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2 papers
Stability criterion for linear differential equations with a delayed argument
S. A. Gusarenko Perm State National Research University, 7 Henkel str., Perm, 614068 Russia
Abstract:
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, Cauchy formula.
UDC:
517.929 Received: 05.03.2022
Revised: 05.03.2022
Accepted: 29.06.2022
DOI:
10.26907/0021-3446-2022-12-34-56
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