# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 3 New Integral Inequalities  
As discussed in the first section, the reduction of conservatism primarily benefits from two factors: the suitable construction of LKF and more accurate estimation for its derivative. Revisiting the literature [14; 17; 23; 26], the most commonly used LKF consists of the following double and triple integral terms:  
\[V_{D}(t)=\int_{t-d_{2}}^{t-d_{1}}\int_{\theta}^{t}\dot{x}^{T}(s)Q_{1}\dot{x}(s )dsd\theta, \tag{5}\]  
\[V_{T}(t)=\int_{t-d_{2}}^{t-d_{1}}\int_{\sigma}^{t-d_{1}}\int_{\theta}^{t}\dot{ x}^{T}(s)Q_{2}\dot{x}(s)dsd\theta d\sigma+\int_{t-d_{2}}^{t-d_{1}}\int_{t-d_{2}}^{ \sigma}\int_{\theta}^{t}\dot{x}^{T}(s)Q_{3}\dot{x}(s)dsd\theta d\sigma, \tag{6}\]  
where \(Q_{i}>0\)\((i=1,2,3)\) are the Lyapunov matrices. The derivatives of \(V_{D}(t)\) and \(V_{T}(t)\) are given by  
\[\dot{V}_{D}(t)=(d_{2}-d_{1})\dot{x}^{T}(t)Q_{1}\dot{x}(t)-\int_{t-d_{2}}^{t-d_ {1}}\dot{x}^{T}(s)Q_{1}\dot{x}(s)ds \tag{7}\]  
\[\dot{V}_{T}(t)=\frac{(d_{2}-d_{1})^{2}}{2}\dot{x}^{T}(t)(Q_{2}+Q_{3})\dot{x}(t )-\int_{t-d_{2}}^{t-d_{1}}\int_{\theta}^{t-d_{1}}\dot{x}^{T}(s)Q_{2}\dot{x}(s )dsd\theta-\int_{t-d_{2}}^{t-d_{1}}\int_{t-d_{2}}^{\theta}\dot{x}^{T}(s)Q_{3} \dot{x}(s)dsd\theta \tag{8}\]  
In order to obtain a linear matrix inequality (LMI)-based stability condition and less conservatism, it is required to provide closer estimations for the integrals of quadratic functions in Equations (7) and (8). For this purpose, a new set of integral inequalities is developed in the following lemmas. **Lemma 2**. : _For a given matrix \(Q>0\), the following inequality holds for all continuously differentiable function \(x\) in \([\alpha,\beta]\rightarrow\Re^{n}:\)_  
\[\int_{\alpha}^{\beta}\dot{x}^{T}(s)Q\dot{x}(s)ds\geq\frac{1}{\beta-\alpha} \Big{(}\psi_{1}^{T}Q\psi_{1}+3\psi_{2}^{T}Q\psi_{2}+5\psi_{3}^{T}Q\psi_{3}+7 \psi_{4}^{T}Q\psi_{4}\Big{)}, \tag{9}\]  
_where_  
\[\psi_{1}=x(\beta)-x(\alpha),\,\,\psi_{2}=x(\beta)+x(\alpha)-\frac{2}{\beta- \alpha}\int_{\alpha}^{\beta}x(s)ds,\]  
\[\psi_{3}=x(\beta)-x(\alpha)+\frac{6}{\beta-\alpha}\int_{\alpha}^{\beta}x(s)ds -\frac{12}{(\beta-\alpha)^{2}}\int_{\alpha}^{\beta}\int_{\theta}^{\beta}x(s) dsd\theta,\]  
\[\psi_{4}=x(\beta)+x(\alpha)-\frac{12}{\beta-\alpha}\int_{\alpha}^{\beta}x(s) ds+\frac{60}{(\beta-\alpha)^{2}}\int_{\alpha}^{\beta}\int_{\theta}^{\beta}x(s) dsd\theta-\frac{120}{(\beta-\alpha)^{3}}\int_{\alpha}^{\beta}\int_{\sigma}^{ \beta}\int_{\theta}^{\beta}x(s)dsd\theta d\sigma.\]  
Proof. : For any differentiable function \(x\), define a function \(\varpi_{1}(s)\) for all \(s\in[\alpha,\ \beta]\) given by:  
\[\varpi_{1}(s)=\dot{x}(s)-\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}\dot{x}( s)ds-\delta_{1}(s)\vartheta_{1}-\delta_{2}(s)\vartheta_{2}-\delta_{3}(s) \vartheta_{3}, \tag{10}\]  
where \(\vartheta_{i}\)\((i=1,2,3)\) are constant vectors to be determined and  
\[\delta_{1}(s)=\frac{2s-\alpha-\beta}{2},\,\,\delta_{2}(s)=\frac{12s^{2}-12( \alpha+\beta)s+2\alpha^{2}+2\beta^{2}+8\alpha\beta}{12},\]  
\[V_{T}(t)=\int_{t-d_{2}}^{t-d_{1}}\int_{\sigma}^{t-d_{1}}\int_{\theta}^{t}\dot{ x}^{T}(s)Q_{2}\dot{x}(s)dsd\theta d\sigma+\int_{t-d_{2}}^{t-d_{1}}\int_{t-d_{2}}^{ \sigma}\int_{\theta}^{t}\dot{x}^{T}(s)Q_{3}\dot{x}(s)dsd\theta d\sigma, \tag{6}\]0  
For a matrix \(Q>0\), integrating \(\varpi_{1}^{T}(s)Q\varpi_{1}(s)\) from \(\alpha\) to \(\beta\) leads to \[