Regularity Coefficients for Impulsive Differential Equations

For a linear impulsive differential equation, we introduce a Lyapunov regularity coefficient following as far as possible the non-impulsive case. We recall that a regularity coefficient is a quantity that characterizes the Lyapunov regularity of the dynamics. In particular, we obtain lower and upper bounds for the Lyapunov regularity coefficient and we show that its computation can always be reduced to that of the corresponding coefficient of an impulsive dynamics defined by upper triangular matrices. We also relate the Lyapunov regularity coefficient with the Grobman regularity coefficient. Finally, we combine all the former results to establish a criterion for tempered exponential behavior in terms of the Lyapunov exponents and of the Lyapunov regularity coefficient.

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 Perron’s 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.