Sufficient conditions for the existence of an asymptotic quiescent position in time-delay systems

In the present paper we study motions of time-delay systems. In particular, the case is studied, when the system has zero limit position which may not be an invariant set with respect to initial differential-difference equations. The concept of an asymptotic quiescent position for the trajectories of time-delay systems is introduced. Sufficient conditions for existence of an asymptotic quiescent position and an asymptotic quiescent position in the large are obtained. The method of proof is based on the modification of the second Lyapunov method, which was proposed by Razumikhin. Its idea is to use the classical Lyapunov functions, but to evaluate their derivatives along the solutions of the system not on the entire set of integral curves of the system, but on its certain subset. The article considers examples of non-linear time-delay equations that have an asymptotic quiescent position illustrating the theory being developed.