A sufficient negative-definiteness condition for cubic functions and application to time-delay systems

This article studies the delay-dependent stability of systems with a time-varying delay. To get a stability criterion described as linear matrix inequalities (LMIs) from a positive definite Lyapunov-Krasovskii functional (LKF), the negative-definiteness condition, guaranteeing the negative definiteness of the derivative of the LKF, is necessary. This article proposes a negative-definiteness lemma for the cubic function with respect to the time-varying delay, which achieves the negative-definiteness requirement without introducing any decision variable and extending the size of the LMIs contained in a stability criterion. Then benefiting from this lemma, an LKF with more delay information is constructed and applied to the stability analysis. After that, a delay-dependent stability criterion is derived based on the LKF and the negative-definiteness lemma. Finally, the contribution of the proposed lemma and the stability criterion is demonstrated with two examples.