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In this paper, the stability analysis of linear systems with time-varying delays is studied. A novel Lyapunov method is presented, in which positive definiteness of the matrices in common Lyapunov functionals is relaxed by adding what is referred to as a zero-integral functional (ZIF). A general form of auxiliary polynomial-based functionals that contains such ZIF is given. Choosing polynomials of different order as well as exploring double-delay-product (DDP) terms, novel Lyapunov functionals are constructed, which contribute to a set of improved stability conditions expressed in terms of linear matrix inequalities. Finally, numerical examples are provided to corroborate the merits of the proposed method relative to a number of existing methods, and in particular, the effectiveness of the proposed ZIFs and DDP terms in reducing the conservatism of stability conditions.

Improved stability conditions for time-varying delay systems via relaxed Lyapunov functionals