Research of the asymptotic equilibrium of time-delay systems by junction of Lyapunov — Krasovskii and Razumikhin approaches

The nonlinear time-delay systems are considered and the limiting behavior of their solutions is investigated. The case in which the solutions have a trivial equilibrium that may not be an invariant set of the system is studied. The junction of Lyapunov — Krasovskii and Razumikhin approaches is applied to obtain sufficient conditions for the existence of an asymptotic quiescent position in the large. In the case when a general system has a trivial solution, new sufficient conditions for its asymptotic stability are obtained. Examples, that illustrate the application of the obtained results, are given.