Lyapunov-Krasovskii-Based conditions for <i>h</i> -stability and <i>H</i> _∞ performance in mixed time-varying delay systems

This paper investigates the [Formula: see text]-stability and [Formula: see text] control design for a class of systems characterized by mixed time-varying delays (MTVDs). A novel analysis framework is developed based on a Lyapunov-Krasovskii functional (LKF), tailored to capture the dynamics of such systems more effectively than conventional direct methods. A set of sufficient conditions expressed in terms of linear matrix inequalities (LMIs) is derived to guarantee [Formula: see text]-stability and achieve a prescribed [Formula: see text] performance bound [Formula: see text] in the absence of external inputs. Furthermore, a state-feedback control strategy is formulated to ensure the closed-loop system maintains [Formula: see text]-stability while satisfying the [Formula: see text] performance criterion. The proposed method is validated through numerical case studies involving various forms of the function [Formula: see text], illustrating the practical applicability and effectiveness of the theoretical results.