Hierarchical stability conditions of systems with time-varying delay

•This paper studies the stability problem of time-varying delay systems.•New stability criterion is derived based on a delay-product Lyapunov-Krasovskii functional.•A hierarchical negativity condition of quadratic polynomials is presented.•The derived stability criterion is expressed in terms of LMIs by using the presented negativity condition.•Two examples are solved so as to demonstrate the effectiveness of the proposed approach. This paper studies the stability problem of time-varying delay systems. Firstly, By constructing a delay-product Lyapunov-Krasovskii functional, a stability criterion is established, where the advantages of convexity properties are being exploited. This stability criterion in terms of linear matrix inequality (LMI) can be checked exactly using an improved negativity condition of quadratic polynomials. It is demonstrated in the numeric examples that the proposed method can greatly reduce the conservativeness without leading to increase the number of decision variables.