On the asymptotic stability of systems with impulses by the direct method of Lyapunov
type: Item
instance of: scholarly article
title: On the asymptotic stability of systems with impulses by the direct method of Lyapunov
zbMATH Open document ID: 0681.34042
DOI: 10.1016/0022-247X(89)90067-X
author: G. K. Kulev
author: Drumi Bainov
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: The paper deals with motions (x(t),y(t)) which satisfy a system of differential equations of the type (1) \(\dot x=f(t,x)+g(t,y),\) \(\dot y=h(t,x,y)\) for all \(t\neq t_ i(x,y)\) \((i=1,2,...)\), but perform certain jumps \((2)\quad \Delta x|_{t=t_ i(x,y)}=A_ i(x)+B_ i(y),\) \(\Delta y|_{t=t_ i(x,y)}=C_ i(x,y)\) when passing the hypersurfaces \(t=t_ i(x,y)\). In six theorems sufficient conditions for (local or global) asymptotic stability are given making use of the concepts of Lyapunov's direct method for the trajectories of (1). A crucial assumption is that the distance of a motion from the origin (and also the value of its Lyapunov function) is non-increasing at each jump (2).
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 4115061
zbMATH Keywords: jumps
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Lyapunov's direct method
zbMATH Keywords: Lyapunov function
MaRDI profile type: Publication
cites work: Stability of measure differential equations
cites work: Stability of differential systems with impulsive perturbations in terms of two measures
cites work: On the stability of impulsively perturbed differential systems
cites work: Stability of impulsively perturbed systems
cites work: Q3782084
cites work: Stability with respect to part of the variables in systems with impulse effect
cites work: Sufficient conditions for absence of ”beating” in systems of differential equations with impulses
cites work: Q3700080
cites work: Q5844517
description: scientific article; zbMATH DE number 4115061