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Uniform asymptotic stability of impulsive delay differential equations. (English) Zbl 0989.34061

The paper has been written by famous scientists in the area of differental equations with impulsive effect. The system

$\frac{dx\left(t\right)}{dt}=f\left(t,{x}_{t}\right),\phantom{\rule{2.em}{0ex}}t\ne {\tau }_{k},\phantom{\rule{2.em}{0ex}}\left(\mathrm{a}\right)$
${\Delta }x\left(t\right)=I\left(t,{x}_{{t}^{-}}\right),\phantom{\rule{2.em}{0ex}}t={\tau }_{k},\phantom{\rule{2.em}{0ex}}\left(\mathrm{b}\right)$

is considered. Here, $t\in {ℝ}_{+}$, $0={\tau }_{0}<{\tau }_{1}<{\tau }_{2}<\cdots$, ${lim}_{k\to \infty }{\tau }_{k}=+\infty$, ${\Delta }x\left(t\right)=x\left(t\right)-x\left({t}^{-}\right)$, $x\left({t}^{-}\right)={lim}_{s\to {t}^{-}}x\left(s\right)$; (a) is a system of functional-differential equations with delay. It is assumed that $f\left(t,0\right)\equiv 0$, $I\left({\tau }_{k},0\right)=0$ for all ${\tau }_{k}\in {ℝ}_{+}$, and the system (a), (b) possesses a trivial (zero) solution $x\left(t\right)\equiv 0$. Two theorems on the uniform asymptotic stability of the trivial solution to (a), (b) are proved by means of Lyapunov functions and Razumikhin techniques. It is shown that impulses do contribute to yield stability properties even when the system (a) does not enjoy any stability behavior. The theorems are illustrated by some examples.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K45 Functional-differential equations with impulses 34A37 Differential equations with impulses
##### Keywords:
impulsive delay differential equations; stability