# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 4 Stability Analysis Criteria for Interval Time-Delay Systems
 \tag{39}\]  
By combing Equations (27)-(39), one has  
\[\dot{V}(t)\leq\upsilon^{T}(t)\Pi\upsilon(t) \tag{40}\]  
where  
\[\Pi=\bigg{(}\frac{d(t)-d_{1}}{d}\Omega_{1}^{T}+\frac{d_{2}-d(t)}{d}\Omega_{2} ^{T}\bigg{)}P\Lambda+\sum_{j=1}^{5}\Phi_{j}+\Gamma+\sum_{l=1}^{4}\Theta_{l}.\]  
Since \(\Pi\) is convex in \(d(t)\in[d_{1},\ d_{2}]\), \(\Pi_{1}<0\) and \(\Pi_{2}<0\) ensure \(\dot{V}(t)<0\), which means that the system (1) is asymptotically stable. This completes the proof. **Remark 3. ** By establishing a novel LKF, a stability criterion is derived in terms of LMIs. Various functionals in triple integral forms are introduced in the proposed LKF, which is effective in improving the feasible stability region [14]. Unlike the existing literatures [14; 20; 21; 22; 23; 26], \(\int_{t-d_{2}}^{t-d_{1}}x(s)ds\), \(\int_{t-d_{1}}^{t}\int_{\theta}^{t}x(s)dsd\theta\) and \(\int_{t-d_{1}}^{t}\int_{\theta}^{t}x(s)dsd\theta d\sigma\) are utilized as elements of different augmented vectors such that the information on the lower bound of delay is fully exploited. Thus, the derived criterion could result in less conservatism, especially for the systems with larger lower bounds of delay. **Remark 4. ** More accurate bounds for the cross terms are obtained by the improved integral inequalities. As a result, it can be expected that the resulting condition has the potential to achieve more desirable performance. Unlike [14; 15], handling the double integrals as a whole, more of a relationship between time-varying delay and delay range is taken into account by elaborately dividing the derivatives of triple integrals into several parts in Equation (33). By the Lemma 3, triple integrals of the state are obtained in the final result to provide more flexibility in finding the solutions of matrix variables than the approaches in [19; 20; 22; 23; 26]. Consequently, a great many of cross terms among double and triple integrals are obtained in the criterion, which are conducive to reducing the conservatism. **Remark 5. ** Applying Lemma 3 to double integrals divided as shown in Equation (33) results in the functions weighted by \((d(t)-d_{2})/(d(t)-d_{1})\) and \((d(t)-d_{1})/(d(t)-d_{2})\). In [28], they are enlarged as \((d(t)-d_{2})/d\) and \((d_{1}-d(t))/d\), respectively. In Theorem 1, using \((d_{2}-d(t))/(d(t)-d_{1})=d/(d(t)-d_{1})-1\), \((d(t)-d_{1})/(d_{2}-d(t))=d/(d_{2}-d(t))-1\), the above terms and the terms with inverses of convex parameters are handled as a combination, avoiding direct approximation (see Equation (39)), which is similar to the treatment in [26]. The advantage of Theorem 1 over [26] is the substituting the Auxiliary-function based inequality by the Lemma 2. On the other hand, when the information on the delay derivative is unavailable, by setting \(Q_{1}=0\), one can easily arrive at the following corollary. **Corollary 1. **_For given scalars \(d_{2}>d_{1}>0\), the system (1) with an interval time-varying delay \(d_{1}\leq d(t)\leq d_{2}\) is asymptotically stable if there exist matrices \(P=[P_{ij}]_{2\times 2}\!\!>\!0\), \(R=[R_{ij}]_{2\times 2}\!\!>\!0\), \(Q_{2}\!\!>\!0\), \(U_{2}\!\!>\!0\), \(S_{1}\!\!>\!0\), \(W_{i}\!\!>\!0\) (\(i=1\), \(2\); \(j=1\), \(2\)), and any matrices \(Z_{l}\) (\(l=1\), \(2\), \(3\), \(4\)) of appropriate dimensions such that the LMIs (25)-(26) hold with \(Q_{1}=0\)._  
**Proof. ** Eliminating the term with \(Q_{1}\) in LKF Equation (27) and proceeding the similar way as Theorem 1 yield Corollary 1. As the deriving process is directly following Theorem 1, it is omitted for the sake of simplicity. **Remark 6. ** When \(d_{1}=0\), the interval \([0,\;d_{1}]\) is missing. It yields different selection of the LKF from the one in Theorem 1, as the \(d_{1}\) in integral limit in Equation (27) becomes zero. Thus, it is more reasonable to reevaluate the Theorem 1 for \(d_{1}=0\) and the following corollary is derived. **Corollary 2.