Geodesic
On the limiting behavior of a time-delay system’s solutions
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 2, pp. 173-182

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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.
Keywords: time-delay systems, asymptotic stability, asymptotic quiescent position, Lyapunov functions. 
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     title = {On the limiting behavior of a time-delay system{\textquoteright}s solutions},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
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     url = {http://geodesic.mathdoc.fr/item/VSPUI_2018_14_2_a9/}
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S. E. Kuptsova; S. Yu. Kuptsov; N. A. Stepenko. On the limiting behavior of a time-delay system’s solutions. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 2, pp. 173-182. http://geodesic.mathdoc.fr/item/VSPUI_2018_14_2_a9/