On the stability of impulsively perturbed differential systems

Abstract

This paper deals with the study of uniform asymptotic stability of the measure differential systemDx=F(t, x) +G(t, x)Du, where the symbolDstands for the derivative in the sense of distributions. The system is viewed as a perturbed system of the ordinary differential systemx'=F(t, x), where the perturbation termG(t, x)Duis impulsive and the state of the system changes suddenly at the points of discontinuity ofu. It is shown, under certain conditions, that the uniform asymptotic stability property of the unperturbed system is shared by the perturbed system. To do this, the well-known Gronwall integral inequality is generalized so as to be applicable to Lebesgue-Stieltjes integrals.