On Jack Hale's problem for impulsive systems

An interesting problem of Jack Hale deals with the existence of a maximal compact invariant set in discrete dynamical systems. A solution for this problem is known for locally bounded dynamical systems. Following this line of research, we consider in this paper a class of systems whose continuous dynamics are interrupted by abrupt changes of state and we present sufficient conditions to obtain the existence of a maximal compact invariant set for a system in this class. We use the theory of asymptotic compactness to get the results.