Exponential dichotomies for linear systems with impulsive effects

Abstract

In this paper we give conditions for the existence of a dichotomy for the impulsive equation $$displaylines{ mu(t,varepsilon) x'= A(t)x, ; t eq t_k,cr x(t_k^+ )= C_k x(t_k^-),, }$$ where $mu(t,varepsilon)$ is a positive function such that $limmu(t,varepsilon)=0$ in some sense. The results are expressed in terms of the properties of the eigenvalues of matrices $A(t)$, the properties of the eigenvalues of matrices ${C_k}$ and the location of the impulsive times ${t_k}$ in $[0, infty)$.

Keywords

• Impulsive linear systems
• singularly perturbed impulsive systems
• dichotomies
• splitting of impulsive systems.