# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 4 Stability Analysis Criteria for Interval Time-Delay Systems
\begin{array}{cc}e_{1}\Gamma_{2}\\ e_{1}\Gamma_{3}\end{array} \right]^{T}\!\left[ \begin{array}{cc}U_{2}&Z_{l}\\ *&U_{2}\end{array} \right]\left[ \begin{array}{cc}e_{1}\Gamma_{2}\\ e_{1}\Gamma_{3}\end{array} \right]\left(l=1,2,3,4\right).\]  
Proof. : Consider the following LKF candidate,  
\[V(t)=V_{1}(t)+V_{2}(t)+V_{3}(t)+V_{4}(t)+V_{5}(t)+V_{6}(t), \tag{27}\]  
where the individual functionals are defined as follows:\[V_{1}(t) =\left[ \begin{array}{cc}x(t)\\ f_{t-d_{2}}^{t-d_{1}}\,x(s)ds\end{array} \right]^{T}\left[ \begin{array}{cc}P_{11} &P_{12}\\ *&P_{22}\end{array} \right]\left[ \begin{array}{cc}x(t)\\ f_{t-d_{1}}^{t-d_{1}}\,x(s)ds\end{array} \right],\] \[V_{2}(t) =\int_{t-d(t)}^{t}x^{T}(s)Q_{1}x(s)ds+\int_{t-d_{2}}^{t-d_{1}}x^{T }(s)Q_{2}x(s)ds,\] \[V_{3}(t) =\left[ \begin{array}{cc}\int_{t-d_{1}}^{t}\int_{0}^{t}x(s)dsd \theta\\ \int_{t-d_{1}}^{t}\int_{\sigma}^{t}\int_{\sigma}^{t}x(s)dsd\theta d\sigma\end{array}  \right]^{T}\left[ \begin{array}{cc}R_{11}&R_{12}\\ *&R_{22}\end{array} \right]\left[ \begin{array}{cc}\int_{t-d_{1}}^{t}\int_{ \sigma}^{t}x(s)dsd\theta\\ \int_{t-d_{1}}^{t}\int_{\sigma}^{t}\int_{\sigma}^{t}x(s)dsd\theta d\sigma\end{array}  \right],\] \[V_{4}(t) =d_{1}\int_{t-d_{1}}^{t}\int_{\hat{q}}^{t}\dot{x}^{T}(s)U_{1}\dot {x}(s)dsd\theta+d\int_{t-d_{2}}^{t-d_{1}}\int_{\sigma}^{t}\dot{x}^{T}(s)U_{2} \dot{x}(s)dsd\theta,\] \[V_{5}(t) =\int_{t-d_{1}}^{t}\int_{\sigma}^{t}\int_{\hat{q}}^{\hat{x}^{T}}( s)S_{1}\dot{x}(s)dsd\theta d\sigma+\int_{t-d_{1}}^{t}\int_{\sigma-d_{1}}^{ \sigma}\int_{\theta}^{t}\dot{x}^{T}(s)S_{2}\dot{x}(s)dsd\theta d\sigma,\] \[V_{6}(t) =\int_{t-d_{2}}^{t-d_{1}}\int_{\sigma}^{t-d_{1}}\int_{\theta}^{ \hat{x}^{T}}(s)W_{1}\dot{x}(s)dsd\theta d\sigma+\int_{t-d_{2}}^{t-d_{1}}\int_ {\sigma-d_{2}}^{\sigma}\int_{\theta}^{t}\dot{x}^{T}(s)W_{2}\dot{x}(s)dsd\theta d\sigma.\]  
Differentiating \(V(t)\) along the solution of system (1) leads to  
\[\dot{V}_{1}(t)=\
u^{T}(t)sym\bigg{\{}\bigg{(}\frac{d(t)-d_{1}}{d}\Omega_{1}^{ T}+\frac{d_{2}-d(t)}{d}\Omega_{2}^{T}\bigg{)}P\Lambda\bigg{\}}\
u(t), \tag{28}\]  
\[\dot{V}_{2}(t)=x^{T}(t)Q_{1}x(t)-(1-\dot{d}(t))x^{T  
\[ \begin{array}{l}-d_{1}\int_{t-d_{1}}^{t-d_{1}}\dot{x}^{T}(s)U_{2}\dot{x}(s)ds \leq(-d/(d(t)-d_{1}))\upsilon^{T}(t)\Gamma_{2}^{T}\varphi_{1}(U_{2})\Gamma_{2} \upsilon(t),\\ \\ -d\int_{t-d_{2}}^{t-d(t)}\dot{x}^{T}(s)U_{2}\dot{x}(s)ds\leq(-d/(d_{2}-d(t))) \upsilon^{T}(t)\Gamma_{3}^{T}\varphi_{1}(U_{2})\Gamma_{3}\upsilon(t),\\ \\ -(d_{2}-d(t))\int_{t-d(t)}^{t-d_{1}}\dot{x}^{T}(s)W_{1}\dot{x}(s)ds\leq(-(d_{2}- d(t))/(d(t)-d_{1}))\upsilon^{T}(t)\Gamma_{2}^{T}\varphi_{1}(W_{1})\Gamma_{2} \upsilon(t),\\ \\ -(d(t)-d_{1})\int_{t-d_{2}}^{t-d(t)}\dot{x}^{T}(s)W_{2}\dot{x}(s)ds\leq(-(d(t)-d _{1})/(d_{2}-d(t)))\upsilon^{T}(t)\Gamma_{3}^{T}\varphi_{1}(W_{2})\Gamma_{3} \upsilon(t).\end{array}  \tag{34}\]  
By Lemma 1, if there exist matrices \(Z_{l}\)\((l=1,2,3,4)\) satisfying Equation (25), the following inequality is obtained, which is similar to the treatment in [26],  
\[ \begin{array}{l}-d\int_{t-d(t)}^{t-d_{1}}\dot{x}^{T}(s)U_{2}\dot{x}(s)ds-d \int_{t-d_{1}}^{t-d(t)}\dot{x}^{T}(s)U_{2}\dot{x}(s)ds-(d_{2}-d(t))\int_{t-d(t )}^{t-d_{1}}\dot{x}^{T}(s)W_{2}\dot{x}(s)ds-(d(t)-d_{1})\int_{t-d_{2}}^{t-d(t) }\dot{x}^{T}(s)W_{2}\dot{x}(s)ds\\ \leq\upsilon^{T}(t)\Big{(}-\frac{d}{d(t)-d_{1}}\Big{(}\sum_{(2l-1)(e_{l}\Gamma _{2})}^{T}(U_{2}+W_{1})(\alpha_{l}\Gamma_{2})\Big{)}+\Big{(}\sum_{(l-1)(e_{l} \Gamma_{2})}^{T}W_{1}(\alpha_{l}\Gamma_{2})\Big{)}\Big{)}\upsilon(t)\\ +\upsilon^{T}(t)\Big{(}-\frac{d}{d_{2}-d(t)}\Big{(}\sum_{(l-1)(e_{l}\Gamma_{ 3})}^{T}(U_{2}+W_{2})(\alpha_{l}\Gamma_{3})\Big{)}+\Big{(}\sum_{l-1}^{4}(2l-1) (e_{l}\Gamma_{3})^{T}W_{2}(\alpha_{l}\Gamma_{3})\Big{)}\Big{)}\upsilon(t)\\ =\upsilon^{T}(t)\Big{(}\sum_{(l-1)}^{4}\Theta_{l}\Big{)}\upsilon(t).\end{array}