Geodesic
Stability criterion for linear differential equations with a delayed argument
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2022), pp. 34-56.

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A semi-effective criterion for the stability of linear differential equations $\mathcal{L} x=f$ with retarded argument is proposed, the general solution of which is represented by the Cauchy formula $$ x(t)=C(t,a)x(a)+\int\limits_a^tC(t,s) f(s) ds. $$ The Cauchy function satisfies the integral identity $$ C(t,s) = U(t,s)U(s,s)^{-1} - \int\limits_s^tC(t,\varsigma)\mathcal{L}_s U(\cdot, s)(\varsigma)U(s,s)^{-1} d\varsigma, $$ where $\mathcal{L}_s$ is the contraction of the operator $\mathcal{L}$ by the interval $[s,\infty)$. Choosing the function $U$ so that the function is $\mathcal{L}_s U(\cdot, s) U(s,s)^{-1}$ is small enough, it is possible to obtain estimates of the Cauchy function $C(t,s)$, which guarantee the stability of the differential equation.
Keywords: stability of differential equations with a delayed argument, stability criterion of differential equations, signs of stability of differential equations, Cauchy function
Mots-clés : Cauchy formula. 
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     author = {S. A. Gusarenko},
     title = {Stability criterion for linear differential equations with a delayed argument},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {34--56},
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     number = {12},
     year = {2022},
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     url = {http://geodesic.mathdoc.fr/item/IVM_2022_12_a2/}
}
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S. A. Gusarenko. Stability criterion for linear differential equations with a delayed argument. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2022), pp. 34-56. http://geodesic.mathdoc.fr/item/IVM_2022_12_a2/

[1] Neimark Yu.I., “Struktura D-razbieniya prostranstva kvazipolinov i diagrammy Vyshnegradskogo i Naikvista”, DAN SSSR, 60 (1948), 1503–1506

[2] Pinni E., Obyknovennye diffrentsialno-raznostnye uravneniya, In. lit., M., 1961, 249 pp.

[3] Tsypkin Ya.Z., “Ustoichivost sistem s zapazdyvyuschei obratnoi svyazyu”, Avtomat. i telemekhanika, 7:2–3 (1946), 107–129

[4] Krasovskii N.N., “O primenenii vtorogo metoda Lyapunova dlya uravnenii s zapazdyvaniyami vremeni”, PMM, 20:2 (1956), 315–327

[5] Razumikhin B.S., “Ob ustoichivosti sistem s zapazdyvaniem”, PMM, XX:4 (1956), 500–512

[6] Berezanskii L.M., “Razvitie W-metoda N.V. Azbeleva v zadachakh ustoichivosti reshenii lineinykh funktsionalno-differentsialnykh uravnenii”, Diffrents. uravneniya, 22:5 (1986), 739–750

[7] Azbelev N.V., Simonov P.M., Ustoichivost reshenii uravnenii s obyknovennymi proizvodnymi, Izd-vo Permsk. un-ta, Perm, 2001

[8] Barbashin E.A., Vvedenie v teoriyu ustoichivosti, Nauka, M.

[9] Azbelev N.V., Maksimov V.P., Rakhmatullina L.F., Elementy sovremennoi teorii funtsionalno-diffrentsialnykh uravnenii. Metody i prilozheniya, In-t komp. issledov., M., 2002

[10] Corless R.M., Gonnet G.H., Hare D.E., Jeffrey D.J., Knuth D.E., “On the Lambert W function”, Adv. Comput. Math., 5 (1996), 329–359