Geodesic
Approximations in the stability problem for linear periodic systems with aftereffect
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 2, Tome 191 (2021), pp. 29-37

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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. 
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     author = {Yu. F. Dolgii and R. I. Shevchenko},
     title = {Approximations in the stability problem for linear periodic systems with aftereffect},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {29--37},
     publisher = {mathdoc},
     volume = {191},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_191_a3/}
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Yu. F. Dolgii; R. I. Shevchenko. Approximations in the stability problem for linear periodic systems with aftereffect. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 2, Tome 191 (2021), pp. 29-37. http://geodesic.mathdoc.fr/item/INTO_2021_191_a3/