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Zbl 0837.34076
Anokhin, A.; Berezansky, L.; Braverman, E.
Exponential stability of linear delay impulsive differential equations.
(English)
[J] J. Math. Anal. Appl. 193, No.3, 923-941 (1995). ISSN 0022-247X

This article deal with a linear delay impulsive differential equation $x'(t) + \sum^m_{i = 1} A_i (t)x [h_i(t)] = r(t)$ $(0 < t < \infty,\ t \ne \tau_j)$, $x (\tau_j) = B_j x (\tau_j - 0)$ $(j = 1,2, \dots)$ under natural conditions for $A_i (t)$, $h_i (t)$, $B_j$, $\tau_j$ and $r(t)$. The main results are a theorem about integral representations of solutions to the Cauchy problem for the above equation and a variant of the Bohl-Perron theorem about the exponential stability of this equation under assumptions that each its solution with the first derivative are bounded for each bounded right hand side $r(t)$. The simple explicit condition of exponential stability in terms of coefficients $A_i (t)$ and $B_j$ is also presented. In the end of the article some illustrating examples are presented.
[P.Zabreiko (Minsk)]
MSC 2000:
*34K20 Stability theory of functional-differential equations
34A37 Differential equations with impulses

Keywords: linear delay impulsive differential equation; integral representations of solutions to the Cauchy problem; Bohl-Perron theorem about the exponential stability

Cited in: Zbl 0901.34068

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