Robustness for Stable Impulsive Equations via Quadratic Lyapunov Functions

摘要

For a linear impulsive differential equation, we give a complete characterization of the existence of a nonuniform exponential contraction in terms of quadratic Lyapunov functions and of the operators defining them. This corresponds to consider a nonuniform exponential stability of the dynamics, which is typical for example in the context of ergodic theory. As an application, we use this characterization to establish in a very simple manner the robustness property of a nonuniform exponential contraction under sufficiently small linear perturbations. In addition, we obtain versions of the robustness property for perturbations of the jumping times and of a strong nonuniform exponential contraction. The latter corresponds to consider not only an upper bound for the dynamics but also a lower bound.