Approximations in the stability problem for linear periodic systems with aftereffect

Yu. F. Dolgii^ab, R. I. Shevchenko<sup>a

a</sup> Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
^b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: The asymptotic stability of a linear periodic system of differential equations with aftereffect is determined by the location of the spectrum of the infinite-dimensional, compact monodromy operator. Analytical representations of such operators can be obtained only for systems of a special type. In numerical simulations, finite-dimensional approximations of the monodromy operators are used. In this paper, we examine a procedure for approximating a system of differential equations with aftereffect by systems of ordinary differential equations of large dimension proposed by N. N. Krasovskii. Finite-dimensional approximations for monodromy operators are constructed in the Hilbert space of states of a periodic system with aftereffect. We prove that increasing of the dimension of finite-dimensional approximations leads to increasing of the approximation accuracy.

Keywords: system with aftereffect, stability of motion, finite-dimensional approximation.

UDC: 517.929

MSC: 39B82

DOI: 10.36535/0233-6723-2021-191-29-37

 English version: Journal of Mathematical Sciences (New York), 2025, 288:6, 695–703