  Linear impulsively controlled systems are suitable to describe a venue of
real-life problems, going from disease treatment to aerospace guidance. The
main characteristic of such systems is that they remain uncontrolled for
certain periods of time. As a consequence, punctual equilibria
characterizations outside the origin are no longer useful, and the whole
concept of equilibrium and its natural extension, the controlled invariant
sets, needs to be redefined. Also, an exact characterization of the admissible
states, i.e., states such that their uncontrolled evolution between impulse
times remain within a predefined set, is required. An approach to such tasks --
based on the Markov-Lukasz theorem -- is presented, providing a tractable and
non-conservative characterization, emerging from polynomial positivity that has
application to systems with rational eigenvalues. This is in turn the basis for
obtaining a tractable approximation to the maximal admissible invariant sets.
In this work, it is also demonstrated that, in order for the problem to have a
solution, an invariant set (and moreover, an equilibrium set) must be contained
within the target zone. To assess the proposal, the so-obtained impulsive
invariant set is explicitly used in the formulation of a set-based model
predictive controller, with application to zone tracking. In this context,
specific MPC theory needs to be considered, as the target is not necessarily
stable in the sense of Lyapunov. A zone MPC formulation is proposed, which is
able to i) track an invariant set such that the uncontrolled propagation
fulfills the zone constraint at all times and ii) converge asymptotically to
the set of periodic orbits completely contained within the target zone.
