# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 1 Introduction  
Because of the finite speed of data transmission, time delays are unavoidably encountered in a variety of real-world systems, such as multi-agent systems [1; 2], active suspension systems [3; 4], chemical engineering systems [5], and so on. The time delays frequently cause undesirable dynamic behaviors [6; 7]. Consequently, a considerable amount of attention has been paid to the stability analysis for time-delay systems [8; 9; 10; 11; 12; 13; 14; 15]. In this area, the Lyapunov-Krasovskii functional (LKF) is the most efficient mathematical tool [4; 5]. It is the main idea of the LKF based approach to establish a positive definite functional such that its derivative along a solution of the considered system is a definite negative [7]. However, a certain degree of conservatism is inevitably introduced, since only sufficient conditions can be obtained. As is well known, the reduction of conservatism depends on the construction of LKF to a considerable extent, which aims at making use of more information about the delay [11]. In this trend, a great deal of effort has been contributed, including augmented LKF [11], delay decomposition/fractioning [12; 13], triple-integral terms [14; 15] and functional including quadratic terms multiplied by a higher degree scalar function [16]. The reduction of conservatism also benefits from the techniques utilized to estimate the cross terms when differentiating the LKF [17]. Compared with the choice of LKF, the accurate bounding technique has been considered as a more effective manner to relax the criteria. In this regard, most of the contributions are derived via Jensen inequality thanks to its convenient tractability [13; 14; 15]. However, as discussed in [18], the Jensen inequality often leads to undesirable conservatism. Thus, reducing the estimation gap of the Jensen inequality has become an open issue. Recently, an alternative inequality based on the Wirtinger inequality has been proposed in [19] to achieve a potential gain with respect to Jensen inequality. In [20], by using information on the double integral of state, a new free-matrix-based integral inequality is developed, which includes the Wirtinger one at the cost of computational burden [21]. In [17], a novel modification method is introduced to derive a new integral inequality. If the LKF with triple-integral terms are established, the search for how to provide tighter bounds for double integral functionals is becoming a stringent task. In [22], the Wirtinger-based inequality is extended to the double integral form. In [23], the Jensen-based inequalities are refined for both single and double integrals and their potential capacity exhibits obvious advantages over the previous ones. In [24], by using Wirtinger-based single and double integral inequalities and delay decomposition technique, the stability analysis of neural systems is investigated. Combining advanced integral inequalities and slack variables, a new delay-dependent stability condition for time delay systems is developed in [25]. Among the literature on this subject, the most noticeable technique is the auxiliary function-based integral inequalities performed in [26], which are suitable for the LKF in triple integral forms and encompass the Jensen and Wirtinger ones as special cases. Therefore, how to develop new integral inequalities to provide more accurate bounds for both single and double integrals than [26] motivates the present study. Based on the above discussions, the main contribution of this paper is that a novel series of integral inequalities independent of slack variables is proposed, which shows significant improvements over the Wirtinger-based, refined Jensen and auxiliary function-based ones. By constructing an augmented LKF with triple-integral terms, the information on the delay range, especially on the lower bound of delay, is fully taken into consideration. By virtue of the improved inequalities, the derived stability conditions are less conservative than some existing ones.