Lyapunov regularity and stability of linear non-instantaneous impulsive differential systems

In this paper, we discuss Lyapunov regularity and stability for linear non-instantaneous impulsive differential systems. In particular, we give sufficient conditions to guarantee any non-trivial solution has a finite Lyapunov exponent and we prove an impulsive system is stable using the Lyapunov exponent for the solution. A new version of Perrons theorem is given by introducing the associated adjoint impulsive system and some criteria for the existence of non-uniform exponential behaviour are given. In addition, we present a stability result for a small perturbed nonlinear impulsive system when the linear impulsive system admits a non-uniform exponential contraction. Finally, we give a bound for the regularity coefficient.