## Exponential stability of linear delay impulsive differential equations.(English)Zbl 0837.34076

This article deal with a linear delay impulsive differential equation $$x'(t) + \sum^m_{i = 1} A_i (t)x [h_i(t)] = r(t)$$ $$(0 < t < \infty,\;t \neq \tau_j)$$, $$x (\tau_j) = B_j x (\tau_j - 0)$$ $$(j = 1,2, \dots)$$ 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.
Reviewer: P.Zabreiko (Minsk)

### MSC:

 34K20 Stability theory of functional-differential equations 34A37 Ordinary differential equations with impulses
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