ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

Abstract

In this paper, we investigate the qualitative behavior of an evolutionary problem consisting of a hyperbolic dissipative equation whose trajectories undergo instantaneous impulsive discontinuities at the moments when the energy functional reaches a certain threshold value. The novelty of the current study is that we consider the case in which the entire infinite-dimensional phase vector undergoes an impulsive disturbance. This substantially broadens the existing results, which admit discontinuities for only a finite subset of phase coordinates. Under fairly general conditions on the system parameters, we prove that such a problem generates an impulsive dynamical system in the natural phase space, and its trajectories have nonempty compact ω-limit sets.