Perturbations of impulsive semigroups and existence of impulsive sets for finite and infinite-dimensional dynamical systems

Abstract

This work explores the robustness of impulsive semigroups under perturbations, focusing on the upper and lower semicontinuity of attractors and, more importantly, on the topological structural stability. We establish that, under suitable perturbations of an impulsive semigroup, the resulting attractor remains dynamically equivalent, in some sense, to the original attractor. Furthermore, regarding the existence of impulsive sets for impulsive dynamical systems, we generalize the results of \cite{BBCC-Acta} to semigroups in finite-dimensional spaces and present sufficient conditions to obtain impulsive sets satisfying condition \ref{T} (in the sense of \cite{BonotoPiotr2020}) in the infinite-dimensional framework. We illustrate the infinite-dimensional results with examples of linear problems in Hilbert spaces and a semilinear reaction-diffusion equation on the real line.