Boundedness and Stability of Impulsively Perturbed Delay Differential Equations

Suppose any solution of a linear impulsive delay differential equation $$\dot{x}(t)+\sum _{i=1}^m{A}_i(t)x[h_i(t)]=0,t\ge 0,x(s)=0,s<0,$$ $$x({\tau }_j+0)=B_{j}x({\tau }_j-0)+{\alpha }_j,j=1,2,\mathrm{.}\mathrm{.}\mathrm{.},$$ is bounded for any bounded sequence $\{{\alpha }_i\}$. The conditions ensuring exponential stability of this equation are presented. The behavior of solutions of the non-homogeneous equation is analyzed.