The construction of augmented Lyapunov-Krasovskii functionals and the estimation of their derivatives in stability analysis of time-delay systems: a survey

The Bessel-Legendre inequality plays an important role in stability analysis of linear systems with time-varying delays. However, various integral vectors inherited from the Bessel-Legendre inequality bring some challenging issues, such as the construction of suitable augmented Lyapunov-Krasovskii functionals (LKFs); high-degree polynomial estimation on the derivative of the chosen LKF. This paper offers an overview of recent developments on these issues. First, an instructive review on the construction of suitable augmented LKFs catering for the use of the Bessel-Legendre inequality is given in detail. Second, an in-depth analysis and insightful understanding of high-degree polynomial inequalities on closed intervals are made. Third, recent results on reciprocally convex combination inequalities are briefly discussed. Finally, several challenging problems are presented for future research.