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. 
@article{IVM_2022_12_a2,
     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},
     publisher = {mathdoc},
     number = {12},
     year = {2022},
     language = {ru},
     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/