Exponential stability of some scalar impulsive delay differential equations. (English) Zbl 0901.34068

Berezansky, L.; Idels, L.

Commun. Appl. Anal. 2, No. 3, 301-308 (1998).

The authors obtain exponential stability results for impulsive delay differential equations \[ \dot{x}(t)+ a(t)x(t)+ \sum_{k=1}^m b_k(t) x [h_k(t)]= f(t), \tag{1} \]
\[ x(\mu_j)= B_j x(\mu_j-0), \qquad j =1,2,\dots, \tag{2} \] where \(h_k(t) \leq t, t \in [0,\infty)\) and \(\lim_{j \to \infty} \mu_j= \infty\). The approach is based on a reduction of the stability of (1), (2) to the solvability of a certain operator equation of second kind [cf. A. Anokhin, L. Berezansky and E. Braverman, J. Math. Anal. Appl. 193, No. 3, 923-941 (1995; Zbl 0837.34076)].

Reviewer: D.Bainov (Sofia)

MSC:

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

Keywords:

exponential stability; impulsive delay differential equations

Citations:

Zbl 0837.34076