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

Anokhin, A.; Berezansky, L.; Braverman, E.

J. Math. Anal. Appl. 193, No. 3, 923-941 (1995).

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

Keywords:

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