Direct Lyapunov approach to the stability analysis of differential-difference systems with linearly increasing time delay

This article is about the stability analysis of linear differential-difference systems with linearly increasing time delay. The necessary and sufficient conditions of the asymptotic stability are given in the article. At the same time the posititive definiteness condition is weakened comparing with classic Krasovskii condition. The quadratic Lyapunov–Krasovskii functional with a given derivative is constructed for a wide class of stable systems with linearly increasing time delay. This functional has a quadratic lower bound on the sets of certain type. The functional is defined by the Lyapunov matrix, its main properties are given. It is shown that after some modifications this functional can be used to analyze the stability with respect to the time-dependent perturbations in coefficients and delay. Such analysis requires upper bounds of a special type on the norm of the Lyapunov matrix. The required bounds for the scalar case were obtained. The sufficient conditions of the asymptotic stability of scalar equation with linearly increasing time delay and the time-dependent perturbations in coefficients are derived based on these bounds. Modification of the functional in this case consists in adding the integral term that assures negativity of a derivative and preserves the quadratic lower bound on the sets of a certain type. Bibliogr. 11.