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Razumikhin-type theorems on exponential stability of impulsive delay systems. (English) Zbl 1128.34047

Consider an impulsive delay system of the form

$\stackrel{˙}{x}\left(t\right)=f\left(t,{x}_{t}\right),\phantom{\rule{1.em}{0ex}}t\in \left[{t}_{k-1},{t}_{k}\right),$
${\Delta }x\left(t\right)={I}_{k}\left(t,{x}_{{t}^{-}}\right),t={t}_{k},\phantom{\rule{1.em}{0ex}}k\in N,$
${x}_{{t}_{0}}=\phi ·$

It is assumed that all necessary conditions for global existence and uniqueness of solutions for all $t\ge {t}_{0}$ are satisfied. The proof of the results on exponential stability is based on Razumikhin’s technique. A more general form of impulsive delay systems

$\stackrel{˙}{x}\left(t\right)=g\left(t,x\left(t\right),x\left(t-{h}_{1}\left(t\right)\right),\cdots ,x\left(t-{h}_{m}\left(t\right)\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left[{t}_{k-1},{t}_{k}\right],$
${\Delta }x\left(t\right)={I}_{k}\left(t,x\left({t}^{-}\right)\right),\phantom{\rule{1.em}{0ex}}t={t}_{k},\phantom{\rule{1.em}{0ex}}k\in N,$
${x}_{{t}_{0}}=\phi$

is considered too. The exponential stability conditions of general system was obtained. And the result is based on previous theorem for the first system. The particular case of system is considered for linear functions

$\stackrel{˙}{x}\left(t\right)=Ax\left(t\right)+Bx\left(t-h\left(t\right)\right),t\in \left[{t}_{k-1},{t}_{k}\right]$
${\Delta }x\left(t\right)={C}_{k}x\left({t}^{-}\right),t={t}_{k},k\in N,$
${x}_{{t}_{0}}=\phi ·$

Examples are presented.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K45 Functional-differential equations with impulses