Abstract:
In the present paper, we study motions of time-delay systems that have limiting behavior for
an unbounded increase in time in the case when the limit sets might not be invariant with
respect to initial differential-difference equations. The concept of an asymptotic quiescent
position for the trajectories of time-delay systems is introduced. By the use of the Lyapunov
functionals method, sufficient conditions for the existence of an asymptotic quiescent position
for systems of differential-difference equations were obtained. In the case when a general
system has a trivial solution, new sufficient conditions for its asymptotic stability are
obtained. Namely, the condition of the negativity of the time-derivative of Krasovskii
functionals is weakened.