Despite its potential use in several meaningful problems linear impulsive systems have not been extensively studied to account for dynamics in which the equilibria are out of the origin. In this work a novel characterization of the system equilibria and invariant regions-derived from the definition of two underlying discrete-time systems-is given, and based on this characterization, an unconstrained feedback control and a zone Model Predictive Control strategies are proposed. The controllers are tested in two drug administration problems: An intravenous bolus administration of Lithium ions and a nonlinear human immunodeficiency virus infection dynamics under Zidovudine treatment.