Two relaxed quadratic function negative-determination lemmas: Application to time-delay systems

Complicated construction of Lyapunov–Krasovskii functional (LKF) makes the quadratic correlation terms of time-delay appear in its derivative. This paper proposes two relaxed negative-determination lemmas to deduce the negative definite condition of a quadratic function with respect to a time-varying delay, which contain some popularly lemmas as their special cases. Combining a novel augmented LKF and generalized free-matrix-based integral inequality (GFMBII), the developed lemmas are respectively applied to derive the stability criteria for nominal and uncertain systems. Three numerical examples are given to prove the potential gain of the two lemmas and the superiority of the criteria over the previous work.