen inequality [23]), respectively, \(\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds\) can be estimated as follows:  
\[\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds\geq\frac{1}{\beta-\alpha} \left(\psi_{1}^{T}Q\psi_{1}+3\psi_{2}^{T}Q\psi_{2}\right), \tag{14}\]  
\[\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds\geq\frac{1}{\beta-\alpha} \left(\psi_{1}^{T}Q\psi_{1}+5\psi_{3}^{T}Q\psi_{3}\right), \tag{15}\]  
\[\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds\geq\frac{1}{\beta-\alpha} \left(\psi_{1}^{T}Q\psi_{1}+3\psi_{2}^{T}Q\psi_{2}+5\psi_{3}^{T}Q\psi_{3} \right), \tag{16}\]  
Compared with Equation (14)-(16), one can see that Equation (9) produces a more accurate bound for a single integral term. Thus, the stability criteria by Lemma 2 tend to be less conservative. Unlike the existing ones, not only the relations between \(x(\beta)\), \(x(\alpha)\), \(\int_{\alpha}^{\beta}x(s)ds\), \(\int_{\alpha}^{\beta}\int_{\beta}^{\beta}x(s)dsd\theta\), but also between them and \(\int_{\alpha}^{\beta}\int_{\sigma}^{\beta}\int_{\beta}^{\beta}x(s)dsd\theta d\sigma\) are all taken into account, which allows one to make use of extra information on time delay. Moreover, no extra slack matrices are introduced in Equation (9). **Lemma 3.