Impulsive Stabilization of Linear Delay Differential Equations

Abstract

The paper is concerned with stabilization of a scalar delay differention equation $$ {\dot x}(t) - \sum_{k=1}^m A_k(t)x[h_k(t)] = 0,~t\geq 0,~ x(\xi)=\varphi (\xi), \xi <0, $$ by introducing impulses in certain moments of time $$ x(\tau_j) = B_j x(\tau_j -0), ~j=1,2, \dots ~. $$ Explicit stability results are presented both for the equation with positive coefficients and for the equation with $A_k$ being of arbitrary sign.